energy scaling of infrared nanosecond optical parametric …1210318/... · 2018-05-28 · energy...

Energy scaling of infrared nanosecond optical

parametric oscillators and amplifiers based on

Rb:KTiOPO4

Riaan Stuart Coetzee

Doctoral Thesis in Physics

Laser Physics

Department of Applied Physics

School of Engineering Science

KTH

Stockholm, Sweden 2018

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Energy scaling of infrared nanosecond optical parametric oscillators

and amplifiers based on Rb:KTiOPO4

© Riaan Stuart Coetzee, 2018 Laser Physics Department of Applied Physics KTH – Royal Institute of Technology 106 91 Stockholm Sweden ISBN: 978-91-7729-826-7 TRITA-FYS: 2018:27 ISSN: 0280-316X ISRN: KTH/FYS/ - 18:27-SE Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i fysik, torsdag 14 June 2018 kl. 10:00 i sal FB42, Albanova, Roslagstullsbacken 21, KTH, Stockholm. Avhandlingen kommer att försvaras på engelska. Cover picture: 1 µm pumped high-energy, 2 µm nanosecond MOPA based on periodically poled Rb:KTiOPO4. Printed by Universitetsservice US AB, Stockholm 2018.

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Abstract High-energy, narrowband, nanosecond pulsed mid-infrared sources centred on 2 µm are required in applications in remote sensing, standoff detection and pollution monitoring using LIDARs. Currently, space-borne LIDAR missions are under development by major space agencies around the world for active measurements of the atmospheric gas constituents and their dynamics. The spectral range around 2 µm is one of the windows of operation for these instruments. Optical parametric oscillators (OPOs) and amplifiers (OPAs) operating at 2 µm are often used as pump sources for cascaded down-conversion schemes to enable generation of wavelengths deeper into the mid-infrared. In order to fulfil these purposes, they are required to have high-energy output with good overall efficiency, while maintaining a narrowband or tailored spectrum. Moreover, for space-based instruments, the radiation hardness of the parametric sources needs to be assessed. The central objectives of this thesis were the scaling of the energy and efficiency of 2 µm based OPOs and OPAs, tailoring their spectral brightness and assessing their suitability for applications in space-borne active gas detection systems. Specifically, we investigated OPOs and OPAs based on periodically-poled Rb:KTiOPO4 (PPRKTP), an engineered nonlinear material which can be fabricated with large optical apertures and sub-µm periodicities. One of the key limitations to energy scaling of these devices is the laser-induced damage threshold (LIDT) of the nonlinear material used. In down-conversion schemes, the devices are subject to both high intensity 1 µm and 2 µm radiation. Prior to the work in this thesis, no LIDT value at 2 µm of KTP and Rb:KTP had been reported. Furthermore, the work in this thesis provides data on effects of different dosage of gamma radiation on the optical properties of this material and the ways to mitigate the damage induced by the ionizing radiation. To demonstrate energy scaling with narrow bandwidth and tunability, a nanosecond, master oscillator power amplifier (MOPA) system operating around 2 µm and based on large-aperture PPRKTP was built. The MOPA system demonstrated dual narrowband spectrum, tunable over 1.5 THz by means of a transversally chirped volume Bragg grating, while delivering tens of mJ in output. The output from this MOPA system will be further used for tunable THz generation. Even narrower spectra can be generated employing backward-wave optical parametric oscillators (BWOPO) based on PPRKTP with the periodicity of 509 nm. This work demonstrates for the first time an efficient, millijoule-level BWOPO with backward propagating signal. The device possessed narrowband spectrum with stable output, making it an excellent seed source in MOPA arrangements.

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Sammanfattning Smalbandiga, nanosekundspulsade ljuskällor med hög energi centrerade kring 2 µm behövs i tillämpningar såsom fjärranalys, avståndsdetektering och föroreningsövervakning med LIDAR. Ledande rymdstyrelser runtom i världen arbetar nu med att utveckla rymdbaserad LIDAR för att aktivt kunna mäta atmosfärens gassammansättning samt dess dynamik. Ljus kring 2 µm täcker ett av spektralområdena som dessa system används vid. Optiskt parametriska oscillatorer (OPOs) samt optiska parametriska förstärkare (OPAs) kring 2 µm används ofta som pumpkällor för kaskaderade nedkonverteringssystem för att generera våglängder djupare in i det mellaninfraröda spektralområdet. Detta kräver att de har en bra omvandlingseffektivitet och ger upphov till utsignaler med hög effekt, medan de samtidigt måste bibehålla smal bandbredd eller given spektral form. För rymdbaserade instrument måste även de parametriska källornas optiska tolerans karaktäriseras. Huvudmålen med denna avhandling var att skala upp energin samt omvandlingseffektiviteten för OPOs och OPAs kring 2 µm, att optimera deras spektrala ljusstyrka och att utvärdera deras användbarhet i tillämpningar för rymdbaserade system för aktiv gasdetektion. Vi undersökte OPOs och OPAs baserade på periodiskt polade Rb:KTiOPO4 (PPRKTP), vilket är ett ickelinjärt material som kan skräddarsys för att generera en viss våglängd och tillverkas med stora tvärsnittsytor och med perioder mindre än 1 µm. En av de största begränsningarna, vad gäller uppskalning av energin i dessa system, är det ickelinjära materialets laserinducerade skadetröskel (LIDT). I nedkonverteringssystem utsätts de ickelinjära materialen för ljus med hög intensitet vid både 1 µm samt 2 µm. Innan arbetet som presenteras I den här avhandlingen så fanns det inget rapporterat LIDT-värde för KTP eller Rb:KTP vid 2 µm. Arbetet som presenteras i denna avhandling innehåller dessutom data för hur olika doser av gammastrålning påverkar de optiska egenskaperna för dessa material, samt hur skador från joniserade strålning kan minskas. För att påvisa energiuppskalning med smalbandigt och inställbart spektrum byggdes ett nanosekunds-master-oscillator-effektförstärkarsystem (MOPA) kring 2 µm, vilket var baserat på PPRKTP med stora tvärsnittsytor. MOPA-systemet hade ett två-våglängdsspektrum, vars avstånd kunde justeras mer än 1.5 THz med hjälp av ett transversellt chirpat volym-Bragg-gitter och levererade pulser med tiotals mJ. Dessa pulser kommer vidare att användas för att generera justerbar THz-ljus. Ännu smalare spektrum kan genereras genom att använda baklängesvågs-optiskt parametriska oscillatorer (BWOPO) baserade på PPRKTP med en periodicitet på 500 nm. I detta arbete har vi för första gången visat en effektiv, BWOPO med baklängespropagerande signal i millijouleskalan. Det systemet hade ett smalbandigt spektrum med stabil utsignal, vilket gör den till en utmärkt källa för MOPA-system.

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Preface

The research presented in this thesis has been performed in the Laser Physics group, at the Applied Physics department of the Royal Institute of Technology (KTH), in Stockholm from 2013 to 2018.

This work was partly funded by the Swedish Foundation for Strategic Research (SSF) and the Swedish Research Council (VR) through its Linnaeus Center of Excellence ADOPT.

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List of Publications This thesis is based on the following journal articles:

I. Coetzee, R. S., Thilmann, N., Zukauskas, A., Canalias, C., Pasiskevicius, V., “Nanosecond laser induced damage thresholds in KTiOPO4 and Rb:KTiOPO4 at 1 µm and 2 µm,” Opt. Mat. Express, Vol. 5, Issue 9 (2015).

II. Coetzee, R. S., Duzellier, S., Dherbecourt, J. B., Zukauskas, A., Raybaut, M., Pasiskevicius, V., “Gamma irradiation-induced absorption in single-domain and periodically-poled KTiOPO4 and Rb:KTiOPO4,” Opt. Mat. Express, Vol. 7, Issue 11 (2017).

III. Coetzee, R. S., Zheng, X., Fregnani. L., Laurell, F., Pasiskevicius, V., “Narrowband, tunable, 2 µm optical parametric master-oscillator power amplifier with large-aperture periodically poled Rb:KTP,” (Accepted: Appl. Phys. B (2018)).

IV. Coetzee, R. S., Zukauskas, A., Canalias, C., Pasiskevicius, V., “Low-threshold, mid-infrared backward-wave parametric oscillator with periodically poled Rb:KTP,” (Submitted: Applied Physics Photonics Letters (2018)).

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Description of Author Contribution My contributions to the original papers in this thesis were the following:

Paper I

I designed the experimental layout with assistance from N. Thilmann. I built and performed the experimental measurement as well as the numerical analysis of the obtained results. I wrote the manuscript with assistance from N. Thilmann.

Paper II

I assisted with the technical design of the experiment. I analysed and interpreted the obtained experimental results with assistance from V. Pasiskevicius. I also performed supplementary experimental measurements found in the paper. I wrote the manuscript with assistance from V. Pasiskevicius.

Paper III

I designed the experimental setup. I performed the experimental measurements with X. Zheng and L. Fregnani. I wrote the numerical model used in the manuscript. I wrote the manuscript with assistance from the co-authors.

Paper IV

I designed the experimental setup. I performed the experimental measurements with assistance from A. Zukauskas. I wrote the manuscript with assistance from the co-authors.

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Other Publications A. Coetzee, R. S., Thilmann, N., Zukauskas, A., Canalias, C., Pasiskevicius, V., “Laser

Induced Damage Thresholds of KTP and RKTP,” Optics and Photonics Sweden, Chalmers University of Technology, Gothenburg, Sweden, 11-12 November (2014).

B. Coetzee, R. S, Thilmann, N., Zukauskas, A., Canalias, C., Pasiskevicius, V., “Investigations of Laser Induced Damage in KTiOPO4 and Rb:KTiOPO4 at 1 µm and 2 µm,” PACIFIC RIM LASER DAMAGE: Optical Materials For High-Power Lasers, 2015, Vol. 9532, article id 95320B (Invited, 2015).

C. Coetzee, R. S., Thilmann, N., Zukauskas, A., Canalias, C., Pasiskevicius, V.,

“Investigations of Laser Induced Damage in KTiOPO4 and Rb:KTiOPO4 at 1 µm and 2 µm,” Conference on Lasers and Electro-Optics Europe (2015).

D. Coetzee, R. S., Thilmann, N., Zukauskas, A., Canalias, C., Pasiskevicius, V., “Laser

Induced Damage thresholds at 1 µm and 2 µm for undoped and R-doped KTiOPO4,” 4th French-German Oxide Crystals Workshop, Saint-Louis, France, September 10-11 (Invited, 2015).

E. Coetzee, R. S., Zukauskas, A., Pasiskevicius, V., “High-energy optical parametric

amplifiers in the mid-infrared with large-aperture periodically poled Rb:KTiOPO4,” 7th EPS-QEOD EUROPHOTON CONFERENCE "Solid State, Fibre, and Waveguide Coherent Light Sources" Vienna, Austria, 21-26 August (2016).

F. Coetzee, R. S., Zukauskas, A., Melkonian, J-M., Pasiskevicius, V., “An efficient 2 µm

optical parametric amplifier based on large-aperture periodically poled Rb:KTP,” Proceedings Volume 10562 International Conference on Space Optics (ICSO), Biarritz, France (2016) | 18-21 OCTOBER 2016, SPIE - International Society for Optical Engineering, (2017), Vol. 10562, article id 105620L.

G. Duzellier, S., Coetzee, R. S., Zukauskas, A., Dherbecourt, J-B., Raybaut, M.,

Pasiskevicius, V., “Color Centers Induced in KTiOPO4 Family Nonlinear Crystals by Exposure to Gamma-Radiation,” International Conference on Space Optics (ICSO), Biarritz, France (2016).

H. Armougom, J., Melkonian, J-M., Dherbercourt, J-B., Raybaut, M., Godard, A., Coetzee,

R.S., Zukauskas, A., Pasiskevicius. V., “70 mJ single-frequency parametric source tunable between 1.87-1.93 µm and 2.37-2.47 µm for difference frequency generation in the LWIR,” European Conference on Lasers and Electro-Optics and European Quantum Electronics Conference (CLEO), Munich, Germany (2017).

I. Armougom, J., Melkonian, J-M., Raybaut, M., Dherbecourt, J-B., Gorju, G., Godard, A.,

Coetzee, R.S., Pasiskevicius, V., Kadlčák, J., “7.3-10.5 µm Tunable Single-frequency Parametric Source for Standoff Detection of Gaseous Chemicals,” Mid-Infrared Coherent Sources (MICS) 2018.

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J. Pasiskevicius, V., Smilgevicius, V., Butkus, R., Coetzee, R. S., Laurell, F., “Spatial and Temporal Coherence in Optical Parametric Devices Pumped with Multimode Beams,” Lithuanian Journal of Physics, Vol. 58, No. 1, pp. 62–75 (2018).

K. Armougom, J., Melkonian, J-M., Raybaut, M., Dherbecourt, J-B., Gorju, G., Godard, A.,

Coetzee, R. S., Pasiskevicius, V., Kadlčák, J., “Longwave Infrared Lidar Based on Parametric Sources for Standoff Detection of Gaseous Chemicals,” Conference on Lasers and Electro-Optics (CLEO), San Jose, California (2018).

L. Coetzee, R. S., Zukauskas, A., Canalias, C., Pasiskevicius, V., “Efficient Backward-Wave

Optical Parametric Oscillator with 500 nm-periodicity PPRKTP,” Conference on Lasers and Electro-Optics (CLEO), San Jose, California (2018).

M. Canalias, C., Zukauskas, A., Viotti, A-L., Coetzee, R. S., Liljestrand, C., Pasiskevicius,

V., “Periodically Poled KTP with Sub-wavelength Periodicity: Nonlinear Optical Interactions with Counter Propagating Waves,” Advanced Photonics Congress, ETH Zurich, Zurich, Switzerland (Invited, 2018).

N. Coetzee, R. S., Zheng, X., Fregnani, L., Laurell, F., Pasiskevicius, V., “Narrowband, tunable, 2 µm optical parametric master-oscillator power amplifier with large-aperture periodically poled Rb:KTP,” 8th EPS-QEOD Europhoton Conference, Barcelona, Spain, 2-7 September (2018).

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Acknowledgements This thesis is the culmination of my time in the Laser Physics group at KTH. I would like to express my gratitude to the following people that contributed to its fruition.

First I would like to thank my main supervisor, Professor Valdas Pasiskevicius. Thank you for inviting and welcoming me into the group in order to pursue PhD studies. I have always admired your depth and breadth of knowledge in physics and always tried to strive towards it as an ideal. Lastly, thank you for always being available to me whenever I needed guidance. I would also like to thank my co-supervisor, Professor Fredrik Laurell. For inviting me into the group and always believing in me. For always giving valuable insight into many of the projects I was working with. Your good sense of humour and energy was always re-assuring during some of the more difficult times of the PhD.

I am grateful to Professor Jens Tellefsen, for all your support and great kindness you shared with me and for being such a beautiful soul. I also thank Lars Gunnar for all your administrative and personal support over the course of my studies.

I am particularly indebted to Nicky Thilmann, who taught me much about nonlinear optics during the beginning of my studies. I also am indebted to Andrius Zukauskas for teaching me much about crystals and poling, providing many of them for my experiments and for just being a great friend.

I would also like to thank all my colleagues in the Laser Physics group for your friendship and good times shared over the years. To my office mates Le Anne-Lise, Jingyi and Robert, for all the laughs and good times. Many thanks to Robert, for helping me with numerical related things, even when the code was looking murky. To my other colleagues, Hoda, Staffan, Peter, Hoon, Charlotte, Gustav and Neils who have all contributed both academically and personally to me over the years, I thank you all.

To my mother and father, thank you for always believing in and loving me. For giving so much of yourself so that I could reach where I am today. To my brothers Andre and Markie, for your trust, love and belief in me as your brother. To my family here in Sweden, Acs, Sithi, Uncle and Aunty and two funny little girls, Sasha and Bokkamma for all your support and love you give me. Finally, I would like to dedicate this thesis to my beautiful wife, Tharany. For always being there

and loving me as I am. For simply bringing out the best in me and always inspiring me to be a

greater man than I was the day before. My love, this would not have been possible without you.

Nāṉ uṉṉai kātalikkiṟēṉ.

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Contents Abstract .......................................................................................................................................... iii

Sammanfattning .............................................................................................................................. iv

Preface .............................................................................................................................................. v

List of Publications .......................................................................................................................... vi

Description of Author Contribution .............................................................................................. vii

Other Publications ........................................................................................................................ viii

Acknowledgements .......................................................................................................................... x

Contents .......................................................................................................................................... 11

1 Introduction ............................................................................................................................. 13

1.1 The Mid-infrared spectrum .............................................................................................. 13

1.2 Energy scaling considerations ......................................................................................... 15

1.3 Aims ................................................................................................................................. 16

1.4 Outline of the Thesis ........................................................................................................ 16

2 Introductory Nonlinear Optics & Three-wave mixing ............................................................ 19

2.1 Background ...................................................................................................................... 19

2.2 χ(2) Processes .................................................................................................................... 20

2.2.1 Energy conservation & photon number conservation .............................................. 23

2.2.2 Phase matching ......................................................................................................... 24

2.2.3 Coupled-wave equations .......................................................................................... 25

2.3 KTiOPO4 and Rubidium doped KTiOPO4 ...................................................................... 26

References to Chapter 2 .............................................................................................................. 29

3 Radiation damage of nonlinear crystals .................................................................................. 31

3.1 Laser induced damage of materials ................................................................................. 31

3.1.1 Nanosecond Laser Damage of KTP and RKTP ....................................................... 32

3.1.2 Experimental procedures for measuring LIDTs. ...................................................... 34

3.1.3 Statistical models for nanosecond laser damage ...................................................... 36

3.1.4 LIDT measurements of KTP and RKTP .................................................................. 39

3.2 Gamma irradiation of KTP and RKTP ............................................................................ 46

3.2.1 Gamma irradiation experiments ............................................................................... 47

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References to Chapter 3 ................................................................................................................. 54

4 Optical Parametric Oscillators and Amplifiers ....................................................................... 60

4.1 OPO/OPA theory ............................................................................................................. 60

4.2 Nanosecond OPOs ........................................................................................................... 64

4.2.1 Energy characteristics .............................................................................................. 64

4.2.2 Pulsed OPO design considerations ........................................................................... 65

4.2.3 Bandwidth and temperature tuning .......................................................................... 65

4.3 Nonlinear mixing with spatially structured fields ........................................................... 69

4.4 High-energy dual-wavelength MOPA ............................................................................. 71

4.4.1 MOPA experimental layout ..................................................................................... 71

4.4.2 TCVBG OPO characterization ................................................................................. 73

4.4.3 MOPA characterization ............................................................................................ 75


5 Nanosecond Mirrorless OPOs ................................................................................................. 81

5.1 Introduction ..................................................................................................................... 81

5.2 MOPO experiments ......................................................................................................... 84

5.2.1 Experimental layout & description........................................................................... 84

5.2.2 BWOPO characterization and measurements .......................................................... 86


Conclusion and Outlook ................................................................................................................. 92

Appendix A .................................................................................................................................... 94

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1 Introduction The work in this thesis is centred on energy scaling of mid-infrared optical parametric oscillators and amplifiers (OPOs and OPAs), particularly at 2 µm. These devices were based on Rubidium-doped KTiOPO4 (RKTP). The work included investigation of factors which limit the energy from such sources, namely the laser-induced damage threshold of KTP and RKTP. High-energy, narrowband, nanosecond pulsed mid-infrared sources centred on 2 µm are required in applications in remote sensing, standoff detection and pollution monitoring using LIDARs. Currently, space-borne LIDAR missions are under development by major space agencies around the world for active measurements of the atmospheric gas constituents and their dynamics. The spectral range around 2 µm is one of the windows of operation for these instruments. Optical parametric oscillators (OPOs) and amplifiers (OPAs) operating at 2 µm are often used as pump sources for cascaded down-conversion schemes to enable generation of wavelengths deeper into the mid-infrared. In order to fulfil these purposes, they are required to have high-energy output with good overall efficiency, while maintaining a narrowband or tailored spectrum. A major limitation in obtaining high-energies from these devices is the laser-induced damage threshold (LIDT) of the nonlinear crystals used. In down-conversion schemes, the devices are subject to both high intensity 1 µm and 2 µm radiation. For space-borne applications, the nonlinear crystals are also subject to high incident levels of gamma and proton radiation. With this in mind, the central objectives of this thesis were the scaling of the energy and efficiency of 2 µm based OPOs and OPAs, tailoring their spectral brightness and assessing their suitability for applications in space-borne active gas detection systems. Specifically, we investigated OPOs and OPAs based on periodically-poled Rb:KTiOPO4 (PPRKTP), an engineered nonlinear material which can be fabricated with large optical apertures and sub-µm periodicities.

