Light diffraction and nonlinear image processing in electrooptic Bi12SiO20 crystals

Light diffraction and nonlinear image processing in electrooptic Bi12SiO20 crystals

Volume 31, number 3 OPTICS COMMUNICATIONS December 1979 LIGHT DIFFRACTION AND NONLINEAR IMAGE PROCESSING IN ELECTROOPTIC Bi12SiO20 CRYSTALS M.P. P...

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Volume 31, number 3

OPTICS COMMUNICATIONS

December 1979

LIGHT DIFFRACTION AND NONLINEAR IMAGE PROCESSING IN ELECTROOPTIC Bi12SiO20 CRYSTALS

M.P. PETROV, S.V. MIRIDONOV, S.I. STEPANOV and V.V. KULIKOV A.F. Ioffe Physieo-Technical Institute, Leningrad, 194021, USSR Received 24 August 1979

Some characteristic peculiarities of the volume phase holograms in a high-sensitive electrooptic Bil2 SiO20 crystals such as polarization properties of the diffracted light beam and nonlinear phenomena in the stationary stage of the hologram formation are investigated. Spatial frequency mixing or combinational hologram writing by means of non-linearity of the holographic process is discussed as a new promising technique of the volume holography. In particular two important problems of Bi12SiO20 crystals: nondestructive readout at the wavelength different from that of the writing beams and correlation analysis of complex two-dimensional pictures can be solved by this method.

The photoconductive electrooptic crystal Bil2SiO20 , that is widely known due to its application in PROM devices [1 ], is also intensively studied for the purposes of volume hologram recording [2]. The high sensitivity of Bi12SiO20 enables applications of this crystal in real-trine coherent systems of optical information processing [3]. The purpose o f this paper is to analyze some important peculiarities of the volume phase holograms in these crystals and in particular: polarization properties o f the reconstructed light beam and nonlinear phenomena in the stationary stage of the hologram formation. Some possibilities to use them in practical applications will be also discussed. 1. Let us begin our analysis with the stage of the hologram reconstruction, the simpliest case of that is a plane-wave Bragg diffraction from the phase grating due to a sinusoidal spatial distribution of the electric field 6~(r): 6 ~ ( r ) = 8~ • cos ( K . r ) ,

(1)

where K is the wave vector o f the grating under consideration. Due to linear electrooptic effect the electric field 6 6(r) induces a sinusoidal modulation of the crystal dielectric tensor [4] : 6~ 0 , ) = 6 ~ . c o s ( K - r ) = - n 4 ( ~ . ~ C ) c o s ( K . r ) ,

(2)

where n is the refractive index, l: the linear electrooptic

tensor (n = 2.5,r41=r52=r63 = r = 5 × 10-10 c m V -1 for X = 0.63/am in Bil2SiO20 [2]) and symmetrical tensor 8~ is an amplitude of the phase grating. The main difficulties in the theoretical treatment of light diffraction phenomena in Bil2SiO20 arise from its natural optical activity (P = 22 deg mm -1 for X = 0.63/lm [2]) and birefringency induced by an external electric field 6 0 usually applied to the crystal to increase the sensitivity to the writing of the hologram [2]. The influence of these two factors can be roughly described by corresponding differences in the crystal refractive indices Ancirc for circularly polarized light waves, and Anli n for linearly polarized ones. For the typical value of C 0 = 20 kV c m - 1, Ancir c and Anli n = ( n 3 r 6 0 ) × 0.5 are of the same order of magnitude 10 - 4 and the characteristic types of the light waves in the crystal are elliptically polarized. As the difference in the characteristic wave refractive indices results in a difference in Bragg conditions, optical activity and electrically induced birefringence are to be taken into account both for dynamic treatment of the diffraction process [5] and for kinematic one (when the diffraction efficiency o f the grating is small). Nevertheless for comparatively thin samples with a thickness d <~ X/2 (Ancirc , Anlin) -1 (that is about 3 mm for X = 0.63 ktm in Bil2SiO20 ) both the birefringency and the optical activity of the crystal 301

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can be neglected, and that is the case we shall discuss below. In a typical transverse electrooptic orientation of the Bil2SiO20 crystal ([110] cut sample [2]), the tensor 6@(2) in the ((1T0), (001), (110)) coordinate system has the following form [5]:

,1 o~ o LO_ ' 0__1' 0 I. . . .

-n4rl6~[

I

,

0 0 --1

'r-o--1-i o ol ' 0

- n 4 r l S ~ l '',1

,

(3)

-0-O- 0

for the cases KII(001) and KI[(ll0), respectively. In the practical interesting case of small refraction angles, the (110) components of electric fields E_I, 2 in the readout and reconstructed light waves can be neglected and only the 2 × 2 matrices outlined in (3) by dashed squares can be taken into account. As follows from (3) for the case of K I[(001) orientation, the amplitude of the phase grating (2) is maximal for the light waves linearly polarized along the (1T0) axis and is equal to zero for the orthogonal polarization. The diffraction efficiency in the former case is equal to:

r7 sin2(Trnrl6 61d/2~) , =

(4)

that follows directly from the well known Kogelnik's formula [6] and the simple equation for the effective amplitude of the refractive index modulation fin = 2nfe. As to the case ofKH (110) orientation it can be shown from (3) that the diffraction efficiency of the grating for the light waves linearly polarized along and perpendicular to the bisector between the (001) and (170) axes is also expressed by equation (4). However the signs of the phase grating for the polarizations mentioned above are opposite, and that is why this hologram acts like the usual halfwave plate. Really, its diffraction efficiency does not depend on the polarization of the incident beam, however it changes the direction of the rotation of the circularly polarized readout wave to the opposite. If one uses a linearly polarized readout wave and rotates its polarization, than the linear polarization of the diffracted wave rotates in the opposite direction with the same speed. Some experiments confirming the main theoretical conclusions were performed with [110] oriented Bil2SiO20 crystals with the thickness d = 1.7 mm at X = 0.63/~m without external field 6 0 for both of the crystal orientations discussed above. For the case of 302

