Acceptable composition-ratio variations of a mixed crystal for nonlinear laser device applications

Acceptable composition-ratio variations of a mixedcrystal for nonlinear laser device applications

Jin Jer Huang, Yuri M. Andreev, Grigory V. Lanskii, Anna V. Shaiduko,Subhasis Das, and Udit Chatterjee

A theory is developed to predict some crucial parameters that optimize the performance of mixednonlinear crystals in nonlinear devices. These include acceptable variations of the composition ratio of theparent crystals and the optimal as well as acceptable interaction lengths for any interaction. The theoryis successfully applied to make necessary predictions for the newly developed LiIn�SexS1�x�2 crystal forsecond-harmonic and optical parametric generation. © 2005 Optical Society of America

OCIS codes: 160.4330, 190.2620, 190.4360, 190.4400, 190.4740.

1. Introduction

It is a well-known fact that mixed crystals grown bystandard crystal growth techniques have variationsof the composition ratio (x) of the two parent crystalsalong the direction of growth. For example, inAgGa1�xInxSe2 crystal a minimal variation of 1.5%has been found along the 40.7 mm long central partsof ingots 80 mm long.1,2 A composition-ratio variation(CRV) of as much as 15% or even 20% was foundthroughout the growth length of Hg1�xCdxGa2Se4 in-gots grown by the Bridgman method.3 Such a CRVcan change the optical properties of a nonlinear fre-quency converter, which may subsequently have anadverse effect on its conversion efficiency. No theoryor algorithm has been available for estimation ofcrystal CRV to maintain frequency conversion effi-ciency at any predetermined level. In this paper wedescribe a method for such an estimation and apply itto calculate the acceptable variations of Se and S inLiIn�SexS1�x�2 single crystals for second-harmonicgeneration (SHG) and optical parametric oscillation

(OPO). In the calculations, a decrease of conversionefficiency to approximately 2�5th of its optimizedvalue is taken as the final acceptable limit for x in thecomposition ratio (CR) of the two parent crystals.

2. Modeling

In our analysis, scattering and absorption effects thatare due to crystallographic defects that may accom-pany CRV are excluded because, owing to consider-able development in methods of crystal growth, theoccurrence of such defects has been drastically re-duced. To estimate the influence of CRV of two parentcrystals in a mixed crystal and hence on the conver-sion efficiencies of nonlinear frequency mixing in thatcrystal, a simple model as detailed below can be used.

The phase-matching condition for three-waveinteraction in a nonlinear crystal is4,5

�k �2�

�1n1 �

2�

�2n2 �

2�

�3n3 � 0, �1 � �2 � �3.

(1)

For a solid solution with CR � x, the refractive indi-ces of mixed crystal are linked to those of the parentcrystals by the equation6

nj�x� � ��njA�2x � �1 � x��nj

B�2�1�2, j � X, Y, Z,(2)

where the superscripts A and B identify the relatedparent crystals. This relation presents a good ap-proximation of the refractive indices of a mixedcrystal along the crystallographic coordinate axes

J. J. Huang is with the Department of Applied Physics, HarbinUniversity of Science and Technology, West Campus, P.O. Box 125,150080 Harbin, China. Y. M. Andreev, G. V. Lanskii, and A. V.Shaiduko are with the Department of Ecological Devices, Instituteof Monitoring of Climate and Ecological Systems SB RAS,Academicheskii Avenue, 10�3, 634055 Tomsk, Russia. S. Das andU. Chatterjee ([email protected]) are with the Laser Labora-tory, Department of Physics, Burdwan University, Burdwan713104, India.

Received 5 May 2005; revised 17 August 2005; accepted 17 Au-gust 2005.

0003-6935/05/357644-07$15.00/0© 2005 Optical Society of America

7644 APPLIED OPTICS � Vol. 44, No. 35 � 10 December 2005

�nX, nY, nZ�.7 Now, if a small change in the CR,i.e., x � x � x�, occurs, where x� is an average orexpected mixing ratio at which Eq. (1) is satisfied, aphase mismatch will appear, i.e., �k 0. By differ-entiating Eq. (1), one gets a relation between devia-tion x and mismatch constant �k:

2� �j�1

3

�j x � �k. (3)

