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Higher harmonic generation by cascade processes in focused beams
Volume
22, number
1
OPTICS COMMUNICATIONS
July 1911
HlGHERHARMONICSGENERATIONBYCASCADEPROCESSESINFOCUSEDBEAMS V.V. ROSTOVTSEVA, A.P. SUKHORUKOV, V.G. TUNKlN Physics Department, Moscow University, Moscow V-234, USSR
and SM. SALTIEL Physics Department, Sofia University, Sofia-1126, Bulgaria Received 7 February 1977 Revised manuscript received
30 March
1977
Two types of phase-matched conditions for higher harmonics generation in focused beam are treated. In the case of direct process kN = Nkl the dependence of harmonic power as a function of the beam waist position in the crystal is found to be symmetrical, while in the second case of the cascade processes the function is asymmetrical, compared to the crystal center. The results are verified experimentally in the case of third harmonic generation.
1. Introduction A great progress is reached now in the field of higher harmonics generation. After a theoretical paper of Harris [ 11, where he discussed the problem of obtaining vacuum ultraviolet in gases by higher order harmonic generation, some experiments have been performed. Phase-matched fourth harmonic was obtained in LFM crystal [2], fifth harmonic in calcite crystal [3]. Experiments of fifth harmonic generation in gases have been reported recently [4,5]. The main problem, which appears in the nonlinear optical effects of higher order is the problem of the possible interference of the cascade and direct processes. As it is well known, synchronous higher harmonics generation in one crystal can go with approximately equal efficiency not only by direct processes on the nonlinearity of the same order as the harmonic number but also by cascade processes on the nonlinearities of the lower orders [3,6]. The case of third harmonic generation was treated in [6,10], the fifth harmonic one in [3]. For example third harmonic (TH) in crystals without center of inversion can be synchronously generated in the following directions: 1) A3 = k3 - 3k, = 0; 2) A, = k, - 2kl = 0; 3) A2k = k3
56
- k, - k, s= 0. Both the direct process (w + w + w = 3w) on the nonlinearity xc3) and the cascade processes (w + w = 2w, w + 2w = 3w) on the nonlinearity xc2) make their contributions into synchronous TH generation in the first direction A3 z 0. For two other directions only cascade processes on ~(~1 contribute to the synchronous TH generation. Such directions (in which only lower order nonlinearities contribute) are often utilized as reference interactions in measuring of higher order nonlinearities [2,3]. In some cases, when the effect is very weak or if we want to obtain higher conversion, it is necessary to work in focused beams. TH generation on nonlinearity ~(~1 and second harmonic (SH) generation in focused gaussian beams has been considered in [7, 81. Expressions for harmonics power as a function of the beam parameters and wave mismatch P,+, b, 1, AN) have been obtained, where N is the harmonic number; b the confocal parameter; m = L/b the focusing parameter; L the length of the nonlinear medium; 1 = zo/L the normalized beam waist position, z. the beam waist center (nonlinear medium center is at z = 0); AN = kN - Nk, the wave mismatch. In particular, it follows from these expressions that harmonics power does not depend on the sign of parameter 1.
July 1977
OPTICS COMMUNICATIONS
Volume 22, number 1
When we generate the TH in a crystal without inversion center or higher harmonics in focused beams, the cascade processes will complete the picture. In the only work, where this problem is discussed [9], a contribution of the nonlinear susceptibility ~(~1 to the direct process of fifth harmonic generation in the direction A, = 0 is studied. In this work the dependence is considered of the power of synchronous higher harmonics generated by cascade processes in the focused gaussian beam on focusing parameter and beam waist position in the crystal; TH generation in crystals without inversion center is taken as an example. It is obtained theoretically that in the direction A3 a 0 this dependence is the same as for the direct process. It is shown theoretically and experimentally that for other directions of synchronous harmonic generation by cascade processes the harmonic power depends on the sign of parameter 1. It is also shown that in this case the difference in harmonic power for opposite signs of parameter 1 becomes more significant when the parameter m is increasing.