1.1 The Mid-infrared spectrum

With the invention of the laser in 1960 [1], much research has been undertaken into the development of coherent sources. This is largely attributed to the properties that light from these sources possess which in turn allows a myriad amount of applications to be served. Of particular interest to applications such as surgery, LIDAR, directed infrared counter measures (DIRCM) and spectroscopy [2,3] are sources that deliver tunable, high-energy within the mid-infrared band of the electromagnetic spectrum. The mid-infrared band is defined as the region corresponding to wavelengths between approximately ~ 2 – 10 µm of the electromagnetic spectrum, as shown in Fig. 1.1.

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Fig. 1.1: The electromagnetic spectrum with visible region inset.

For energies exceeding multiple milli-joules within the mid-infrared, there are two candidates which are currently the most promising, solid-state lasers based on crystals or fibers [4] and optical parametric oscillators based on periodically poled nonlinear crystals [5,6]. Solid-state lasers doped with Tm3+ and Ho3+ ions provide high brightness radiation in the band 1.9 - 2.1 µm, often with excellent energies, efficiencies and diffraction limited beam qualities. As such, they act as very good pump sources for down-conversion schemes further into the infrared. However, they are limited in tunability and are therefore restricted within applications such as spectroscopy [7]. With limited tunability available, the device may typically only target a single molecule or single species.

Fig. 1.2: Typical wavelength emission range for different coherent sources [8].

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On the other hand, 1 µm pumped OPOs based on periodically poled nonlinear crystals are capable of delivering tunable radiation which spans the entire transparency range of the nonlinear crystal used [6]. This tunability ensures a large degree of flexibility for different applications over the mid-infrared band. With regard to spectroscopy, these devices may target multiple molecular species in LIDAR applications. To realize wavelengths deeper in the mid-infrared band, cascaded down-conversion schemes are utilized, where nonlinear crystals such as ZnGeP2 (ZGP) and orientation-patterned GaAs (OP-GaAs) facilitate longer wavelength generation [9,10]. The transparency range of these crystals often start at 2 µm due to two-photon absorption, thus requiring a high energy 2 µm source to act as the pump for the process. Each step in the cascaded chain carries an efficiency loss and therefore it is imperative to reach as high energy as possible in the start of the 2 µm pumping chain. Furthermore, high energy, tunable and narrowband 2 µm radiation is required for atmospheric gas measurements of compounds such as CO2 [11]. Two-stage down-conversion schemes pumped with well-established 1 µm lasers, are attractive to reach deeper into the mid-infrared [12]. Down-conversion schemes to the mid-IR often employ quasi-phase matched (QPM) KTP or RKTP as the gain material owing to its high-damage threshold, high nonlinearity, good transparency and possibility to fabricate such crystals with large optical apertures [13]. Thus the materials used in this thesis were KTP and RKTP.

1.2 Energy scaling considerations

In order to obtain high energies required for long wavelength generation schemes, these sources are typically employed in master oscillator power amplifier (MOPA) schemes [14, 15]. In essence a MOPA consists of a seed source, such as an OPO, followed by the amplification of the seed output in an amplifier stage. The seed or oscillator is designed in such a way to have tailored properties such as tunability and a narrowband spectrum via volume Bragg gratings (VBGs) [16], which is then further amplified in energy. Broadband OPOs based on periodically poled nonlinear crystals are capable and have reached pulsed output energies of multiple mJ, with a current record of 0.54 J [17]. However, obtaining a narrowband spectrum often involves a loss in the output energy. Thus, MOPA configurations are favourable for reaching high-energies in QPM devices with narrowband spectral output. Employing such an OPO with an amplifier stage allows much higher output energy to be generated, while still maintaining desired spectral properties. A few factors set a limit on how much energy may be obtained from amplification, such as back-conversion [18] and the laser-induced damage threshold (LIDT) [19] of the optical elements and nonlinear crystals within the setup. It is therefore crucial to accurately know the LIDT for both 1 µm and 2 µm, since these wavelengths represent the largest energies within the overall down conversion scheme. LIDT results for KTP at 1 µm in the nanosecond regime, are well known [20] however the LIDT at 2 µm was not previously measured. The laser damage mechanism for nonlinear crystals differs with the pulse duration. For nanosecond pulses, laser damage is a statistical process in comparison to ultrashort pulse damage which is deterministic. The devices in this thesis employed large aperture crystals with high fidelity which allowed further energy scaling of the MOPAs. The large aperture allows the pump energy to be increased while also increasing

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the beam size on the crystal, keeping the incident fluence below that of the damage threshold of the material [21]. As was mentioned, space-borne applications are subject to high degrees of proton and gamma radiation. This radiation can induce color-centers or grey-tracking within the nonlinear material [22]. This may lead to an increase in absorption of 1 µm and 2 µm radiation and thus will lead to damage, inhibiting further energy scaling. Therefore, a study and circumvention of this effect for high gamma radiation doses, is paramount for high-energy space-borne applications.

1.3 Aims

In this thesis, the aim was the energy scaling of high-energy 2 µm parametric sources based on PPRKTP, with narrowband and tunable output while maintaining high conversion efficiencies and excellent beam qualities. The two configurations where energy scaling was investigated was in MOPA arrangements as well as backward-wave oscillator approaches, where narrowband spectral output is inherently obtained. In the MOPA arrangements, narrowband output was obtained utilizing a chirped volume Bragg grating (VBG) as the output coupler of the seed source. Both MOPA and backward-wave devices are applicable for LIDARs, which include space-borne platforms. Supplementary to the energy scaling of both of these devices, the LIDT of KTP and RKTP were measured in detail in the nanosecond regime at both 1 µm and 2 µm. As was mentioned, LIDT results at 1 µm and 2 µm for RKTP and 2 µm for KTP had not been measured prior to the work in this thesis. Also, gamma radiation induced color centers were studied in both these nonlinear materials for potential use in space-borne applications.

1.4 Outline of the Thesis

First some background theory in nonlinear optics and parametric processes is discussed. Following this, radiation damage in the nanosecond regime is discussed and results of gamma as well as laser damage of KTP and RKTP is presented. Based on these damage measurements, we develop a high-energy wavelength locked 2 µm MOPA system as well as a high-energy sub-µm mirrorless optical parametric oscillator (MOPO). To summarize, we discuss prospects for further research.

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References to Chapter 1

1. T. H. Maiman, “Stimulated Optical Radiation in Ruby,” Nature, 187, 493-494, 6 August (1960).

2. V. Petrov, “Parametric down-conversion devices: The coverage of the mid-infrared spectral range by solid-state laser sources,” Opt. Materials, 34, 536–554 (2012).

3. J. Peng, “Developments of mid-infrared optical parametric oscillators for spectroscopic sensing: a review,” Opt. Engineering, 53, 6, 061613 (2014).

4. A. Godard, “Infrared (2–12 µm) solid-state laser sources: a review,” Comptes Rendus Physique, 8, 1100–1128 (2007).

5. L. E. Myers and W. R. Bosenberg, “Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators,” IEEE Journal of Quantum Electronics, 33, 10, October (1997).

6. A. Dergachev, D. Armstrong, A. V. Smith, T. Drake and M. Dubois, “3.4-µm ZGP RISTRA nanosecond optical parametric oscillator pumped by a 2.05-µm Ho:YLF MOPA system,” Optics Express, 15, 22, 29 October (2007).

7. Q. Wang, J. Geng and S. Jiang, “2 µm fiber laser sources for sensing,” Optical Engineering, 53, 6, 061609, June (2014).

8. C. W. Rudy, “Mid-IR Lasers: Power and pulse capability ramp up for mid-IR lasers,” Laser Focus World, 2 May (2014).

9. G. Stoeppler, N. Thilmann, V. Pasiskevicius, A. Zukauskas, C. Canalias and M. Eichhorn, “Tunable Mid-infrared ZnGeP2 RISTRA OPO pumped by periodically-poled Rb:KTP optical parametric master-oscillator power amplifier,” Optics Express, 20, 4, 4509-4517, 13 February (2012).

10. K. L. Vodopyanov, I. Makasyuk and P. G. Schunemann, “Grating tunable 4 - 14 µm GaAs optical parametric oscillator pumped at 3 µm,” Optics Express, 22, Issue 4, 4131-4136 (2014).

11. M. Raybaut, T. Schmid, A. Godard, A. K. Mohamed, M. Lefebvre, F. Marnas, P. Flamant, A. Bohman, P. Geiser and P. Kaspersen, “High-energy single-longitudinal mode nearly diffraction-limited optical parametric source with 3 MHz frequency stability for CO2 DIAL,” Optics Letters, 34, Issue 13, 2069-2071 (2009).

12. M. Henriksson, M. Tiihonen, V. Pasiskevicius and F. Laurell, “ZnGeP2 parametric oscillator pumped by a linewidth-narrowed parametric 2 µm source,” Optics Letters, 31, 12, 15 June (2006).

13. J. D. Bierlein and H. Vanherzeele, “Potassium titanyl phosphate: properties and new applications,” J. Opt. Soc. Am. B, 6, 4, April (1989).

14. G. Arisholm, Ø. Nordseth and G. Rustad, “Optical parametric master oscillator and power amplifier for efficient conversion of high-energy pulses with high beam quality,” Optics Express, 12, 18, 4189-4197, 6 September (2004).

18

15. M. W. Haakestad, H. Fonnum and E. Lippert, “Mid-infrared source with 0.2 J pulse energy based on nonlinear conversion of Q-switched pulses in ZnGeP2,” Optics Express, 22, 7, 8557-8564, 7 April (2014).

16. J. Saikawa, M. Fujii, H. Ishizuki and T. Taira, “High-energy, narrow-bandwidth periodically poled Mg-doped LiNbO3 optical parametric oscillator with a volume Bragg grating,” Optics Letters, 32, 20, 2996-2998, 15 October (2007).

17. H. Ishizuki and T. Taira, “Half-joule output optical-parametric oscillation by using 10-mm-thick periodically poled Mg-doped congruent LiNbO3,” Optics Express, 20, 18, 20002-20010, 27 August (2012).

18. G. Arisholm, G. Rustad and K. Stenersen, “Importance of pump-beam group velocity for backconversion in optical parametric oscillators,” J. Opt. Soc. Am. B, 18, 12, 1882-1889, 12 December (2001).

19. F. R. Wagner, A. Hildenbrand, H. Akhouayri, C. Gouldieff, L. Gallais, M. Commandré and J-Y. Natoli, “Multipulse laser damage in potassium titanyl phosphate: statistical interpretation of measurements and the damage initiation mechanism,” Optical Engineering, 51, 12, 121806, December (2012).

20. A. Hildenbrand, F. R. Wagner, H. Akhouayri, J-Y. Natoli, M. Commandré, F. Théodore and H. Albrecht, “Laser-induced damage investigation at 1064 nm in KTiOPO4 crystals and its analogy with RbTiOPO4,” Applied Optics, 48, 21, 4263-4269, 20 July (2009).

21. A. Zukauskas, N. Thilmann, V. Pasiskevicius, F. Laurell and C. Canalias, “5 mm thick periodically poled Rb-doped KTP for high energy optical parametric frequency conversion,” Optics Materials Express, 1, 2, 201-206, 1 June (2011).

22. M. V Alampiev, O. F. Butyagin, and N. I. Pavlova, "Optical absorption of gamma-irradiated KTP crystals in the 0.9 — 2.5 µm range," Quantum Electronics, 30, 255–256 (2000).

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2 Introductory Nonlinear Optics & Three-wave mixing

2.1 Background

Following the development of the laser in 1960, the field of nonlinear optics began in 1961 with the discovery of second harmonic generation [1]. This was due to the fact that the laser was the first coherent source capable of sufficient intensity to allow nonlinear optical effects within a material to become apparent. Following this, the central theory governing the physics behind nonlinear optics, was outlined in 1962 [2]. The electromagnetic wave equation, under the assumption of no free charges and no magnetization, is given by [3]:

∇ × ∇ × ��, � + 1�� , �� = −�� , �� 2.1� Where, c is the speed of light in vacuum, µ0 is the vacuum permeability, E is the electric field and P is the polarization of the material. Solutions to Equation 2.1 can be expressed in plane-wave form, where the electric field is commonly given as: �� = 12 ��, �, �, �� !"� + c. c. �2.2�

Where k is the wave-vector given by |%| = 2&'/) and ω is the angular frequency of the wave, A

is the complex envelope amplitude of the electric field, ��, �, �, � = ��, �, ��*�"�, where E0 is the electric field strength and + is the temporal phase of the wave. In the general sense, nonlinear optics involves light-matter interactions where the incident electric field undergoes a modulation due to the nonlinear polarization present within a material. Typically, the electric dipole approximation is employed to express the polarization as a Taylor expansion around the electric field:

�� = , -� .�/��/� �2.3�1/23

Where, -� is the permittivity of free space and χ(n) is the n-order susceptibility. The linear term of the equation, � = -�.�, corresponds to linear optics responsible for effects such as reflection and

refraction, which is described by the refractive index ' = 4Re�.�3�� + 1 and linear

absorption 7 = Im:.�3�; × !/<. In general there also exists a magnetic field present in the material,

however for nonlinear crystals, this is orders of magnitude weaker than the electric field [4]. Thus, the magnetization of the medium is assumed to be zero and therefore the dipole expansion is only around the electric field. For higher electric field intensities such as those found in lasers, the

20

material properties are altered by this incident field. This in turn affects the electric field itself, where the nonlinear polarization term acts as the driving force for new frequencies to arise. In this thesis the χ(2) processes are of central importance, as they are what are used in frequency conversion schemes. χ(3) processes, such as four-wave mixing [5], are also important especially when field intensities become large. Higher order terms beyond χ(3) are typically very weak and therefore we

do not explicitly discuss them in this thesis. In centrosymmetric media, .�� = 0 and therefore processes arising from the second-order nonlinearity do not occur. χ(3) processes on the other hand can occur in media regardless of any centrosymmetry consideration.

2.2 χ(2) Processes

We may illustrate the most important χ2 processes by considering two linearly polarized, monochromatic plane waves (with >� > >@) which are incident upon some nonlinear material along the z direction. The plane waves are described by Equation 2.2. The total field is then a linear

superposition of the two fields: �� = �� cos�C�� + �@cos �C@�, where C� = %�� − >� + +� and C@ = %@� − >@ + +@. Inserting this expression into Equation 2.3 and only considering the second-order term, gives a response of the form: �� = -�.�� D�� cos��C�� + �@� cos��C@� + 2��@ cos�C�� cos �C@�E �2.4� Expanding this via trigonometric identities, one can arrange the resulting equation as follows: �� = -�.�� G+ 12 �� cos�2C�� SHG of >�

+ 12 �@� cos�2C@� SHG of >@ + 12 �� + �@�� Optical Rectification + ��@ cos�D%� + %@E� − D>� + >@E + D+� + +@E� SFG + ��@ cos�D%� − %@E� − D>� − >@E + D+� − +@E� DFGU �2.5�

The resulting equation illustrates the emergence of nonlinear optical effects and thus new frequencies due to the presence of the two fields interacting within the nonlinear medium, such as difference frequency generation (DFG) and sum frequency generation (SFG). In the case of DFG where there is a pump field present, the process is initiated from quantum noise [6]. It is convenient to write the second order polarization expression above in Cartesian form as: �W��>X� = -�Y�� , , .W�Z��−>X; >\ + >]� ��>\��Z�>]� �2.6�\]�Z

21

Here, D(2) is a degeneracy factor which is equivalent to two if the electric fields are distinguishable. If not, the factor is equal to one. The indices j,k,l represent the Cartesian axes. The argument in the χ2 are generated frequencies >\ and >], with an input frequency >X. The negative factor in front

of the input frequency allows energy conservation to be fulfilled. The nonlinear susceptibility here .W�Z�� is a tensor of rank 2 and thus is comprised of 27 elements. However, one may reduce or

simplify the number of non-zero elements in the matrix by applying symmetry considerations. In

isotropic media where there is an inherent centrosymmetry, all of the elements in the .W�Z�� tensor

will be zero. For non-isotropic media this is not the case and other symmetry considerations can be used to simplify. First one may use the fact the tensor possess full permutation symmetry, thus: .W�Z��:−>X; >\ + >]; = .WZ��:−>X; >] + >\; = .ZW��−>]; −>X + >\� �2.7�

One may also apply Kleinman symmetry in the case where the dispersion of .W�Z�� is neglected [7].

This is a reasonable approximation if one considers the nonlinear process to be almost instantaneous and the frequencies involved in the interaction are far from absorption lines in the media. The means that the indices of the tensor may be permuted without permuting the frequencies involved. Thus we may write this as: .W�Z��:−>X; >\ + >]; = .WZ��:−>X; >\ + >]; = .ZW��−>X; >\ + >]� �2.8�

The implication of this is that the same nonlinear coefficient or nonlinear strength interaction is

experienced by all the .�� processes described earlier. Due to the tensor nature of the susceptibility, electric fields with different Cartesian components may interact with one another.

Therefore it is convenient to express the .W�Z�� tensor in d-matrix form, where aWb = 3� .W�Z��. Here

the contracted index m and the corresponding kl indices are related as shown in Table 2-1 below, where x-y-z refer to the optical axes of the nonlinear crystal.

kl index Contracted index m

xx 1 yy 2 zz 3

yz, zy 4 xz, zx 5 xy, yx 6

Table 2-1: Indexing convention for d-matrix formalism.

As a simple example one can consider the case of a .�� process in KTP. The d-matrix of KTP, after applying the symmetry considerations above, is given as [8]:

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aWb = c 0 0 0 0 a@3 00 0 0 a@� 0 0a@3 a@� a@@ 0 0 0 d �2.9�

Assuming that the interacting electric fields are all linearly polarized along the z-axis of the crystal and making use of Equation 2.6, we may write:

f �g��>X��h��>X��>X� i = -�Y�� j a3@��>\��>]�a�@��>\��>]�a@@��>\��>]� k �2.10�

Since a3@ = a�@ = 0, due to the symmetry considerations above, Equation 2.10 further simplifies to give the expression for the second order polarization: ��>X� = -�Y��almm��>\��>]� �2.11�

Here, almm is the effective nonlinearity. One may visualize the parametric process above in terms of energy level diagrams, which is shown in Fig. 2.1. The incident pump photon ‘excites’ the nonlinear material to a virtual state. The nonlinear response is due to this excitation of valence electrons in the material, however the state is called virtual due to the fact that they are not eigenstates of the Hamiltonian of the system and therefore cannot be assigned a specific lifetime. Following the excitation of the material with the pump field output photons are generated, a signal and idler. This parametric process is known as optical parametric generation (OPG), which was first studied in 1961 [9]. Typically, there is little to no absorption of the fields in the material, provided that the frequencies involved fall within the transparency range of the nonlinear medium. Due to the fact that the quantum state of the material is identical before and after the interaction, the process is commonly referred to as parametric [10].