Rearranging relation (3), we have

x ��k2��

j�1

3

�j�1

, �j � ��nj�x��j�x

x�x�, (4)

where � is used for j � 1, 2 and � is taken for j� 3. Relation (4) is valid for a small deviation, i.e.,x �� x, and can be used to define an acceptable pa-rameter, e.g., the acceptance limits of the CRV or themixing acceptance bandwidth. It is assumed that therefractive indices vary weakly with x, which is quiteacceptable for almost all practical purposes. Underthis assumption, relation (4) is also weakly varying inCR, and a linear approximation for ��j�1

3 �j��1 isreadily acceptable to describe its dependence on x�.And an expansion according to a reference CR � x0(arbitrary) such as

��j�1

3

�j�1

� [1 � �(x��x0)],

� � �j�1

3�j��1

�x�x0,

� � � ��j�13

�j�13

�j�

�j

x�x0

(5)

can readily be made. Here, the prime means a de-rivative of �j taken with respect to x�, coefficients and � depend on the wavelengths, and � �� 1 holdsbecause of the assumption stated above. It is appar-ent that � is a slowly varying function of wavelengths,in contrast to , as the denominator and the numer-ator in the expression for � show such behavior withvariations of wavelengths. From expression (5) wehave

x ��k

2� �1 � ��x� � x0��. (6)

If �k � ��L (L is the crystal length), which indi-cates an �59% decrease of conversion efficiency bythe dephasing factor sinc2 ��kL�2� � 4��2 � 41%�,there will be a simple definition of mixing acceptancebandwidth at x�, i.e., |�x|, where �x is defined by

�x 1 � ��x��x0�

2L . (7)

If x0 is equal to x�, then the second term vanishes, asis the case for the study reported in this paper. Equa-tion (7) shows that acceptable variation �x is directlyproportional to x� and inversely proportional to thecrystal length. It is reasonable to emphasize that thecondition �x �� 1 should be fulfilled first. In fact, ifthe phase-matching characteristics (angle versuswavelength) of all compositions create a fixed or anearly fixed crossing point, then, near that point, thebandwidths of the composition ratio will be so largethat they can reach 1.0. However, this is unimportantbecause the mixing of parent crystals will loose itsmeaning for phase matching near that point. In thiscase the following equations, which refer to Eq. (1),should be used for a better estimation:

�k�x� � �x�, �1, �2, �3, �� L,

�k�x� , �1, �2, �3, �� 0, (8a)

�k�x� � �x�, �1, �2, �3, �� L,

�k�x� , �1, �2, �3, �� 0, (8b)

where x� is the expected CR, which ensures perfectphase matching at angle �, and �x� and �x� denoteacceptable positive and negative variations, respec-tively. An average of |�x�| and |�x�| should betaken as the acceptance bandwidth. However, mixedcrystals are generally used for noncritical phasematching (NCPM). In such a case the acceptancebandwidth of the CR is quite small; it is different withangular acceptance. So there is no need to discrimi-nate between �x� and �x� because of the approxi-mation |�x�| � |�x�|.

3. Optimal and Acceptable Crystal Lengths

It is known that the phase mismatch caused by aCRV will turn conversion efficiency back at a cer-tain crystal length. So it is reasonable to defineoptimal and acceptable lengths of a mixed crystal.Two typical situations are considered here. One isthat there is no local CRV but a gradual deviationfrom an expected value x� in the direction of crystalgrowth, which in general is not along the beam axis.It can be proved that the induced longitudinal non-uniformity of the refractive index dominates thephase mismatch, whereas the transverse index vari-ation, which deflects the incident beam slightly, canbe ignored in a plane-wave model. The other one isthat there is a nonnegligible local distortion of theCR, which varies in a highly random manner. In thefirst case, to define the geometrical relations we sim-ply set x to be the varied mixing ratio (perfect phasematching occurs at x�, which is located at the centralpart of the crystal). This does not involve a limitation,as one can always tune the angle of incidence to asmall extent in an experiment to offset the phasemismatch that arises from the deviation of the CR atthe center. Let parameter � be the gradient of the CR

10 December 2005 � Vol. 44, No. 35 � APPLIED OPTICS 7645

along the propagation direction of crystal length L.Then

x � x� � ��z � L�2�, x � ��z � L�2�, (9)

where z is the longitudinal coordinate inside the crys-tal with z � 0 for the incidence surface and z � L forthe output surface. In the low-conversion limit, let usconsider plane-wave interaction. Because of a CRV[Eq. (9)], the refractive index of the mixed crystalvaries slightly with z in an approximately linear pat-tern, where transverse variation is ignored. Then acommon plane-wave transmission in the z directionto this kind of medium is E0 exp�i�k*�z�z � �t��,where E0 is a constant amplitude vector, k*�z� is aquasi-wave constant, and t is time. With respect tothe plane-wave form, some concepts of a wave con-stant should be introduced. For three-wave interac-tion, the jth �j � 1, 2, 3� wave vector (instant) is

kj�z� � ��kj*�z�z��z � kj*�z� � z��kj*�z��z,

j � 1, 2, 3. (10)

