Picosecond nonlinear absorption and phase conjugation in BSO and BGO crystals

Optics Communications 90 (1992) 391-398 North-Holland

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F u l l length article

Picosecond nonlinear absorption and phase conjugation in BSO and BGO crystals M. Sylla, D. R o u ~ d e , R. Chevalier, X. N g u y e n P h u a n d G. R i v o i r e Laboratoire des Proprietes Optiques des Materiaux et Applications, U.A. CNRS D0780, 4 Boulevard Lavoisier, B.P. 2018, 49016 Angers Cedex, France

Received 9 October 1991

We have observed a very strong nonlinear absorption in BSO and BGO crystals when they are illuminated by picosecond laser pulses with optical intensity on the order of 108 W/cm 2. In addition, a phase conjugate refleetivity of 2 × 10-3 is obtained in a degenerate four-wave mixing (DFWM) experiment using the same optical power. We derive the nonlinear coupled-wave equations to explain the experimental results. We show that the creation of the conjugate wave can be explained by the presence of a strong nonlinear absorption.

1. Introduction Optical phase conjugation induced in photorefractive BSO and BGO crystals has been extensively studied [ 1 ]. Most of the work has been done using cw lasers with optical power less than 1 W / c m 2. The mechanism o f phase conjugation is based on the diffraction of the optical waves on an induced phase grating. The modulation of the index is related to displacement o f charges and to the linear electro-optic effect. Few people have studied nonlinear interaction induced with pulse lasers. Lesaux et al. [ 2 ] and Jonathan et al. [ 3 ] have shown that the response time o f the photorefractive effect could be very fast, in the nanosecond range. The purpose o f this paper is to reveal that in addition to the standard electro-optic effect, very fast effects can be present in these materials when they are excited by picosecond pulses with an intensity o f approximately 10 s W / c m 2. We observe the creation o f a phase conjugate beam in a four-wave mixing experiment with a reflectivity o f 10-3. This nonphotorefractive effect is local, due to electronic excitations [ 4 ] and is independent of an external electric field. We derive the coupled-wave equations using the formalism o f third-order nonlinear polarization in order to explain the experimental results. We show that the reflectivity o f phase conjugation can be explained by the presence of a very strong nonlinear absorption.

2. Experimental setup The degenerate four-wave mixing ( D F W M ) geometry is shown in fig. I. Three waves o f equal frequency to are generated by a mode-locked N d : Y A G laser delivering 35 ps pulses at 2 = 532 nm. We choose to polarize the three beams circularly. The two p u m p beams 1 and 2 have equal input intensities, I <~>( z = 0 ) = 1 <2> ( z = L ) , and opposite directions. Beam 3 is the probe beam and I <3>(z = 0) = 10 - 21 <~>(z = 0 ) in our experiment. The direction o f propagation o f each beam outside the crystal is at a slight angle 0 ( = 6 ° ) in relation to the z direction. Beam 4 is generated in the interaction and is phase conjugated with beam 3. We use optical delays in 0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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Fig. 1. Diagram of the four-wave mixing experiment: all the incident beams ( 1, 2, 3) are generated by a mode-locked Nd: YAG laser delivering 35 ps pulses at 2 = 532 nm. The two pump beams 1 and 2 have equal input intensities (I (z = 0) = I <2>(z= L) ) and opposite directions. Beam 3 is the probe beam (I <3>(z = 0) = 10 - 21( z = 0 ) ). 0 is the half-interacting angle. All three incident beams are made to be circularly polarized. Beam 4 is generated in the interaction and is phase conjugated with beam 3. The crystallographic orientation is shown in the figure and the BGO sample used in the experiment is holographically cut with L=2 ram.

order to control the temporal coincidence o f the three beams. The crystallographic orientation o f BSO is shown in fig. 1 and is the holograpl: ic configuration c o m m o n l y used [ 5 ].

3. Theory We solve the p r o b l e m in tl te steady state a p p r o x i m a t i o n so that the a m p l i t u d e A o f each b e a m ( j = 1, 2, 3, 4) m a y be taken time ind,~ pendent. The justification o f this a p p r o x i m a t i o n will be discussed later. We also use the plane wave approxim~.tion. The coupled-wave equations in the slowly varying a m p l i t u d e a p p r o x i m a t i o n are given, for the colinear ge ~metry along the z direction, by [ 6] dA /dz= (k} j > / I k } j> I ) [ -- ½o_d + i [ G ] A + i (2~k/n2)p (3) (o9j) ] .

