December 1, 2000 / Vol. 25, No. 23 / OPTICS LETTERS 1735

Frequency doubling of femtosecond pulses inwalk-off-compensated N -(4-nitrophenyl)-L-prolinol

Juan P. Torres, Silvia Carrasco, and Lluis Torner

Laboratory of Photonics, Department of Signal Theory and Communications, Universitat Politecnica de Catalunya,Gran Capitan UPC-D3, Barcelona ES 08034, Spain

Eric W. VanStryland

Center for Research and Education in Optics and Lasers, University of Central Florida, Orlando, Florida 32826

Received June 29, 2000

We show how to exploit the high quadratic nonlinear coefficient of the organic crystal N-(4-nitrophenyl)-L-prolinol for generation and parametric mixing of ultrashort pulses by use of tilted-pulsetechniques. The effective crystal length for subpicosecond operation is shown to be enhanced fromtens of micrometers to tens of millimeters. Efficient frequency doubling of 100-fs pulses is predicted inwalk-off-compensated geometries with peak intensities of a few megawatts per square centimeter. © 2000Optical Society of America

OCIS codes: 190.2620, 190.7110, 320.2250, 160.4890.

N-(4-nitrophenyl)-L-prolinol (NPP) is an organicmolecular crystal developed by molecular engineering1

that exhibits one of the highest phase-matchable sec-ond-order susceptibilities reported so far in the near-infrared spectral range �deff � 56 pm�V�.2 Becauseof such large nonlinearity, NPP holds promise for theimplementation of parametric devices for frequencyconversion and mixing with pump signals with signifi-cantly reduced intensities relative to those needed withconventional quadratic materials. An important stepin this direction was reported recently by Banfi et al.,who realized a highly efficient frequency shifter in a2.8-mm-thick NPP crystal.3 The device operated in anearly degenerate frequency-mixing configuration inwhich a frequency-doubled pump light interacted witha synchronous signal to produce a frequency-shiftedoutput. To avoid walk-off effects they performed theexperiments with pump and signal wavelengths setnear the noncritical phase-matching wavelength ofthe material �lncpm � 1.15 mm�, with pulses havinga duration of 20 ps. However, as was pointed out byWang et al.,4 the large spatial and temporal walk-offthat exist in NPP severely limit the usefulness of thematerial away from the noncritical phase-matchingwavelength and for shorter pulses. Such limitationsare twofold. First, walk-off causes the fundamentalfrequency (FF) and the second-harmonic (SH) signalsto walk off from each other while propagating insidethe crystal, thus reducing the useful nonlinear crystallength in which eff icient interaction takes place. Sec-ond, the presence of group-velocity mismatch (GVM)can largely distort the shape and duration of the SHpulses generated in crystals that are thicker than thepulse walk-off length. As a consequence, extremelyshort crystal thicknesses must be employed withsubpicosecond pulses.

Here we show that subpicosecond pulses can be eff i-ciently frequency doubled and mixed in NPP with mod-erate pump intensities. This method benefits fromthe large Poynting vector walk-off exhibited by NPPcrystals outside noncritical phase matching, because

0146-9592/00/231735-03$15.00/0

it employs tilted-pulse techniques. This techniquespectral disperses the pump FF signals in differentdirections by passing them through a medium withangular dispersion, such as a prism5 or a properlydesigned diffraction grating.6,7 One needs to tailorthe amount of angular dispersion to obtain effectiveGVM compensation. Tilted-pulse techniques havebeen used extensively in recent years for eff icientdoubling of ultrashort pulses8 and the observationof quadratic temporal and spatiotemporal solitonsmediated by cascading nonlinearities in b-bariumborate (BBO) and LiIO3, respectively.9,10

Here we study SH generation pumped by pulsedFF light. In the wavelength band 1 2 mm, type Iooe �eeo� phase matching in NPP is achieved whenlight propagates in the yz �xz� plane. In the proposedscheme the pump light that is input into the crystal isa highly elliptical beam, so diffraction along one of thetransverse axes is negligible. The evolution of the en-velopes of the FF and the SH signals is described by11

i≠A1

≠z2

k100

2≠2A1

≠t21

12k1

≠2A1

≠x2 1 iG1

2A1

1 K1A1�A2 exp�2iDkz� � 0 , (1)

i≠A2

≠z2

k200

2≠2A2

≠t21

12k2

≠2A2

≠x2 2 irt≠A2

≠x2 irx

≠A2

≠x

1 iG2

2A2 1 K2A1

2 exp�iDkz� � 0 , (2)

where An , with n � 1, 2, are the amplitudes of theFF and the SH waves, respectively; z is the lon-gitudinal coordinate; t is the retarded time; x isthe spatial transverse coordinate; and Gn are ab-sorption coefficients. The parameters kn are thelinear wave numbers, kn