Inasmuch as kj�z� depends weakly on z, it can bewritten as

kj�z� � kj0 � kj�z� (11)

when kj0 is constant and kj�z� � 0 when x � x�, at thecenter of the mixed crystal. Likewise, kj*�z� can bedefined, and their relation readily obtained, throughEq. (10):

kj*�z� kj*�z� � kj0, (12a)

kj�z� � kj*�z� � z�kj*�z��z � ��zkj*�z��z ,(12b)

kj*�z� �1z�

0

z

kj�z�dz. (12c)

Now we are set to conceive three-wave couplingequations in this gradual index medium. For brevitywe omit simple intermediate approaches and knownresults that were reported in Ref. 8. The typical non-linear wave equation in a charge-neutral and non-conducting medium is

� � � � Ej � �0�j2��z� · Ej � �0�j

2PjNL, (13)

where �j is the angular frequency; � is a linear sus-ceptibility tensor; and Ej and Pj

NL are electric fieldand nonlinear polarization vectors, respectively. Theform of solution that we used is

Ej�z, t� � ½ aj Aj�z� exp�i�kj*�z� z � �j t�� c.c.,

j � 1, 2, 3; (14a)

hence the intensity is

Ij�z, t� � ½ nj�z�c�0�Aj�z��2, (14b)

where aj is unit constant vector and c is the velocityof light in vacuum. In a straightforward mathemati-cal exercise, the following simple coupling equationresults:

2ikj�z�aj

dAj

dz � �0�j2Pj

NL exp��ikj*�z�z�. (15)

In relation (15), known relation kj�z� � kj�z� � aj

� �0�j2��z� · aj � � 0 for a plane wave as well as the

slowly varying approximation are applied. With theapproximation kj�z� � kj0 on the left-hand side ofrelation (15), the set of coupling equations is obtainedas

dA1

dz � iB1 A3 A2* exp��i�k*�z�z�, (16a)

dA2

dz � iB2 A3 A1* exp��i�k*�z�z�, (16b)

dA3

dz � iB3 A1 A2 exp�i�k*�z�z�, (16c)

where

Bj ��j djeff

nj0c cos �j,

�k* k10 � k20 � k30 � k1* � k2* � k3*; (17)

nj0 comes from kj0; and �dj�eff and �j are the effectivenonlinear coefficient and the walk-off angle, respec-tively.

Hence the generated harmonic amplitude is givenas

A3 �i�3

n30cd3eff A1 A2 �

0

L

exp�i�k*�z�z�dz

� �0

L

exp�i�k*�z�z�dz. (18)

From relations (6) and (9) and choosing x0 � x� andK � 1��2 �, we have

k�z� k1�z� � k2�z� � k3�z�

� 2�x� � 4�Kx � 4�K��z � L�2�. (19)

With Eqs. (12) and basic phase matching, �k0 � k10� k20 � k30 � 0, the phase factor in relation (16)


becomes

�k*�z� �1z �

0

z

k�z�dz � 2�K��z � L�. (20)

Then the right-hand side of relation (18) gives

where � � L|K�|1�2 ; S�� and C�� are the Fresnelsine and cosine integral functions, defined as�0

� sin��x2�2�dx and �0� cos��x2�2�dx, respectively;

and the plus in the last line of Eq. (21) is for positiveK�, whereas the minus is for negative K�. With iden-tity (21), the intensity of the generated beam reads as

I3 �I1I2

I1 � I2

1

�K��C2�� S2��

LNL2 � � (I1 � I2),

(22a)

where

LNL � �8�2deff2�I1 � I2�

n10n20n30�0c�32 ��1�2

, (22b)

� is the conversion efficiency and LNL is the nonlinearor characteristic length.

