manifestation of active medium astigmatism at transverse mode locking in a diode end-pumped stable...

Manifestation of active medium astigmatismat transverse mode locking in a diodeend-pumped stable resonator laser

Victor V. Bezotosnyi,1 Evgeniy A. Cheshev,1 Mikhail V. Gorbunkov,1

Pavel V. Kostryukov,2 and Vladimir G. Tunkin2,*1Lebedev Physical Institute, Leninskiy Prospect 53, Moscow 119991, Russian Federation

2Lomonosov Moscow State University, International Laser Center, Leninskie Gory 1, Moscow 119991, Russian Federation

*Corresponding author: [email protected]

Received 11 March 2008; revised 16 May 2008; accepted 3 June 2008;posted 12 June 2008 (Doc. ID 93724); published 9 July 2008

Transverse mode locking in a diode end-pumped Nd:YAG laser with up to ∼140 cm resonator length wasinvestigated. It was found that for each resonator degeneracy, there are two degenerate lengths wherethe fundamental mode is very different from the Gaussian mode. Fundamental mode intensity patternsfor these lengths expand in directions perpendicular to each other. Experimental results are in a goodagreement with numerical calculations, taking into account active medium (AM) astigmatism and in-homogeneous gain. Optical powers of astigmatic AM can be found directly from measurements of degen-erate lengths without using numerical modeling. © 2008 Optical Society of America

OCIS codes: 140.3300, 140.3410, 140.3480.

1. Introduction

Diode end-pumped solid-state lasers have advan-tages of high beam quality and high efficiency dueto an appropriate pump beam and resonator modeoverlapping provided by inhomogeneous pumping[1]. It has been shown in Ref. [2] that the efficiencyincreases as the pump beam size decreases. In somepapers [3–7], CW diode end-pumped Nd : YVO4 la-sers with a resonator formed by high-reflection planesurface of the active medium (AM) and output cou-pler (OC) with radius of curvature ROC < 10 cm wereinvestigated theoretically and experimentally. Acomplicated ring pattern of the output radiationwas observed in the so-called critical resonator con-figurations under a pumping spot size less than thatof the empty resonator mode. The existence of the cri-tical configurations is connected with the frequencydegeneracy of the empty resonator modes, which arelocked under the given pumping conditions [7]. For

degenerate resonators the g parameters satisfy thecondition [8]

arccos� ffiffiffiffiffiffiffiffiffiffig1g2

pπ ¼ r

s; ð1Þ

where r and s are integers and r=s is an irreduciblefraction specific for each degenerate configuration.The sign before the radical coincides with the signsof g1 and g2. Under a sufficiently narrow pump beamand degeneracy condition [Eq. (1)], the fundamentalmode is a superposition of the degenerate modes ofthe empty resonator. As it has been shown in [9],the degenerate resonator length shifts due to ther-mal lens in AM, and this effect was used to measurethe thermal lens focal length. Transverse mode lock-ing in an axially symmetric stable resonator laserpumped by a narrow pump beam was also theoreti-cally investigated in Ref. [10]. The axial symmetry ofa laser results in a circular symmetry of output ra-diation intensity patterns. Intensity patterns withsymmetry close to circular were observed experimen-tally in Refs. [3–7], at least for degenerate resonatorswith ROC < 10 cm. Resonators of ∼100 cm or longer

0003-6935/08/203651-07$15.00/0© 2008 Optical Society of America

10 July 2008 / Vol. 47, No. 20 / APPLIED OPTICS 3651

in length are used in many types of picosecond andfemtosecond lasers [for example, see Refs. [11–13].Transverse mode locking in resonators of ∼100 cmin length has, to the best of our knowledge, not beenconsidered yet.We investigate both experimentally and numeri-

cally the transverse mode locking in a CW diodeend-pumped Nd:YAG laser with resonator lengthsup to 140 cm.