( 1)

The electric field E assoc:i i t e d with each b e a m j is defined as E = ½{A exp [ i ( o 9 t - k ( j > - r ) } + c . c . , where k} j> is the z c o m p o r ent o f the wave vector k for each b e a m j. k =k=-k}2>=-kz<4> = 2nn/2 a n d the cos (0) depem lence is neglected for such a small angle, ot is the linear absorption (or = 1.1 c m at 2 = 532 n m ) , n is the line~ r refractive index and ;t the optical wavelength. [G] is the a n t i s y m m e t r i c tensor [ 7 ] specifying the optical ac :ivity. In BSO a n d B G O crystals the optical activity is very strong ( g ~ 31 ° / m m at 2 = 532 n m ) a n d has to be taken into account in the coupling process [ 8 ]. P <3> is the nonlinear polarization o f order 3 resulting from the n t e r a c t i o n o f three waves. The F o u r i e r c o m p o n e n t o f the third order polarization at the frequency o9 is given t a r each b e a m j ( / = 1, 2, 3, 4) by [9]

P<3> (O.)j=(,Ok +O.)l--O.)m) 3 = ~ 6.~(3)(O9j=O9k+O91-'O9m)!AAA*+ ~., 3X(3)(Ogj=O9j+O9t--O9t)i .'4.4A* , klm I= 1 for k, 1, m = 1, 2, 3, 4 a n d j : a k # l # m .

(3)

Z (3) is the complex third-or,c ter susceptibility tensor. The subscript in o9 indicates which wave is involved in the interaction. The first terta couples three different waves a n d is responsible for the creation o f a phase conjugate wave. The second t e n a couples two different waves a n d describes nonlinear change o f refractive index or absorption. F o r crystals ~ ith the 23 symmetry, the third order susceptibility tensor, when expressed in the crystallographic coordinates, has 21 nonzero coefficients. Only 7 coefficients, n a m e d a~ to a7, are i n d e p e n d e n t and are given by [ 10] 392

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The subscripts 1, 2, 3 in j( refer to the spatial crystallographic directions ( 100 ), (010 ), (001 ). BGO is a cubic crystal with optical activity, consequently we have chosen to derive the coupled-wave equations using circularly polarized beams. Following the method described in ref. [ 11 ], we can derive the coupledwave equations for the circularly polarized amplitude of each beam.

4. Propagation of the pump beams If we consider that the pump beams have an intensity much greater than the probe and conjugate beams, the first term in eq. (3) is neglected and only the effects between the pumps are taken into account in the second term. The equations relative to the pump beams can be solved separately and the amplitude A ~> (the plus and minus signs stand for right-hand and left-hand circularity respectively) of beam j (j = l, 2 ) can be derived from eq. ( l ) using the usual transformations of coordinates. In this paper, we always will take beam 1 as having a right polarization and in that case the propagation of the pump beams is described by the following set of equations: dA <+~>/dz= [ - ½a + i g + ½iH(C~ I + Co I<2> ) ]a ~+1)

(5)

and dA ~2>/dz= [ ½ot+_ig-½iH( C~ I + C ~ l<2> ) ]A ~2> .

(6)

C o = C ~ or Cff depending on the circularity, right or left, of beam 2. I<~>=½A<+I>A~_1>* and 1<27= ½A~_2>A~_2>. are the intensities of beam 1 and 2. The last term in brackets in eq. (5) is relative to nonlinear effects induced by the two strong pumps. H = 127t2/An. C~ and Cff are two constants depending on the seven independent coefficients of the third order susceptibility tensor Xt3). They are calculated for the geometry described in fig. 1 and are given by C~ --0.5 [3al + 2.5 (a2 + a 3 +a4 +a5) - 1.5 (a6 +aT) ] , Cff --0.5 [3al + 2.5 (a2 +a3 +a6 +a7) - 1.5 (a4 +a5) ] •

(7)

We now will consider the propagation of beam 1 only. The differential equation for the intensity of the pump, resulting from eq. (5), reduces to d//dz=- [ct+HIm(C~

)I ]l .