Furthermore, we can define optimal length LM asthe value of crystal length L at which � reaches itsfirst local maximum. Therefore, by differentiatingconversion efficiency � or intensity I3 in Eq. (22a), wecan obtain the optimal length from the followingequation:

C�� cos��2��2� � S�� sin��2��2� � 0. (23)

It is easy to find the solution of Eq. (23) by using atypical numerical method to find the root that gives�0 1.21. Thus

LM � �0�K��1�2. (24)

The absolute average deviation of the CR can beshown from Eq. (9) to be |�xd| � L|�|�4, andfurthermore we have |�| � 4|�xd|�L. Substituting|�| and the second equality of relation (19) intoEq. (24), and setting L to be LM, we obtain

LM � �02� ��xd��1�2. (25)

�0

L

exp�i�k*z�dz ��0

L

exp�i2�K��z � L�z�dz

��0

L

exp�i2�K��z � L�2�2 � L2�4��dz

� exp��i�L2K��2��0

L

exp�i2�K��z � L�2�2�dz

� exp��i�L2K��2��K��1�2�C�L�K��1�2� � iS�L�K��1�2��

� exp��i�L2K��2�L�C�� iS��, (21)

Table 1. Sellmeier Coefficients of LiInS2 and LiInSe2 Crystals

Crystal

Coefficient

A1 A2 A3 A4 A5

LiInS2

X 5.59849 530.77914 �636.96513 0.05037 �0.13168Y 6.07171 645.92403 �991.01494 0.03743 �0.16904Z 5.14293 311.40611 �182.42666 0.06601 �0.13359

LiInSe2

X 5.6650 448.1772 �282.8189 0.0863 �0.2175Y 5.6417 366.1032 �160.3973 0.0931 �0.2377Z 6.2066 639.0029 �620.4328 0.0914 �0.2441


The simple relation in Eq. (25) expresses a kind ofbalance between the chosen crystal length and theCRV, as is important for when one wishes to cut acrystal for a practical device. When the CRV in acrystal is appreciably large, the crystal length shouldbe shortened appropriately. The walk-off effect andabsorption will be small, as will the conversion effi-ciency; hence Eq. (25) presents a considerably goodestimation of the optimal crystal length.

When a local CRV occurs randomly inside a crystal,however, we generally can just estimate an averageresult of a phase mismatch or a reduction in conver-sion efficiency. If an absolute average deviation of x is|�xd|, we use it to determine the acceptable crystallength LA from Eq. (7). It is easy to obtain

LA � � ��xd��1�2. (26)

Equation (26) can be used to obtain a fairly accurateestimation of LA, provided that the CRV of the mixedcrystal is not extremely large. Comparing Eq. (25)with Eq. (26), we can see that the optimal length hasthe same form (an inverse relation to |�xd|) as theacceptable length, except for a constant factor �0

2

� 1.46. However, it should be emphasized that thelow conversion limit that we used does not permit along optimal length or acceptable length. And char-acteristic length LNL [Eq. (22b)] can be taken as acriterion with which SHG will produce �58% conver-sion efficiency in the high conversion model.7 So LM

� LNL and LA � LNL should be satisfied, and, if theinequalities are inverse, the variation of the compo-sition ratio can certainly be neglected if one is notaiming at a high conversion efficiency (beyond 58%).At the same time, the inequalities imply the existenceof minimal variations of composition ratio by Eq. (25)and Eq. (26), i.e.,

��xd� �2�0

2� �

LNL�optimization�,

��xd� �2� �LNL

�acceptance�. (27)

The results given above are suitable for generalthree-wave parametric interactions, sum-frequencygeneration, difference-frequency generation, and OPO.Though the two kinds of CRV are frequently encoun-tered, Eqs. (25) and (26) can be taken as referenceswith which to determine an effective length of mixedcrystal associated with an arbitrary CRV.

4. Applications in the Mixed Crystal LiIn(SexS1�x)2

The mixed crystal LiIn�SexS1�x�2 is developed frombiaxial parent crystals LiInS2 and LiInSe2. Thoughthe optical properties of LiInS2 were studied threedecades ago,9 the low nonlinear coefficients limit itsfurther application. However, recently its ability forfrequency conversion of an ultrashort pulse and abil-ity to convert in the mid-infrared range make it afavored crystal once again.10–13 Only recently was thecrystal LiInSe2 synthesized14 to improve frequencyconversion ability compared with that of the earlierknown LiInS2. Certainly the mixing of the two crys-

Fig. 1. (a) Phase-matching diagrams and (b) spectral dependenceof the mixing acceptance bandwidth on wavelengths in the XYprincipal plane of Type I SHG with 1 cm LiIn(SexS1�x)2 for severalvalues of mixing ratios: x � 0, 0.2, 0.4, 0.6, 0.8, 1.0.