2. Experiment

The experimental setup of a CW diode end-pumpedNd:YAG laser is shown in Fig. 1. The Nd:YAG lasercrystal with a 4mm diameter and a 14mm lengthwas coated at its surface facing the pumping beamfor 4% reflection at the pump wavelength and99.8% reflection at the laser wavelength. The otherface of the crystal was angled 3° and antireflection-coated at the laser wavelength. The pump source wasa 4WCW laser diode emitting at 808nm. The cylind-rical lens with a focal length of 0:2mm was used toreduce pump beam divergence along the diode fastaxis. The pump beam was collimated and focusedinto the laser crystal by two lenses, L1 and L2, withequal focal lengths of 8:8mm. The pump radiationwas attenuated by a chopper. The pump pulse dura-tion was ≈1ms. The average pump power in the crys-tal was 10mW, therefore we can assume the thermallens optical power to be negligible. The laser resona-tor OC was a ∅40mm concave mirror with a radiusof curvature ROC ¼ 150 cm and 99% reflection at thelaser wavelength. The screen was placed just behindthe OC. An objective lens was used to image outputradiation intensity pattern on the screen onto a CCDcamera. The OC, the screen, the objective lens, andthe CCD camera were mounted on a translationstage that offers the possibility to adjust the resona-tor length continuously.The pump beam spot size on the front surface of

the crystal was measured to be 0:04mm × 0:12mm.At the same time the diameter, 2w, of the emptyresonator Gaussian mode in the laser crystal for asemiconfocal resonator with a length of L ¼ ROC=2 ¼75 cm is ≈1:0mm. Thus the pump beam spot size inthe laser crystal was much smaller than that of theGaussian mode of the empty resonator.The resonator length was varied in the experiment

from 45 to 140 cm. As a result, 15 resonator degener-

ate configurations were observed. Their r=s were5=12, 2=5, 3=8, 4=11, 5=14, 1=3, 4=13, 3=10, 2=7,3=11, 1=4, 3=13, 2=9, 3=14, and 1=5, respectively.Away from the critical configurations, the intensitypatterns are close to the Gaussian pattern (seeFig. 2). Values of M2 for noncritical configurationswas measured to be 1:3� 0:1.

At the critical configurations, complicated intensitypatterns are observed, but in contrast with the shortlength resonators [3–7], they do not exhibit circularsymmetry (see Figs. 3 and 4). In the vicinity of eachobserved degenerate configuration, one can clearlysee two so-called degenerate resonator lengths, Ldeg

xandLdeg

y , where the intensity patterns are very differ-ent from the Gaussian pattern. At these lengths theintensity patterns expand in principal directions thatare perpendicular to each other. The left-hand col-umns inFigs. 3 and4 show the experimental intensitypatterns for the degenerate configurations, wherer=s ¼ 1=3 and 1=4, respectively. For r=s ¼ 1=3, Ldeg

x ¼1084mm and Ldeg

y ¼ 1113mm. For r=s ¼ 1=4, Ldegx ¼

700mm and Ldegy ¼ 734mm.

The principal directions do not change with sphe-rical mirror rotation but rotate with Nd:YAG crystalrotation around the resonator axis. The output radia-tion intensity patterns obtained for L ¼ 734mm anddifferent orientations of the Nd:YAG crystal areshown in Fig. 5.

Degenerate configurations with even s and odd sdiffer in their intensity pattern structures in a similarway as the case of axial symmetry [10]. Intensity pat-terns for odd s consist of a narrow central maximumand a broad pedestal with some structure. In the caseof even s, intensity patterns have several maxima.

As the pump power increases, Ldegx and Ldeg

y be-come shorter due to the influence of a thermal lensin the laser crystal. But we did not find the distancebetween them to be changed. Intensity patterns ob-served within the region between Ldeg

x and Ldegy are

complicated, but they exhibit certain symmetry.We also investigated four other Nd:YAG crystal

samples of the same geometry, and the degeneratelengths Ldeg

x and Ldegy were observed for all of them.

The orientation of principle directions relative tothe plane of the laser beam incidence on the angledcrystal surface and the values of Ldeg

x and Ldegy were

specific for each of the five investigated crystals. As

Fig. 1. Experimental setup of a diode end-pumped Nd:YAG laser. LD is the laser diode; L1 and L2 are the collimating and focusing lenses;Ch is the chopper; S is the screen; and OL is the objective lens.