where A,, A,, A3 are the amplitudes of the pump wave, SH and TH waves correspondingly; AI = a2/ax2 + a21ay2 the transverse laplacian; oN the nonlinear coupling coefficients, proportional to ~(~1. The pump wave is a gaussian beam: A,(x,Y,v)=C
l
-_.L!--
a(1 +iQ)
where a is a beam radius; TJ= 2(z - zn)/b; C, is a constant. The upper equations can be solved by means of Green’s function and the expression for integral TH power +OO
c P3 = cr
Jr
A3A;dxdy
--o
can be found
P3 : C, iu3
2m(1+1/2) exp(ibA3v/2) ~--~-s
dq
(1 + i?7j2
2m(l-l/2)
2m(1+1/2) exp(ibA%q/2) + bu2u2k
2. Theory Let us consider a focused pump beam entering a nonlinear crystal without inversion center. The second and third harmonics are generated simultaneously; in the common case they are non phase-matched. As we have mentioned above the general TH appears as a result of direct tripling of the pump frequency and as a result of mixing of pump and SH waves too. These two TH waves interfere. If we neglect the walk-off of the Poynting vector and the depletion of the fundamental amplitude the parabolic equations of ray amplitudes will have the form : aA, T+
1. tiq,
s
2m(Z-l/2)
. $I- AlA2 = -io2AT 2
exp(iA2z) ,
2
exp(ibA2r/2) 2m(Z-l/2)
dg dr
I ti7
,
(1)
where C2 is a constant. Here and further on we meet integrals of the type b
sf(d
exp(ibAd2)drl .
Q
If IbAl > 1, i.e. if coherence length is much smaller than the confocal parameter b, then such integrals can be expanded in series using consecutive integration by parts: ,’ P
a‘43
,+&A,a,
s c1
=
f(v) exp(ibAr)D)
dv = (2)
-io3A: A,(w,
exp(iA3z) - ia,A1A2 -L/2)
= A,(x,y,
-L/2)
exp(iA,,z) =0 ,
,
&Af(q) exp(ibAd2) /B- -_!- f(o) a
exp(ibAd2)
(ibA)2
57
r+ (Y
July 1977
OPTICS COMMUNICATIONS
Volume 22. number 1
Let us consider TH generation near the direction for synchronous direct process: bA3 = 0, IbA21 = IbA,,I > 1. Taking the first term in the expansion (2) we will have:
P3(A3 = 0) = C, ( a3 + yr
IF3(A3)12 9
2
(a)
7 ’
(3)
3 a”
where
0
L
H2
2m(1+1/2) exp(ibA&2) FN(AM) =
I I-
dg .
s 2m(l-l/2)
(1 t ivy-’ (b)
We see that for A3 = 0 the dependence of cascade process on focusing is the same as for direct one. In particular the harmonic power does not depend on the parameter 1 sign: F(Z) = F(- I>; TH power will be just the same when the beam waist center is on entrance or exit face of the crystal. Using the relation (2) it is possible to find the expression for TH power for other synchronous cascade processes: P3(A2 E 0) =
1 [I +4rnqP3(A,,
IF2(A2>12 >
(4)
l/2)2]
= 0) =
1 [l +4m2(1+
IF2(A2k)12
. (5)
o
1
2
3
4
5
6
Fig. 1. The power of the third harmonic versus focusing parameter m: a) in the direction A3 = 0; b) in the direction A2 = 0 (in the direction A2k = 0 the upper curve corresponds to
I = -0.5 and lower one to I= +0.5). by pumping amplitudes on the front and rear faces of the nonlinear medium. On fig. 1 theoretical curves for TH power are plotted as a function of the focusing parameter m for two directions of synchronous interaction and three values of normalized beam waist position: 1 = 0, l/2, -l/2. For each m and 1 the wave mismatch A has been optimized for maximum harmonic power.
1/2)2]
It can be seen from (5) that TH power for A2 = 0 is greater when the beam is focused on the rear face (I = l/2) of the nonlinear medium, than on the front one (1= -l/2): P3(Z = l/2)/P3(Z = -l/2) = (1 t 4m2). For direction As = 0 the relations are opposite. The dependence of TH power on the I sign when A2k = 0 can be explained by taking into account that in this case synchronous summing of the SH “free” wave and the fundamental wave occurs and that amplitude of the “free” wave is determined by fundamental wave amplitude on the front face of nonlinear medium. When the case A2 = 0 is considered we should take into account that here the nonsynchronous summing of synchronous SH and fundamental waves occurs. so according to (2) the harmonic power is determined 58
1
3. Experiment Experimental investigation of TH generation in crystal without center of symmetry was performed with KDP crystal. The main elements of the experimental setup are an YAG : Nd 3+ laser operating in a Q-switched mode (peak power 1 MW, pulse-duration 10 ns) and a recording system, which include photomultiplier FEU39A, amplitude-time converter and X-Y recorder. The crystals were mounted on a two-coordinate table, which permitted to move them along the direction of the laser beam. The crystal lengths used are 4 cm. The interaction oloie2 (phase-matched angle es = 4 1o 11’) was used for investigation of TH generation in the direction A2 = 0, and the interaction e1 oTe3 (0, =
July 1977
OPTICS COMMUNICATIONS
Volume 22, number 1
1
IF3(A3)12 ;
[I t 4m2(I - 1/2)2]2
WA31 B 1,
C)A,,=k,-k&k+,
1 &(A3k)12 [l t4m2(1+ g CSI” I-
.
