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Degenerate four-wave mixing via shifted phase holograms in cubic photorefractive crystals
Volume 53, number 1
OPTICS COMMUNICATIONS
1 February 1985
DEGENERATE FOUR-WAVE MIXING VIA SHIFTED PHASE HOLOGRAMS IN CUBIC PHOTOREFRACTIVE CRYSTALS
S.I. STEPANOV and M.P. PETROV A.F. Ioffe Physical Technical Institute o f the USSR Academy of Sciences, Leningrad, 194021, USSR
Received 2 May 1984 Revised manuscript received 3 October 1984
It is shown that it is possible to increase dramatically the efficiency of stationary four-wave mixing in cubic photorefractive crystals of the sillenite family. In order to do this one has to use a holographic arrangement of the cubic crystal where the same volume phase hologram has opposite contrasts for orthogonally polarized light waves traversing the sample in opposite directions. In combination with a new efficient nonstationary mechanism of shifted hologram formation, this arrangement can enable phase conjugation of amplified waves and stationary selfoscillation without an external resonator cavity.
1. Introduction
Degenerate four-wave mixing in electrooptic photorefractive crystals is now considered to be one of the promising techniques for phase conjugation [ 1 - 3 ] . The most important feature of these crystals as nonlinear media is extremely low threshold light intensities which allow experiments with low-power CW helium-neon and helium-cadmium lasers [2,4,5]. From other photorefractive crystals known up to now, cubic crystals of the sillenite family [6,7] (Bil2SiO20(BSO), Bi12GeO20(BGO)) differ favourably by their photosensitivity, high optical quality, and low commercial cost. It was mainly an insufficiently high diffraction efficiency o f the hologram recorded that prevented wide practical use o f these crystals in real-time holographic devices. However, we have recently demonstrated [8,9] a new nonstationary technique of holographic recording in the BSO which gives rise to a remarkable growth in the diffraction efficiency o f the hologram as compared with a standard drift mechanism of recording in an external electric field E0" The practical realization o f the new mechanism resulting in a shifted [10, 11 ] stationary hologram can be achieved by different 64
methods, in particular by means of recording an interference pattern "running" with a certain resonance speed along the external electric field. But, in any case, the long drift length of photoelectrons in the external electric field E 0 (as compared with the fringe spacing A) turns out to be responsible for hologram enhancement. So below we shall refer to this nonstationary mechanism of hologram recording as the long drift length (LDL) mechanism. One has to keep in mind that, because of low diffraction efficiencies of the holograms recorded in the BSO crystal by means of usual (diffusion or drift) methods, its practical applications for amplification and phase conjugation were of limited importance [7]. From our point of view, it is only the LDL mechanism that gives a real possibility to get an amplified phase conjugate stationary wavefront in these crystals. That is why the theoretical analysis of stationary four-wave mixing via shifted phase holograms in cubic photorefractive crystals has become the problem of major interest.
0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 53, number 1
OPTICS COMMUNICATIONS
I February 1985
2. Theoretical analysis In a conventional four-wave mixing experiment, the photorefractive crystal is illuminated by two collinear counterpropagating plane pump waves R1, R 2 and a weak signal wave S I (fig. la). From a simple holographic consideration, the process of four-wave interaction can be divided into two stages: the hologram recording by waves R1, S 1 and the hologram reconstruction by R 2 resulting in a fourth wave S 2 which is a complex conjugate of S 1 . In fact, it is more complicated because of diffraction of R 1 resulting in a change of S 1 amplitude and an additional process o f hologram recording due to interference of R 2 and S 2. The main equation describing the interaction o f waves R1, S 1 (the so-called two-wave mixing (fig. lb) at R 2 = 0) due to selfdiffraction of R 1 from the hologram being recorded is well known from literature (see e.g. [12]). For the stationary regime of the hologram formation in an undepleted pump wave approximation (ISll ~ IR 11 = const(z))
OS1/8z = ½ r S 1 .
(1)
Here the gain-factor of the photorefractive crystal P is real for a shifted phase hologram. For usual boundary conditions Sl (Z = O) = S 0, eq. (1) results in
S 1(cO Sok 1 =
=
S O exp(Fd/2),
(2)
which predicts amplification (for P > 0) or depression (for F < 0) of the signal wave emerging from the crystal. Amplification is observed when the recorded phase grating (Sn(x)) is shifted from the recording interference pattern (I1 (x)) in the direction of signal beam S 1 (fig. 2).
