- Email: [email protected]
Quantum correlation between the signal modes in forward three-wave mixing
15 December
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
144 (1997) 237-240
Quantum correlation between the signal modes in forward three-wave mixing ’ Yajun Lu a32,Ling-An Wu b, Meijuan Wu a, Peilin Zhang
a, Shiqun
Li
a
a Department of Modem Applied Physics. Tsinghua Uniuersity. Beijing 100084, China h Institute of Physics, Academia Sinica, Beijing 100080, China Received 30 April 1997; accepted
1 August 1997
Abstract A quantum theory of forward three-wave mixing is developed for photorefractive nonlinear media in the drift regime or Kerr media. A good agreement between the quantum and classical theories is demonstrated. Quantum noise suppression of the difference between the intensities of the two signal modes is discussed. 0 1997 Published by Elsevier Science B.V.
The strong optical nonlinearity of photorefractive media manifests itself in many interesting and important phenomena. Among these phenomena are weak signal amplification, phase conjugation, mirrorless oscillation, and photoinduced scattering (see, for example, Refs. [ 1.21). The coupling of optical waves in photorefractive media has been widely studied over the past two decades. The traditional amplification mechanism relies on the imaginary part of the coupling constant, or expressed differently, the out of phase component of the grating relative to the interference pattern of the waves. A new gain mechanism was proposed in Refs. [2-51 in which amplification relies on parametric three-wave mixing, or in other words, on the in-phase grating with respect to the intensity pattern. Experiments in liquid crystals by Sanchez et al. [5] and in silicon by Eichler et al. [6] have shown that amplification is achievable in media in which no phase shift takes place. Experiments in photorefractive BSO by Webb et al. [7] were performed by three-wave mixing. Much attention has been focused on beam amplification. The classical theory explaining the nonlinear coupling is already well established. but a quantum approach is not available and nonclassical effects have not been discussed. In this paper we
I
Foundation
of
present, to our knowledge for the first time, a quantum treatment of three-wave mixing (3WM) in photorefractive media. We study the 3WM geometry in Fig. 1. Two signal modes 1 and 2 intersect with the pump symmetrically at a small angle in a photorefractive nonlinear media in the drift regime or in a Kerr medium. The energy coupling due to two-wave mixing (2WM) is absent. In addition, we only consider gratings generated by adjacent pairs of modes. disregarding the direct interaction between the two signal modes. This is justifiable because we can adjust the experimental parameters to optimize the space charge field associated with those gratings so that the strength of the grating induced by the nonadjacent pair will be very weak. Unlike in 2WM, with this geometry the gain will depend on the real part of the space charge field. The - 1 order diffracted wave of mode 1 is just mode 2 and vice versa. The two signal modes will grow at the same rate due to the scattering of the pump mode by the gratings created by beams 1 and 2 with the pump, (1. pl and (2. p), respectively. This can be physically explained by the following argument. The two interference patterns that contribute efficiently to the energy transfer are: (i) interference of the fields associated with (1, p). (ii) interference of the fields associated with (p, 2). In photorefractive media, as a result of the drift mechanism or other media with local response. the photoinduced index gratings are exactly in phase with the corresponding fringe pattern. However the index grat-
Supported by the National Natural China. ’ E-mail: [email protected].
Science
0030-4018/97/$17.00 0 1997 Published PZI SOO30-40 18(97)00466-5
by Elsevier Science B.V. All rights reserved
238
equations. The coupled-mode equations (2) can be derived from the following effective interaction Hamiltonian
P
H&i~&i;rr,,fujo,
+&iZ)
+(~~nnbe”b:+~~~~cr,nze-“L;)],
Fig. 1. Schematic of the setup for three-wave mixing.
ing (1, p) is generally out of phase with respect to the interference pattern (p, 21, so energy coupling occurs via three-wave interaction. Considering the assumptions mentioned above and following the approach in Refs. [2,4,5], we can derive the coupled wave equations
(la)
(4)
where i’ is the speed of light in the medium. The first term in (4) leads to phase modulation and the second is responsible for the energy coupling and phase conjugation. In quantum language. such a process can be expressed as: two pump photons are scattered into two weak mode photons during the four-photon interaction and the weak modes are amplified equally [8]. Or in other words, this process can be described as the abso~tion of two photons from the pump modes and the creation of a pair of signal photons. Temporal differential equations may be obtained from (4) using da,
-
dt
= -$“;,HJ,
i== 1,2,
and converted into spatial differential equations by the change : = 1’1. Taking the pump fields to be so strong that they may be treated as classical fields, and assuming phase matching, the quantized equations (2) then become da,,, -C?= d:
-“x(E,‘E,n,,,+E~~~.,)).