1/2)212
It can be seen from these expressions that the dependence of fifth harmonic power on parameter I sign is more pronounced than in the case of TH.
4’0 w-e,
u-
0.2 -
5. Conclusion 0
-05
a25
0
025
a5
Fig. 2. The third harmonic power as a function of normalized beam waist position I in the crystal. The focusing parameter m = 5.4. Solid curve - theory, points - experiment.
59OO6’) in the direction A2k = 0. The pump beam was focused in the crystal by a short-focus lens; the focusing parameter was m = 5.4. In fig. 2 experimental curves of TH power dependence on mutual disposition of the crystal and beam waist center are present. The directions A2 = 0 and A2k * 0 are investigated. For each value of 1 the wave mismatch was optimized by slightly changing the crystal orientation. In the same figure the theoretical dependence calculated by using (4) and (5) for given m are plotted. Comparison of theoretical and experimental dependences was made by its values in point I= 0.
4. The case of fifth harmonic generation Carrying out analogous calculations in a media with a center of inversion we obtain following expressions for the fifth harmonic power: a)A5=k5-5klz0,
Ps
I&l
=$(a5 +yrIFs(
b)A3=k3-3kl=0,
s 1,
;
IbA3klE+l,
In summary, we have shown that the picture of higher order harmonic generation in focused beams in the phase-matched directions for direct processes (kN = Nkl) and for cascade processes is different. The experiment confirms the predicted dependences. The effective nonlinear susceptibilities for the treated processes in KDP are Xi2 = 153 ld14(KDP)12 = 65 X lo-l6 esu for the direction 2kp = ks and ‘dj _ Xer - 681d14(KDP)12 = 0.74 X lo-l6 esu for the direction ki + k! = k5. Such values are too small to achieve a high conversion efficiency in KDP for THgeneration. But in crystals with higher quadratic nonlinear susceptibilities as LiNbO,, LiIO,, BaNaNb,O,, and others, one can expect to obtain a conversion efficiency more then four orders of magnitude larger then that in KDP, where we estimate 17- lo-*. The largest conversion can be achieved when the laser beam is focused on one of the faces of the nonlinear media. In such a way a greater conversion efficiency than n - 10e6 [4] in the process of synchronous fifth harmonic generation in metal vapor can be obtained also. One can also utilize these results in the case of measuring higher order nonlinearities in focused beams.
References [l] S. Harris, Phys.Rev. Letters 31 (1973) 341. [ 21 S.A. Akhmanov, A.N. Dubovik, S.M. Saltiel, I.V. Tomov and V.G. Tunkin, ZhETP Pis. Red. 20 (1974) 264.
Volume 22, number 1
OPTICS COMMUNICATIONS
[ 31 S.A. Akhmanov, V.A. Martinov, SM. Saltiel and V.G. Tunkin, ZhETF Pis. Red. 22 (1975) 143. [4] D.I. Metchkov, V.M. Mitev, L.I. Pavlov and K.V. Stamenov, Optics Commun. 21 (1977) 391. [5] J. Reintjes, R.C. Eckardt,C.Y. She, N.E. Karangelen, R.C. Elton and R.A. Andrews, Phys. Rev. Letters 37 (1976).
60
July 1977
[6] E. Yablonovitch, C. Flytzanis and N. Bloembergen, Phys. Appl. Letters 29 (1972) 865. [7] G.D. Boyd and D.A. Kleinman, J. Appl. Phys. 39 (1968) 3597. [8] J.F. Ward and G.H.C. New, Phys. Rev. 185 (1966) 57. [9] I.V. Tomov, IEEE J. Quant. Electr. QE-12 (1975) 521. [lo] C. Wang and E.L. Baarsden, Appl. Phys. Letters 15 (1969) 396.