5
/IX
/'x
~,la)
Fig. 1. Traditional holographic schemes for four-wave (a) and two-wave (b) mixing experiments in photorefractive crystals. Rt, R2 plane pump waves, St, S2 weak signal waves, d thickness of the sample.
P t
Fig. 2. Illustration of the energy transfer via selfdiffraction of the light waves recording the phase hologram. It,2(x) the interference patterns, 6n(x) the refractive index distribution in the recorded phase hologram.
To consider the four-wave mixing scheme, eq. (1) is to be supplemented with another equation for selfdiffraction o f R 2 into $2, and the contribution of R 2 and S 2 into hologram recording is to be taken into account. For the simplest case of strictly anticollinear pump waves o f equal intensity, the set of equations for the stationary four-wave mixing in an undepleted pump wave approximation is (see e.g. [13])
8S 1/Sz = I p ( S I + S'~),
aS~/Sz = ¼F(S 1 + S~).
(3)
Here we have neglected a possible contribution from the refraction holograms, which is possible for the crystals under consideration because of their limited spatial frequency range [7]. Under usual boundary conditions (S 1 (z = 0) = S O ; S2(z = at) = 0), set o f equations (3) yields the following transmission (ki-) and reflection (k~) coefficients [13] : ki_ . Sl(d) . . . . 2exp(Pd/2) SO 1 + exp(rd/2)' k~-
R,
X
[!-exp(Fd/2)1 S O - ~ 1 +exp(rd/2)] "
$2(0)
-
(4)
So, even for the optimal initial orientation of the crystal (F > 0), the transmission coefficient turns out to be lower than that for two-wave mixing (eq. (2)). This is consistent with the well known fact that, for a given orientation of the sample (and light polarization), 65
1 February 1985
OPTICS COMMUNICATIONS
Volume 53, number 1
one of the counterpropagating signal beams S 1 and S 2 is amplified and the other is depressed due to a corresponding two-wave mixing process (fig. 2). In the four-wave mixing scheme under consideration, this means that the appearance of the counterpropagating conjugate wave S 2 results in depression of the hologram recorded in the volume of the sample. This enables one to call this case of four-wave mixing a negative feedback case. In cubic photorefractive crystals, however, it turns out to be possible to overcome this restriction and to convert the negative feedback into a positive one. Indeed, there are some holograp~c arrangements in these crystals where the sign of the phase hologram amplitude can be changed (i.e., the contrast of the grating reversed) by a simple change of the polarization of the readout beam [14]. In particular, for a typical transverse electrooptic (110) orientation of the cubic crystal when the grating vector K II [1T0], the same phase hologram turns out to have opposite contrasts for the readout beams polarized linearly at +45 ° to the incidence plane [14] (fig. 3). For such orthogonal polarizations of the beams traversing the crystal in opposite directions one has to use the minus sign in the second eq. (3), which leads to the following coefficients (see fig. 4) k~ = SI(CO/S 0 = 4/(4 - rd), k~ = $2(0)/S 0 = (rd/(4 - Fd))*.
(5)
These equations predict that at P > 0 (a) Ik~l > Ik~-I; (b) k~ = 1 at P d = 2; and (c) k~ and k~ go to infinity btom]
I
2
~ ~d
Fig. 4. Transmission and reflection coefficients for a weak signal beam in two- and four-wave mixing experiments• Twowave mixing: (a) kl. Four-wave mixing: (b) kl; (c) k~; (d)
-k~; (e) k~. at I'd -~ 4, which means the selfoscillation regime [3, 4]. It is worth mentioning also that at Fd > 4 set of equations (3) also gives some stationary solution (with limited values of k~',2 ), but it is unstable. In addition to the reflection coefficients k~, the value of a possible angular misalignment of the anticollinear plane pump waves is also of significance. In practice, this problem is closely related to the quality of pump fronts and the phase homogeneity of the sample. Here we shall briefly discuss the effect of angular misalignment of the plane pump waves in the incidence plane (fig. 5). In this case a small angular misalignment 50 results in an additional purely imaginary term -iz~S~ = -i(4nn/X)60 sin 0 S~ in the right part of the second eq. (3) similar to that in the coupled wave equations (see e.g. [11]) for light diffraction from a volume hologram. Using more suitable variables S 1 = exp(i ½Az)S 1 , S~ = exp(-i½ Az)S 2 one can transform set of eqs. (3) into a more symmetric form •
,
t
tt~01 Fig. 3. Special transverse electrooptic orientation of the cubic non-centrosymmetric crystal ((110) cut) used in the fourwave mixing experiment with a positive feedback. The solid circle and dashed ellipse are the initial optical indicatrix cross section and that distorted by the electric field of the hologram ~ ( x ) UK LI [1]-0]. Orthogonal linear polarizations of the light beams used in the four-wave mixing with a positive feedback are shown by arrows.