(61
where E,, E1 and f& are the field amplitudes of the signal modes and pump mode, respectively, x is the real nonlinear coupling constant and I$ is the phase mismatch factor. No analytical solutions seem to exist. Numerical results show that modes I and 2 will gain at the same rate at the expense of the pump mode. Particularly. the forward phase conjugate wave of mode 1 (2) will be generated simultaneously if E,(O) = 0 (E,(O) = 0) (refer to Refs. [2,5]). These results remind us that there exists strong quantum correlation between the two signal modes. Moreover, the threewave mixing process can be a possible way for generating non-classical effects such as quadrature squeezing or intensity difference squeezing. We now consider the quantum treatment. Following the standard procedure we quantize the signal modes and the pump mode. replacing E, and E,” with the photon annihilation and creation operators a; and G:, respectively, for i = I.2 and p. Eqs. (1) then become
where y= /f$,~‘x and C$ is the phase of the pump field with E;, = lf$Iexp(i~). From the coupled field equations we can see that two pump wave photons are scattered into two signal wave photons during the four-photon interaction, i.e.. the pump photon is scattered off the gratings induced by Ep*u, + ~2:E, and Ep*a, + ~7:E,. The solutions demonstrate that a,,( ,-I. and a?(;) are conjugate waves if A n,(O) = O(a,fO) = 0) (0 means the state is in the vacuum state 1911,which is in agreement with the above discussion. We also derive the expectation values of the photon number operators N, and IV?,
where the field operators rules
When the input signal fields are the coherent states \ aI > and \ a,) (we shall concentrate on such a case), numerical results show that the energy gain curves are consistent with those of the classical theory. Now let us examine the quantum correlation between the two signal modes. Strong quantum correlation between the signal and idler intensities in optical parametric downconversion was discussed by Reynaud et al. [IO]. Twin
Now we assume
satisfy the Boson commutation
that Eqs. (2) are the correct
quantum
The solutions of the field operators are found to be ~1,,~(:)=(1
- iy:)a,,,(O)- iyr:e-?'+uS.,(O),
N,,, -ci~~,(O)~,~~(O)(l + y'?)
(7)
+ &,(0)uz&3)y'?
f[uj(0)cr~(O)e"m+a,(O)nl(O)e-"~b]y'~~
Y. Lu et nl. / Optics Communicutions
photons are produced simultaneously in such a process. As the photons are created in pairs, the intensities of the two modes have the same intensity fluctuations. This intensity correlation corresponds to a squeezing of the intensity difference between the twin modes. Quantum correlation between fundamental and second harmonic for second harmonic generation was also discussed by Horowitz [ 1I]. The fluctuations in the sum of the intensities are reduced relative to the corresponding shot-noise level. The physical mechanism responsible for this correlation is that for second harmonic generation the creation of one up-converted photon corresponds to the simultaneous annihilation of two photons from the fundamental mode, which therefore acts as a selective filter for bunched photons. A similar case occurs in three-wave mixing. Since the signal photons are generated in pairs in the process, it is not surprising that the two signal modes are strongly correlated in a way similar to optical parametric generation. Since the detection of a signal photon of mode 1 corresponds to the creation of a signal photon of mode 2, both modes follow the fluctuations of the pump mode, resulting in a positive correlation, We can evaluate the degree of correlation by the correlation factor
(AN,( z)AN2( C( N,.Nz) =
\:‘(A@(
:)>
:)XAN:(
(9)
T.)>
The cross term (A N,A N2) is the difference between the conditional probability of detecting two photons and its respective uncorrelated probability. C( N, , N3) = 0 corresponds to the no correlation case and C( N,, Nz) = 1 corresponds to strong positive correlation. Using Eq. (8) the variances of the photon number and the correlation term can be obtained
144 (10971 -737-240
o!U -______
0
239
---I~~~
0.2
--i
0.4 Length
0.6
0.8
1
z
Fig. 2. Correlation factor C(N,.N2) versus the effective interaction length y: ( y = 5) for different intensity ratios fl = N,(0)/N2(0). Curve a: p = 0; Curve b: p = I. For other cases. the curves will fall between the two curves.
two modes using the Fano factor or the Mandel Q-parameter. The twin modes have the same intensity fluctuation (above the shot noise limit), which is independent of the initial intensity ratio p = N,(O)/N,(O). The super-Poissonian property of the twin modes is due to the energy coupling from the pump mode. The more signal photons exist, the stronger the coupling, resulting in bunching of the signal photons. This is consistent with the common mechanism that the increase in quantum noise is related to amplification. The sub-Poissonian characteristic of the intensity sum/difference is due not only to the sub-Poissonian character of each individual mode. but also to the correlation between the two modes. which can be shown to be (A(N,iNJ2)=(AN,‘)+(AN,‘)k2(AN,AN,).