0 are the inverse group veloci-ties, kn

00 are the group-velocity dispersion (GVD’s),and Dk � 2k1 2 k2 is the wave-vector mismatch.The walk-off parameters are given by rt � k1

0 2 k20

and rx � tan�r2� 2 tan�r1�, where r1 and r2 are the

© 2000 Optical Society of America

1736 OPTICS LETTERS / Vol. 25, No. 23 / December 1, 2000

Poynting vector walk-off angles. Finally, Kn �4pdeff�lnn, where l is the FF wavelength, nn arethe refractive indices and deff is the second-ordersusceptibility.

Tilted-pulse techniques are based on diffraction ofthe input FF wave, A1�z;x, t�, by a grating so thateach spectral component is dispersed in a differentdirection.6 –10 The resulting signal is a tilted pulse,A1�z;x, t 2 x tan�c��c�, where c is the tilt angle andc is the velocity of light, so that its peak intensity islocated at a different time for each value of x. Fornormal incidence on the nonlinear crystal, one has6

tan�c� � 2ml

d cos b, (3)

where d is the groove spacing of the grating, m is thediffraction order, and b is the diffraction angle. Whenthe beam width in the x direction of the incident sig-nal on the grating is large enough (i.e., a few millime-ters), the effect of diffraction can be accounted for byconsideration of temporal evolution only but with neweffective inverse group velocities and GVD’s,

un � kn0 1 tan�c�tan�rn��c , (4)

gn � kn00 2

1kn

�tan�c��2

c, (5)

respectively. The SH wave is assumed to acquire thesame tilt as the FF. Therefore, the spatiotemporallight evolution can be described by

i≠A1

≠z2

g1

2≠2A1

≠t21 i

G1

2A1

1 K1A1�A2 exp�2iDkz� � 0 , (6)

i≠A2

≠z2

g2

2≠2A2

≠t22 iru

≠A2

≠t1 i

G2

2A2

1 K2A12 exp�iDkz� � 0 , (7)

with ru � u1 2 u2. Notice that the contribution of thetilt angle to the effective GVD is always negative.

To elucidate whether input light and materialconditions suitable for efficient SH generation ex-ist, we proceed as follows: First, we calculate thephase-matching angle upm at the pump wavelength,using the Sellmeier equations for NPP as measured byDatta et al.12 Second, we find the value of the tilt, c0,required for GVM cancellation �u1 � u2�. This givesthe effective GVD’s, gn . Figure 1 shows the outcomefor t � 100 fs pulses (t is the FWHM duration ofthe signal intensity). Figure 1(a) shows the tilt c0required as a function of pump wavelength in the bandl � 1 2 mm, and Fig. 1(b) shows the correspondingeffective dispersion lengths, Ldn � 0.161t2�gn , expe-rienced by the FF and the SH signals. The effectiveGVD at the FF is found to be anomalous �g1 , 0� overthe whole wavelength band that is displayed, whereasthe effective GVD at the SH frequency turns out tobe anomalous inside the band l � 1.07 1.23 mmand normal otherwise. At lncpm the Poynting vectorwalk-off vanishes, hence GVM cancellation is notpossible. Thus, both the tilt, c0, and the GVD’s,

gn, are undefined. It is worth noticing the narrowwavelength band around lncpm, where both the FFand the SH signals experience very high effectiveanomalous dispersion. Such huge dispersion mayallow, for the first time, the observation of quadraticsolitons over many dispersion lengths, provided thatthe corresponding diffraction gratings can be made.

The important conclusion revealed by Fig. 1 isthat suitable conditions for eff icient SH generation,namely, negligible GVM and dispersion lengths ofseveral millimeters, exist in a wide pump wavelengthband centered at the third telecommunication win-dow. Importantly, the required tilt c0 is shown tobe a smooth function of the pump wavelength. Thissmooth function is an indication of the robustness ofthe proposed scheme. Such robustness is confirmedby Fig. 2(a), which shows the variation of the dis-persion lengths and of the walk-off length, definedas Lw � 0.567t��u1 2 u2�, versus deviations of thenominal tilt required for exact walk-off cancellationat a given wavelength. In Fig. 2(a), l � 1.6 mm, butsimilar results are obtained for other wavelengthsinside the band. Notice that tilt acceptance of severaldegrees is obtained with walk-off lengths in excess of5 mm. Similarly, Fig. 2(b) shows that for a fixed tilta wavelength bandwidth of several tens of nanometersis obtained. Naturally, all tolerances increase wheneither the acceptable walk-off length is reduced or thepulse width is increased.