66
Fig. 5. Drawing in the Fourier space illustrating the effect of angular misalignment of the plane pump waves.
Volume 53, number 1
O S ] / O z -- 1 ~ F ( S
OPTICS COMMUNICATIONS
1 February 1985 d ÷ ("")0.s
1. , 1, "1"S ~ * ) + ~1z2~81,
~s'2*/~ = +-¼r(sl + s'2*)
- - ]' 1 ~ ~ '* 2 ,
-----.j.,,
3 . . . . . . . .
t,
o
0-
(6)
which yields the following reflection coefficients k~ = ((i'd/2)(1 - e x p [ [ ( i ' d / 2 ) 2 - (A~/) 2] 1/21) × ([ [(Fd/2)2 _ (Ad)2l 1/2 _ iAd] × exp[[(Pd/2) 2 - (zXd)2] ]1/21
~
N
Fig. 7. Theoretical curve for the halfwidth o f Ik~l (at the level ~ 0x/0~-.5)as a function o f rd.
+ [[(rd/2)2 _ (Ad)2 ] 1/2 + iAd] }-1 }*, k-~ = {(Pd/2) { 1 - exp [ [irzXd 2 - (zXa92 ] 1/2 ]) × ([Pd/2 + iAd - [ii'Ad 2 - (Ad2)] 1/2]
case, in the most interesting region 2 < Fd < 4, the halfwidth (at the level ~¢/b--.5.5)of the Ik~l curve turns out to be (Ad)~.5 ~ ~(4 - I'd) that results in
× exp[[iFAd 2 _ (ADZ)] 1/2]
~ ( a = 0X~d)~. 5/(ad)0. 5 ~ 2,
- [Pd/2 + iAd + [ii'~xd 2 - ( ~ / 2 ) ] 1 / 2 ] } - 1 },.
(7)
Normalized to their maximum values (at A = 0), the theoretical curves for [k~-I and Ik~l as functions of I Adl are presented in fig. 6. At F ~ 0 both curves transform into a usual curve for the angular selectivity of the volume hologram with the same fringe spacing and thickness. As I'd grow, the selectivity curve Ik~l has a tendency to broadening and Ik~l to narrowing. A detailed analysis shows that in the latter
~
i',,',,2\..\
where (Ad)0.5 is the halfwidth of a simple volume hologram (fig. 7). This formula reflects the fact which is important from the practical point of view, i.e., the higher reflection coefficients one wants to get the better plane pump waves and samples one has to use.
3. Conclusion The theoretical analysis presented above predicts efficient four-wave mixing in cubic photorefractive crystals of the sillenite family. To get it one has to use the positive feedback holographic arrangement and provide the gain-thickness product Fd ~ 2. So the theoretical estimation of the gain value I" turns out to be of major practical importance. It can be shown that for the simplest case of a monopolar photoconductive electrooptic crystal with one partly compensated donor level unsaturated during hologram formation [11 ] F ~ 2 ~'eff(Ulr/rr) - 1 .
Fig. 6. Theoretical curves for normalized Ik~l (dashed lines: (b) I'd = 3; (c) I'd = 3.4; (d) I'd = 3.8) and Ik21 (dot-and-dash lines: (e) Fd = 5; (f) I'd = 6) as functions o f (,at/). Solid curve (a), Isinc(zXd/2)[, corresponds to those o f Ik~l and Ik21 at Fd--, 0.
(8)
(9)
Here Ceff is the effective electric field relevant to a given mechanism of holographic recording, and the characteristic electrooptic voltage U~ = X/n3r equals nearly 8 kV in the BSO and BGO crystals. So for the case of recording of a shifted hologram b y means of the usual diffusion mechanism ( l e f t = 6D = (27r/A) × k T / e ~ 1.5 kV cm -1 at A ~- 1/am) the gain-factor 67
Volume 53, number 1
OPTICS COMMUNICATIONS
F in the photorefractive crystals o f the sillenite family is as small as 1 cm - 1 . As mentioned above, much more efficient stationary shifted phase holograms can be obtained by means o f the LDL mechanism. In this case ~ elf can exceed the external electric field 6 0 which is restricted by surface breakdowns and is about ~ 1 0 kV cm -1 . From our experiments on the LDL mechanism in the BSO [8,9] the possible value o f F can be estimated as 1 0 - 2 0 cm -1. This gives a real hope not only to get Ik~l > 1, but also to provide the selfoscillation regime o f stationary four-wave mixing without an external resonator cavity.