23

Fig. 2.1: Virtual energy level diagram for OPG (left); the associated momentum conservation of the interaction

(right).

The two main important factors governing this interaction are energy conservation and momentum conservation which are discussed further in the following sections.

2.2.1 Energy conservation & photon number conservation

For the case of OPG, for one input pump photon, one signal and one idler photon are generated. Energy is conserved and since the interaction is parametric and thus ℏ>o = ℏ>p + ℏ>�. Assuming

a lossless medium we may write: 1>p

aqpa� = 1>�aq�a� = − 1>o

aqoa� �2.12�

where, I denotes the intensity of the respective field. If we consider the fields passing through some

unit area A, over a time ∆t, the above equation can be re-arranged to the following: aa� r qpℏ>p × � × ∆t = aa� r q�ℏ>� × � × ∆t = − aa� u qoℏ>o × � × ∆v �2.13�

The quantities within the brackets are now equivalent to the number of photons N. Thus we can re-write them simply as: awpa� = aw�a� = − awoa� �2.14�

24

Equations 2.12 and 2.14 are known as the Manley-Rowe relations and describe the energy flow between the fields during the interaction [7,11]. Thus in the case of OPG, for every pump photon that is annihilated, one signal and one idler photon are created.

2.2.2 Phase matching

The χ(2) parametric processes mentioned previously involve interacting fields which propagate through the same crystal medium. Since the wavelength of each field is different, each field will experience a different refractive index and therefore will propagate at different phase and group

velocities within the crystal. This temporal dispersion leads to a phase mismatch factor (∆k)

between the interacting fields. If no phase matching is present, that is: Δ% = %o − %p − %� ≠ 0 for

OPG, no successive build-up of the signal/idler field will take place. In order to circumvent this issue, one may employ birefringent phase matching or quasi-phase matching (QPM). Birefringent phase matching is commonly employed in non-ferroelectric materials such as ZGP [12] and involves orientation of the crystal along certain axes to reduce the phase mismatch factor. Utilizing QPM for the interaction tends to lead to higher efficiencies in comparison to birefringent phase matching. By utilizing QPM, one is able to choose the nonlinear coefficient one would like to use. QPM, which was first proposed in 1961 [2], involves the periodic inversion of the second order nonlinear susceptibility along the propagation direction. In ferroelectric materials this is achieved by alternating the direction of the spontaneous polarization after one coherence length. This effectively amounts to reversing the sign of the 2nd order nonlinear susceptibility and thus the direction of the energy flow between the interacting fields. This can be done in a crude sense by stacking layers of the nonlinear material with each subsequent layer orientation rotated by 180º. This method is commonly used for materials that are non-ferroelectric (such as orientated-patterned GaAs [13]) and thus the orientation of their spontaneous polarization cannot be reversed by electric field poling. At present, electric field poling or periodic poling is the most mature method to achieve QPM and thus is preferred. Periodic poling allows one to achieve high-quality domain structures in the material for poling periods as short as 500 nm. In general, QPM introduces a term, %z into the

phase mismatch factor: {% = %o − %p − %� − %z. This term is given by %z = �|b} , where m is the

QPM order, and Λ is the poling period. By knowing the pump and desired signal/idler wavelengths and refractive indices, one can calculate the needed poling period in order to ensure the phase mismatch factor is zero. For first order QPM, This is given simply by: Λ = 1'o/�� − 'p/�� − '�/�� 2.15�

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2.2.3 Coupled-wave equations

In the case of OPG where >o = >p + >�, it is useful to see how the respective fields evolve in the

absence of diffraction, walk-off and other temporal effects. We assume the small signal limit, thus the pump is un-depleted and remains constant over the crystal propagation. For OPG, the nonlinear mixing terms become: ��:>o; = 2-�almm�p�� 2.16 a� ��>p� = 2-�almm�o��∗��:�� ;� �2.16 b� ��>�� = 2-�almm�o�p∗��:�� ;� �2.16 c�

Using these expressions of the second-order polarization, for OPG, as well as a simplified form of the wave equation, it is possible to derive the coupled-wave equations which are given as: ��o�� = − 72 �o + �>oalmm'o� �p�� 2.17 a� ��p�� = − 72 �p + �>palmm'p� �o��∗�� 2.17 b� �� = − 72 �� + �>�almm'�� o�p∗�� 2.17 c�

Exact solutions for the above equations exist and are found by making use of Jacobi-elliptic functions [2,14]. By making some assumptions it is possible to find a more simple solution to demonstrate some of the characteristics of the interaction. For this case, if we assume Ap and As are constant and only Ai evolves, Ai may be solved for analytically, by relating the field amplitude to the plane-wave intensity. The intensity of the idler may be expressed as: q� = 8>��'o'p'�-�� almm� qoqp sinc� rΔ%�2 t �2.18�

where, L is the length of the nonlinear medium. We plot Equation 2.18 as a function of the argument Δ%�/2, which is given below in Fig. 2.2. The function reaches its first minimum

when �� = ± &. This is double the coherence length which is given by �< = &/Δ%. The coherence

length in this case denotes the propagation length over which the fields propagate and exchange energy, before back-conversion begins to occur.

26

Fig. 2.2: Phase matching efficiency plot for the DFG interaction plotted against multiples of &.

2.3 KTiOPO4 and Rubidium doped KTiOPO4

The nonlinear crystals primarily used throughout the work in this thesis are KTP and RKTP. KTP was first synthesized in 1890 [15], but only began to draw attention in the 1970s due to its potential for frequency conversion [16]. KTP and its isomorphs are orthorhombic and are in the point and space groups 2mm and Pna21 respectively. KTP has properties which make it an excellent candidate for frequency conversion schemes, particularly those involving high energies. It has broad transparency from the visible to the mid-infrared, typically in the range 350 - 4000 nm. Absorption values for 1.064 µm and 2.128 µm are typically lower than < 0.5 % cm-1 [17,18]. It possesses a high nonlinearity for efficient parametric interactions, with the highest nonlinear coefficient being d33 = 16.9 pm/V. The nonlinear coefficient matrix for KTP is given as [8]:

a = � 0 0 0 0 1.91 00 0 0 3.64 0 0 2.54 4.35 16.9 0 0 0 � pm/V �2.19�

For low energy applications, LiNbO3 is often preferred for frequency conversion due to larger available nonlinearity. However, due to its low damage threshold, energy scaling with LiNBO3 is inherently limited. This is somewhat mitigated by the doping these crystals with Mg to avoid photo-refraction, but typically the damage threshold remains below that of KTP [19]. On the other hand, KTP has previously been found to possess high damage thresholds, around 10 J/cm2 for nanosecond pulses at 1 µm [20]. KTP is also ferroelectric, which allows it to be periodically poled via electric field poling leading to highly efficient conversion. In order to energy scale devices based on these crystals, the aperture of the crystal needs to be increased (Fig. 2.3). This allows

27

larger beams incident on the crystal and therefore lower the fluence of the pump beam. In general, electric field poling becomes more challenging for large aperture crystals often leading to instabilities in the domain structure and in particular, the emergence of domain broadening [21]. This leads to a loss in the overall conversion efficiency of the crystals.

Fig. 2.3: 5, 3 and 1 mm thick PPRKTP crystals (a); Domain structure for 5 mm thick PPRKTP poled for degeneracy

(b).

The magnitude of this domain broadening can be approximated by the product of the ionic conductivity of the material as well as the time of the switching pulse used. The switching pulse duration is typically fixed and therefore lowering the ionic conductivity is necessary to prevent domain broadening. In KTP, due to the vacancies present in the material, a fairly large ionic conductivity exists. However doping with Rubidium ions to fill these vacancies has been shown to lower the ionic conductivity substantially. In turn this has led to RKTP crystals with apertures of 5 mm with excellent domain structure [22]. The refractive indices of KTP and RKTP are governed by their respective Sellmeier equations. For KTP, the most accurate fit the refractive indices take the form [23]: '�� = � + �1 − �) � + Y1 − �) � − �)� �2.20�

Here, λ is the vacuum wavelength. Since the refractive index is also temperature dependent, corrections to the value given by the Sellmeier equation above need to be included. The temperature correction to the refractive index can be expressed as:

Δ' = '3�� − 25�� + '�� − 25��; with '3,� = , �b)b@

b2� �2.21�

In the case of KTP, the constants in the above equation vary depending on the wavelength involved. For wavelengths shorter than 1 µm, the best fit coefficients are found in [23] and the temperature

28

correction coefficients, �b, found in [24]. For wavelengths longer than 1 µm, the fitting coefficients can be found in [25], and the temperature corrections in [26]. Using these refractive index values and combining Equations 2.18, 2.20 and 2.21, it is possible to obtain a DFG phase matching curve for KTP pumped at 1.064 µm which is shown in Fig. 2.4. Degenerate operation is approximately at the QPM period of 38.86 µm.

Fig. 2.4: Phase matching curve for RKTP at room temperature.

29


1. P. A Franken, A. E. Hill, C. W. Peters and G. Weinreich, “Generation of Optical Harmonics,” Physical Review Letters, 7, 4, 15 August (1961).

2. J. A. Armstrong, N. Bloembergen, J. Ducing and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Physical Review, 127, 6, 15 September (1962).

3. J. D. Jackson, “Classical Electrodynamics,” 3rd Edition, ISBN: 978-0-471-30932-1, August (1998).

4. O. Svelto, “Principles of Lasers,” 4th Edition, pp 37, ISBN: 978-1-4419-1302-9 (2010).

5. R. L. Carman, R. Y. Chiao and P. L. Kelley, “Observation of Degenerate Stimulated Four-Photon Interaction and Four-Wave Parametric Amplification,” Physical Review Letters, 17, 26, 26 December (1966).

6. C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Reviews of Modern Physics, 68, 801 (1996).

7. P. E. Powers, “Fundamentals of Nonlinear Optics,” 1st Edition, ISBN 9781420093513, 25 May (2011).

8. M. V. Pack, D. J. Armstrong and A. V. Smith, “Measurement of the .�� tensors of KTiOPO4, KTiOAsO4, RbTiOPO4 and RbTiOAsO4 crystals,” Applied Optics, 43, 16, 1 June (2004).

9. W. H. Louisell, A. Yariv and A. E. Siegman, “Quantum Fluctuations and Noise in Parametric Processes. I,” Physical Review, 124, 6, 15 December (1961).

10. R. W. Boyd, “Nonlinear Optics,” 3rd Edition, ISBN: 978-0-12-369470-6, 28 March (2008).

11. J. M. Manley and H. E. Rowe, “Some General Properties of Nonlinear Elements - Part I. General Energy Relations,” Proceedings of the IRE, 46, 850-860, May (1958).

12. V. Petrov, “Frequency down-conversion of solid-state laser sources to the mid-infrared spectral range using non-oxide nonlinear crystals,” Progress in Quantum Electronics, 42, 1-106 (2015).

13. P. S. Kuo, K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, X. Yu, J. S. Harris, D. Bliss, and D. Weyburne, “Optical parametric generation of a mid-infrared continuum in orientation-patterned GaAs,” Optics Letters, 31, 1, 1 January (2006).

14. R. A. Baumgartner and R. L. Byer, “Optical Parametric Amplification,” IEEE Journal of Quantum Electronics, QE-15, 6, June (1979).

15. J. Nordborg, “Non-linear Optical Titanyl Arsenates,” Doctoral Thesis, Department of Inorganic Chemistry, Chalmers University of Technology, Göteborg, Sweden (2000).

16. F. C. Zumsteg, J. D. Bierlein and T. E. Gier, “KxRb1-xTiOPO4: A new nonlinear optical material,” Journal of Applied Physics, 47, 4980-4985 (1976).

17. G. Hansson, H. Karlsson, S. Wang and F. Laurell, “Transmission measurements in KTP and isomorphic compounds,” Applied Optics, 39, 27, 5058-5069, 20 September (2000).

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19. D. A. Bryan, R. Gerson and H. E. Tomaschke, “Increased optical damage resistance in lithium niobate,” Applied Physics Letters, 44, 847 (1984).


21. G. Rosenman, Kh. Garb, A. Skliar, M. Oron, D. Eger and M. Katz, “Domain broadening in quasi-phase-matched nonlinear optical devices,” Applied Physics Letters, 73, 7, 17 August (1998).


23. T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer and R. S. Feigelson, “Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOPO4,” Applied Optics, 26, 2390-2394 (1987).

24. W. Wiechmann, S. Kubota, T. Fukui and H. Masuda, “Refractive-index temperature derivatives of potassium titanyl phosphate,” Optics Letters, 18, 1208-1210 (1993).

25. K. Fradkin, A. Arie, A. Skliar and G. Rosenman, “Tunable mid-infrared sources by difference frequency generation in bulk periodically poled KTiOPO4,” Applied Physics Letters, 74, 914-916 (1999).

26. S. Emanueli and A. Arie, “Temperature-Dependent Dispersion Equations for KTiOPO4 and KTiOAsO4,” Applied Optics, 42, 6661-6665 (2003).

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3 Radiation damage of nonlinear crystals This section deals with the laser induced and gamma radiation damage threshold measurements of KTP and RKTP. Both are important due to the use of KTP in high energy down-conversion devices. In particular, it is also used in LIDAR schemes involving a parametric device which is in orbit around the Earth, where high incident proton and gamma radiation levels are encountered. As was mentioned, for such schemes operating around 2 µm and pumped with well-established 1 µm lasers, type-II phase matched KTP is widely used as the gain material owing to its high-damage threshold, narrow gain bandwidth, relatively high nonlinearity and broad transparency. Owing to the critical phase matching, multi-crystal walk-off compensating arrangements [1,2] or more elaborate OPO cavities with image-rotation [3,4] are usually employed to increase the output beam quality. Tuning in such arrangements involves precise rotation of pair of crystals in opposite direction and represents an additional and unwelcome complication in the LIDAR systems which need to operate unattended. Periodically poled KTP and RKTP show similar or higher laser induced catastrophic damage threshold as single-domain KTP and RKTP [5,6,7]. The catastrophic optical damage can be readily avoided by judiciously choosing optical intensities in the system design. The high effective nonlinearity in PPKTP and RPPKTP provides for large design margins. For long-term unattended system operation within a space environment it is important to consider other effects which could degrade the system performance.

3.1 Laser induced damage of materials

As mentioned earlier in this work, one of the key limitations for energy scaling is the damage threshold of the optical elements and materials within the setup. The nonlinear crystals which are routinely used in frequency conversion experiments are often of limited aperture and therefore the maximum fluence or intensity needs to be optimized to avoid damage. Furthermore, the damage threshold of nonlinear crystals and other materials is crucial to their micro/nano structuring, allowing one to create waveguides and other exotic structures from them [8]. In general laser damage of materials consists of two types, namely, Catastrophic and Transient damage. Catastrophic damage is characterized by a permanent alteration in the crystal structure and in the context of high energy systems, is undesirable [9]. On the other hand transient damage is non-permanent and can be mitigated as well as removed. It is well known that high intensity green and blue radiation may induce absorbing color centers within nonlinear crystals. This is known as GLIIRA and BLIIRA respectively [10,11]. As has been previously shown, these color centers may be removed with temperature annealing [11]. For KTP, LIDT measurements have been performed for different pulse durations and wavelengths. Since KTP is typically pumped by 1 µm sources for SHG or down-conversion applications, many LIDT results at this pump wavelength, for CW and pulsed experiments, have been reported [12,13,14,15]. However, no LIDT measurements of KTP and RKTP at 2 µm and RKTP at 1 µm have been previously reported. Since down-conversion stages may be subject to multi-millijoules of 2 µm radiation, the LIDT of these materials at 2 µm is important.

32

The damage mechanism within materials differs according to the pulse duration (�o�. Typically

one may expect a square root dependence on the LIDT [16] of the form LIDT ~� ∙ 4�o . The

constant C is specific to the material in question. This relation breaks down however for pulses approaching picoseconds, as shown in Fig. 3.1. This is due to the underlying mechanisms responsible for the laser damage in each pulse time regime. For nanosecond LIDT measurements, the pulse fluence is used to quantify the LIDT value, whereas the intensity value is used for shorter pulse durations owning to multiphoton ionization dominating this pulse regime. The central mechanism responsible for laser damage in ultrashort pulses is avalanche ionization [17,18], however has also been shown play a role in the nanosecond regime [19] .This is a deterministic process and can be modelled via the appropriate rate equations of the avalanche. On the other hand damage effects with longer pulses tend to exhibit a change in structure due to thermal processes, which is primarily initiated by multiphoton absorption.

Fig. 3.1: LIDT dependence on pulse duration for fused silica and CaF2 (left) [16]; Principle damage processes for

varying pulse durations (right) [8].

3.1.1 Nanosecond Laser Damage of KTP and RKTP

As was mentioned, for nanosecond and long pulse durations the underlying mechanism is related to thermal processes which are generally mediated by phonons within the crystal lattice [20]. In general for laser damage breakdown of materials, damage begins with the generation of a plasma. The steps leading to the generation of the plasma varies according to the pulse regime being used. Typically, there is a photo-ionization of the material which leads to the presence of free-electrons. Here, free-electrons refer to those electrons excited into the conduction band of the material. These free-electrons may then absorb photons in a non-resonant process (free carrier absorption) exciting it further into the conduction band [21]. The electrons gain more kinetic energy as more photons are absorbed. Energy and momentum need to be conserved for the interaction between the electrons and photons thus the electron has to transfer momentum by either absorbing or emitting a phonon as a result of absorbing a photon [22]. Once the energy of the electron exceeds the bandgap and

33

ionization energy of the material, the electron can then ionize another electron from the valence band. The process then continues to repeat until an avalanche of electrons is formed within the conduction band. This leads to the formation of the plasma and additional heat deposited into the material via phonons and thermal diffusion. The plasma itself is heated via further free-electron absorption. As a result thermal effects further alter the structure of the material. A graphical illustration of this is shown in Fig. 3.2. Laser damage is in this case a probabilistic or statistical affair. The damage morphology in the nanosecond regime is typically characterized by so-called melting and boiling of damage points [17]. For laser damage with nanosecond pulses, the initial intensity may not be large enough to directly ionize electrons in the material. Thus, in order to initiate damage in this pulse regime, the process relies on the presence of seed impurity electrons already present in the conduction band, or thermally excited from the valence band [23]. This is one reason why damage in the long-pulse regime is statistical rather than deterministic.

Fig. 3.2: Graphical illustration of catastrophic laser damage in materials subject to long pulse radiation [24].