Fig. 1. (a) Tilt angle required for exact GVM cancellationin phase-matched type I SH generation in NPP as a func-tion of pump wavelength. (b) Corresponding effective dis-persion lengths at the FF and the SH. Solid curves, FF;dashed curves, SH. Pulse duration t � 100 fs. A and Nstand for anomalous and normal dispersion, respectively.

Fig. 2. (a) Effective dispersion lengths at the FF and theSH and (dashed curves) temporal walk-off length as a func-tion of tilt angle for a fixed pump wavelength, l � 1.6 mm.(b) Dispersion and walk-off lengths as a function of pumpwavelength for a fixed tilt angle c0 � 24.9±. In both cases,u � upm and t � 100 fs.

December 1, 2000 / Vol. 25, No. 23 / OPTICS LETTERS 1737

Fig. 3. Simulated evolution of the peak intensity of theFF and the SH signals as a function of crystal length.Solid curves, �1 1 1� temporal evolution with GVM com-pensation; dotted–dashed curves, �2 1 1� spatiotemporalfull evolution with GVM compensation; dotted curves, evo-lution without GVM compensation; dashed curves, �1 11� temporal evolution with GVM compensation includingabsorption. Inset, output SH pulses. Conditions: phasematching, I0 � 10 MW�cm2, w0 � 3 mm, and t � 100 fs.

Fig. 4. Detailed evolution of the SH pulse under the condi-tions of Fig. 3 (a) with and (b) without GVM compensation.

To show that eff icient SH generation occurs underthe conditions suggested by Figs. 1 and 2, we solvedexpressions (6) and (7) numerically with a split-stepFourier method. The input conditions contain onlyFF light with the shape A1�z � 0, t� � I01�2 sech�t�t0�,where I0 is the input peak intensity and t0 � t�1.763.The derivation of expressions (6) and (7) from Eqs. (1)and (2) assumes a plane wave that is incident onthe diffraction grating. Actually, in practice theinput beam is a Gaussian beam. To verify the ac-curacy of expressions (6) and (7) we also solved thefull system of equations (1) and (2) for input condi-tions corresponding to light distribution producedby the diffraction grating, namely, A1�z � 0, x, t� �I01�2 sech��t 2 x tan�c��c��t0exp�2x2�w0

2�, where w0is the waist of the Gaussian beam that is incidenton the grating. Because of the departures of theGaussian shape from a plane wave, the characteristicslength at which the system of expressions (6) and (7)fails is L w0�tan�r�. Otherwise, perfect agreementbetween the predictions of expressions (6) and (7) andthose of Eqs. (1) and (2) was always obtained.

Figure 3 shows the typical outcome of the simu-lations. The figure shows phase-matched upcon-version of a l � 1.6 mm pump with peak intensityI0 � 10 MW�cm2 in a 3-mm-long NPP crystal. Thepulse width is set to t � 100 fs in a beam with widthw0 � 3 mm. In the simulations we took into accountthe absorption of NPP owing to its molecular structurebe letting G1 � 1.5 cm21 and G2 � 3.0 cm21. Thedotted curves correspond to the SH that was gener-ated without GVM compensation. Figure 4 shows

the detailed SH pulse evolution with and withoutGVM compensations. A drastic difference is clearlyvisible: Although no useful SH is generated withoutwalk-off compensation, a clean, narrow high-intensitySH pulse is obtained with the tilted pulse. Losseswould reduce the efficiency of SH generation in thickcrystals, but a high-quality output SH pulse would al-ways be obtained. Results analogous to those shownhere were obtained with other comparable values ofthe peak intensities and signal shapes, as well asoutside phase matching.

In conclusion, we predict that highly efficient sec-ond-harmonic generation of subpicosecond pulses canbe accomplished in walk-off-compensated NPP organiccrystals with peak pump intensities in the megawattsper square centimeter range, in a wide wavelengthband centered around the third telecommunicationwindow. Available diffraction gratings with suitablegroove spacing can produce the required tilted pulses.The results presented here are believed to expandgreatly the potential applications of NPP to theimplementation of parametric devices and cascadingphenomena in general.13

This work was supported by the Generalitat deCatalunya. Important remarks by J. Zyss and W. E.Torruellas are gratefully acknowledged. J. Torres’se-mail address is jperez@tsc.upc.es.

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