22, number
1
OPTICS COMMUNICATIONS
July 1911
HlGHERHARMONICSGENERATIONBYCASCADEPROCESSESINFOCUSEDBEAMS V.V. ROSTOVTSEVA, A.P. SUKHORUKOV, V.G. TUNKlN Physics Department, Moscow University, Moscow V-234, USSR
and SM. SALTIEL Physics Department, Sofia University, Sofia-1126, Bulgaria Received 7 February 1977 Revised manuscript received
30 March
1977
Two types of phase-matched conditions for higher harmonics generation in focused beam are treated. In the case of direct process kN = Nkl the dependence of harmonic power as a function of the beam waist position in the crystal is found to be symmetrical, while in the second case of the cascade processes the function is asymmetrical, compared to the crystal center. The results are verified experimentally in the case of third harmonic generation.
1. Introduction A great progress is reached now in the field of higher harmonics generation. After a theoretical paper of Harris [ 11, where he discussed the problem of obtaining vacuum ultraviolet in gases by higher order harmonic generation, some experiments have been performed. Phase-matched fourth harmonic was obtained in LFM crystal [2], fifth harmonic in calcite crystal [3]. Experiments of fifth harmonic generation in gases have been reported recently [4,5]. The main problem, which appears in the nonlinear optical effects of higher order is the problem of the possible interference of the cascade and direct processes. As it is well known, synchronous higher harmonics generation in one crystal can go with approximately equal efficiency not only by direct processes on the nonlinearity of the same order as the harmonic number but also by cascade processes on the nonlinearities of the lower orders [3,6]. The case of third harmonic generation was treated in [6,10], the fifth harmonic one in [3]. For example third harmonic (TH) in crystals without center of inversion can be synchronously generated in the following directions: 1) A3 = k3 - 3k, = 0; 2) A, = k, - 2kl = 0; 3) A2k = k3
56
- k, - k, s= 0. Both the direct process (w + w + w = 3w) on the nonlinearity xc3) and the cascade processes (w + w = 2w, w + 2w = 3w) on the nonlinearity xc2) make their contributions into synchronous TH generation in the first direction A3 z 0. For two other directions only cascade processes on ~(~1 contribute to the synchronous TH generation. Such directions (in which only lower order nonlinearities contribute) are often utilized as reference interactions in measuring of higher order nonlinearities [2,3]. In some cases, when the effect is very weak or if we want to obtain higher conversion, it is necessary to work in focused beams. TH generation on nonlinearity ~(~1 and second harmonic (SH) generation in focused gaussian beams has been considered in [7, 81. Expressions for harmonics power as a function of the beam parameters and wave mismatch P,+, b, 1, AN) have been obtained, where N is the harmonic number; b the confocal parameter; m = L/b the focusing parameter; L the length of the nonlinear medium; 1 = zo/L the normalized beam waist position, z. the beam waist center (nonlinear medium center is at z = 0); AN = kN - Nk, the wave mismatch. In particular, it follows from these expressions that harmonics power does not depend on the sign of parameter 1.
July 1977
OPTICS COMMUNICATIONS
Volume 22, number 1
When we generate the TH in a crystal without inversion center or higher harmonics in focused beams, the cascade processes will complete the picture. In the only work, where this problem is discussed [9], a contribution of the nonlinear susceptibility ~(~1 to the direct process of fifth harmonic generation in the direction A, = 0 is studied. In this work the dependence is considered of the power of synchronous higher harmonics generated by cascade processes in the focused gaussian beam on focusing parameter and beam waist position in the crystal; TH generation in crystals without inversion center is taken as an example. It is obtained theoretically that in the direction A3 a 0 this dependence is the same as for the direct process. It is shown theoretically and experimentally that for other directions of synchronous harmonic generation by cascade processes the harmonic power depends on the sign of parameter 1. It is also shown that in this case the difference in harmonic power for opposite signs of parameter 1 becomes more significant when the parameter m is increasing.