References [1 ] A. Yariv, IEEE QE-14 (1978) 650. [2] N. Kukhtarev and S. Odulov, Optics Comm, 32 (1980) 183. [3] J. Feinberg and R.W. Hellwarth, Optics Lett. 5 (1980) 519.
68
1 February 1985
[4] J.O. White, M. Cronin-Golomb, B. Fischer and A. Yariv, Appl. Phys. Lett. 40 (1982) 450. [5] S.G. Odulov and M.S. Soskin, Pis'ma v JETF 37 (1976) 591, in russian. [6] J.P. Huignard and F. Micheron, Appl. Phys. Lett. 29 (1976) 591. [7 ] J.P. Huignard, J.P. Herriau, G. Rivet and P. Gunter, Optics Lett. 5 (1980) 102. [8] S.I. Stepanov, V.V. Kulikov and M.P. Petrov, Pis'ma v JETF 8 (1982) 527, in russian. [9] S.I. Stepanov, V.V. Kulikov and M.P. Petrov, Optics Comm. 44 (1982) 19. [10] V.L. Vinetskii, N.V. Kukhtarev, S.G. Odulov and M.S. Soskin, Sov. Phys. Usp. 22 (1979) 742. [11 ] M.P. Petrov, S.I. Stepanov and A.V. Khomenko, Photosensitive electrooptic media in holography and optical information processing (Nauka, Leningrad, 1983), in russian. [12] J.P. Huignard and A. Marrakchi, Optics Comm. 38 (1981) 249. [13] B. Fisher, M. Cronin-Golomb, J.O. White and A. Yariv, Optics Lett. 6 (1981) 519. [14] M.P. Petrov, T.G. Pencheva and S.I. Stepanov, J. Optics 12 (1981) 287.
OPTICS COMMUNICATIONS
1 February 1985
DEGENERATE FOUR-WAVE MIXING VIA SHIFTED PHASE HOLOGRAMS IN CUBIC PHOTOREFRACTIVE CRYSTALS
S.I. STEPANOV and M.P. PETROV A.F. Ioffe Physical Technical Institute o f the USSR Academy of Sciences, Leningrad, 194021, USSR
Received 2 May 1984 Revised manuscript received 3 October 1984
It is shown that it is possible to increase dramatically the efficiency of stationary four-wave mixing in cubic photorefractive crystals of the sillenite family. In order to do this one has to use a holographic arrangement of the cubic crystal where the same volume phase hologram has opposite contrasts for orthogonally polarized light waves traversing the sample in opposite directions. In combination with a new efficient nonstationary mechanism of shifted hologram formation, this arrangement can enable phase conjugation of amplified waves and stationary selfoscillation without an external resonator cavity.
1. Introduction
Degenerate four-wave mixing in electrooptic photorefractive crystals is now considered to be one of the promising techniques for phase conjugation [ 1 - 3 ] . The most important feature of these crystals as nonlinear media is extremely low threshold light intensities which allow experiments with low-power CW helium-neon and helium-cadmium lasers [2,4,5]. From other photorefractive crystals known up to now, cubic crystals of the sillenite family [6,7] (Bil2SiO20(BSO), Bi12GeO20(BGO)) differ favourably by their photosensitivity, high optical quality, and low commercial cost. It was mainly an insufficiently high diffraction efficiency o f the hologram recorded that prevented wide practical use o f these crystals in real-time holographic devices. However, we have recently demonstrated [8,9] a new nonstationary technique of holographic recording in the BSO which gives rise to a remarkable growth in the diffraction efficiency o f the hologram as compared with a standard drift mechanism of recording in an external electric field E0" The practical realization o f the new mechanism resulting in a shifted [10, 11 ] stationary hologram can be achieved by different 64
methods, in particular by means of recording an interference pattern "running" with a certain resonance speed along the external electric field. But, in any case, the long drift length of photoelectrons in the external electric field E 0 (as compared with the fringe spacing A) turns out to be responsible for hologram enhancement. So below we shall refer to this nonstationary mechanism of hologram recording as the long drift length (LDL) mechanism. One has to keep in mind that, because of low diffraction efficiencies of the holograms recorded in the BSO crystal by means of usual (diffusion or drift) methods, its practical applications for amplification and phase conjugation were of limited importance [7]. From our point of view, it is only the LDL mechanism that gives a real possibility to get an amplified phase conjugate stationary wavefront in these crystals. That is why the theoretical analysis of stationary four-wave mixing via shifted phase holograms in cubic photorefractive crystals has become the problem of major interest.