3652 APPLIED OPTICS / Vol. 47, No. 20 / 10 July 2008

mentioned above, the principal directions do notchange with spherical mirror rotation but rotate withtheNd:YAGcrystal rotationaroundtheresonatoraxis(seeFig. 5). Thus theelliptical pumpprofile shouldnotbe considered as a source of the principal directions.The intensitypatternsobtaineddonotexhibit circu-

lar symmetry like those calculated in Ref. [10]. Weassume that breaking the intensity pattern’s circu-lar symmetry is connected with the astigmatism ofthe AM, i.e., the AM wavefront correction functionΔφðx; yÞ is not cylindrically symmetric. Principle di-rections obtained experimentally are supposed to co-incide with the principal directions of the surfacez ¼ Δφðx; yÞ,whichareperpendicular toeachotherac-cording to Euler’s theorem [14]. To verify our assump-tions we have developed a numerical model of thefundamentalmodeof theresonatorwithanastigmaticAMwith different lateral and vertical optical powers.

3. Numerical Modeling

For numerical modeling, we used an approach imple-mented in Ref. [15], though the beam propagationmethod can also be used. An intracavity fieldcan be considered as a superposition of astigmaticHermite–Gaussian (HG) beams, which are eigen-modes of an unpumped resonator. We regard theAM as an element that affects the relative energydistribution among the set of the HG modes.The laser scheme used in this modeling consists of

a thin astigmatic AMwith a high-reflectionmirror onits back surface and OC. The AM has different lateraland vertical optical powers, Dx and Dy, and a is theAM radius. The OC is an infinite spherical mirrorwith a radius of curvature ROC. The AM amplifica-tion profile was taken in the following form:

Kðx; yÞ ¼ 1þ ðK0 − 1Þ exp�−x2

ρ2x−y2

ρ2y

�: ð2Þ

The modes of the unpumped resonator are de-scribed by astigmatic HG beams [16]:

umnðx; yÞ ¼ffiffiffi2π

r1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2mþnm!n!p 1ffiffiffiffiffiffiffiffiffiffiffiffi

wxwyp Hm

� ffiffiffi2

px

wx

�

Hn

� ffiffiffi2

py

wy

�exp

�−i

kx2

2Rx−x2

w2x

�

exp�−i

ky2

2Ry−y2

w2y

�; ð3Þ

where k ¼ 2π=λ is the wave number, wx and wy are

the lateral and vertical spot sizes, respectively, andRx and Ry are the wavefront curvature radii. Rx,Ry, wx, and wy can be expressed using complex

Fig. 2. (Color online) Intensity pattern obtained away from thedegeneracy.

Fig. 3. (Color online) Intensity patterns near degeneracy r=s ¼1=3. The left-hand column is experimental, and the right-hand col-umn is calculated.


curvature parameters qx and qy:

wx;y ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−

2kImð1=qx;yÞ

s; Rx;y ¼

1Reð1=qx;yÞ : ð4Þ

As in the case of axial symmetry, the resonator gparameters are introduced as

gx1 ¼ 1 −DxL2; gy1 ¼ 1 −Dy

L2;

gx2 ¼ gy2 ¼ 1 −L

ROC: ð5Þ

Three planes are outlined in the resonator underconsideration: 1 and 2 are the input and outputplanes of the AM, respectively, and 3 is the plane justbefore the OC. According to Eq. (4), complex curva-ture parameters qx and qy at planes 1–3 are deter-mined by the following expressions:

1qx;y1

¼ þ 1Lð1 − gx;y1 Þ −

iL

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigx;y1 ð1 − gx;y1 gx;y2 Þ

gx;y2

s;

1qx;y2

¼ −1

Lð1 − gx;y1 Þ −iL


gx;y2

s;