Other testing parameters are important to determining the actual LIDT value of a material. Besides the wavelength and pulse duration, these may be the polarization [25], repetition rate [13] and beam size of the incident beam [26,27]. In the context of KTP, the polarization orientation may lead to an increase of other nonlinear processes such as SHG. This may lead to a lower LIDT value due to color centre generation at these wavelengths and increasing the optical absorption in the material [28]. Another consideration with regard to damage of materials is if damage occurs in the bulk of the material or on its surface. Surface damage is more susceptible to damage which originates from defects which are already present, i.e. due to poor polishing. Bulk damage on the other hand may originate from not only defects present in the material but also intrinsic effects such as grey tracking. The process depicted in Fig. 3.2 is how plasma damage of the material generally takes place. However, the events leading up to the generation of the plasma will differ according to pulse duration and the material in question. In the context of KTP which is pumped by 1 µm, the materials electrons are typically excited from the valence band into the conduction band via two and three photon absorption of 532 nm and 1 µm photons respectively. This is assisted by the presence of an intrinsic Urbach-tail in KTP [29]. Proposed damage precursors in this case

34

include self-trapped excitons and Frenkel-pair defects, since they may both give rise to electronic states in the bandgap of the material [30]. Frenkel-pair lifetimes have been estimated to be around 1-10 ms, making them suited as precursors for laser damage [31]. Damage then proceeds in a similar fashion described in the beginning of this section. The material then releases some of this energy via phonons throughout the crystal lattice, leaving it in a state which precedes damage. Further single photons of both 532 nm and 1 µm are absorbed by the material in this state leading to more energy released in the material via phonons. Eventually this generates significant heat in the material which leads to a plasma within the crystal which proceeds to cause catastrophic damage [20]. Bulk damage is especially important for devices which involve focused beams in the material. However, for high-energy parametric devices the pump and generated fields tend to be collimated and un-focused in order to scale the energy without damaging the nonlinear crystal. Thus in this case, damage typically begins on the surface of the crystal. It is for this reason that the LIDT measurements in this thesis were performed on the surface of the KTP and RKTP crystals.

3.1.2 Experimental procedures for measuring LIDTs.

Due to there being only a probability of damage on the sample within the nanosecond regime, S-on-1, R-on-1 and 1-on-1 procedures are commonly employed for LIDT measurements. These are set out in the respective ISO standards in order to allow a valid comparison between different experimental results [32,33]. The ISO standard 21254, consists of two parts with part 1 containing an outline of different testing setups and equipment used for LIDT measurements. Part 2 entails a description of the actual LIDT procedure. The central objective of the ISO standard is to outline a protocol for the LIDT testing of materials such that the result may be easily compared between different equipment and experiments. 1-on-1 measurements are relevant for samples that would be employed in single pulse systems. The Raster scan or R-on-1 measurement is beneficial for measuring the LIDT where there are limited available testing sites upon the sample. The S-on-1 procedure (where S is equal to the number of incident pulses) is naturally situated for crystals that will be employed in multi-pulse frequency conversion schemes, and it is the same procedure used in the experiments within this thesis.

A typical S-on-1 measurement involves a material sample of appropriate size, a coherent source or laser, with accurately known pulse duration as well as wavelength, and a detection scheme to confirm if damage has taken place. A control mechanism via feedback and an optical shutter are also frequently used in order to select the number of pulses per damage site, as well as block the pulse train after damage has occurred. Given below in Fig. 3.3 is a depiction of the S-on-1 testing procedure. The sample space is divided into a number of lattice sites which are evenly spaced by a

value ∆x. This value needs to be chosen well, as damage in the nanosecond regime often leads to

the creation of debris around the lattice site. If ∆x is too small, damage may initiate on debris which

would lead to an inaccurate result. The sample is translated by the factor ∆x for each measurement at a set fluence value. A new row of lattice sites would represent a higher/lower fluence value.

35

Fig. 3.3: General S-on-1 LIDT experimental scheme (left); Typical LIDT curve for two different beam sizes (right)

[34]. If damage occurs, scattered light from the site is picked up by a photo-detector. The number of damage and un-damaged lattice sites are recorded for each fluence value and from these a LIDT curve is obtained. The LIDT value then is deduced from the generated LIDT curve, as seen in Fig. 3.3 and is defined as the fluence where �� 0, where P is the probability for damage and F is the applied laser fluence. The LIDT curve tends to be linear in between the approximate probability range of 0.1 - 0.7. Thus, a linear regression fit may be applied to this portion of the curve to then estimate the LIDT [9].

Since the measurement relies on probabilites in the nanosecond regime, the results obtained are statistical in nature [35,36]. Assuming that there are n sites available for testing and k of those sites were damage during testing, one arrives at a measured probability for damage at a given fluence. This is given as �� = %/'. This is a counting problem which conforms to that described by a binomial distribution. Thus we may write that the probability to obtain k number of positive incidents after n trials (with success probability denoted by p) is described by the binomial probability density function (PDF):

�Y� = � '% � ��1 − ��/ � �3.1�

To further illustrate this, we plot the PDF as a function of the measured probability, which is shown below in Fig. 3.4. The measured probability is given as � = 1/5 , for each case below. The difference is the number of lattice sites available for testing and those that are damaged. As can be seen, by increasing the value of n and k, while maintaining the same ratio p, the PDF curve becomes narrower. In other words, there is more certainty that the value p is the actual probability.

36

Fig. 3.4: Plot of the probability density function for constant p value, but varying values of k and n.

From the PDF curve and the empirical rule, it is also possible to establish an error estimation protocol. The empirical rule states that for a normal distribution, 99.7 % of the data obtained will

fall within three standard deviations (±3σ) of the mean value (p). The PDF curve is normalized so that integration of the curve would yield unity. By selecting an appropriate confidence interval and knowing integration should yield the corresponding value from the empirical rule (i.e. 95% in the

±2σ case), we can determine the associated probability width values. These width values are what give the vertical errorbars on the measured data points. It is also worth noting that the PDF is non-symmetric in general, and therefore the errorbars obtained are also non-symmetric. Horizontal errorbars on data points may be inferred if necessary from general uncertainties in the fluence value, such as flucations in the pulse energy.

3.1.3 Statistical models for nanosecond laser damage

One way of considering how laser damage occurs is through the idea of nanoprecursors or damage defects, which was first proposed in 1972 [37]. The crystal structure is not perfect and damage may arise by initiating on a weak point within the crystal or on its surface. The imperfect structure in the bulk and surface may be attributed to errors during crystal growth and polishing on the surface. These are commonly referred to as nanoprecursors, small defects which are the precursors for catastrophic laser damage. Polishing has been found to significantly influence the overall surface LIDT of optical materials [38], further validating the defect/nanoprecursors idea. We follow the notation outlined in [36], to describe some fundamental statistics and theory behind the defect damage idea. For a Gaussian laser beam, we may express the fluence as: � = �� ¡ �3.2�

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Where, F0 is the peak fluence and w is the 1/e2 beam radius. The area A which now exceeds the intensity damage threshold of a defect (�¢lml£¤) is traced out by a circle coinciding within radius of the beam. The size of this area will vary according to the peak fluence value, and it is given as: � = &¥�ln r ��¢lml£¤ t �3.3�

We can now calculate the average amount of defects which lead to damage within the irradiated area, which is simply given by: )¦ = � × §, where M is the average surface density of defects. As the peak fluence F0 increases, the area A and therefore average number of defects, λd, will also increases. The probability to irradiate a defect with sufficient fluence (to cause damage) within the area A during single shot laser exposure is expressed as �� and can be calculated from Poisson statistics: ��a� = �)¦�¦a � � �3.4�

Here, ��a� is the probability value to hit d number of defects inside an area A when the average

amount of d is )¦. Since the probability to either damage (D) or not damage (ND) the material must sum to unity, we may write: ��Y� + ��wY� = 1 �3.5� The probability to not damage the material can be calculated from Equation (3.4) by simply choosing a value d = 0. That is, there are no defects present so damage in this framework is not possible. Using this assumption and re-arranging Equation (3.5), we may calculate the value of P(D): ��Y� = 1 − ��wY� = 1 − ��0� = 1 − � �¨ �3.6� By now taking into account the dependence of the area A on the applied input fluence �� and relating this to the value )¦, it is possible to determine the probability of damage that is expected to be experimentally measured:

�� = ©0 �ª �� < �¢lml£¤¬.1 − exp r−§® ln ��¢lml£¤¬t �ª �� ≥ �¢lml£¤¬. �3.7�

where ® = &¥� is the area of the irradiated spot. This solution is valid for defects which are degenerate or identical and have therefore same fluence damage threshold. This is a two parameter solution, with the probability of damage depending on the beam size w and the defect density d.

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Assuming that the defects are degenerate is often not a valid approximation due to the intrinsic randomness of their formation and the varying amount of processes leading up to it. Therefore, some models assume that the defects can damage around a certain distribution of fluences with some mean fluence value [39]. This defect distribution has been assumed to take on varying forms, the most commonly used being power and Gaussian distributions. The power distribution leads to a three parameter model, which often leads to an accurate estimation of the LIDT [40]. However, a Gaussian distribution leads to a four-parameter model, offering higher accuracy than the former models and it is this model we will discuss [41].

Fig. 3.5: Varying damage threshold of defects which are ordered as a Gaussian distribution.

A defect distribution with a Gaussian profile is shown in Fig. 3.5. This distribution °�� has a

mean threshold fluence given by T0 and a distribution width of ∆T. This implies that the majority of the non-degenerate defects will initialize damage at the input fluence T0, with some initiating damage at slighter smaller/higher input fluence in the approximate range �� ± ∆�/2. The distribution is subject to the condition:

± °�� a� = a1� �3.8�

Assuming a Gaussian profile with width ∆T and centred on T0, we integrate the above equation which leads to the normalized distribution function:

°�� = r 2a∆�√2&t exp ³− 12 r� − ��∆�/2 t�´ �3.9�

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This distribution is used in a similar derivation of Equation 3.7. We omit the full derivation, which can be found in [41] and only show the final probability result for this model. This probability is given as:

�� = 1 − exp µ− √&a¥�∆�√2 × ± exp µ− 12 j� − ��∆�2 k�¶ ln r��t a�·� ¶ �3.10�

Equation (3.10) is fitted to the experimentally measured LIDT values to obtain an LIDT curve. From the best fit curve, the parameters ¥, a, �� and Δ� may be obtained.

3.1.4 LIDT measurements of KTP and RKTP

The setups for the LIDT measurements at 1 µm and 2 µm are depicted in Fig. 3.6 and Fig. 3.7 respectively. For the measurements at 1.064 µm, a diode pumped, Nd:YAG, seeded master oscillator power amplifier system was employed providing ~8 ns (FWHM) long single longitudinal mode pulses at a repetition rate of 100 Hz. The power delivered to the sample was controlled by a wave-plate/thin-film polarizer (TFP) arrangement. A small percentage of the pump power was reflected in order to monitor the power incident on the sample. The beam was focused with a f = 200 mm spherical lens as well as a f = 150 mm cylindrical lens to account for astigmatism in the beam. The beam waist radius (1/e2) was determined with the 90-10 travelling knife-edge technique and was measured to be approximately 40 µm and 50 µm in the x and y directions, respectively. The M2 value of the beam was measured to be approximately 3.2 for both transversal directions.

Fig. 3.6: Experimental layout for the 1 µm LIDT measurements with pump beam inset.

For the 2 µm measurements, the above described Nd:YAG laser was employed as a pump source for a degenerate OPO. The OPO consisted of a dielectric mirror as input coupler, a periodically poled RKTP crystal and a volume Bragg grating (VBG) as output coupler. The use of the VBG as an output coupler narrowed the output spectrum and improved the output power stability, thus

40

reducing any deviation which would be present in the horizontal error bars. The RKTP crystal was AR-coated for both 1 µm and 2 µm and was periodically poled with a period of Λ = 38.86 µm, resulting in a signal/idler wavelength of 2.128 µm. The un-depleted pump was dumped and the 2 µm output was reflected towards two CaF2 spherical lenses with focal lengths of f = 100 mm and f = 50 mm. The beam waist radius was measured to be approximately 40 µm and 50 µm in the x and y directions respectively. The M2 value of the 2 µm output beam was measured to be approximately 8 and 4.2 for the x and y directions respectively.

Fig. 3.7: Experimental layout for the 2 µm LIDT measurements with signal/idler beam inset.

The dependence of the LIDT on the number of pulses, S, was investigated in detail in [13]. It was found that the damage threshold at 1 µm in bulk KTP decreases until S reaches a value of about 200 and then stays approximately stable and independent of S. This is a, so called, quasi-fatigue effect. Therefore in the S-on-1 configuration, we choose an S value of 200 pulses. Additionally, this number is frequently used in literature and allows (to some extent) a comparison to literature values and is suitable to represent realistic applications and not just single shot experiments. In both setups an electrically controlled shutter was used to allow an opening time of 2 seconds corresponding to 200 pulses. The scattered light was detected by an InGaAs photo-diode (Hamamatsu) in the case of 1.064 µm light and PbSe photoconductive detector (Hamamatsu) in the case of 2 µm radiation. In addition, the crystals were inspected under a microscope after the damage experiments. The samples were polished to optical finish on both x-faces using industry-standard Logitech equipment and employing SF1 Syton chemo-mechanical polishing suspension. All samples were cleaned with isopropyl alcohol and flushed with dry air prior to LIDT testing. The samples were orientated such that the beam waist was positioned on and perpendicular to the crystal x-face. The samples where left at room temperature As previously mentioned, laser ablation of materials tends to leave a debris ring centered around the damage site. This debris needs to be avoided in subsequent damage measurements in order to avoid inaccuracies. The maximum distance that this debris tends to reach is dependent on the delivered fluence. We measured this distance for a pulse energy of 10 J/cm2 and beam size radius (1/e2) of w0x = 40 µm and w0y = 50 µm respectively.

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Fig. 3.8: Debris ring around a damage site on the KTP surface.

As can be seen in Fig. 3.8, the majority of the debris accumulated circularly around the damage site. In our measurement, we found the maximum distance the debris reached to be approximately

350 µm away from the damage site. Thus, we chose 350 µm as the distance between sites (∆x) for

our LIDT measurements. Our crystals had varying dimensions and using this value of ∆x we typically had n = 15-20 sites available for testing per fluence. In some initial experiments we tried to cause bulk damage with a collimated pump beam as well as with a beam focused into the crystal. It was reported elsewhere [42] that a collimated beam with a wavelength of 1064 nm damages the bulk of KTP prior to the (uncoated) surface. However, we were not able to obtain isolated bulk damage in any of our samples. In the cases where the bulk of the crystal was damaged, the damage actually started at the surface and propagated further into the bulk. Previous work has found that the 1-on-1 bulk damage threshold of KTP, with beams polarized parallel to the z-axis is around five times larger than the bulk damage threshold with beams polarized perpendicular to this axis, with 1-on-1 LIDT values > 45 J/cm2 [43].

Fig. 3.9 shows the measured data sets for un-coated KTP and RKTP at 1 µm. Here we present three measurement sets for KTP and three for RKTP. As can be seen from these data sets, and using the definition of the LIDT, we measured a surface LIDT value of ~ 12 J/cm2 for KTP and ~ 10 J/cm2 for RKTP samples. The damage morphology is typical of that caused by nanosecond pulses, with crater formation and melting attributed to damage arising from thermal and acoustic shock wave effects

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Fig. 3.9: Consolidated data set for KTP and RKTP measurements at 1 µm (left); S-on-1 damage layout on the crystal surface (right).

As described in the previous section, we made use of a theoretical model that was developed by H. Krol, et al [41] that builds on the idea that damage initiates due to the presence of nano-defects which could create electronic states below the bandgap energy. Using the model fitting given by Equation (3.10), which assumes that the damage threshold for an ensemble of nano-defects is normally distributed, one can obtain fitting model parameters, such as the defect density = d, the mean damage threshold = T0 and the threshold standard deviation = ∆T. For the particular data set in Fig. 3.10 showing the damage probability as a function of fluence, we obtain values: d = 4 × 104 /cm2, T0 = 12 J/cm2, ∆T = 1 J/cm2.

Fig. 3.10: Single data set for 1 µm LIDT measurement on KTP with fitted model (left); Damage site morphology on the KTP surface at the max testing fluence of 24 J/cm2 (right).

The error-bars in Fig. 3.10 are calculated using the error-method described in the previous section. We chose a confidence value of 68 % corresponding to a 2σ width of the distribution. The error on the probability value in this case was approximately ± 15 %. The damage site at max fluence (24 J/cm2) is also depicted in Fig. 3.10. The change in structure is typical for nanosecond pulses, with clear melting of the crystal structure present.

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Fig. 3.11: LIDT measurement for AR-coated RKTP at 1 µm (left); Damage sites with cracking of the AR coating visible at low fluence (21 J/cm2) (right).

LIDT measurements for AR coated RKTP samples are also shown in Fig. 3.11. A higher LIDT value was measured in comparison to the uncoated samples, with a surface LIDT of 18 J/cm2. The fitting parameters for the AR coating were determined to be: d = 8 × 104 /cm2, T0 = 18.5 J/cm2 and ∆T ≈ 1 J/cm2. According to the model, the AR coating had a higher defect density value compared to the un-coated samples. Therefore the higher LIDT value in this case may be attributed to the larger band gap of the AR coating material in comparison to the bandgap of RKTP. In this case a larger site spacing needed to be chosen due to severe cracking of the AR-coating. Thus there were fewer lattice sites available for testing resulting in a larger overall error in the probability values. Results for the 2 µm LIDT for KTP and RKTP are presented in Fig. 3.12. The data set contains two measurement sets for KTP and two for RKTP. As can be seen from the graph, we measured a surface LIDT of 10 J/cm2 for both KTP and RKTP. The damage probabilities at different fluence values are similar to those measured in the 1 µm case. The damage morphology was similar to the 1 µm result.

Fig. 3.12: Consolidated date set for KTP and RKTP measurements at 2 µm (left); Single data set for 2 µm LIDT measurement on RKTP with fitted model (right);

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We again applied the model fit to the measured data, shown here as an example is the curve for RKTP (Fig. 3.12). The model yields parameter values of: d = 3×104 /cm2, T0 = 10.5 J/cm2, ∆T = 1.1 J/cm2. The defect density value is very similar to that found in the 1 µm case. This is to be expected since all samples were polished by the same equipment and in the same polishing run. The experimental results show that the LIDT at 2 µm is essentially the same as at 1 µm. This might be somewhat surprising, considering that at the wavelength of 2 µm it would take absorption of 6 photons to cover the bandgap in KTP. This assumes the widely used optical damage physical picture which contains two steps: electron transfer from the valence band into a quasi-continuum of states close to the conduction band and subsequent heating by free-carrier absorption. The free carrier absorption coefficient is proportional to the free carrier concentration and the free carrier absorption cross-section, which, in turn, scales as > � for optical frequencies above the plasma frequency, which is a typically valid approximation [44]. Thus for generating the same heat power for the same optical fluence as at 1 µm one would need 4-times lower free-carrier concentration for 2 µm pumping. This difference of 4-times is way too small to give the same LIDT at 2 µm if 6-photon absorption indeed were the main mechanism generating free carriers. A more plausible scenario for the heat generation and damage both at 1 µm and 2 µm is then by linear absorption and non-radiative relaxation from surface defect states. In conclusion we presented surface S-on-1 LIDT measurements on KTP and RKTP at 1 µm and 2 µm. We measured the LIDT in the QPM configuration, with all fields polarized along the z-axis of the crystal. We summarize the main results in Table 3-1.