(AN,::)
(12)
= ( I + ~‘c~)~N,,~(O) + -y’:‘(
I + 2yZz2)(
+ y’?(
I + y?)(
+ iy:(
I + 2y’:‘)(
+ y’?N,.,(O) (Y,IY?ee21d + (~Tcr: eZi”)
However the super-Poissonian character of each individual mode cannot exclude the possibility of sub-Poissonian
N,(O) + NT(O) + 1) (~,a!? e-“‘-
cr:aT
eZfca), (IO)
(AN,ANz) = (I + y’$( + y7?(
N,.?(O) + N,,,(O)) 1 + 3yy)
x( a,a2e
- 1zdJ+ N,‘a;
+ y’:‘(
I + y?)(
+ iy,-(
I + 2y’:‘)(
e”“)
N,(O)
+ N,(O)
(Y,(Y?e-“+
+ 1)
- (YI*(YTe”“). (II)
At the beginning, the two modes are uncorrelated, which is shown in Fig. 2. With increase of the interaction length, the degree of correlation of the twin modes increases sharply. We investigate the intensity fluctuations of the
Length
z
Fig, 3. Fano factor of the intensity difference F,? versus the effective interaction length y; (y = 5) for different intensity ratios p = N,(O)/N,(O). Curve a: p = 0: Curve b: p = I. For other cases, the curves will fall between the two curves.
K. LA et al. / Optics Communications
240
characteristic of intensity sum/difference. due to the strong co~elation. Using Eq. (8) with the coherent input states \cu, > and 1a,), the fluctuation of the intensity difference can be obtained to be (A(&
- hQ2> = N,(O) + N2(0),
144 f19971237-240
possible for the mode combinations
through phase conju-
gation.
References
(13)
which is constant throughout the three-wave mixing process. This can be explained by the non-depleted approximation. Since we neglect the noise induced by the pump mode, the possible ~uctuation in the intensity difference should be determined by the initial conditions and no external noise is introduced. The fluctuation of the intensity difference is suppressed with increase of the effective interaction length, during which the twin modes are amplified equally. Fig. 3 shows the squeezing of the noise in the intensity difference. We see that once the strong correlation between the twin modes has been es~blished, the degree of squeezing can be large, approaching perfect squeezing. In conclusion we have proposed a new scheme for generating twin modes. We have discussed for the first time the strong quantum correlation and possibility of squeezing of the intensity difference in photorefractive media in the drift regime or Kerr media, which provides an oppo~nity to observe non-~lassicai effects in fairly lowpower lasers. Furthermore, generation of squeezed states is
[I] P. Yeh, Introduction to photorefractive nonlinear optics (Wiley, New York, 1993). 121 L. Solymar, D.J. Webb. A. Grunnet-Jepsen, Prog. Quantum Electron. 18 (1994) 377. [3] L.B. Au, L. Solymar, IEEE J, Quantum Electron. 24 (1988)
162. [4] L.B. Au, L. Solymar. Appl. Phys. B 45 (1988) 125. 151 Ii. Sanchez, P.H. Kayoun, J.P. Huignard. J. Appi. Phys. 64 (1988) 26. 161 H.J. Eichler, M. GIotz, A. Kummrow, K. Richter. X. Yang,
Phys. Rev. A 35 (1987) 4673. [7] D.J. Webb. K.H. Ringhofer, L. Solymar, Electron. Let&.27 (1991) 889. 181 V.I.,. Vinetskii, N.V. Kukhtorev, S.G. Odulov, MS. Soskin, Sov. Phys. Usp. 22 (1979) 742. 191J. Perina, Quantum statistics of linear and nonlinear optical phenomena (Kluwer Academic Publishers, Dordrecht, 1991) p. 318. Ii01 S. Reynaud. C. Fabre, E. Giacobino, J. Opt. Sot. Am. 3 4 (1987) 1520. [I 11 R. Horoeicz, M. Pinard, S. Reynaud, Optics Comm. 61 (1987) 142.