1qx;y3

¼ þ 1Lð1 − gx;y2 Þ −

iL


gx;y1

s: ð6Þ

The laser field uðx; yÞ can be described by matrixβ̂, in which elements βmn are the coefficients in theexpansion:

uðx; yÞ ¼Xm;n

βmnumnðqx; qy; x; yÞ: ð7Þ

With known uðx; yÞ, values of βmn can be found as

βmn ¼ZZ

u�mnðqx; qy; x; yÞuðx; yÞdxdy: ð8Þ

Expansion Eq. (7) is valid for arbitrary values of qxand qy, but in the case of resonator, it is reasonableto use the values of qx and qy defined byEq. (6). Underthenormalization requirement

Pm;njβmnj2 ¼ 1, jβmnj2

represent the part of the fundamental mode energycontained in the HG beam with indices m and n.

The transformation of the matrix β̂ with the propa-gation from plane 2 to plane 3 and then back to plane1 is expressed as [16]

β0mn ¼ βmn exp�2ikL − ið2mþ 1Þ arccos� ffiffiffiffiffiffiffiffiffiffi

gx1gx2

p− ið2nþ 1Þarccos�

ffiffiffiffiffiffiffiffiffiffigy1g

y2

q �; ð9Þ

where þ or − coincides with the sign of gx1 and gy1.The redistribution of beam amplitudes βmn on the

AM with gain profile Kðx; yÞ is given by

β0mn ¼Xj;l

tmnjlβjl; ð10Þ

Fig. 4. (Color online) Intensity patterns near degeneracy r=s ¼1=4. The left-hand column is experimental, and the right-hand col-umn is calculated.


where

tmnjl ¼ZZ

u�mnðqx; qy; x; yÞKðx; yÞujlðqx; qy; x; yÞdxdy:

ð11ÞIt can be noticed that tmnjl ¼ δmjδnl only in the caseof Kðx; yÞ≡ 1.The transformation of beam amplitudes after the

complete resonator round-trip can be written usingEqs. (9)–(11) as

β0mn ¼Xj;l

f mnjlβjl; ð12Þ

where

f mnjl ¼ tmnjl exp�2ikL − ið2jþ 1Þarccos� ffiffiffiffiffiffiffiffiffiffi

gx1gx2

p− ið2lþ 1Þarccos�

ffiffiffiffiffiffiffiffiffiffigy1g

y2

q �: ð13Þ

One should note that the resonator four-dimensionalmatrix f mnjl is nonhermitian. Therefore its eigenvec-tors, which are two-dimensional matrices βmn, do notform an orthogonal set [17]. But any of them (andalso the fundamental mode) can be presented assome superposition of orthogonal HG modes (3).

The matrix elements corresponding to the funda-mental mode can be found as components of the ei-genvector of the equation

βmn ¼ λXj;l

f mnjlβjl; ð14Þ

with maximum eigenvalue jλj. This equation wassolved by the successive iteration method. The solu-tion found is a set of theHGbeamamplitudes at plane2.Amplitudes of theHGbeamsatplane3 canbe foundfrom the solution of Eq. (14) in the following way:

βOCmn ¼ βmn exp½−iðmþ 1=2Þðψx − ψ0

xÞ− iðnþ 1=2Þðψy − ψ0

yÞ�; ð15Þ

Fig. 5. (Color online) Intensity patterns for five orientations of the Nd:YAG crystal.

Fig. 6. Relative energies of HG beams jβmnj2 for r=s ¼ 1=3 [(a) L ¼ Ldegx ¼ 1084mm, (b) L ¼ 1099mm, and (c) L ¼ Ldeg

y ¼ 1113mm] andr=s ¼ 1=4 [(d) L ¼ 700mm, (e) L ¼ Ldeg

x ¼ 717mm, and (f) L ¼ Ldegy ¼ 737mm].


where

ψx;y − ψ0x;y ¼ arctan

�−Reð1=qx;y3 ÞImð1=qx;y3 Þ

�

− arctan�−Reð1=qx;y2 ÞImð1=qx;y2 Þ

�:

Using Eqs. (3) and (7), one can find the fundamentalmode field across the OC.