Table 3-1: Surface LIDTs and fitting parameter values for KTP, RKTP and AR-coated RKTP

Measured values Fitted model parameters

Material LIDT (J/cm2) Defect density (/cm2) T0 (J/cm2) ∆T (J/cm2)

1 µm 2 µm 1 µm 2 µm 1 µm 2 µm 1 µm 2 µm

KTP 12 10 4 x 104 4 x 104 12 10.4 1 1

RKTP 10 10 3.8 x 104 3 x 104 10.5 10 0.9 1.1

AR coated RKTP

18 - 8 × 104 - 18.5 - 1 -

As mentioned, for this QPM configuration no isolated bulk damage was observed. The large surface LIDT values found for both crystals are beneficial for the construction of high-energy, frequency conversion schemes which utilize the QPM configuration. In general, the theoretical and experimental estimations of the LIDT used in these experiments make use of the average fluence of the Gaussian beam utilized. Also, it does not consider the actual time structure of the incident pulses used, i.e. how much of the pulse was incident in the material at the time of damage. In order to circumvent this issue, a recent technique called Spatio-Temporally Resolved Optical LIDT was

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developed (STEREO LIDT) [45]. This technique considers the actual damage location within the test beam as well as the time position of the incident pulse at the time of damage, leading to a more accurate LIDT value. We mention this technique here as a possible avenue for further improving the results obtained in this thesis.

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3.2 Gamma irradiation of KTP and RKTP

Further improvement of climate models and better understanding of the shifting radiative power balance in the Earth’s atmosphere requires increasingly accurate measurements of greenhouse gases (CO2, CH4, etc.) concentration distribution in the atmosphere on the global scale [46,47]. Ground-based and airborne LIDAR schemes, based on differential absorption lidar (DIAL), are currently employed to achieve the required sensitivity [48,49,50]. Determining the global concentration distributions and their dynamics however requires spaceborne LIDAR instruments to improve on the data produced by the passive spaceborne instruments [51,52]. For this reason several DIAL LIDAR missions have been proposed and are under active development [46,53]. Targeting the 2 µm spectral region is beneficial due to the possibility to simultaneously access several gas species, including CO2, CH4 and H2O. A spaceborne DIAL LIDAR instrument typically requires substantial pulse energy of tens of mJ delivered in transform-limited nanosecond pulses with rapidly tunable central wavelength. Although solid state and fiber lasers based on Tm3+, Ho3+ are being investigated for LIDAR applications [54,55], the coherent light sources based on optical parametric oscillators and amplifiers remain the most viable technology for the active spaceborne multispecies gas sensing instruments today. Pulsed parametric sources based on optical parametric oscillators and optical parametric amplifiers are well suited for such applications, due to their high-energy output, high overall efficiencies, broad tunability and access to wavelengths within the mid-infrared. As previously discussed all nonlinear crystals, to a different degree, are susceptible to formation of infrared-absorbing color centers produced by exposure to parasitic blue-ultraviolet optical radiation and ionizing radiation in space environment. The color centers which are responsible for near-infrared absorption in KTP isomorphs are related to polarons formed by self-trapped photo-generated electrons and holes in proximity to Ti4+ and O2- ions, respectively [11,56,57]. It was shown that Rb-doping of KTP with concentrations of about 1% strongly reduces accumulation of the induced color centers, making PPRKTP superior in terms of quality of the large-aperture QPM structures and in terms of performance at high-average and peak-powers [58]. In contrast to catastrophic laser-induced damage, it is well known that the color centers in KTP isomorphs can be removed by annealing at temperatures above ~100ºC. For spaceborne LIDAR systems, the nonlinear crystals are not only subject to laser radiation, but also to the ionizing radiation from energetic electrons, protons and heavier ions originating from the Sun and cosmic rays, as well as gamma rays produced by collisions of the energetic particles with ions in the upper layers of the Earth atmosphere [59]. The radiation can create free carriers and introduce new defects in the nonlinear materials which, in turn give rise to additional linear absorption and leads to degradation of the device performance. The influence of gamma and proton radiation on transmission properties in commercially tradeable KTP and other nonlinear crystals has been previously studied with the focus on second-harmonic generation at 532 nm [60,61]. It was found that the SHG efficiency decreased by about 15% after exposure to 139 krad dose of gamma radiation, primarily caused by the decrease in the linear transmission in the spectral range around 500 nm. On the other hand the transmission of KTP at 1 µm was only weakly modified by doses as high as 1000 krad [62]. Susceptibility of the periodically poled KTP and Rb:KTP has not

47

been investigated so far. Assuming that some of the induced absorption is owing to the color-centers of similar nature as produced by the UV radiation we might expect that it would be possible to devise a procedure for thermal annealing whereby the effects of the ionizing radiation are mitigated.

3.2.1 Gamma irradiation experiments

Before discussing the experimental procedures it is important to estimate the cumulative dose and the dose rate that the LIDAR instrument could be exposed to on board a satellite. Most of the protons and electrons are trapped by the Earth’s magnetic field within van Allen radiation belts, so the total expected radiation dose during the mission strongly depends on the actual orbit of the satellite, shielding, as well as the activity of the Sun and changes in the Earth’s magnetic field. For instance, the radiation dose measured on a satellite in a geostationary orbit (altitude of about 36000 km) was measured to be 50 krad/year by a dosimeter, which was shielded by a 1.89-mm-thick Al sheet [63]. Measurements on the satellite, with the highly elliptical orbit traversing van Allen radiation belts, gave the effective radiation dose of 135 krad/year under 2-mm thick Al shielding [64]. The majority of the Earth observation satellites, including passive atmospheric gas sensors (ENVISAT, OCO-2, GOSAT), are launched in the low Earth close to polar orbits (altitude of 100 -1000 km) where the expected radiation dose is much lower, about 2 krad/year for 2-mm-thick Al shielding [65].

Fig. 3.13: GMA2500 irradiator with lead/aluminum protection wall (left); GR50 irradiator (right).

Although the components in orbit are primarily exposed to energetic electron and proton radiation, it has been shown that gamma radiation exposure emitted by 60Co sources, with photon energies of 1.17 MeV and 1.33 MeV, produces very similar radiation damage effects at equivalent doses [66]. Therefore gamma rays are suitable and a much lower-cost proxy for evaluating radiation hardness of the components. The irradiation experiments were performed at the MEGA and MILGA facilities at the Office National d'Etudes et de Recherches Aérospatiales/The French Aerospace Lab (ONERA). The facilities are equipped with commercial 60Co sources GMA2500 (dose rates 3.6 rad/h -10.8 krad/h) and GR50 (dose rates 0.36 rad/h - 1.8 krad/h) (Fig. 3.13). Both facilities are

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panoramic allowing for large volume irradiation (full room volume). Dose cartography of both facilities is calibrated annually utilizing a Radcal active system, and typical transmission and reflection measurements have an error of ± 0.01 %. In order to determine the dosage rate, the distance between the sample and gamma source was varied according to calibration charts. Three dosages were chosen, specifically 10, 30 and 100 krad in accordance to the testing recommendations by the European Space Agency [67]. Moreover, similar doses have been used in the previous experiments reported in the literature [60, 61]. Commercial flux-grown KTP and Rb:KTP wafers were cut to the sizes of 12×7×5 mm along x-y-z

crystal axes, respectively. The optical surfaces corresponding to [100] planes were optically polished and left uncoated. One KTP and one Rb:KTP crystal was then periodically poled with the ferroelectric domain periodicity of 38.86 µm for type-0 degenerate parametric down-conversion from 1.064 µm to 2.128 µm. The crystals were irradiated in three steps each time approximately tripling the radiation dose, as recommended by the ESA testing protocol [67]. Following the gamma irradiation of the nonlinear crystals with a predetermined dose, the transmission of each was measured. The transmission was measured using a Lambda 1050 Perkin-Elmer spectrophotometer coupled with an integrating sphere with a diameter of 150 mm. The spectrometer used has a maximum resolution of 0.05 nm in the visible and 0.2 nm in the near-infrared ranges. A Si-photodiode was used in the visible while an InGaAs photodiode in the near infrared. The longest accessible wavelength for this instrument was 1.4 µm. After the maximum irradiation dose of 100 krad, and transmission characterization, single domain KTP and RKTP crystals were annealed at 150ºC for 2 hours. The remaining crystals were kept at room temperature and their transmission characterized again after 24 hours. After the annealing, the transmission of the samples was measured again in the spectral range from visible to mid-infrared. For the transmission measurements in the mid-infrared we employed a FTIR spectrometer (Perkin-Elmer Spotlight 400). In all cases the transmission was measured for the optical beam propagating along the crystal x-axis. All samples were irradiated in three steps with the calibrated dose rate of 400 rad/hour to reach the dose of 10 krad. The first step lasted 25 hours, following which the samples were removed and their transmission immediately measured. After this, the irradiation continued for another 50 hours at the same dose rate to reach the cumulative dose of 30 krad. Again the transmission was measured immediately. Then the exposure continued for another 183 hours to reach the final cumulative dose of 100.32 krad. The dose rate is on the high-side of the “low-rate window” recommended for the radiation hardness assurance tests [67]. It is much higher than the dose rates of about 4.8 rad/hour and 0.3 rad/hour measured on satellites in the geostationary and low-earth orbit, respectively [63,65]. The optical transmission spectra in the visible and near-infrared for the single-domain KTP and RKTP are shown in Fig. 3.14. The corresponding spectra for the periodically poled crystals PPKTP and PPRKTP are shown in Fig. 3.15.

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Fig. 3.14: Measured transmission spectra for KTP (a) and RKTP (b) for varying gamma radiation doses.

The black curves in these figures represent the spectra measured before gamma irradiation. The kink seen in the spectra at 860 nm is an artifact related to the change of the detectors/gratings in the spectrophotometer. The strongest decrease in absorption in all samples was in the visible spectral range ranging from 380 nm, i.e. close to the bandgap and down to the wavelength of 800 nm. The largest increase in absorption happened after initial irradiation step by 10 krad. Increasing the dosage we observed that the transmission tended to saturate at dosage of 30 krad. Thus, further increasing the dosage to 100 krad showed a minor effect on the overall transmission. After gamma irradiation single-domain KTP and RKTP crystals were thermally annealed at 150ºC for 2 hours and the transmission of each sample was re-measured. Both samples showed improved transmission in the visible range, approaching the measured initial transmission values prior to gamma irradiation. The transmission spectra before and after gamma irradiation in PPKTP and PPRKTP are shown in Fig. 3.15 (a) and (b), respectively. The conditions of the irradiation were equivalent to those used in the case of single-domain samples. The qualitative behaviour of the irradiated periodically structured crystals is very similar to that observed in the single-domain samples. Namely, the absorption is induced in the same spectral range and it saturates after irradiation with the dose of 30 krad. After reaching the target cumulative dose of 100 krad and transmission characterization the samples where left at room temperature for 24 hours and their transmission was measured again. The results show that the induced absorption does not decrease appreciably, i.e. color centre relaxation is very slow at the temperature of 24ºC.

50

Fig. 3.15: Measured transmission spectra for PPKTP (a) and PPRKTP (b) for varying gamma radiation doses.

For greater clarity, we have summarized the irradiation results for all samples which are shown in Fig. 3.16. We show the measured transmission difference between the initial and 30 krad irradiated samples, since this is where saturation of the transmission was found. For KTP and RKTP we measured a maximum transmission difference approaching 2 % and 1.5 % respectively at around 600 nm. For PPKTP and PPRKTP, we measure a maximum transmission difference of 1.5 %. Although radiation-induced absorption is systematically lower in poled crystals, the difference is too small to warrant strong conclusions.

Fig. 3.16: Measured transmission difference from the initial to 30 krad irradiated samples (a); post irradiation FTIR

scan over the infrared range of the samples (b). The difference might be related to the different concentration of native defects in the crystals. Although single domain and periodically poled samples came initially from the same crystal wafers, for poling we always select parts of the wafers with the lowest ionic conductivity, which directly related to the parts of the wafer with the lowest concentration of potassium and oxygen

51

vacancies. It has been shown that these vacancies play a crucial role in stabilizing color-centers induced in KTP isomorphs by UV light [11]. Finally, the sample transmission at wavelengths longer than 1 µm was measured using a FTIR spectrophotometer. The results are shown in Fig. 3.16 (b). The single domain samples have been annealed at 150ºC for 2 hours before this measurement while the PPKTP and PPRKTP were kept at 24ºC without annealing step. We paid particular interest to any long term changes around 2 µm, due to the relevance to 1 µm-pumped down-conversion schemes for spaceborne LIDAR instruments. The measurement showed that there is no discernible change in transmission in this spectral region even in the periodically poled samples without high-temperature annealing. The transmission spectra correspond to those reported in KTP isomorphs without any ionizing irradiation [68]. There are several possible mechanisms by which gamma rays can interact with dielectrics, like KTP isomorphs. To be more specific we used the XCOM program by NIST, to calculate gamma photon interaction cross sections for different processes in KTP [69]. As can be seen from the calculated dependencies in Fig. 3.17, the main contributing process for the gamma photons emitted by the 60Co source would be Compton scattering with some small contribution from electron-positron pair production.

Fig. 3.17: Interaction cross sections for different processes in KTP as a function of gamma-photon energy.

Compton electron energy spans a wide energy spectrum with energies of up to 60% of the gamma quantum for scattering at 180º. From the cross section data in Fig. 3.17 we can estimate that the Compton electron production rate will be about 4×108 s-1cm-3 at the incident gamma dose rate of

52

400 rad/h. Some of those electrons at high energies will leave the crystal; the others will produce secondary electrons which eventually will end up building up a space charge in the crystal volume. It is well-established that Compton scattering of gamma rays can produce a space charge and associated strong internal electric fields in dielectrics [70, 71]. The space charge field will relax with a characteristic Maxwell relaxation time, � = --�/¸, where -, -�, ¸ are the dielectric constant, the permittivity of free space, and the electrical conductivity of the crystal, respectively. In KTP isomorphs used in this study with the typical ionic conductivity of 10-6 S/m and the low-frequency dielectric constant in the range of 103 [72], the relaxation time will be in the range of 10 ms. First it should be stressed that the electric field produced by the space charge in our periodically poled samples was not strong enough to modify the QPM structure. This has been verified by not finding any difference in the quasi-phase matched second harmonic generation efficiency before and after radiation exposure with the dose of 100 krad. The spatial charge relaxes in KTP and RKTP by redistribution of potassium ions K+ and vacancies, (VK-) as well as free charge carriers both, intrinsic and induced by the radiation in the conduction and the valence bands. Part of the secondary electrons will have low enough energy (electron work function in KTP is about 4.5 eV [73]) to be bound in the valence orbitals of the TiO6 structural groups, which were ionized before by the gamma radiation or the Compton electrons. In KTP the electronic states of these structural groups are mostly contributing to the states at top of the valence band and at the bottom of the conduction band [74]. The electrons with higher energy could be captured with simultaneous generation of electron-hole pairs in the process similar to impact ionization in solids. The fact that the gamma-induced absorption centers can be readily annealed at the temperatures as low as 150ºC, as well as similarity of the absorption spectra, indicate that these centers could be of the same origin as those produced by optical irradiation with intense blue-UV pulses. These centers are related to self-trapped free electrons and holes at the Ti4+/Ti3+ and O2-/O-

, respectively, with stabilization provided by mobile potassium vacancies [11,75,76]. Comparison of the induced absorption measured in our experiments with the measurements reported in the literature [60,61,62] indicates that the absolute value of the induced absorption depends much stronger on the gamma radiation dose rate than on the cumulative dose. That would be the case if the processes of free carrier generation, recombination as well as color center self-annealing by the generated phonons are competing. For different radiation dose rates the balance among these processes will be reached at different total radiation doses and at these doses one would expect saturation of the induced absorption. To summarize this chapter, we have investigated the LIDT of KTP and RKTP at 1 and 2 µm in the nanosecond regime. The LIDT of KTP and RKTP at 1 µm were measured to be 12 J/cm2 and 10 J/cm2 respectively, while at 2 µm were both measured to be around 10 J/cm2. Furthermore, we also studied the optical absorption induced by gamma radiation in KTP, RKTP, PPKTP and PPRKTP nonlinear crystals with the aim of radiation hardness assurance tests for possible application in spaceborne 2 µm LIDAR missions. The cumulative radiation doses up to 100 krad with the dose rate of 400 rad/hour which were employed in this study significantly exceed the levels expected in a low Earth orbit where such instruments will be located. The measurements

53

reveal that the main effect of the gamma radiation at the levels employed here is to induce additional absorption in the visible spectral range, while the changes in optical transmission in the range between 1 µm and 4 µm were not measurable. Moreover, the structure of periodically poled crystals was not impacted by the radiation. Investigation shows that the induced color centers could be readily annealed at temperatures of 150ºC and higher, while they remained stable at room temperature. The characteristics of the induced color centers seem to be very similar to those generated by high intensity picosecond optical pulses in blue-ultraviolet spectral region and are attributed to self-trapped excess electrons and holes in the vicinity to Ti4+/Ti3+ and O2-/O- centers in the TiO6 structural groups and possibly self-trapped excitons.

54


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4 Optical Parametric Oscillators and Amplifiers Development of high-energy, narrowband coherent mid-infrared sources has rapidly increased over the past few decades. This is largely motivated by the applications where these sources may serve, such as in LIDAR, surgery/medical and defence to name a few [1,2]. These applications require tunable high energy sources, while still maintaining excellent spectral and temporal characteristics. Typical sources which fulfil these criteria include 2 µm Tm3+/Ho3+-doped solid-state and fiber lasers, as well as parametric devices such as OPOs [3]. OPOs are of considerable interest due to the ability to deliver high-energies over a very broad tuning range, which is impossible or limited with other coherent sources. This radiation can be further amplified in OPA stages, yielding the high energies required. Such a MOPA scheme allows for efficient energy extraction from the pump without sacrificing spectral and spatial quality of the generated radiation [4]. Two-stage down-conversion schemes pumped with well-established 1 µm lasers, are attractive to reach deeper into the mid-infrared [5,6]. Down-conversion schemes to the mid-IR often employ quasi-phase matched KTP as the gain material owing to its high-damage threshold, high nonlinearity, good transparency and possibility to fabricate such crystals with large optical apertures [7]. In this chapter we introduce some background theory and characteristics related to OPOs and OPAs. Following this we report on a highly efficient MOPA configuration with two tunable narrowband outputs at 2 µm by employing periodically poled RKTP. We describe the optimization of the OPO stage in the MOPA system in order to avoid cascaded four-wave mixing (FWM), which has tendency to emerge in type-0 QPM OPOs operating close to degeneracy [8,9]. In particular, we show that such FWM processes can be suppressed by limiting non-collinear interactions in the OPO stage.

4.1 OPO/OPA theory

OPOs are .�� devices which share many similarities with lasers. They consist of a pump source, cavity and gain medium in the form of a nonlinear crystal. First designed in 1965 [10], OPOs have attracted considerable attention due to a number of favourable properties in comparison to lasers. They provide coherent radiation with broad tunability over the entire transparency range of the nonlinear crystal. Tuning is done through either temperature or angle tuning depending on the crystal used. Since the interacting fields fall within this transparency range, OPOs typically experience negligible absorption and therefore are somewhat immune to thermal lensing issues encountered in lasers.

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Fig. 4.1: Doubly resonant Fabry-Perot OPO layout.