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
144 (1997) 237-240
Quantum correlation between the signal modes in forward three-wave mixing ’ Yajun Lu a32,Ling-An Wu b, Meijuan Wu a, Peilin Zhang
a, Shiqun
Li
a
a Department of Modem Applied Physics. Tsinghua Uniuersity. Beijing 100084, China h Institute of Physics, Academia Sinica, Beijing 100080, China Received 30 April 1997; accepted
1 August 1997
Abstract A quantum theory of forward three-wave mixing is developed for photorefractive nonlinear media in the drift regime or Kerr media. A good agreement between the quantum and classical theories is demonstrated. Quantum noise suppression of the difference between the intensities of the two signal modes is discussed. 0 1997 Published by Elsevier Science B.V.
The strong optical nonlinearity of photorefractive media manifests itself in many interesting and important phenomena. Among these phenomena are weak signal amplification, phase conjugation, mirrorless oscillation, and photoinduced scattering (see, for example, Refs. [ 1.21). The coupling of optical waves in photorefractive media has been widely studied over the past two decades. The traditional amplification mechanism relies on the imaginary part of the coupling constant, or expressed differently, the out of phase component of the grating relative to the interference pattern of the waves. A new gain mechanism was proposed in Refs. [2-51 in which amplification relies on parametric three-wave mixing, or in other words, on the in-phase grating with respect to the intensity pattern. Experiments in liquid crystals by Sanchez et al. [5] and in silicon by Eichler et al. [6] have shown that amplification is achievable in media in which no phase shift takes place. Experiments in photorefractive BSO by Webb et al. [7] were performed by three-wave mixing. Much attention has been focused on beam amplification. The classical theory explaining the nonlinear coupling is already well established. but a quantum approach is not available and nonclassical effects have not been discussed. In this paper we
I
Foundation
of
present, to our knowledge for the first time, a quantum treatment of three-wave mixing (3WM) in photorefractive media. We study the 3WM geometry in Fig. 1. Two signal modes 1 and 2 intersect with the pump symmetrically at a small angle in a photorefractive nonlinear media in the drift regime or in a Kerr medium. The energy coupling due to two-wave mixing (2WM) is absent. In addition, we only consider gratings generated by adjacent pairs of modes. disregarding the direct interaction between the two signal modes. This is justifiable because we can adjust the experimental parameters to optimize the space charge field associated with those gratings so that the strength of the grating induced by the nonadjacent pair will be very weak. Unlike in 2WM, with this geometry the gain will depend on the real part of the space charge field. The - 1 order diffracted wave of mode 1 is just mode 2 and vice versa. The two signal modes will grow at the same rate due to the scattering of the pump mode by the gratings created by beams 1 and 2 with the pump, (1. pl and (2. p), respectively. This can be physically explained by the following argument. The two interference patterns that contribute efficiently to the energy transfer are: (i) interference of the fields associated with (1, p). (ii) interference of the fields associated with (p, 2). In photorefractive media, as a result of the drift mechanism or other media with local response. the photoinduced index gratings are exactly in phase with the corresponding fringe pattern. However the index grat-
Supported by the National Natural China. ’ E-mail: [email protected].
Science
0030-4018/97/$17.00 0 1997 Published PZI SOO30-40 18(97)00466-5
by Elsevier Science B.V. All rights reserved
238
equations. The coupled-mode equations (2) can be derived from the following effective interaction Hamiltonian
P
H&i~&i;rr,,fujo,
+&iZ)
+(~~nnbe”b:+~~~~cr,nze-“L;)],
Fig. 1. Schematic of the setup for three-wave mixing.
ing (1, p) is generally out of phase with respect to the interference pattern (p, 21, so energy coupling occurs via three-wave interaction. Considering the assumptions mentioned above and following the approach in Refs. [2,4,5], we can derive the coupled wave equations
(la)
(4)
where i’ is the speed of light in the medium. The first term in (4) leads to phase modulation and the second is responsible for the energy coupling and phase conjugation. In quantum language. such a process can be expressed as: two pump photons are scattered into two weak mode photons during the four-photon interaction and the weak modes are amplified equally [8]. Or in other words, this process can be described as the abso~tion of two photons from the pump modes and the creation of a pair of signal photons. Temporal differential equations may be obtained from (4) using da,
-
dt
= -$“;,HJ,
i== 1,2,
and converted into spatial differential equations by the change : = 1’1. Taking the pump fields to be so strong that they may be treated as classical fields, and assuming phase matching, the quantized equations (2) then become da,,, -C?= d:
-“x(E,‘E,n,,,+E~~~.,)).