(8)

The second term in the square brackets represents a self-induced nonlinear absorption. The transmission T <~>= I <~>(z = L ) / I <~>(z = 0 ) resulting from the integration of eq. ( 8 ) is T<~> exp(-otL) - 1+ [ H I m ( C ~ ) I < l > ( z = O ) / o t ]

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(9)

The experiment was performed to demonstrate the intensity dependence of this transmission. Only one beam is incident on the crystal and is filtered with a maximum incident intensity I <~>(z = 0 )max-- 170 M W / c m 2. The beam diameter is 2.5 mm. The theoretical expression and the experimental results of the transmission are presented in fig. 2 for a 2 m m thick BGO crystal. The effective transmission T <1> shows that the absorption increases with intensity. The theoretical expression given by eq. (9) is used to fit the experimental data. We find H I m ( C d ) I ( z = 0) = 2 c m - 1 and H Im(Cg- ) I ( z - - O ) / a = 1.8 for 1 <1> ( z = O ) = 170 M W / c m 2, which shows that the nonlinear absorption is 393

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greater than the linear one fi: r such input intensity. The theoretical expression and the experimental values of the transmission for a left ciL"cularly polarized beam would give the same results. The analytical expression ii" <,2> of the transmission for beam 1 in the presence of beam 2 can also be derived using eqs. (5) and (6). Bottl theoretical and experimental values are presented in fig. 2 and are in good agreement. T is also indepen lent of the relative circularity of the two beams.

5. P r o p a g a t i o n

o f t h e p r o b e II c a m in p r e s e n c e o f t h e p u m p b e a m s

We now consider the tran,'~ mission of the probe beam in presence of the pump beams. The equation relative to the propagation of the pro )e can also be derived using eq. ( 1 ). Taking into account that the phase conjugate reflectivity is low (of the or,~ er of 10-3), we neglect the phase conjugate term in eq. (3). In addition, the intensity of the probe is chose]: sufficiently low to eliminate a self-induced absorption. In that case, the equation of propagation for the ampl tude A ~3> of the probe is

dA _~3>/dz=[-½a+_ig+½iHIC~ I+ C ~ I<2>) ]A ~3>

(10)

for two right circularly pola~t [zed pump beams. If beam 2 is left circularly polarized then the upper subscripts plus and minus reverse in th second coefficient Cff. As in the previous section, the last term in brackets represents a nonlinear change o: absorption of refractive index and is due to the presence of the two pump beams. We now will consider the )articular case of the transmission of the probe in presence of beam 1 only. Consequently, using the analyti( al expression of i given by the integration of eq. (8), we can solve eq. (10) immediately. The resulting ransmission T <31>=/<3> (z=L)/l<3>( z = 0 ) of the probe beam is given by the same equation as eq. (9) if beam 3 is right circularly polarized. Im(C~- ) is replaced by Im(Cff ) if the circularity of beam 3 is left. In ag. 3 we present the experimental and theoretical results of the transmission. The upper curve is the experimel tal transmission of the probe without a pump. We can notice from this curve that there is no self-induced non inear absorption. The curve with black circles represents the experimental values of the probe transmission int the presence of pump 1 when both beams are fight circularly polarized. If we use the theoretical expression o:1 T <3,> and the value of H I m (C~-) determined previously, the theoretical curve and the experimental values ffthe transmission are in perfect agreement, as is shown in the figure. If we change the circularity of the probe, :he experimental results remain unchanged, which means that the transmission is independent of the relative )olarization of the two beams and consequently Im(C~" ) = I m ( C f f ). We have performed the s a n e experiment using pump 2 instead of pump 1. The transmission of beam 3 does not change for an equal pun p intensity. This result shows that the temporal coincidence of all the three beams is correct in the experiment The lower curve in fig. 3 is the transmission of the probe in the presence of the two pump beams and is 394

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Fig. 3. Transmission of the probe beam as a function of the incident pump intensity I <~>( 0 ). The open stars (-A-) represents the experimental values for beam 3 alone. The experimental transmission of the probe in presence of beam 1 or beams 1 and 2 are represented by black circles ( • ) and triangles ( A ) respectively. The solid curves are relative to the theoretical results when the probe is in presence of one pump ( T <3~>) or two pumps ( T <312>).

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i n d e p e n d e n t o f the circularity o f the three beams. The theoretical transmission T <312> can also be derived using eqs. ( 5 ) , ( 6 ) and ( 11 ) a n d is in good agreement with the experimental values. In o r d e r to check the steady state a p p r o x i m a t i o n , we have m e a s u r e d the transmission o f b e a m 3 in the presence o f a constant p u m p intensity but with an increasing delay. We noticed [ 12 ] that the i n d u c e d absorption goes up rapidly a n d is still partially present after a few h u n d r e d picoseconds, which justifies our a p p r o x i m a t i o n . More details will be given in an u p c o m i n g publication in o r d e r to i m p r o v e the model. F r o m all the results presented above, we can conclude that this strong nonlinear a b s o r p t i o n does not d e p e n d on the polarization o f the different waves and also is consistent with the two-photon absorption model.