0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 53, number 1
OPTICS COMMUNICATIONS
I February 1985
2. Theoretical analysis In a conventional four-wave mixing experiment, the photorefractive crystal is illuminated by two collinear counterpropagating plane pump waves R1, R 2 and a weak signal wave S I (fig. la). From a simple holographic consideration, the process of four-wave interaction can be divided into two stages: the hologram recording by waves R1, S 1 and the hologram reconstruction by R 2 resulting in a fourth wave S 2 which is a complex conjugate of S 1 . In fact, it is more complicated because of diffraction of R 1 resulting in a change of S 1 amplitude and an additional process o f hologram recording due to interference of R 2 and S 2. The main equation describing the interaction o f waves R1, S 1 (the so-called two-wave mixing (fig. lb) at R 2 = 0) due to selfdiffraction of R 1 from the hologram being recorded is well known from literature (see e.g. [12]). For the stationary regime of the hologram formation in an undepleted pump wave approximation (ISll ~ IR 11 = const(z))
OS1/8z = ½ r S 1 .
(1)
Here the gain-factor of the photorefractive crystal P is real for a shifted phase hologram. For usual boundary conditions Sl (Z = O) = S 0, eq. (1) results in
S 1(cO Sok 1 =
=
S O exp(Fd/2),
(2)
which predicts amplification (for P > 0) or depression (for F < 0) of the signal wave emerging from the crystal. Amplification is observed when the recorded phase grating (Sn(x)) is shifted from the recording interference pattern (I1 (x)) in the direction of signal beam S 1 (fig. 2).
5
/IX
/'x
~,la)
Fig. 1. Traditional holographic schemes for four-wave (a) and two-wave (b) mixing experiments in photorefractive crystals. Rt, R2 plane pump waves, St, S2 weak signal waves, d thickness of the sample.
P t
Fig. 2. Illustration of the energy transfer via selfdiffraction of the light waves recording the phase hologram. It,2(x) the interference patterns, 6n(x) the refractive index distribution in the recorded phase hologram.
To consider the four-wave mixing scheme, eq. (1) is to be supplemented with another equation for selfdiffraction o f R 2 into $2, and the contribution of R 2 and S 2 into hologram recording is to be taken into account. For the simplest case of strictly anticollinear pump waves o f equal intensity, the set of equations for the stationary four-wave mixing in an undepleted pump wave approximation is (see e.g. [13])
8S 1/Sz = I p ( S I + S'~),
aS~/Sz = ¼F(S 1 + S~).
(3)
Here we have neglected a possible contribution from the refraction holograms, which is possible for the crystals under consideration because of their limited spatial frequency range [7]. Under usual boundary conditions (S 1 (z = 0) = S O ; S2(z = at) = 0), set o f equations (3) yields the following transmission (ki-) and reflection (k~) coefficients [13] : ki_ . Sl(d) . . . . 2exp(Pd/2) SO 1 + exp(rd/2)' k~-
R,
X
[!-exp(Fd/2)1 S O - ~ 1 +exp(rd/2)] "
$2(0)
-
(4)
So, even for the optimal initial orientation of the crystal (F > 0), the transmission coefficient turns out to be lower than that for two-wave mixing (eq. (2)). This is consistent with the well known fact that, for a given orientation of the sample (and light polarization), 65
1 February 1985
OPTICS COMMUNICATIONS
Volume 53, number 1
one of the counterpropagating signal beams S 1 and S 2 is amplified and the other is depressed due to a corresponding two-wave mixing process (fig. 2). In the four-wave mixing scheme under consideration, this means that the appearance of the counterpropagating conjugate wave S 2 results in depression of the hologram recorded in the volume of the sample. This enables one to call this case of four-wave mixing a negative feedback case. In cubic photorefractive crystals, however, it turns out to be possible to overcome this restriction and to convert the negative feedback into a positive one. Indeed, there are some holograp~c arrangements in these crystals where the sign of the phase hologram amplitude can be changed (i.e., the contrast of the grating reversed) by a simple change of the polarization of the readout beam [14]. In particular, for a typical transverse electrooptic (110) orientation of the cubic crystal when the grating vector K II [1T0], the same phase hologram turns out to have opposite contrasts for the readout beams polarized linearly at +45 ° to the incidence plane [14] (fig. 3). For such orthogonal polarizations of the beams traversing the crystal in opposite directions one has to use the minus sign in the second eq. (3), which leads to the following coefficients (see fig. 4) k~ = SI(CO/S 0 = 4/(4 - rd), k~ = $2(0)/S 0 = (rd/(4 - Fd))*.