Similarly to lasers, OPOs are subject to boundary conditions due to the cavity and therefore possess a threshold. That is, the resonant fields within the cavity must reproduce themselves in amplitude as well as phase after a single cavity round trip. As shown in Fig. 4.1, the process begins with the

insertion of the pump field �)o� into the OPO cavity. This leads to the formation of signal ()p� and

idler �)�� photons through parametric fluoresence and OPG [11]. Provided a sufficient pumping rate and that the cavity reflection is high enough to overcome the losses experienced by the fields, the amplitudes of the resonant fields will grow and the device will overcome threshold. After a number of cavity roundtrips, the device will reach steady-state similar to that found in lasers [12]. In the case of OPG, if we consider our fields to be monochromatic plane-waves the evolution of the field amplitudes along the crystal length is described by the coupled wave equations which are given by [13]: ��o�� = − 7o2 �o + �>oalmm'o� �p�� 4.1 a� ��p�� = − 7p2 �p + �>palmm'p� �o��∗�� 4.1. b� �� = − 7�2 �� + �>�almm'�� o�p∗�� 4.1. c�

where, α is the linear absorption of the respective field. These equations describe the evolution of the fields beyond the small-signal limit. Beyond the small-signal limit implies that the pump field is sufficiently intense to cause saturation in the gain of nonlinear medium. However, it is instructive to solve the above equations for a single pass through the nonlinear crystal and in the regime of no pump depletion. In this case, one attempts a trial solution of the form: �p~�b� , which is inserted into a second-order differential equation, which proceeds the coupled wave-equations above. This

leads to a general solution of the form: �p = ��b¹� + Y�b¡�. The coefficients are determined from the boundary conditions imposed on the system. In this case, the boundary conditions are �p�� = 0� = �p�0� and �� = 0� = ��0�. We define two variables in order to re-write the exponential factors in Equations 4.1 a- c, which are: Γ� = %p%�»�o� » = 8&�almm�'o'p'�)p)�-�� qo = ¼qo ; ° = 4Γ� − �Δ%/2�� 4.2�

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The m coefficients can be re-written in these variables and give ½3,� = � �¾� − X�� ± °. Making these

substitutions, applying boundary conditions and assuming � = 1, gives solutions of the field amplitudes as a function of the crystal length L [14]: �p�� = ³�p�0� rcosh�°�� − � ∆%2° sinh�°��t + � %p�o��∗�0�° sinh�°��´ ��∆��/�� X��/� �4.3�

�� = ³��0� rcosh�°�� − � ∆%2° sinh�°��t + � %��o�p∗�0�° sinh�°��´ ��∆��/�� X��/� �4.4�

It is now possible to derive an expression for the parametric gain. For the signal field, this is simply defined as: ¿p�� = |�p��||�p��0�| − 1 �4.5�

For simplicity we assume that the fields experience no absorption, eliminating the latter exponential term. Applying the parametric gain definition to the Equations 4.3 and 4.4, and simplifying yields the following expression for the parametric gain:

¿p�� = ³1 + r∆%2°t� ´ sinch��°�� = 1 + Γ� sinh�:�4Γ� − �Δ%/2��;Γ� − �Δ%/2�� 4.6�

The gain experienced for the signal field is thus a maximum when Δ% = 0. If we assume perfect phase matching Δ% = 0 and that Γ� ≫ 1 i.e. the high gain limit, Equation 4.6 reduces to ¿p�� = �1/4��Á. Conversely, in the low gain limit where Γ� ≪ 1, the expression for the gain

takes the form ¿p�� Γ��. This expression shows that the gain of the signal evolves exponentially throughout the length of the crystal in the high gain limit, but has a quadratic dependence in the low gain limit. This solution is not valid beyond the small signal regime, where the pump field starts to experience significant depletion. In this case, the pump amplitude cannot be assumed to be constant. Solutions of this exist in the form Jacobi-elliptic functions [15]. By measuring the signal gain as a function of the pump intensity and then making use of Equations (4.2), (4.3) and (4.4), it is possible to determine the associated almm value of an OPA experiment [14]. The χ2 process responsible for parametric amplification [16] is difference frequency generation (DFG), which is shown in Fig. 4.2.

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Fig. 4.2: Illustration of the difference frequency generation process.

DFG involves inserting a pump field with frequency >o and seed field (typically the signal field) >p into a nonlinear crystal. This leads to amplification of the inserted seed field, as well as the

generation of a new frequency, the idler field >� = >o − >p. For non-degenerate OPAs, the

amplification process is not sensitive to the input phase of the seed field. In this case the phase of the DFG field (>p above) will automatically adjust its phase in accordance with that of the pump,

still leading to overall amplification of the seed. However, in the degenerate case, where >p =>� , the process is phase-sensitive. For multi-mode pumping it is challenging to maintain a fixed phase relationship between the seed and pump fields during propagation through the medium and thus one would expect a lower amplification factor for the seed. Thus degenerate amplification is typically undesirable for high-energy down conversion schemes. However, degenerate OPAs do have some use in the generation of so-called squeezed states of light and avoiding quantum noise in sensitive applications [17,18]. Ultimately, OPAs are limited in scalability by the damage threshold of the nonlinear material, back-conversion and gain-guiding [19].

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4.2 Nanosecond OPOs

4.2.1 Energy characteristics

The equations above also assume no time structure inherent in the fields. For pulsed OPOs, the solution depends on the temporal regime encountered, i.e. nanoseconds, picoseconds, etc. For nanosecond OPOs which are singly resonant, it has been shown that the threshold energy value takes the form [12]:

-"Ã = 0.6�:>o� + >p�;¼�� G 25�£ÄÅ�� + 12 ln 2Æ�1 − �� U� �4.7�

where, � is the FWHM pulse duration, >o and >p are the pump and signal beam radii (1/e2)

respectively, L is the crystal length, �£ÄÅ is the optical cavity path length, R is the reflectivity of the output coupler and A is the roundtrip loss factor. By experimentally measuring the threshold of a pulsed OPO, it is possible obtain an estimate of the almm value. This expression is valid for singly resonant OPOs which are treated in this thesis. Threshold values for doubly and triply resonant OPOs for different time regimes can be found in [20]. For nanosecond based OPOs, the pulse length is typically much longer than the cavity round trip time. This means that the same laser pulse interacts with itself each round trip, leading to very high intra-cavity powers. After multiple round trips, the OPO approaches a steady-state similar to that found in lasers [21]. Since the interaction is also phase sensitive, the temporal phases of the fields will also reach some steady-state value where the energy transfer between the fields is most efficient. This relative phase value has been shown to be where +o − +p − +� = ±½&/2, where m is an odd integer value [22]. In order to

quantify the performance of the OPO, it is useful to introduce some parameters. The first is the conversion efficiency of the OPO, which is given as: Çlmm = -pÈÉ¤ + -�ÈÉ¤-oÊË �4.8�

where, -pÈÉ¤ and -�ÈÉ¤ are the measured signal and idler energies out of the OPO and -oÊË is the input

pump energy. In practice, high conversion efficiencies are often difficult to reach due to the presence of back-conversion [23]. Back-conversion is an undesirable effect and occurs when the generated signal and idler fields become strong enough to start converting back into the pump field. For this reason, the crystal length and mode areas of the fields need to be chosen accordingly, so as to scale the generated energy without any back-conversion taking place. Back-conversion also leads to further distortion in the generated beam profiles. In the plane-wave approximation and for a single pass pump, Fabry-Perot singly resonant OPO, the maximum conversion efficiency can reach 100 %, with w = �&/2��. Here N is the number of times the OPO is pumped above

threshold, i.e. w = -o�//-ÌÍ. In the same situation, for a doubly resonant OPO, the maximum

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conversion efficiency which can be reached is 50 %, with w = 4. In reality, Gaussian pump distributions are used and this needs to be considered in determining the efficiency limit of these devices. Taking this into account gives, for the singly and double resonant OPO cases, 71 % at w = 6.5 and 41 % at w = 12.5 respectively [20,24, 25 ]. Another parameter which is routinely used to quantify the OPO is the pump depletion efficiency, given as: Ç¢lÎ = 1 − -oÈÉ¤-oÊË Ï �4.9�

where -oÈÉ¤ is the measured depleted pump energy and -oÊË is the input pump energy. The factor X

is a linear loss factor of the OPO, which can be measured by simply measuring the pump energy before and after the OPO while the OPO is below the oscillation threshold. In general, for a lossless OPO, the pump depletion and conversion efficiency will be equal.

4.2.2 Pulsed OPO design considerations

There are some design considerations to consider in order to realize a nanosecond OPO with high energies. In general, the cavity length of the OPO should be kept as short as possible. From Equation 4.7 this clearly implies that the threshold of the device will be minimized. The reason for this is that a single pump pulse will then undergo multiple round-trips, essentially allowing the same pump pulse to interact with itself within the crystal. This leads to higher intra-cavity pump energies which in turn lowers the threshold of the OPO. Another way to phrase the same argument is that the build-up time of the OPO must be shorter than the pump pulse duration, which is achieved by keeping the cavity length short. On the other hand, by shortening the cavity length to ensure low operating thresholds one increases the cavity Fresnel number, which is given as: w· = Ð¡�� . Here, a is the beam mode radius, λ is the wavelength of the beam and L is the cavity

length. Additionally, in order to scale energies within these devices, large beam sizes need to be used to prevent laser damage. This further increases the value of the Fresnel number. A high Fresnel number implies that there is little diffraction taking place on higher-order modes of the beam. Parts of the beam will tend to de-couple and oscillate independently from one another. This leads to un-correlated phase and amplitude variations across the beam profile which in turn results in less stable output and poor beam qualities from the OPO. Typical methods to circumvent this issue involve the use of Rotated Image Singly Resonant Twisted RectAngle (RISTRA) cavities or spatial walk-off compensation geometries [26].

4.2.3 Bandwidth and temperature tuning

OPOs have the advantageous property of being able to undergo temperature tuning, effectively allowing varying signal and idler wavelengths to be generated for a fixed poling period. In order to obtain an expression for the tuning rate of the signal with respect to temperature, we take the

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derivative of the phase matching condition with respect to temperature T and applying the chain rule, one obtains the following: �� ∆%�� = u�%o�>o

�>o�� − �%p�>p�>p�� − �%��>�

�>�� v � �4.10�

Since the pump wavelength for OPOs tends to remain fixed, we can assume that Ñ!�ÑÌ = 0. We also

re-write >� = >o − >p using the fact that the energy is conserved. Applying these assumptions and

simplifying yields the following expression: 1Ò �� ∆%�� = �>p�� , where Ò = − �%p�>p + �%��>� �4.11�

Since the wave vector k contains both refractive index n and wavelength which are both temperature dependent, we take the derivative of Equation 4.11 to finally obtain: 1Ò� u>o �'o�� − >p �'p�� − >� �'�� v = �>p�� 4.12�

This gives the expected temperature tuning rate of the signal. It is worthwhile to note that close to degeneracy where >p � >� , the parameter b tends to zero. Thus the tuning rate of the OPO becomes very fast near degeneracy. This may lead to instabilities in the OPO output. For this reason, it is often preferred to operate the OPO slightly off degeneracy. As Equation 4.12 suggests, at degeneracy the tuning rate becomes infinite, however this is not true as one would also need to account for higher order derivatives, responsible for effects such as group-velocity dispersion. Temperature tuning curves for RKTP poled at degeneracy have been previously measured [27]. The experimentally measured curves as well as those predicted in other works is shown below in Fig. 4.3.

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Fig. 4.3: Theoretical and measured temperature tuning curves for RKTP [27].

Bandwidth considerations are also important for OPO design. In particular, some applications require radiation with tailorable bandwidth. As was mentioned narrowband sources are required for spectroscopy and other meteorological applications. For OPOs with high energy outputs, narrowband spectral output is typically achieved with intracavity etalons or volume Bragg gratings [28, 29]. In essence, a Bragg grating consists of a periodic modulation of the refractive index which is written into some photo-thermal-refractive material. As the word ‘volume’ suggests, VBGs consist of a Bragg grating written in a bulk material, as opposed to fiber-Bragg gratings which are written within an optical fiber core. VBG technology is well established, and is routinely used in nanosecond high-energy systems where narrow bandwidths are required, owing to their high damage threshold in this regime (~ 10 J/cm2 [30]) amd ability to deliver linewidths in the order of < 1 nm. By utilizing a VBG as the cavity element for an OPO, the gain bandwidth is narrowed substantility. Thus the OPO output will also narrow and oscillate on fewer modes, also improving the overall stability of the device. In order to gain a better understanding of bandwidth effects within an OPO it is useful to first define a gain bandwidth from the generated OPG. In order to do this we make use of Equation 2.18 and find the associated ∆% value where amplification is halved. This is given as:

∆%3/� = 2Ôln 2 Γ� �4.13�

We can then perform a Taylor expansion of ∆% around the point ∆% = 0. Assuming the interaction is collinear, this yields:

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Δ% = u 1ÕÖp − 1ÕÖ�v Δ>p + 12 �×�p + ×��Δ>p�� 4.14�

where ÕÖ = 1/��%/�>� is the group velocity and × = ��%/�>� is the group velocity dispersion.

This leads us to a FWHM gain bandwidth expression given by:

∆Ø = 2& ÔΓ ln 2� × Ù 1ÕÖp − 1ÕÖ�Ù 3 �4.15�

For PPKTP the OPG bandwidth can be a few hundred nanometers long as show in Fig. 4.4. From the above expression, it is clear the bandwidth is inversely proportional to the crystal length L. Thus, besides efficiency concerns, the choice of crystal length is also important if one is seeking a specific bandwidth . Four-wave mixing as well as Raman processes may also emerge for sufficiently high OPO intensities as seen Fig. 4.4. Therefore a wavelength locking element employed in the OPO is often required to eliminate gain for these processes.

Fig. 4.4: Spectral output for broadband OPO based on RKTP poled for degeneracy [27].

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4.3 Nonlinear mixing with spatially structured fields

Frequency conversion schemes realistically do not employ plane-wave beams as governed by Equations 4.1 (a)-(c), but rather with a defined spatial structure such as Gaussian or top-hat beams. In other words, the amplitude value has a dependence on the transversal coordinates x and y. In order to properly account for these beams the wave equation given by Equation 2.1 needs to be

extended. We achieve this by making use of the identity ∇ × ∇ × ��, � = ∇ �∇ ∙ ��, �� −∇��, � upon the wave equation and solve each term with the electric field form given by

Equation 2.2. It can be shown by simple calculation that the ∇�� term is equivalent to: ∇�� = 12 u∇Ú� � − %�� + 2�%� a�a� + a��a��v ��Û� !"��Ü + c. c �4.16�

In order to calculate the divergence term ∇ :∇ ∙ ��;, we consider the adjacent figure. The Poynting

vector of the electric field is often not parallel to the propagation direction, but offset by some angle, known as the walk-off angle (Ý�. If we assume that an extraordinary-wave propagates through the crystal, with walk-off direction shown in the adjacent figure, we may write the electric field direction as: �Ü = cos �Ý��Þ + sin �Ý��̂ �4.17� This is shown graphically via a Cartesian reference frame in Fig. 4.5 below:

Fig. 4.5: Diagram illustrating walk-off angle of the Poynting vector.

Using this it is now possible to calculate the divergence term ∇ ∙ ��, which gives: ∇ ∙ �� = 12 r �� cos�ρ� + G �� + �%�U sin�Ý�t ��Û� !"� + c. c �4.18�

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In order to fully calculate the last term ∇:∇ ∙ ��;, we re-arrange Equation 4.17 in terms of �Þ and

apply the slowly varying amplitude approximation (SVEA) which is given by: |∇��| ≪»%�� ∙ ∇��» ⇒ â¦¡ã¦�¡â ≪ â%� ¦ã¦�â. The SVEA approximation allows us to ignore second-order partial

derivative contributions to the amplitudes. This yields the following: ∇:∇ ∙ ��; � 12 r2�%� tan�Ý� �� Û� !"� + �. �t �4.19�

Inserting Equations 4.19 and 4.18 into Equation 2.1, we obtain the following envelope equation: �� = �2%� ∇Ú� � + tan�Ý� �� − 72 � + ��>�2%� �ä�� Û� �4.20�

This is the full envelope wave equation for structured beams and contains term describing diffraction, walk-off, absorption and nonlinear mixing in that order. Equation 4.20 can be extended to account for fields that are time-dependent and broadband, leading to dispersion and group velocity terms [31]. However, in the case of near monochromatic fields with long pulse durations exceeding a few hundred picoseconds, Equation 4.20 accurately describes the evolution of the field envelope through the nonlinear medium. It has no analytical solution and has to be solved numerically, which is typically done by split-step Fourier methods (SSFM) or finite-element schemes [32,33]. For a specific χ(2) process, one determines the associated �ä� term and inserts this into Equation 4.20. Thus, in the case of OPG, the full coupled wave-equations describing the evolution of the pump, signal and idler amplitudes are: ��o�� = �2%o ∇Ú� �o + tan:Ýo; ��o�� − 7o2 �o + � almm >o�'o �p�� 4.21 �� p�� = �2%p ∇Ú� �p + tan�Ýp� ��p�� − 7p2 �p + � almm >p�'p �o��∗�� 4.21 b�� = �2%� ∇Ú� �� + tan�Ý�� − 7�2 �� + � almm >��'� �o�p∗�� 4.21 c�

As mentioned above, the coupled wave equations can be extended further to account for temporal structure in the fields, particularly for laser pulses. The extended equations incorporate terms which account for dispersion and group-velocity effects, which are crucial for short pulses below a few picoseconds. Injection seeded or narrowband systems utilizing nanosecond pulses, which are encountered in this thesis, are not drastically influenced by temporal effects and therefore the equations above yield reasonable approximations for the field amplitudes.

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4.4 High-energy dual-wavelength MOPA

In this section we describe the realization of a dual-wavelength, tunable, narrowband, high-energy, nanosecond MOPA that is based on large aperture PPRKTP. Owing to the results obtained, this setup could be used as pump source for ns DFG THz generation.

4.4.1 MOPA experimental layout

The MOPA system was pumped by a Q-switched Nd:YAG laser at 1.064 µm (InnoLas Laser GmbH). The laser delivered Gaussian pulses with an available energy of 150 mJ, a repetition rate of 100 Hz and a pulse duration of ~10 ns (FWHM). The laser was injection-seeded with a distributed-feedback fiber laser to yield single longitudinal mode operation. The power stability of the laser was measured to have an RMS value of 0.3 %. The beam quality of the pump beam was measured with the 90-10 travelling knife edge method and M2 values of 3.2 and 3.3 were obtained for the transverse x and y axes, respectively. The pump beam profile was measured after beam shaping, and prior to the OPO, and is shown as an inset in Fig. 4.6. The pump field was linearly polarized along the z-axis of the crystals in order to utilize the d33 nonlinear coefficient of PPRKTP.

Fig. 4.6: Experimental layout of the OPO and two crystal OPA with pump beam profile inset.