(61
where E,, E1 and f& are the field amplitudes of the signal modes and pump mode, respectively, x is the real nonlinear coupling constant and I$ is the phase mismatch factor. No analytical solutions seem to exist. Numerical results show that modes I and 2 will gain at the same rate at the expense of the pump mode. Particularly. the forward phase conjugate wave of mode 1 (2) will be generated simultaneously if E,(O) = 0 (E,(O) = 0) (refer to Refs. [2,5]). These results remind us that there exists strong quantum correlation between the two signal modes. Moreover, the threewave mixing process can be a possible way for generating non-classical effects such as quadrature squeezing or intensity difference squeezing. We now consider the quantum treatment. Following the standard procedure we quantize the signal modes and the pump mode. replacing E, and E,” with the photon annihilation and creation operators a; and G:, respectively, for i = I.2 and p. Eqs. (1) then become
where y= /f$,~‘x and C$ is the phase of the pump field with E;, = lf$Iexp(i~). From the coupled field equations we can see that two pump wave photons are scattered into two signal wave photons during the four-photon interaction, i.e.. the pump photon is scattered off the gratings induced by Ep*u, + ~2:E, and Ep*a, + ~7:E,. The solutions demonstrate that a,,( ,-I. and a?(;) are conjugate waves if A n,(O) = O(a,fO) = 0) (0 means the state is in the vacuum state 1911,which is in agreement with the above discussion. We also derive the expectation values of the photon number operators N, and IV?,
where the field operators rules
When the input signal fields are the coherent states \ aI > and \ a,) (we shall concentrate on such a case), numerical results show that the energy gain curves are consistent with those of the classical theory. Now let us examine the quantum correlation between the two signal modes. Strong quantum correlation between the signal and idler intensities in optical parametric downconversion was discussed by Reynaud et al. [IO]. Twin
Now we assume
satisfy the Boson commutation
that Eqs. (2) are the correct
quantum
The solutions of the field operators are found to be ~1,,~(:)=(1
- iy:)a,,,(O)- iyr:e-?'+uS.,(O),
N,,, -ci~~,(O)~,~~(O)(l + y'?)
(7)
+ &,(0)uz&3)y'?
f[uj(0)cr~(O)e"m+a,(O)nl(O)e-"~b]y'~~
Y. Lu et nl. / Optics Communicutions
photons are produced simultaneously in such a process. As the photons are created in pairs, the intensities of the two modes have the same intensity fluctuations. This intensity correlation corresponds to a squeezing of the intensity difference between the twin modes. Quantum correlation between fundamental and second harmonic for second harmonic generation was also discussed by Horowitz [ 1I]. The fluctuations in the sum of the intensities are reduced relative to the corresponding shot-noise level. The physical mechanism responsible for this correlation is that for second harmonic generation the creation of one up-converted photon corresponds to the simultaneous annihilation of two photons from the fundamental mode, which therefore acts as a selective filter for bunched photons. A similar case occurs in three-wave mixing. Since the signal photons are generated in pairs in the process, it is not surprising that the two signal modes are strongly correlated in a way similar to optical parametric generation. Since the detection of a signal photon of mode 1 corresponds to the creation of a signal photon of mode 2, both modes follow the fluctuations of the pump mode, resulting in a positive correlation, We can evaluate the degree of correlation by the correlation factor
(AN,( z)AN2( C( N,.Nz) =
\:‘(A@(
:)>
:)XAN:(
(9)
T.)>
The cross term (A N,A N2) is the difference between the conditional probability of detecting two photons and its respective uncorrelated probability. C( N, , N3) = 0 corresponds to the no correlation case and C( N,, Nz) = 1 corresponds to strong positive correlation. Using Eq. (8) the variances of the photon number and the correlation term can be obtained
144 (10971 -737-240
o!U -______
0
239
---I~~~
0.2
--i
0.4 Length
0.6
0.8
1
z
Fig. 2. Correlation factor C(N,.N2) versus the effective interaction length y: ( y = 5) for different intensity ratios fl = N,(0)/N2(0). Curve a: p = 0; Curve b: p = I. For other cases. the curves will fall between the two curves.
two modes using the Fano factor or the Mandel Q-parameter. The twin modes have the same intensity fluctuation (above the shot noise limit), which is independent of the initial intensity ratio p = N,(O)/N,(O). The super-Poissonian property of the twin modes is due to the energy coupling from the pump mode. The more signal photons exist, the stronger the coupling, resulting in bunching of the signal photons. This is consistent with the common mechanism that the increase in quantum noise is related to amplification. The sub-Poissonian characteristic of the intensity sum/difference is due not only to the sub-Poissonian character of each individual mode. but also to the correlation between the two modes. which can be shown to be (A(N,iNJ2)=(AN,‘)+(AN,‘)k2(AN,AN,).