6. Phase conjugate beam In this case, all the b e a m s are incident on the crystal. The equation o f p r o p a g a t i o n for the a m p l i t u d e A ~4> o f the conjugate wave can be d e r i v e d in the same way a n d takes the following form when the two p u m p b e a m s are right circularly polarized:

dA~4>/dz=[½a+ig-½iI-[(C~I+C~l<2>)]A~4> - ~u,,~ l;ur'-+A . , ,. -a <2>A ~ +<3>*

(11)

If b e a m 2 is left polarized then the u p p e r subscripts plus and minus are reversed in the second coefficient C ~ . A is A ~> or A , d e p e n d i n g if b e a m j ( j = 2 , 3) is right or left circularly polarized. C + as C~ is a c o m b i n a t i o n o f the seven i n d e p e n d e n t coefficients o f the t h i r d - o r d e r susceptibility tensor Z (3) a n d the values o f this coefficient are presented in fig. 4. The last t e r m on the right in ¢q. ( 1 1 ) is the phase conjugate term responsible for the creation o f b e a m 4. As above, we have only considered the effect o f the p u m p b e a m s in the i n d u c e d absorption term and the coupling between A <+4> a n d A <_4> has been omitted. In o r d e r to estimate the reflectivity o f phase conjugation, I<~> +i<2> is taken constant (i<1> (z=O)= i<2> (z=L) a n d the m u t u a l a b s o r p t i o n o f the p u m p b e a m s is a p p r o x i m a t e l y identical in that case). In the resolution o f eq. ( 1 1 ), Re ( C ~ ) can also be neglected because we have estimated the corresponding phase change using a M a c h - Z e h n d e r interferometer a n d we have found that 395

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H Re(C~ ) (I 4..1<2> ) +l ) A ! 2> is simp ¢ given by A (+l >A ~2) = 2

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expl :(g-T-g)z],

(12)

where I < ' > I <2> is an exact c, nstant of propagation.

Using the same approxima tions, eq. (10) is easily integrated and the analytical expression A <3>(z) can be derived. Finally, with the bc mdary condition, A ~4> ( z = L ) = 0 , the reflectivity of phase conjugation R+ has the following form:

Re =I~4>(z=O)/I<3>(z=Oi =HZIC -+12L2I <1>( z = 0 ) 1 <2>( z = 0 ) ~,

(13)

with ~=2 c o s h ( a + a . t ) L - c o s ( ~ : L) i - - exp[ - ( a + ct,~)L] . (t~ 4. O~nl)2L2+ (o+_L

(14)

The reflectivity of phase con ugation given by eq. ( 13 ) with ~= 1 is the classical expression for a lossless medium. ~ is a corrective facto: taking into account both nonlinear absorption and optical activity. a n , = H I m ( C +) [I ( z = ,) +1 <2>( z = 0 ) ] is the total nonlinear absorption induced by the two pump beams. O+ is a constant, the value ot which is mostly determined by the rotatory power. All the values of ¢+ are listed in fig. 4 for different circula ities of the four beams. In eq. ( 13 ), the driving cc .'fficient responsible for the creation of beam 4 is C-+. As we have shown above, I m ( C + ) > R e ( C +) and the lonlinear absorption is here, unlike phase conjugation using an index variation via the electro-optic effect, t Le main effect involved in the creation of the conjugate beam. In the special ease of low al sorption (o6 + oent)L << 1, then ~= sine 2( ½¢ +L ) is a reducing factor mainly related to the presence of the optic~ activity. Its effect has been noted [ 13 ] before in a standard two-wave mixing experiment using cw lasers. In fig. 5 we present the e~ perimental results for the reflectivity. The two eigen modes of polarization are simultaneously present in b~ lm 4 and are separately detected with circular analyzers. From these curves and in elation to the results above, it is important to note that beam 4 and the induced absorption appear for the sa ne input intensity and therefore are strongly related. The value of the reflectivir mainly depends on the C -+ coefficient but the corrective factor can be significant. For example, in case ( + - - + ) of fig. 5d, ~=0.09 explains the very low reflectivity. By comparing the value o! the reflectivity in the different configurations of polarization of fig. 5, we obtain IlCoU~ IlC21l. 3llC~ll is of the ;ame order of magnitude. We can estimate the refect :ity in case ( 4- + 4, + ) using the value of Im ( C + ) as measured above. I <2>(z = 0 ) is approximately ½I(z= )) for I < l > ( z = 0 ) = 1 7 0 M W / c m 2 and consequently ~=0.45 for g L = 6 2 °. n21l C~- II2L21< l> (2=0) 1 <2> z = 0 ) = 8 × l0 -2 and finally R = 3 . 6 X 10 -2. The experimental value of the reflectivityunder the same co: ditions is 3.1 × 10- 3. This difference can be attributed to the complexity of the DFWM experiment using pi :osecond pulses. In the same setup, we have tested CS2, which is well known for its strong nonlinear effects, at :l we have found that the values of the nonlinear susceptibility given by the DFWM experiment are always lower Lhan the usual values determined by other methods [ 14 ]. In the same conditions of experimentation we find hat Rcs2 = 10RnGo. The experimental results 1 resented here are relative to an undoped crystal. We have tested other BSO and BGO crystals from differen! sources and with various dopings. The same quantitative results relative to the induced absorption and refl, ctivity are obtained.