where A,, A,, A3 are the amplitudes of the pump wave, SH and TH waves correspondingly; AI = a2/ax2 + a21ay2 the transverse laplacian; oN the nonlinear coupling coefficients, proportional to ~(~1. The pump wave is a gaussian beam: A,(x,Y,v)=C
l
-_.L!--
a(1 +iQ)
where a is a beam radius; TJ= 2(z - zn)/b; C, is a constant. The upper equations can be solved by means of Green’s function and the expression for integral TH power +OO
c P3 = cr
Jr
A3A;dxdy
--o
can be found
P3 : C, iu3
2m(1+1/2) exp(ibA3v/2) ~--~-s
dq
(1 + i?7j2
2m(l-l/2)
2m(1+1/2) exp(ibA%q/2) + bu2u2k
2. Theory Let us consider a focused pump beam entering a nonlinear crystal without inversion center. The second and third harmonics are generated simultaneously; in the common case they are non phase-matched. As we have mentioned above the general TH appears as a result of direct tripling of the pump frequency and as a result of mixing of pump and SH waves too. These two TH waves interfere. If we neglect the walk-off of the Poynting vector and the depletion of the fundamental amplitude the parabolic equations of ray amplitudes will have the form : aA, T+
1. tiq,
s
2m(Z-l/2)
. $I- AlA2 = -io2AT 2
exp(iA2z) ,
2
exp(ibA2r/2) 2m(Z-l/2)
dg dr
I ti7
,
(1)
where C2 is a constant. Here and further on we meet integrals of the type b
sf(d
exp(ibAd2)drl .
Q
If IbAl > 1, i.e. if coherence length is much smaller than the confocal parameter b, then such integrals can be expanded in series using consecutive integration by parts: ,’ P
a‘43
,+&A,a,
s c1
=
f(v) exp(ibAr)D)
dv = (2)
-io3A: A,(w,
exp(iA3z) - ia,A1A2 -L/2)
= A,(x,y,
-L/2)
exp(iA,,z) =0 ,
,
&Af(q) exp(ibAd2) /B- -_!- f(o) a
exp(ibAd2)
(ibA)2
57
r+ (Y
July 1977
OPTICS COMMUNICATIONS
Volume 22. number 1
Let us consider TH generation near the direction for synchronous direct process: bA3 = 0, IbA21 = IbA,,I > 1. Taking the first term in the expansion (2) we will have:
P3(A3 = 0) = C, ( a3 + yr
IF3(A3)12 9
2
(a)
7 ’
(3)
3 a”
where
0
L
H2
2m(1+1/2) exp(ibA&2) FN(AM) =
I I-
dg .
s 2m(l-l/2)
(1 t ivy-’ (b)
We see that for A3 = 0 the dependence of cascade process on focusing is the same as for direct one. In particular the harmonic power does not depend on the parameter 1 sign: F(Z) = F(- I>; TH power will be just the same when the beam waist center is on entrance or exit face of the crystal. Using the relation (2) it is possible to find the expression for TH power for other synchronous cascade processes: P3(A2 E 0) =
1 [I +4rnqP3(A,,
IF2(A2>12 >
(4)
l/2)2]
= 0) =
1 [l +4m2(1+
IF2(A2k)12
. (5)
o
1
2
3
4
5
6
Fig. 1. The power of the third harmonic versus focusing parameter m: a) in the direction A3 = 0; b) in the direction A2 = 0 (in the direction A2k = 0 the upper curve corresponds to
I = -0.5 and lower one to I= +0.5). by pumping amplitudes on the front and rear faces of the nonlinear medium. On fig. 1 theoretical curves for TH power are plotted as a function of the focusing parameter m for two directions of synchronous interaction and three values of normalized beam waist position: 1 = 0, l/2, -l/2. For each m and 1 the wave mismatch A has been optimized for maximum harmonic power.
1/2)2]
It can be seen from (5) that TH power for A2 = 0 is greater when the beam is focused on the rear face (I = l/2) of the nonlinear medium, than on the front one (1= -l/2): P3(Z = l/2)/P3(Z = -l/2) = (1 t 4m2). For direction As = 0 the relations are opposite. The dependence of TH power on the I sign when A2k = 0 can be explained by taking into account that in this case synchronous summing of the SH “free” wave and the fundamental wave occurs and that amplitude of the “free” wave is determined by fundamental wave amplitude on the front face of nonlinear medium. When the case A2 = 0 is considered we should take into account that here the nonsynchronous summing of synchronous SH and fundamental waves occurs. so according to (2) the harmonic power is determined 58
1
3. Experiment Experimental investigation of TH generation in crystal without center of symmetry was performed with KDP crystal. The main elements of the experimental setup are an YAG : Nd 3+ laser operating in a Q-switched mode (peak power 1 MW, pulse-duration 10 ns) and a recording system, which include photomultiplier FEU39A, amplitude-time converter and X-Y recorder. The crystals were mounted on a two-coordinate table, which permitted to move them along the direction of the laser beam. The crystal lengths used are 4 cm. The interaction oloie2 (phase-matched angle es = 4 1o 11’) was used for investigation of TH generation in the direction A2 = 0, and the interaction e1 oTe3 (0, =
July 1977
OPTICS COMMUNICATIONS
Volume 22, number 1
1
IF3(A3)12 ;
[I t 4m2(I - 1/2)2]2
WA31 B 1,
C)A,,=k,-k&k+,
1 &(A3k)12 [l t4m2(1+ g CSI” I-
.