(5)
These equations predict that at P > 0 (a) Ik~l > Ik~-I; (b) k~ = 1 at P d = 2; and (c) k~ and k~ go to infinity btom]
I
2
~ ~d
Fig. 4. Transmission and reflection coefficients for a weak signal beam in two- and four-wave mixing experiments• Twowave mixing: (a) kl. Four-wave mixing: (b) kl; (c) k~; (d)
-k~; (e) k~. at I'd -~ 4, which means the selfoscillation regime [3, 4]. It is worth mentioning also that at Fd > 4 set of equations (3) also gives some stationary solution (with limited values of k~',2 ), but it is unstable. In addition to the reflection coefficients k~, the value of a possible angular misalignment of the anticollinear plane pump waves is also of significance. In practice, this problem is closely related to the quality of pump fronts and the phase homogeneity of the sample. Here we shall briefly discuss the effect of angular misalignment of the plane pump waves in the incidence plane (fig. 5). In this case a small angular misalignment 50 results in an additional purely imaginary term -iz~S~ = -i(4nn/X)60 sin 0 S~ in the right part of the second eq. (3) similar to that in the coupled wave equations (see e.g. [11]) for light diffraction from a volume hologram. Using more suitable variables S 1 = exp(i ½Az)S 1 , S~ = exp(-i½ Az)S 2 one can transform set of eqs. (3) into a more symmetric form •
,
t
tt~01 Fig. 3. Special transverse electrooptic orientation of the cubic non-centrosymmetric crystal ((110) cut) used in the fourwave mixing experiment with a positive feedback. The solid circle and dashed ellipse are the initial optical indicatrix cross section and that distorted by the electric field of the hologram ~ ( x ) UK LI [1]-0]. Orthogonal linear polarizations of the light beams used in the four-wave mixing with a positive feedback are shown by arrows.
66
Fig. 5. Drawing in the Fourier space illustrating the effect of angular misalignment of the plane pump waves.
Volume 53, number 1
O S ] / O z -- 1 ~ F ( S
OPTICS COMMUNICATIONS
1 February 1985 d ÷ ("")0.s
1. , 1, "1"S ~ * ) + ~1z2~81,
~s'2*/~ = +-¼r(sl + s'2*)
- - ]' 1 ~ ~ '* 2 ,
-----.j.,,
3 . . . . . . . .
t,
o
0-
(6)
which yields the following reflection coefficients k~ = ((i'd/2)(1 - e x p [ [ ( i ' d / 2 ) 2 - (A~/) 2] 1/21) × ([ [(Fd/2)2 _ (Ad)2l 1/2 _ iAd] × exp[[(Pd/2) 2 - (zXd)2] ]1/21
~
N
Fig. 7. Theoretical curve for the halfwidth o f Ik~l (at the level ~ 0x/0~-.5)as a function o f rd.
+ [[(rd/2)2 _ (Ad)2 ] 1/2 + iAd] }-1 }*, k-~ = {(Pd/2) { 1 - exp [ [irzXd 2 - (zXa92 ] 1/2 ]) × ([Pd/2 + iAd - [ii'Ad 2 - (Ad2)] 1/2]
case, in the most interesting region 2 < Fd < 4, the halfwidth (at the level ~¢/b--.5.5)of the Ik~l curve turns out to be (Ad)~.5 ~ ~(4 - I'd) that results in
× exp[[iFAd 2 _ (ADZ)] 1/2]
~ ( a = 0X~d)~. 5/(ad)0. 5 ~ 2,
- [Pd/2 + iAd + [ii'~xd 2 - ( ~ / 2 ) ] 1 / 2 ] } - 1 },.