The layout of the MOPA system is depicted in Fig. 4.6. The pump beam is split into two arms, one for the OPO and one for the amplifier stages. The pump beam for the OPO was shaped with spherical and cylindrical optics to correct for astigmatism and produce the desired beam size within the OPO cavity. The OPO gain medium consisted of a single PPRKTP crystal with x-y-z dimensions of 12 × 7 × 5 mm. Rubidium doped KTP (RKTP) was exploited for fabrication of the periodically poled RKTP samples thanks to advantageous poling properties [34]. The Rb-doping also allows large-aperture PPRKTP crystals to be fabricated, while obtaining good homogeneity of the parametric gain [27]. Large-aperture crystals in turn allow utilizing larger incident beam sizes

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which is required for high energy scaling. The crystals were flux grown and periodically poled with a duty cycle of 50 %, for first-order QPM. The poling period was 38.85 µm in order to achieve degenerate type-0 down-conversion from 1.064 µm. All crystals used were AR-coated for 1 and 2 µm and temperature controlled at ~55 ºC to operate near degeneracy. The input coupler of the OPO was a broadband, plane-concave mirror with a radius of curvature of 150 mm and highly reflective over the wavelength range 1.85-2.20 µm. The plane side of the mirror was un-coated, with measured transmission of ~ 92 % at 1 µm. The output coupler was a transversely chirped VBG (Optigrate Inc.). The grating dimensions were x-y-z, 2.2 × 19 × 5 mm, respectively, and the grating had a linear chirp rate along the y-direction of 1.08 nm/mm corresponding to a tuning range of approximately 21.5 nm around the OPO degeneracy. The VBG was mounted onto a 3-axis stage for alignment and OPO tuning. The central wavelength of the VBG was specified to be 2131.3 nm. The diffraction efficiency of the VBG at the central wavelength was specified to be 55 ± 5 % and its damage threshold was specified to ~ 5 J/cm2 at 1 µm for 10 ns long pulses. Both the entry and exit sides of the VBG were AR-coated for 1 µm and 2 µm radiation and the VBG was temperature-stabilized at 20 ºC with a Peltier element. The cavity design was simulated and arranged in order to have a signal/idler beam waist radius of 480 µm (1/e2) incident on the VBG. According to calculations this would yield a reflectivity bandwidth of ~ 1 nm. The OPO was singly resonant throughout the VBG tuning range, except at the degeneracy point. The total cavity length of the OPO was approximately 55 mm. To optimize the nonlinear conversion the crystal was placed as close as possible to the VBG, where the size of the pump beam and cavity modes matched. The OPO output was separated from the depleted pump with a dichroic mirror and reflected towards the amplifier stage. This seed beam was shaped appropriately with spherical CaF2 lenses to produce a collimated beam and have the seed beam radius of ~ 2 mm incident on the amplifier. The pump beam of the OPA was collimated and mode-matched to the seed beam. Half-wave plate and thin-film polarizer combinations for 1 µm and 2 µm allowed separately varying the seed and the pump power in the OPA stage. The number of crystals used in the amplifier was varied and the results were compared. Finally, at the exit of the amplifier, the amplified 2 µm light was separated from the depleted pump field with two dichroic mirrors. In order to accurately describe the dynamics of the VBG OPO a numerical model was developed taking into account the spatial distribution of the pump with its non-symmetric structure. Since the

crystal is periodically poled for QPM, we assume that the wave-mismatch vector, given by: ∆k =

kp – ks – ki, is equal to zero at degeneracy, as the crystals are poled for these wavelengths. The mismatch vector is recalculated for different signal and idler wavelengths. The spatial distribution of the pump, measured prior to the OPO, was used as an input to the model with a 128×128 grid. We chose a step-size to yield 100 integration points over the length of the crystal. To match the simulations with experiments we altered the value of the nonlinear coefficient deff to get a best fit (deff is assumed to be uniform over the whole crystal). Since the interaction is noncritical and

collinear, we set the walk-off angle ρ to zero. We omit a full description of the numerical model and refer the reader to Appendix A. In designing of the MOPA system we aimed at high overall

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efficiency without sacrificing spectral or spatial properties of the signal and idler beams while keeping the system as simple as possible. The overall conversion efficiency and simplicity are aided by our choice of not using a high-gain pre-amplifier stage and instead designing the VBG OPO to provide sufficient seed to saturate the OPA. At the same time, the VBG OPO was driven below saturation where the generated output was spectrally narrow and free from unwanted back-conversion processes.

4.4.2 TCVBG OPO characterization

The total output of the VBG OPO, with the largest wavelength difference (signal at 2117 nm and idler at 2141 nm, corresponding to a frequency difference of 1.53 THz), is shown in Fig. 4.7 (a).

Fig. 4.7: Measured and simulated output energies of the OPO (a); Conversion and pump depletion efficiencies (b).

The threshold of the OPO was found to be around 2.4 mJ of input pump energy. By pumping with a maximum energy of 6.5 mJ, we obtained a total output energy of 1.26 mJ. The pump energies shown in Fig. 4.7 were corrected for the transmission loss of the pump mirror. At this pump energy the fluence incident on the VBG was calculated to be 2.5 J/cm2. This corresponds to operation of the OPO at around 2.7 times the threshold. An energy stability measurement yielded an rms value of 1.4 % for the OPO. The numerical model agrees quite well with the experimentally measured values. In the numerical model the linear absorption value was set to 0.5 % for all the interacting fields [35] and the output coupling value of the VBG was set to 0.54 for the resonant idler. In this case the best fit was obtained using a deff value of 7.6 pm/V. The numerical model predicts a slightly higher threshold than what was measured, as well as having a marginally larger slope efficiency. The little difference can be attributed to some losses at the resonant wave which were not accounted for in the model. The maximum conversion efficiency of the OPO was measured to be 20.8 %. It is clear from Fig. 4.7 (b) that the VBG OPO has not reached saturation and back-conversion was hence insignificant. Similarly, a pump depletion of 21.6 % was measured for the OPO at maximum pump energy, which corresponds well with the calculated overall conversion efficiency. The

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measured far field of the output OPO beam had M2 values of 7.7 and 3.4 for the x and y directions, respectively.

The spectral output of the OPO and the OPA was measured with a Jobin Yvon iHR330 diffraction grating monochromator and a PbSe photodetector. The grating had a groove density of 300/mm and was blazed for wavelengths around 2 µm. The slit size was minimized to 80 µm, allowing a maximum resolution <0.5 nm in this wavelength band.

Fig. 4.8: Signal and idler center wavelengths of the OPO for increasing VBG position (a); measured spectra for

different VBG positions (b). The spectrum from the OPO was recorded for varying positions on the VBG. The central wavelengths for the OPO signal and idler as well as the total output energies are shown in Fig. 4.8 (a). These measurements were done at a constant OPO pump energy of 6.5 mJ. The measured tuning of the OPO was linear with respect to the VBG position, in agreement with specifications. With this VBG we could choose to oscillate the signal (VBG positions up to “9 mm”) or idler (for larger VBG positions), with the idler-resonant OPO showing somewhat higher output power, presumably owing the specified variation of the VBG reflectivity of ± 5 %. The variation of the OPO output energy for the idler-resonant OPO was 13.4 %. A few characteristic measured signal and idler peaks are shown in Fig. 4.8 (b). The peaks were symmetric Gaussian and the FWHM bandwidth of each peak was approximately 0.56 nm. No parasitic behaviour or four-wave mixing effects were seen in the spectral output of the OPO even when operating at the maximum pump energy of 6.5 mJ [8]. At the far end of the VBG (position 20 mm), the frequency separation between signal and idler was approximately 1.53 THz. At this position of the VBG, it is the idler which is resonant in the cavity. A broad parametric gain bandwidth experienced by type-0 OPO with VBG spectral narrowing can lead to generation of equi-spaced frequency sidebands owing to cascaded four-wave mixing [9]. The sideband generation process would be enhanced in an OPO with a plane-plane cavity with a large-Fresnel number, i.e. with plane mirrors and a large pump beam waist where noncollinear parametric wave components can be readily amplified. To investigate this we intentionally

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misaligned the OPO cavity about its vertical axis in incremental steps, and measured the output spectra for each angle, as can be seen in Fig. 4.9.

Fig. 4.9: Four-wave mixing peaks in the output spectrum obtained through misalignment of the cavity (a); First

spectral peak magnified by a factor 10 for clarity (b).

For a well-aligned cavity the sidemode suppression ratio was larger than -29 dB at maximum pump energy. The fact that we did not drive the OPO into saturation also efficiently prevented cascaded processes from taking place. If should be mentioned, though that a careful alignment of the cavity is crucial for the sideband suppression, as it is the noncollinear components which will mostly contribute to the sidebands. Indeed by aligning cavity axis 1 degree with respect to the pump we observed decrease in the sideband suppression to about -20 dB. The generated side-band peaks had wavelengths of approximately 2095 nm and 2165 nm, corresponding to the four-wave mixing processes 2ωi – ωs and 2ωs – ωi, respectively. Although the four-wave mixing frequency sidebands generated by the MOPA would be detrimental in some spectroscopic applications e.g. in LIDARs, they would not be harmful but, actually, contribute to the THz generation in the DFG process.

4.4.3 MOPA characterization

The OPO output was steered to propagate collinearly and mode-overlap with the pump in the OPA stage. The measured output energy and conversion efficiency of the different OPA stages are given below in Fig. 4.10. At the maximum pump energy of 140 mJ used in the amplifier and with a seed energy of ~ 1.5 mJ, this corresponded to an incident fluence of 2.25 J/cm2. It is well below the damage threshold of the crystals, which have been measured to be 10 J/cm2 at 1 µm with nanosecond pulses [36], indicating further power scaling would still be possible. As mentioned, the number of crystals in the amplifier stage was varied, from one to three in total, and the energy output of each configuration was measured. The seed energy from the OPO was 1.5 mJ in each case. For a single amplifier crystal with a length of 12 mm, and pumping with 140 mJ we were able to reach a maximum total energy output of 38.3 mJ. This corresponds to an OPA conversion efficiency of 26 %. In this case, the OPA did not reach saturation due to insufficiently available

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pump power. In order to drive the OPA into saturation and increase the extraction efficiency, a longer OPA crystal would be required. Therefore, we added a second 12 mm PPRKTP crystal to the OPA stage and re-measured the output. For two crystals (12 mm + 12 mm) we measured an improvement in the energy yield of OPA, with a maximum output energy of 52 mJ at 140 mJ pump energy, see Fig. 4.10.

Fig. 4.10: MOPA output energy vs input pump energy (a); Conversion efficiency of the respective MOPAs (b).

This corresponded to the OPA power extraction efficiency of 36 %. In this case the amplifier began to saturate at around 60 mJ of the input pump. Extending the OPA length further by adding one more 7 mm-long crystal did not improve the OPA efficiency. This configuration (12 mm + 12 mm + 7 mm), actually showed lower performance than the two crystal amplifier. A maximum output energy of 48.5 mJ was reached in this case, with a corresponding conversion efficiency of 33.5 %, limited by back-conversion. For this reason, we used the two-crystal amplifier for further measurements. The OPA gain as a function of the seed energy was measured at the pump energy of 60 mJ where the OPA just reached saturation (see Fig. 4.11).

Fig. 4.11: Amplifier gain and output energy vs seed energy, at constant pump energy of 60 mJ (a); Spectral output of

the OPA stage for increasing incident pump energy (b).

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The amplifier shows significant saturation for seed energies above 1 mJ. The linear slope of the output energy versus input seed energy for the seed below 0.5 mJ is 12.8, meaning that OPO fluctuations of 1.4% rms would be amplified to 18% rms after the amplifier. At the seed energies above 1 mJ the slope decreases to 2.6 reducing the output fluctuations to 3.6%. By increasing the pump energy to 140 mJ the rms fluctuations of the OPA output approach that of the seed OPO. For this reason we decided to use a seed energy of ~1.5 mJ in the amplifier experiments. The output spectra of the amplifier was also measured (Fig. 4.11 (b)). The spectrum of the OPA stage was measured for increasing pump energies (40, 80 and 120 mJ,). As can be seen in the figure, the amplified signal and idler spectra showed no broadening or wavelength shift even for maximum pump energy. To summarize this chapter, we introduced some theory and central concepts related to OPOs and OPAs. Furthermore we extended the coupled-wave equations derived in Chapter 2, to account for diffraction and spatial walk-off effects. Using the model developed we realized a dual-wavelength, tunable, narrowband high-energy MOPA system. We describe in detail the characterization of the seed source, which in this case was an OPO locked with a transversally chirped VBG. We characterized our MOPA and obtained maximum energies of 52 mJ at 140 mJ input pump, corresponding to an overall conversion efficiency of 36 %.

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9. N. Thilmann, B. Jacobsson, C. Canalias, V. Pasiskevicius and F. Laurell, “A narrowband optical parametric oscillator tunable over 6.8 THz through degeneracy with a transversely-chirped volume Bragg grating,” Applied Physics B, 105, 239-244 (2011).

10. J. A. Giordmaine and R. C. Miller, “Tunable Coherent Parametric Oscillation in LiNbO3 at Optical Frequencies,” Physical Review Letters, 14, 24, 14 June (1965).

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31. A. V. Smith, R. J. Gehr and M. S. Bowers, “Numerical models of broad-bandwidth nanosecond optical parametric oscillators,” Journal of the Optical Society of America B, 16, 4, 609-619, April (1999).

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5 Nanosecond Mirrorless OPOs

5.1 Introduction

First proposed in 1966 [1], the mirrorless optical parametric oscillator (MOPO) was physically realized in 2007 [2]. Since that time further work has been undertaken into the device, owing to its mirrorless nature and its inherent narrowband output among others [3,4,5]. As has been mentioned, narrowband sources in the milli-joule range are often required for numerous applications, such as seeding applications in master-oscillator power amplifier schemes [6]. Another favorable property that these devices possess is their ease of alignment. No mirrors are required and hence the MOPO orientation only needs to be optimized such that the input angle of the pump stays approximately along the grating vector. This leads to stable operation which is somewhat immune to mis-alignment. Such oscillators possess additional properties which are rather unusual for OPOs. First, owing to the fact that the oscillation is established by the distributed feedback and not by any external cavity, the pump intensity at threshold will depend primarily on the length and nonlinearity of nonlinear medium [7,8,9]. Secondly, the parametric wave (signal or idler) which is generated in the opposite direction to that of pump (giving rise to the name backward-wave optical parametric oscillator (BWOPO)), is inherently narrowband and largely insensitive to pump frequency variation. The energy conservation then ensures that this variation is inherited by the complementary parametric wave generated in the forward direction [10,11]. Third, the frequencies of the parametric waves generated in BWOPO are substantially less sensitive to the nonlinear crystal temperature and pump angle variations as compared to the conventional OPOs [5]. Such properties are conductive to achieve narrowband precisely tunable near- and mid-infrared wavelength generation with scalable output energy in a simple and robust arrangement. This would be beneficial in a number of applications including sources of nonclassical light [12], remote sensing and differential absorption LIDARs and others, where seeded or doubly-resonant OPOs are currently employed. On the other hand, in order to fully realize a MOPO device, the chief difficulty to overcome is to create sub-µm QPM periods. Much work has been done in order to better produce these sub-µm domains [13,14,15]. MOPO operation leads to forward and backward propagating fields within the nonlinear medium due to its phase matching condition. This is given as %o = %å − %æ. Here, %å and %æ are the wave-vectors of the forward and backward propagating

fields respectively. In ordinary co-propagating OPOs, feedback is provided by the cavity mirrors, leading to oscillation. However, since there are no mirrors in the MOPO, feedback is provided by the counter propagating parametric interaction itself. By applying energy conservation, that is >o = >å + >æ, with the phase matching equation above, we can deduce the wavelength of the

backward propagating field which is then given as: )æ = )o 'å + 'æ'å − 'o �5.1�

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where, n represents the refractive index of the respective field. In the first order-QPM case however, the phase matching condition is extended to %o = %å − %æ + çè, where çè = 2&/Λ,

with Λ being the poling period. Applying energy conservation with this phase matching condition leads to the relation: 1Λ = 1)o :'o − 'å; + 1)æ �'å + 'æ� �5.2�

From Equation 5.2 it is possible to determine the appropriate poling period in order to generate a desired output wavelength. A phase matching curve utilizing Equation 5.2 is shown below. As can be seen from Fig. 5.1, for a pump wavelength at 1.064 µm it is possible to reverse the direction of the signal and idler fields by selecting an approximate period Λ ≤ 580 nm. For these periods, the signal would then propagate backward instead of the idler.

Fig. 5.1: Phase matching curve for sub-µm PPRKTP with Λ = 500 nm wavelengths indicated.

Another interesting property of MOPOs is revealed by taking the derivative of Equation 5.1 with respect to the pump frequency. This yields: �>å�>o = ØÖå:ØÖæ + ØÖo;ØÖo:ØÖæ + ØÖå; = 1 + ê3, �>æ�>o = ØÖæ:ØÖo − ØÖå;ØÖo:ØÖæ + ØÖå; = −ê3 �5.3�

where ëÖ is the group velocity of the respective field and ê3 is a dimensionless parameter which is

introduced to show that the forward wave follows the frequency of the pump. Thus for a nonlinear medium which exhibits a low dispersion value, we would have |ê3| ≪ 1. In this case, the change of the forward wave with respect to the pump is equal to one, whereas that of the backward wave is close to zero. For down-conversion parametric processes the temporal phases of the fields are

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related to a constant value. For MOPOs, this is given as∶ +o − +å − +æ = −&/2. It can be shown

the time derivatives of these phases are related as: �+æ� � 0, �+å� � �+o� �5.4�

Since the phase of the backward wave remains constant over time, this implies that the backward wave is transform limited for narrowband pulses. On the other hand, the phase of the forward wave is almost equivalent to that of the pump and therefore inherits much of the initial pump structure. This transform limited property of the backward propagating field is highly desirable for applications requiring a narrow linewidth. As mentioned, for small enough poling periods the direction of the idler and signal fields are exchanged. This would allow a signal field which is then transform limited instead of the idler. The MOPO device is suitable with pulses with pico-second duration and longer. This is simply due to the fact the fields propagate at different velocities in the medium and need to have temporal overlap with one another in order to efficiently exchange energy. Due to the longer pulse duration in nanosecond devices relative to typical crystal lengths (~ 10 mm), the backward and forward propagating fields will always see some temporal overlap between the fields and therefore are more immune to some temporal offset. As was also mentioned in throughout this thesis, narrowband, high-energy, nanosecond sources are in great demand. Since MOPO devices inherently exhibit favorable properties such as narrow bandwidth, energy scaling is key for their implementation in many applications.

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5.2 MOPO experiments

In this section we describe the development of a 1 µm pumped, nanosecond, high-energy, mid-infrared BWOPO with a poling period of Λ = 509 nm. For the MOPO experiment, RKTP was chosen as the nonlinear material, owing to its excellent nonlinearity, high-damage threshold and broad-transparency within the mid-infrared. Previous work involved gratings periods exceeding crystal domain periods of Λ > 500 nm [4, 5]. With those periods, due to phase matching considerations, this would result in a counter-propagating idler field relative to forward propagating pump and signal field, as is evident from the phase matching curve in Fig. 5.1.

5.2.1 Experimental layout & description

The experimental setup is shown in Fig. 5.2. The mirrorless OPO was pumped by a Q-switched, injection-seeded, Nd:YAG, 1 µm source. The pump source had a Gaussian pulse duration of 10 ns (FWHM) and a repetition rate of 100 Hz with a maximum output energy of 200 mJ. The amount of pump energy steered towards the MOPO was varied with a half-wave plate and thin-film polarizer combination. The pump beam had a Gaussian spatial profile with M2 values of 3.2 and 3.3 in the x and y directions respectively.

Fig. 5.2: Experimental layout for the 1 µm pumped MOPO.

The crystal used as the gain medium was PPRKTP. The crystal had a poled volume of 7×3×1 mm3

for the x-y-z dimensions respectively. The grating within the crystal had a length of approximately 7.3 mm. The crystal was periodically poled via electric field poling with a period of Λ = 509 nm for first order quasi-phase matching. With this period, and for a 1064 nm pump, the signal and idler wavelengths were 1856 nm and 2495 nm respectively. The crystal was polished to optical quality and was un-coated for the interacting wavelengths.

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Fig. 5.3: Atomic force microscopy scans showing the etched relief of the domain structure in the (a) patterned face,

(b) opposite polar face.