1/2)212
It can be seen from these expressions that the dependence of fifth harmonic power on parameter I sign is more pronounced than in the case of TH.
4’0 w-e,
u-
0.2 -
5. Conclusion 0
-05
a25
0
025
a5
Fig. 2. The third harmonic power as a function of normalized beam waist position I in the crystal. The focusing parameter m = 5.4. Solid curve - theory, points - experiment.
59OO6’) in the direction A2k = 0. The pump beam was focused in the crystal by a short-focus lens; the focusing parameter was m = 5.4. In fig. 2 experimental curves of TH power dependence on mutual disposition of the crystal and beam waist center are present. The directions A2 = 0 and A2k * 0 are investigated. For each value of 1 the wave mismatch was optimized by slightly changing the crystal orientation. In the same figure the theoretical dependence calculated by using (4) and (5) for given m are plotted. Comparison of theoretical and experimental dependences was made by its values in point I= 0.
4. The case of fifth harmonic generation Carrying out analogous calculations in a media with a center of inversion we obtain following expressions for the fifth harmonic power: a)A5=k5-5klz0,
Ps
I&l
=$(a5 +yrIFs(
b)A3=k3-3kl=0,
s 1,
;
IbA3klE+l,
In summary, we have shown that the picture of higher order harmonic generation in focused beams in the phase-matched directions for direct processes (kN = Nkl) and for cascade processes is different. The experiment confirms the predicted dependences. The effective nonlinear susceptibilities for the treated processes in KDP are Xi2 = 153 ld14(KDP)12 = 65 X lo-l6 esu for the direction 2kp = ks and ‘dj _ Xer - 681d14(KDP)12 = 0.74 X lo-l6 esu for the direction ki + k! = k5. Such values are too small to achieve a high conversion efficiency in KDP for THgeneration. But in crystals with higher quadratic nonlinear susceptibilities as LiNbO,, LiIO,, BaNaNb,O,, and others, one can expect to obtain a conversion efficiency more then four orders of magnitude larger then that in KDP, where we estimate 17- lo-*. The largest conversion can be achieved when the laser beam is focused on one of the faces of the nonlinear media. In such a way a greater conversion efficiency than n - 10e6 [4] in the process of synchronous fifth harmonic generation in metal vapor can be obtained also. One can also utilize these results in the case of measuring higher order nonlinearities in focused beams.
References [l] S. Harris, Phys.Rev. Letters 31 (1973) 341. [ 21 S.A. Akhmanov, A.N. Dubovik, S.M. Saltiel, I.V. Tomov and V.G. Tunkin, ZhETP Pis. Red. 20 (1974) 264.
Volume 22, number 1
OPTICS COMMUNICATIONS
[ 31 S.A. Akhmanov, V.A. Martinov, SM. Saltiel and V.G. Tunkin, ZhETF Pis. Red. 22 (1975) 143. [4] D.I. Metchkov, V.M. Mitev, L.I. Pavlov and K.V. Stamenov, Optics Commun. 21 (1977) 391. [5] J. Reintjes, R.C. Eckardt,C.Y. She, N.E. Karangelen, R.C. Elton and R.A. Andrews, Phys. Rev. Letters 37 (1976).
60
July 1977
[6] E. Yablonovitch, C. Flytzanis and N. Bloembergen, Phys. Appl. Letters 29 (1972) 865. [7] G.D. Boyd and D.A. Kleinman, J. Appl. Phys. 39 (1968) 3597. [8] J.F. Ward and G.H.C. New, Phys. Rev. 185 (1966) 57. [9] I.V. Tomov, IEEE J. Quant. Electr. QE-12 (1975) 521. [lo] C. Wang and E.L. Baarsden, Appl. Phys. Letters 15 (1969) 396.