(7)
Normalized to their maximum values (at A = 0), the theoretical curves for [k~-I and Ik~l as functions of I Adl are presented in fig. 6. At F ~ 0 both curves transform into a usual curve for the angular selectivity of the volume hologram with the same fringe spacing and thickness. As I'd grow, the selectivity curve Ik~l has a tendency to broadening and Ik~l to narrowing. A detailed analysis shows that in the latter
~
i',,',,2\..\
where (Ad)0.5 is the halfwidth of a simple volume hologram (fig. 7). This formula reflects the fact which is important from the practical point of view, i.e., the higher reflection coefficients one wants to get the better plane pump waves and samples one has to use.
3. Conclusion The theoretical analysis presented above predicts efficient four-wave mixing in cubic photorefractive crystals of the sillenite family. To get it one has to use the positive feedback holographic arrangement and provide the gain-thickness product Fd ~ 2. So the theoretical estimation of the gain value I" turns out to be of major practical importance. It can be shown that for the simplest case of a monopolar photoconductive electrooptic crystal with one partly compensated donor level unsaturated during hologram formation [11 ] F ~ 2 ~'eff(Ulr/rr) - 1 .
Fig. 6. Theoretical curves for normalized Ik~l (dashed lines: (b) I'd = 3; (c) I'd = 3.4; (d) I'd = 3.8) and Ik21 (dot-and-dash lines: (e) Fd = 5; (f) I'd = 6) as functions o f (,at/). Solid curve (a), Isinc(zXd/2)[, corresponds to those o f Ik~l and Ik21 at Fd--, 0.
(8)
(9)
Here Ceff is the effective electric field relevant to a given mechanism of holographic recording, and the characteristic electrooptic voltage U~ = X/n3r equals nearly 8 kV in the BSO and BGO crystals. So for the case of recording of a shifted hologram b y means of the usual diffusion mechanism ( l e f t = 6D = (27r/A) × k T / e ~ 1.5 kV cm -1 at A ~- 1/am) the gain-factor 67
Volume 53, number 1
OPTICS COMMUNICATIONS
F in the photorefractive crystals o f the sillenite family is as small as 1 cm - 1 . As mentioned above, much more efficient stationary shifted phase holograms can be obtained by means o f the LDL mechanism. In this case ~ elf can exceed the external electric field 6 0 which is restricted by surface breakdowns and is about ~ 1 0 kV cm -1 . From our experiments on the LDL mechanism in the BSO [8,9] the possible value o f F can be estimated as 1 0 - 2 0 cm -1. This gives a real hope not only to get Ik~l > 1, but also to provide the selfoscillation regime o f stationary four-wave mixing without an external resonator cavity.
References [1 ] A. Yariv, IEEE QE-14 (1978) 650. [2] N. Kukhtarev and S. Odulov, Optics Comm, 32 (1980) 183. [3] J. Feinberg and R.W. Hellwarth, Optics Lett. 5 (1980) 519.
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[4] J.O. White, M. Cronin-Golomb, B. Fischer and A. Yariv, Appl. Phys. Lett. 40 (1982) 450. [5] S.G. Odulov and M.S. Soskin, Pis'ma v JETF 37 (1976) 591, in russian. [6] J.P. Huignard and F. Micheron, Appl. Phys. Lett. 29 (1976) 591. [7 ] J.P. Huignard, J.P. Herriau, G. Rivet and P. Gunter, Optics Lett. 5 (1980) 102. [8] S.I. Stepanov, V.V. Kulikov and M.P. Petrov, Pis'ma v JETF 8 (1982) 527, in russian. [9] S.I. Stepanov, V.V. Kulikov and M.P. Petrov, Optics Comm. 44 (1982) 19. [10] V.L. Vinetskii, N.V. Kukhtarev, S.G. Odulov and M.S. Soskin, Sov. Phys. Usp. 22 (1979) 742. [11 ] M.P. Petrov, S.I. Stepanov and A.V. Khomenko, Photosensitive electrooptic media in holography and optical information processing (Nauka, Leningrad, 1983), in russian. [12] J.P. Huignard and A. Marrakchi, Optics Comm. 38 (1981) 249. [13] B. Fisher, M. Cronin-Golomb, J.O. White and A. Yariv, Optics Lett. 6 (1981) 519. [14] M.P. Petrov, T.G. Pencheva and S.I. Stepanov, J. Optics 12 (1981) 287.