Fig. 2. Phase-matching diagrams of OPO for Type I in the XYprincipal plane of a LiIn(SexS1�x)2 crystal pumped by Nd:YAG andHo:YLF lasers.


tals will enhance their potential for frequency con-version and will enable us to achieve NCPM.

Table 1 lists the Sellmeier coefficients of LiInS2 andLiInSe2 with the same refractive index equation:nX,Y,Z

2 � A1 � A3��A2 � �2� � A5��A4 � �2�.15

With Eqs. (1) and (2) and Table 1, phase-matchingcalculations can be made in SHG or sum-frequency–difference-frequency generation. Figure 1(a) showsthe Type I phase matching in the XY principal planefor SHG in a LiIn�SexS1�x�2 crystal with perfect mix-ing. Figure 1(b) shows the maximum acceptable vari-ations �x� and �x� in the XY plane for SHG in a 1 cmthick LiIn�SexS1�x�2 crystal with imperfect mixing,under the assumption that up and down x exists lo-cally inside the crystal. Remember that acceptablevariations of x will become much larger than theusual variations near 3 �m; hence Eqs. (8) combinedwith Eqs. (1) and (2) must be used to get enoughaccuracy. And the intermediate parameter, phase-matching angle � (in the XY plane), should be deter-mined at CR � x by Eq. (1) with �1 � �2 � 2�3.

In Fig. 1(a) there is a fixed phase-matching point at��, �� 2.98 �m, 35.8°� that makes phase match-ing nearly independent of CR at 3.0 �m, as can beseen from Fig. 1(b). And the more the fundamentalwavelength deviates from 3.0 �m, the less the accep-tance bandwidth decreases. So, generally, there is achallenge for NCPM on the phase-matching bound-ary where the smallest acceptance bandwidthfalls. For OPO we consider only two cases: OPOwith a LiIn�SexS1�x�2 crystal pumped by a Nd:YAG�1.06 �m� and a Ho:YLF �2.08 �m� laser. Figure 2shows the phase-matching diagrams in the XY planefor Type I interaction. The results of Type I OPOpumped by the two lasers, all in the XY plane, with a1 cm long crystal are presented in Fig. 3. The behav-ior of the acceptance composition ratio of OPOpumped by a Nd:YAG laser is similar to that of SHGin Fig. 1(b); i.e., there is also a sharp increase but at�1 � 5 �m. This can be explained by the emergence ofa cross point in Fig. 2. However, the phase matchingof OPO pumped by a Ho:YLF laser does not experi-ence the emergence of a cross point, so �x� and �x�

are quite small, and the curves in Fig. 3(b) are gov-erned by Eq. (7). Combining Eq. (7) with Eq. (25) orEq. (26) and neglecting �, we have

LM � �02 L �x

�xd, LA � L �x

�xd. (28)

We let L � 1 cm according to the figures above. So wecan see that the curves in Figs. 1(b) and 3 revealthe values of optimal and acceptable lengths fromEq. (28) for any given |�xd|, where |�x| is themean of |�x�| and |�x�|. For example, if NCPMSHG at 9.6 �m is to be obtained with |�xd| � 0.02 atx � 0.4, the optimal and acceptable lengths will beLM � 7.3 and LA � 5.0 mm, respectively.

5. Conclusions

In conclusion, for the first time to our knowledge thetheory of an acceptance composition ratio to predictthe optimal and acceptable lengths in a low-conversion model of plane waves has been developed.According to the theory, the estimations carried outfor SHG and OPO in mixed LiIn�SexS1�x�2 crystals ofreasonable length �1 cm� show that mixing accep-tance bandwidths are strongly dependent on thestructure of phase-matching diagrams and pumpingwavelengths. The same is true for the optimal andacceptable crystal lengths. However, for ultrashortpulse interaction, CR has less influence on phasematching because a relatively short crystal is com-monly used for group-velocity matching. If a beam’saperture or high conversion efficiency is considered,the mathematics will be more complicated and can-not be treated analytically.

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Fig. 3. Spectral dependence of the mixing acceptance bandwidthon wavelengths �1 for Type I OPO pumped by (a) Nd:YAG and(b) Ho:YLF laser radiation in the XY principal plane with 1 cmLiIn(SexS1�x)2 for several values of mixing ratios: x � 0, 0.2, 0.4,0.6, 0.8, 1.0. ssf, slow-slow-fast combination of refractive indicesreferring to Type I phase matching.


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Acceptable composition-ratio variations of a mixed crystal for nonlinear laser device applications

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