4. Calculation Results and Discussion

Intensity patterns calculated for r=s ¼ 1=3 and 1=4are given in Figs. 3 and 4 (right-hand columns), re-spectively. The calculated patterns correspond to theexperimental patterns, which are given in the left-hand columns. The values of the parameters usedare as follows: K0 ¼ 1:32, ρx ¼ ρy ¼ 0:08mm, Dx ¼0:056m−1, Dy ¼ 0:18m−1, and a ¼ 1:6. Pump-lightdistribution was taken here as circularly symmetric(ρx ¼ ρy). This simplification does not deteriorate theagreement between the numerical and experimentalresults. The values of Dx and Dy were found directlyfrom measurements of Ldeg

x and Ldegy without using

numerical modeling by means of the degeneracy con-dition [Eq. (1)] [9]:

Dx;y ≈ −8ΔLx;y

R2OCsin

2

�2π r

s

� ; ð16Þ

where ΔLx;y ≡ Ldegx;y − L0, and L0 ¼ ROCð1 − cos2

ðπðr=sÞÞÞ is the resonator optical length under the as-sumption of the AM with no optical power (Dx ¼Dy ¼ 0). The values of K0, ρx, ρyð¼ ρxÞ, and a werechosen to provide the best agreement between the ex-perimental and calculated patterns. With thesevalues the experimental and calculated intensitypatterns are close to each other not only for r=s ¼1=3 and 1=4 but also for less pronounced degenera-cies observed in the experiment.Relative energies of HG beams jβmnj2 for r=s ¼ 1=3

(L ¼ 1084 and 1099mm) and r=s ¼ 1=4 (L ¼ 700 and717mm) are given in Fig. 6. One can see that only acertain set of beams have substantial nonzero ener-gies. For r=s ¼ 1=3, the sum of mþ n for these beamsis a multiple of 2s ¼ 6. For r=s ¼ 1=4, mþ n is a mul-tiple of s ¼ 4. As it follows from Eq. (1), these beamsare degenerate modes of the unpumped resonator.Calculations also showed that there are fixed phaserelations between amplitudes βmn of HG beams. Thisgives a possibility to speak about transverse modelocking.The values of Dx and Dy obtained by means of de-

termining Ldegx and Ldeg

y and using Eq. (16) for all fiveNd:YAG crystal samples we investigated fall into theinterval from−0:028 to þ0:20m−1.It should be noted that we investigated the mani-

festation of astigmatism of an AM but not how an as-tigmatic laser beam propagates outside a resonator.For example, the propagation of an astigmatic output

beam from a diode-pumped unstable resonatorNd : YVO4 laser was investigated in Ref. [18]. Theanisotropic index of Nd : YVO4 crystal and the useof an unstable resonator resulted in different wave-front curvatures in the x and y directions (Rx ¼5:34mm and Ry ¼ 5:06mm). This made it possibleto observe different locations of horizontal and verti-cal beamwaists outside the resonator. In our case of astable resonator geometry, an AM astigmatism re-sults in the splitting of a degeneracy point into twolengths Ldeg

x and Ldegy . One can see from Eq. (16) that

the shift of Ldegx and Ldeg

y from the value L0 is propor-tional to R2

OC. According to this the manifestation ofa small astigmatism takes place in the case of suffi-ciently long resonators.

5. Conclusion

For the first time to our knowledge, the manifesta-tion of an AM small astigmatism was observed atthe effect of transverse mode locking in a laserpumped by a narrow beam. The numerical modelhas been developed to calculate the fundamentalmode of an astigmatic resonator with inhomoge-neous gain distribution. A good agreement wasachieved between the calculated and experimentalintensity patterns. It was shown that small opticalpowers of astigmatic AM can be found directly fromthe measurement of degenerate lengths withoutusing numerical modeling.

This work was supported by the program of funda-mental studies from the Department of PhysicalSciences of the Russian Academy of Science, “LaserSystems Based on New Active Materials and Opticsof StructuredMaterials,” and by the Russian Founda-tion for Basic Research RFBR grant 08-08-00108-a.

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manifestation of active medium astigmatism at transverse mode locking in a diode end-pumped stable...

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