The domain structure of the crystals was investigated with the aid of atomic force microscopy (AFM), as shown in Fig. 5.3. Both faces of the crystal were studied via chemical etching (a). The domain structure close to the patterned face showed consistent periodicity and therefore the pump beam was focused in this region of the crystal. The pump beam was polarized parallel to the z-axis of the crystal in order to utilize the d33 component of RKTP for the type-0 parametric interaction, thus both the generated signal and idler were also z-polarized. The pump beam was guided through a CaF2 mirror which was reflective for the signal wavelength. The pump beam was focused by a spherical CaF2 lens with a focal length of f = 250 mm. The resulting beam radius in the crystal was measured to be w0x = 298 µm and w0y = 297 µm (1/e2) with the travelling knife-edge method. The crystal was positioned to have its center coincide the focus of the beam. The crystal was mounted onto a 5-axis translation stage for fine alignment, as well as a Peltier element for temperature stabilization. The Peltier temperature was set to room temperature around 21ºC. Lastly, a mirror that is highly reflective for the idler wavelength, was placed after the crystal to separate the idler from the depleted pump field.

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5.2.2 BWOPO characterization and measurements

The performance of the MOPO in this architecture was investigated. Specifically, the energy output, its spectral and temporal properties as well as the spatial profiles of the generated fields. The energy of the signal, idler and depleted pump fields were all measured and are shown below in Fig. 5.4.

Fig. 5.4: Output energies, conversion efficiency and depleted pump from MOPO for increasing input pump energy.

Since, the crystal and CaF2 lens was un-coated we incorporate the additional linear loss due to Fresnel reflections in Fig. 5.4. Pumping with a maximum input energy of 7 mJ, output energies of 1.96 mJ and 1.46 mJ were reached for the signal and the idler respectively. This corresponded to a fluence of ~ 5 J/cm2 inside the crystal. Since the damage threshold of RKTP in this configuration is reported to be around 10 J/cm2, further energy scaling is still possible. The MOPO threshold was measured to be around ~ 1.5 mJ of input pump energy, corresponding to a threshold intensity of 83 MW/cm2. Using this value, we estimated an effective nonlinearity value of 7.5 pm/V. Thus, the MOPO was pumped 4.6 times above threshold when operating at maximum output. As indicated in Fig. 5.4, at maximum pump energy, the total conversion of the device was measured to be 53 %. Correspondingly, the pump depletion was measured to be 53.8 %. Typically for ordinary OPOs with mirrors providing feedback, back-conversion of the idler/signal to the pump field is a significant problem at high-energies and serves as a general limitation for these devices. Unlike ordinary OPOs however, the MOPO feedback mechanism differs, with a counter-propagating signal. Back-conversion is the product of the field amplitudes and since the fields grow in opposite directions, the one field reaches its maximum value where the other is at its minimum and vice

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versa. This provides some immunity to back-conversion in MOPO devices. Therefore, the laser damage threshold may serve as the final conversion limit for such mirrorless devices. The spectral and temporal features of the MOPO operating two times above threshold were also investigated. The spectra were measured with a grating spectrometer (Yvon-Horiba iHR 550). For the selected slit-width of 80 µm, a maximum resolution of ~ 0.5 nm was possible at the signal and idler wavelengths. The grating utilized was blazed for 2 µm and had a groove density of 300/mm.

Fig. 5.5: Measured spectral output of the generated signal (a) and idler (b) from the MOPO.

The measured data for the signal and idler are shown in Fig. 5.5. The center wavelengths of the signal and idler were measured to be approximately 1856 nm and 2495.5 nm respectively. A Gaussian fit was applied to the measured signal and idler data. From this, the bandwidth (FWHM) of the signal and idler was estimated. The signal and idler bandwidths were found to be ~ 0.54 nm and ~ 0.46 nm respectively. These bandwidths are on the same order as those of OPOs employing some kind of wavelength narrowing element, such as volume Bragg gratings (VBG). This spectral bandwidth is an order of magnitude smaller than that of ordinary, co-propagating OPOs, as shown in Fig. 5.6.

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Fig. 5.6: Normalized idler spectra of the PPRKTP BWOPO (red line) and a singly-resonant OPO using 7 mm –long PPRKTP with period of 38.5 µm (blue line).

The reason for the broad output in ordinary OPOs is due to the fact that the phase mismatch vector varies slowly with respect to some tuning away from the phase-match point. This is not the case for MOPOs, with a counter propagating field. The tuning rate is much larger, and therefore the MOPO is only phase-matched for a very small range of frequencies. This can be understood further by considering that that for single-frequency pumping the parametric gain bandwidth in co-propagating TWM is inversely proportional to the signal and idler group velocity difference, ∆Õíîí ∝ ÕÖpÕÖ�/»ÕÖp − ÕÖ�» [16]. In the counter-propagating TWM case, the parametric gain

bandwidth is inversely proportional to the sum of the group velocities: ∆Õíîí ∝ ÕÖpÕÖ�/:ÕÖp +ÕÖ�;. Spatial intensity profile of the idler were also measured with the aid of a thermal camera

(Pyrocam III), shown in Fig. 5.7. Filters were used to prevent any residual pump light arriving at the camera. The beam profiles were taken for increasing pump energy. The idler beam had a clear Gaussian intensity profile similar in structure to the input pump.

Fig. 5.7: Far-field beam profiles for the idler beam for (a) 4 mJ, (b) 5 mJ and (c) 6 mJ input pump energy

respectively.

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Due to the fact that the pump is injection seeded, the pump beam has a high spatial coherence. Also, transverse gain mapping of the crystal indicated a near-uniform gain of the interaction region. These factors contribute to the achievement of a clean beam profile. Since there is also no back-conversion present, the idler beam profile structure remains consistent over the entire pumping range. To summarize this chapter, we have demonstrated a low-threshold, narrowband BWOPO pumped at 1 µm. With a grating period of Λ = 509 nm, we showed backward propagation of signal and forward propagation of the idler. Pumping with a maximum input energy of 7 mJ, output energies of 1.96 mJ and 1.46 mJ were reached for the signal and the idler respectively. This corresponded to pump depletion and conversion efficiencies of 53.8 % and 53 %. The spectral linewidths of the signal and idler were measured and found to be on the order of 0.5 nm. Owing to the narrowband output, suitable energy and excellent beam profiles we believe that this source is ideal for seeding of amplifiers where high-energy and narrowband light is critical. Also, due to the fact that the BWOPO does not inherently rely on mirrors for feedback, the source itself is more robust and mechanically immune to misalignment errors, making it suitable for rigorous field applications. Further optimization of the BWOPO is still possible. Anti-reflection coatings could be applied to the crystal ends and the crystal could be cut to the exact grating length, further improving the overall efficiency of the BWOPO. Additionally, the energy could be scaled higher since the input fluence of the pump was only ~ 5 J/cm2 inside the crystal, far below reported damage thresholds of RKTP in the nanosecond pulse regime.

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6. G. Stoeppler, N. Thilmann, V. Pasiskevicius, A. Zukauskas, C. Canalias, and M. Eichhorn, “Tunable Mid-infrared ZnGeP2 RISTRA OPO pumped by periodically-poled Rb:KTP optical parametric master-oscillator power amplifier,” Optics Express, 20, 4, 13 February (2012).

7. Y. J. Ding and J. B. Khurgin, “Backward optical parametric oscillators and amplifiers,” IEEE J. Quantum Electron. 32, 1574–1582 (1996).

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9. H. Su, S.-C. Ruan and Y. Guo, “Generation of mid-infrared wavelengths larger than 4.0 m in a mirrorless counterpropagating configuration,” J. Opt. Soc. Am. B 23, 1626–1629 (2006).

10. G. Strömqvist, V. Pasiskevicius, C. Canalias and C. Montes, “Coherent phase-modulation transfer in counterpropagating parametric down-conversion,” Phys. Rev. A 84, 023825 (2011).

11. C. Montes, B. Gay-Para, M. De Micheli and P. Aschieri, “Mirrorless optical parametric oscillators with stitching faults: backward downconversion efficiency and coherence gain versus stochastic pump bandwidth,” J. Opt. Soc. Am. B 31, No. 12, 3186-3192 (2014).

12. C. Chuu and S. E. Harris, “Ultrabright backward-wave biphoton source,”Phys. Rev. A 83, 061803R (2011).

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14. C. Canalias, V. Pasiskevicius and F. Laurell, “Periodic Poling of KTiOPO4: From Micrometer to Sub-Micrometer Domain Gratings,” Ferroelectrics, 340:27-47 (2006).

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15. C. Liljestrand, A. Zukauskas, V. Pasiskevicius and C. Canalias, “Highly efficient mirrorless optical parametric oscillator pumped by nanosecond pulses,” Optics Letters, 42, 13, 2435-2438, 1 July (2017).

16. C. Manzoni and G. Cerullo, “Design criteria for ultrafast optical parametric amplifiers,” Journal of Optics, 18, 103501 (2016).

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Conclusion and Outlook In this thesis I have demonstrated energy scaling within nanosecond, mid-infrared OPOs and OPAs, while delivering and maintaining narrow spectral linewidths. This included the development of a low-threshold, high-energy BWOPO based on RKTP, utilizing sub-µm periodicities. Such sources are suitable candidates for both ground and space-borne LIDAR operations. One of the key limitations in obtaining high-energies from these devices is the laser-induced damage threshold of the nonlinear crystals used as the gain medium. Thus, the LIDT of KTP and RKTP were investigated at 1 µm and 2 µm. For space-borne applications, gamma and proton radiation induced damage in nonlinear crystals is also crucial to understand and therefore, this was investigated in KTP and RKTP. Below, I summarize the main achievements obtained in each publication. In Paper I, we investigated the surface LIDT of KTP and RKTP at 1 µm and 2 µm in the nanosecond regime. Prior, to these measurements, no LIDT values of RKTP at 1 µm or 2 µm were known. Additionally, only 1 µm LIDT results had been previously measured. The crystals were tested in a configuration identical to that used in high-energy systems, with z-polarized light to utilize the highest nonlinearity in the crystal. For KTP, we measured surface LIDT values of 12 J/cm2 and 10 J/cm2 for 1.064 µm and 2.128 µm respectively. For RKTP, 10 J/cm2 was measured for both wavelengths. In Paper II the effect of gamma radiation upon KTP, PPKTP, RKTP and PPRKTP was studied. The crystals were irradiated with gamma radiation provided by a 60Co source with varying levels of incident doses. The samples were irradiated with gamma levels of 10, 30 and 100 krad. The radiation induced absorbing color centers in the crystals which were particularly pronounced in the visible range of 400 – 800 nm. The crystals were actively and passively temperature annealed at 150 ºC for two hours and room temperature respectively. Furthermore, after temperature annealing transmission spectrum measurements over the wavelength range 1 µm – 4.5 µm were undertaken. Results showed no long-term permanent change in the transmission of samples, indicating temperature annealing as an effective counter-measure for any gamma radiation induced effects. Paper III entailed the design of and realization of a high-energy 2 µm MOPA system based on the results obtained in Paper I. The crystals used in this experiment were 5 mm thick PPRKTP which had been poled for type-0 degenerate operation with a pump wavelength of 1.064 µm. The MOPA consisted of an OPO which was wavelength locked by transversally chirped VBG, acting as the seed source for a two-crystal amplifier stage. The OPO allowed a total tuning range of 1.53 THz. Pumping with a maximum energy of 140 mJ and with a seed energy of ~ 1.5 mJ we obtained a total output energy of 52 mJ and conversion efficiency of 36 %. Output linewidths on the order of ~ 0.5 nm were obtained for the amplified signal and idler fields, limited by the spectrometer

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resolution available. Additionally, we developed a numerical model to investigate and optimize the system further. In Paper IV we developed a low-threshold, nanosecond BWOPO based on PPRKTP with a sub-µm grating period of Λ = 509 nm. Poling periods this short result in a backward propagating signal field relative to the forward pump/idler field. This was the first time such a counter propagating scheme was shown in mirrorless OPOs. The signal and idler wavelengths were measured to be 1.856 µm and 2.495 µm respectively. When pumped with ~ 6.48 mJ, the device delivered output energies of 1.96 mJ and 1.46 mJ for the signal and idler fields respectively. This corresponded to pump depletion and conversion efficiencies of 53.8 % and 53% respectively. Spectral output measurements revealed a narrow linewidths on the order of 0.5 nm, again only limited by the resolution of the spectrometer.

The results in this thesis further motivate and demonstrate the suitability of RKTP for use in high-energy applications. Accurately knowing the damage threshold of the material allowed optimal design of the MOPA system within Paper III and the BWOPO in Paper IV. As was mentioned in Chapter 3, in order to improve the LIDT values obtained in Paper I, spatially and temporally resolved LIDT (STEREO-LIDT) measurements could be undertaken. Furthermore, it will useful to know the bulk damage threshold of the nonlinear materials used, at 1 µm and 2 µm. This will indicate a limit for lower energy systems where beams are tightly focused in the crystal. The measured damage thresholds were not reached in the MOPA system described in Paper III, and therefore further energy scaling of the system was still possible in principle, but only limited by the available pump energy. Due to the high-energies reached in this system, as well as the tunable dual-wavelength output obtained, the MOPA is suited for use as a pump source in THz DFG schemes. The BWOPO developed in Paper IV demonstrated low-thresholds with narrowband spectral outputs. With the energies obtained, this source would be suitable as a seed for multi-millijoule MOPA systems or a stand-alone source for low-energy applications. Further work with the BWOPO would be to increase the crystal thickness/aperture while still maintaining excellent domain structures.

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Appendix A

Numerical Modelling of OPOs and OPAs As was discussed in section 4.4.1, a spatially resolved numerical model was developed to predict the OPO output and behaviour. The model is based mainly on references [32, 33] in chapter 4. Equation (A. 1) was used to model propagation in the nonlinear crystal within the OPO and represent three coupled equations for each field. This is given by: ��W�� = �2%W ∇Ú� �W + tan:ÝW; ��W�� − 7W2 �W + w�W��, �, �, � ��. 1�

where the index j = p, s, i, represent the pump, signal and idler fields respectively, �W is the complex

amplitude, %W is the wave-vector, ÝW is the walk-off angle and 7W is the linear absorption value. The

interacting fields take the usual form of: �W = 3� ð�W� ��!ñ" �ñ��ò� + �W∗��!ñ" �ñ��ò�ó. The

nonlinear mixing term w�W for each wave is given as.

w�o��, �, �, � = � almm >o�'o ��, �, �, ��p��, �, �, �� ∆�� . 2.1�

w�p��, �, �, � = � almm >p�'p �o��, �, �, ��∗��, �, �, ��∆�� . 2.2�

w��, �, �, � = � almm >��'� �o��, �, �, ��p∗��, �, �, ��∆�� . 2.3�

Here the phase mismatch vector is given by Δ% = %o − %p − %�. Equation (A. 1) can be separated

into linear and nonlinear (with respect to the amplitude) terms and thus the split-step Fourier method (SSFM) is ideally suited to approximate a solution for the amplitude �W , which allows us

to express the spatial derivatives as spatial frequencies in the Fourier domain. From the definition of the two dimensional Fourier transform we may express the amplitude and nonlinear mixing term as: �W��, �, �, � = ± ± �ôW:õg , õh , �, ;�1

1 ��|:pög�p÷h;aõgaõh ��. 3.1�

w�W��, �, �, � = ± ± w�øW:õg , õh, �, ;�1 1 ��|:pög�p÷h;aõgaõh ��. 3.2�

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where õg and õh are the spatial frequencies and �ùú and w�øW are the Fourier transforms of the

amplitude and nonlinear mixing terms respectively. Substitution of these expressions into equation (A. 1) allows us to rewrite the equation in the following form: ��ôW�õg , õh , �, �� = −� ³2&�%W :õg� + õh�; + 2&õg tan:ÝW;´ �ôW:õg , õh, �, ; + w�øW:õg , õh, �, ; ��. 4�

Equations (A. 2) and (A. 4) are solved iteratively throughout the length of the nonlinear medium. Here, t is to be regarded as an index of time slices of the temporal profile of the respective pulse, with � = �/'. The linear part of the equations handle the propagation of the amplitudes through

the medium. The algorithm proceeds in the following manner: Initial amplitude values :��W; are

inserted into the cavity through the input coupler with reflectivity Æ3. These amplitudes are then

inserted into equations (A. 2) in order to calculate the nonlinear mixing terms w�W��, �, 0, �. These

terms are then Fourier transformed into their Fourier counterparts to yield �ôW �õg, õh , 0, � and w�øW:õg , õh , 0, ;. The two terms are now inserted into equation (A. 4) to calculate the Fourier

amplitudes at a new position in the crystal �ôW �õg, õh , Δ�, �.

Equations (A. 4) are solved via the Cash-Karp Runge-Kutta algorithm. Finally the solution is inverse Fourier transformed to yield the amplitude value at the next step: �W ��, �, Δ�, �. The

algorithm is now repeated until the end of the crystal is reached �� = ��. At the crystal and cavity ends boundary conditions are applied, and these take the form: �ÊË = 41 − �Æ3 ∙ 41 − Æ3 ∙ �� ÈÉ¤ = 41 − �Æ� ∙ 41 − Æ� ∙ �� = �� ülmýl£¤l¢� = 41 − �Æ� ∙ 4Æ� ∙ �� = �� ülmýl£¤l¢3 = 41 − �Æ3 ∙ 4Æ3 ∙ �� = 0�

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where Æ3 and Æ� are the mirror reflectivity the field wavelengths, �Æ3 and �Æ� are the crystal end losses due to Fresnel reflections. In order to obtain accurate results from the model, the time and spatial structure of the fields needs to be initialized properly. First a time profile (Gaussian in this case) is assumed for the pulses. The pulse is then normalized such that the sum of the energy of the time slices will equal the input pump energy. The normalized form of this pulse is given by:

�� = 4�o ∙ exp c− �o� �2 ln2�d ��. 5�

where �� represents the power amplitude of the pulse with units þ3/�, o is the FWHM pulse

duration and �o is the peak power of the pulse. As was discussed in section 4.2, for nanosecond

OPOs the pump pulse duration is longer than the cavity round trip time ��Ì�. Thus the pulses are gridded in time with respect to a number of round trips which is determined beforehand by knowing the optical cavity path length of the OPO. Each pulse slice, which is now one cavity round trip time long, is modelled using equations (A. 2) and (A. 4).

We also assumed that the fields are monochromatic and all travel at the same speed within the medium, which is a reasonable approximation for longer pulse systems. Thus, group-velocity dispersion is ignored and energy does not flow between individual pulse elements/slices. Therefore we can write the complex amplitude as ��, �, �, � = ��, �, �� ∙ ��. The power amplitude value

for a certain time ��∗� is now distributed over a symmetric grid with a particular spatial profile. For Gaussian beams, we can express this intensity as: q��, �, ∗� = ��∗��lÄ� ∙ exp u− 2�� + ��¥�g¥�h v ��. 6�

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where ��lÄ� = :&¥�g¥�h;/2 is the area of the beam with ¥�g and ¥�h being the 1/e2 transversal

beam radii. Each element on this grid is now converted to its respective field amplitude value via the plane-wave equation:

��, �, ∗� = Ô2 q��, �, ∗�-�'� � �*Û ��. 7�

Thus each amplitude element ��W within the spatial grid is modelled using the algorithm outlined

earlier. The amplitudes at new positions can then be used to determine the new intensity and thus

the new power/energy. A random phase factor � �*Û is multiplied to the amplitude value in equation (A. 7). This needs to be a fixed factor for the pump if it were injection seeded. In general, the signal and idler fields start from quantum noise and therefore this phase factor is a random value between 0 and 2&. The signal and idler modes are assumed to also have a Gaussian temporal

profile, but with an initial energy given by � = Ã�� , originating from quantum noise, if unseeded.

energy scaling of infrared nanosecond optical parametric …1210318/... · 2018-05-28 · energy...

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