(AN,::)
(12)
= ( I + ~‘c~)~N,,~(O) + -y’:‘(
I + 2yZz2)(
+ y’?(
I + y?)(
+ iy:(
I + 2y’:‘)(
+ y’?N,.,(O) (Y,IY?ee21d + (~Tcr: eZi”)
However the super-Poissonian character of each individual mode cannot exclude the possibility of sub-Poissonian
N,(O) + NT(O) + 1) (~,a!? e-“‘-
cr:aT
eZfca), (IO)
(AN,ANz) = (I + y’$( + y7?(
N,.?(O) + N,,,(O)) 1 + 3yy)
x( a,a2e
- 1zdJ+ N,‘a;
+ y’:‘(
I + y?)(
+ iy,-(
I + 2y’:‘)(
e”“)
N,(O)
+ N,(O)
(Y,(Y?e-“+
+ 1)
- (YI*(YTe”“). (II)
At the beginning, the two modes are uncorrelated, which is shown in Fig. 2. With increase of the interaction length, the degree of correlation of the twin modes increases sharply. We investigate the intensity fluctuations of the
Length
z
Fig, 3. Fano factor of the intensity difference F,? versus the effective interaction length y; (y = 5) for different intensity ratios p = N,(O)/N,(O). Curve a: p = 0: Curve b: p = I. For other cases, the curves will fall between the two curves.
K. LA et al. / Optics Communications
240
characteristic of intensity sum/difference. due to the strong co~elation. Using Eq. (8) with the coherent input states \cu, > and 1a,), the fluctuation of the intensity difference can be obtained to be (A(&
- hQ2> = N,(O) + N2(0),
144 f19971237-240
possible for the mode combinations
through phase conju-
gation.
References
(13)
which is constant throughout the three-wave mixing process. This can be explained by the non-depleted approximation. Since we neglect the noise induced by the pump mode, the possible ~uctuation in the intensity difference should be determined by the initial conditions and no external noise is introduced. The fluctuation of the intensity difference is suppressed with increase of the effective interaction length, during which the twin modes are amplified equally. Fig. 3 shows the squeezing of the noise in the intensity difference. We see that once the strong correlation between the twin modes has been es~blished, the degree of squeezing can be large, approaching perfect squeezing. In conclusion we have proposed a new scheme for generating twin modes. We have discussed for the first time the strong quantum correlation and possibility of squeezing of the intensity difference in photorefractive media in the drift regime or Kerr media, which provides an oppo~nity to observe non-~lassicai effects in fairly lowpower lasers. Furthermore, generation of squeezed states is
[I] P. Yeh, Introduction to photorefractive nonlinear optics (Wiley, New York, 1993). 121 L. Solymar, D.J. Webb. A. Grunnet-Jepsen, Prog. Quantum Electron. 18 (1994) 377. [3] L.B. Au, L. Solymar, IEEE J, Quantum Electron. 24 (1988)
162. [4] L.B. Au, L. Solymar. Appl. Phys. B 45 (1988) 125. 151 Ii. Sanchez, P.H. Kayoun, J.P. Huignard. J. Appi. Phys. 64 (1988) 26. 161 H.J. Eichler, M. GIotz, A. Kummrow, K. Richter. X. Yang,
Phys. Rev. A 35 (1987) 4673. [7] D.J. Webb. K.H. Ringhofer, L. Solymar, Electron. Let&.27 (1991) 889. 181 V.I.,. Vinetskii, N.V. Kukhtorev, S.G. Odulov, MS. Soskin, Sov. Phys. Usp. 22 (1979) 742. 191J. Perina, Quantum statistics of linear and nonlinear optical phenomena (Kluwer Academic Publishers, Dordrecht, 1991) p. 318. Ii01 S. Reynaud. C. Fabre, E. Giacobino, J. Opt. Sot. Am. 3 4 (1987) 1520. [I 11 R. Horoeicz, M. Pinard, S. Reynaud, Optics Comm. 61 (1987) 142.