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Fig. 5. Experimental values of the rcflcctivityas a function of the incident pump intensity I (z = 0). Both right and left polarizations issued from beam 4 are measured with a circular analyzer. The squares (11) and circles (O) refer to the experimental values of the reflectivityassociated with the right and left part of the conjugate beam. Inside the figure, the four signs in brackets ( + _++ + ) refer to the circularity of the interactive beams (1234). (a) Beams 1, 2 and 3 are right circularlypolarized. (b) Beams 1 and 2 arc right circularly polarized and beam 3 is left, (c) Beams 1 and 3 are right circularly polarized and beam 2 is left. (d) Beam 1 is rigl'itcircularly polarized and beams 2 and 3 are left.

7. Conclusion We have observed a very strong n o n l i n e a r absorption in BSO and BGO crystals when they are illuminated by picosecond laser pulses ( 2 = 532 n m ) with optical intensity o f 108 W ] c m z. In addition, a phase conjugate reflectivity of 3.1 × 1 0 - 3 is o b t a i n e d in a D F W M experiment using the same optical power. We have derived the n o n l i n e a r coupled-wave equations in order to explain the experimental results. We have shown that the creation o f the conjugate wave can be explained by the presence of this strong absorption a n d that the optical activity is an i m p o r t a n t factor which spoils the reflectivity of phase conjugation.

Acknowledgements The authors are very grateful to Pr. L a u n a y from the University of Bordeaux for providing the B G O samples and to J.P. Lecoq for technical help.

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References [ 1 ] P. Gunter and J.P. Huignard, e,., Topics in applied physics, Vol. 61. Photorefractive materials and their applications, I. Fundamental phenomena. (Springer, Berlin, 1988 ); Topics in applied physics, Vol. 62. Photorefractive materials and their applications, II. Survey of applications (Springer, Bed n, 1989). [2 ] G. Lesaux, G. Roosen and A. 1 run, Optics Comm. 56 (1986) 374. [ 3 ] J.M.C. Jonathan, Ph. Roussigr ol and G. Roosen, Optics Lett. 13 (1988) 224. [ 4 ] A. Miller, D.A.B. Miller and S D. Smith, Adv. Phys. 30 ( 1981 ) 697. [5] A. Marrakchi and J.P. Huignm d, Appl. Phys. 24 ( 1981 ) 131. [6] A. Fischer, Optical phase conjl gation (Academic Press, New York, 1983 ). [ 7 ] A. Yariv and P. Yeh, Optical ~ tves in crystals (Wiley, New York, 1984). [8] S. Mallick, D. Rouede and A . ( . Apostolidis, J. Opt. Soc. Am. B 4 (1987) 1247. [ 9 ] H. Rabin and C.L. Tang, Quar turn electronics, Vol. 1. Nonlinear optics, Part A (Academic Press, New York, 1975 ). [ 10] S. Kielich, Podstawy optyki Ni :liniowej (Universit6 A. Mickiewicza, Poznan, 1972 ). [ l 1 ] M. Sylla, Interactions non lin& ires dans les cristaux Bi~2(Ge, Si )020 excitrs par des impulsion laser picosecondes: Absorption non linraire et conjugaison de phas :, Th~se de Doctorat de l'Universit6 d'Angers ( 1991 ). [ 12 ] M. Sylla, R. Chevalier, X.P. N :uyen, D. Rouede and G. Rivoire, Ann. Phys. Coll. 1, Suppl. 2 (1990) 15. [ 13] S. Mallick and D. Rouede, Apl I. Phys. B 43 (1987) 239. [ 14] X.P. Nguyen, J.L. Ferrier, J. G tzengel and G. Rivolre, Optics Comm. 51 (1984) 433.

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