December 1979

K II(001) disappearance of the diffraction maximum for the readout light beam, linearly polarized in the plane of incidence has been observed. As to the orthogonal orientation of the crystal (K 11(1T0)) we have observed opposite rotation of the readout and diffracted beams linear polarizations (fig. 1a). One can see from this figure that for the readout beams polarized in and perpendicular to the plane of incidence, the first order diffraction maxinmm is polarized orthogonally to the readout beam. This fact was experimentally observed independently in [7] and was used to increase the signal to noise ratio in the image reconstruction from Bi12SiO20. In conclusion to this section it is worth to note that even for the crystal thickness d = 1.7 mm used in our experiments, the theoretically predicted behaviour of the diffracted light polarization was observed only for the center of the Bragg maximum. For the linearly polarized readout beam, KII(1To) and d o = 0, the reconstructed beam was linearly polarized in the center and circularly on the shoulders of the diffraction maximum (fig. lb). As it was shown in [5], this effect is due to the crystal optical activity that was neglected in the present simplified analysis. 2. Another characteristic feature of Bi12SiO20 crystals we are going to discuss here is the nonlinear effects in the stationary regime of the hologram writing. It is well recognized now that the presence of high spatial harmonics of the initial sinusoidal interference pattern is characteristic of the hologram refractive index spatial modu(a)

Co)

.g arb.

~\ ~8°F8o o

~: 90°T~T

90~\~ -90

/~i .5

f "+

0

un.

9'~' d e g r .

-10

o

1(3

Fig. 1. Polarization o f the reconstructed beam ([110] cut Bil2SiO2o; KII (1TO)). (a) Reconstructed beam polarizing angle (a) versus that (#) o f the readout beam. (Both a and # are measured from the plane of incidence). (b) Incident angle dependences of the diffracted beam amplitude measured through the analyser oriented in (11) and perpendicular to (±) the plane of incidence. Solid curve is the theoretical one calculated in kinematic approximation.

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OPTICS COMMUNICATIONS

lation in this regime (see for example [2]). For instance, in a [110] oriented Bi12SiO20 crystal we have experimentally observed a second harmonic hologram with an amplitude linear dependent on the external electric field 6 0 (fig" 2a) and quadratic on the interference pattern modulation index m (fig. 2b). For the case m ~ 1 its amplitude was only three times less than that of the first, the basic spatial harmonic. In fact this phenomenon is only a particular case of spatial frequency mixing occuring in the stationary regime analogous to the usual frequency mixing in nonlinear radioelectronic systems. So exposing the volume of the crystal by the complex interference pattern consisting of two spatial frequencies K 1 and K 2 one can observe combinational holograms with the spatial frequencies K± = K 1 + K 2 and amplitudes proportional to the product of the initial sinusoidal interference pattern modulation indices. Below we shall show that by means of this nonlinear method it is possible to solve some typical problems of volume holograms (written in particular in Bi12 SiO20 ) resulting from their high selective properties (see also [8]). A. The reconstruction of the complex image at the wavelength )t2 essentially different from the wavelength X1 the hologram was written at. As was shown earlier by the authors [9] this is practically possible only in Gabor's scheme or in a specially developed holographic arrangement using anisotropic diffration in birefringent

arb. ~n.

December i979

crystals like LiNbO 3 (see also [7]). Unfortunately, both of these schemes satisfying wide-band condition of the complex image reconstruction - collinearity of the writing (at Xl) and reconstructed (at X2) light beams [9] - are not acceptable for Bil2SiO20 crystals. However, by writing in the crystal another auxiliary sinusoidal grating K 2 in addition to the usual "object" hologram K1, one can choose such an arrangement that at the reconstruction of the combinational hologram K± at )k2 :# )t 1 the wide-band condition (collinearity of two object beams) will be fulfilled. A geometrical scheme in the wave vector space, illustrating the main idea of this technique (for more details see [8]) and a typical image reconstructed from the combinational hologram in a Bi12SiO20 crystal are presented in fig. 3. The maximum bandwidth of the spatial frequencies in the reconstructed image is equal to those for the wideband arrangements we have mentioned above and for the typical case X1 = 0.5 gm, ~k2 = 1.0 gm, d = 2 mm, and n = 2.5, it reaches 200-300 lin mm -1 . B. Nonlinear methods enable also performing such an important operation as a correlation analysis of complex two-dimensional pictures by means of volume holographic medium. This type of coherent information processing developed for the case of thin holograms is well known as the Von der Lught method (see [10]). Unfortunately this technique is not applicable to our

/~

arb.

(a)

Co)

um I

)~= 0.51]~m

/]

8°: 1°~v'c'~-1 t/l= 0.5 ~<=3.4 .i04 cm-I

/

~

I

K=3"4" I°4 cm-U

I0.3 g~

0.5 0.1

0.03 /

I

I

5

~C

IL

15 kv.cm-I

0.03

0.1

0.3

~1

Fig. 2. Amplitudes o f the first (6 ~ 1) and the second (6 E 2) harmonics versus external electric field E o (a) and interference pattern modulation index m (b).

303

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December 1979

b) b)

Fig. 3. Nondestructive readout. (a) Ko, K1, K2 and K_ are the wave vectors of the object beams, the basic hologram, the auxiliary grating and the combinational hologram. (Large and small circles correspond to wave vector surfaces for hi and X2 ; here and fig. 4a is hi < X2). (b) Image reconstructed from the combinational hologram in Bi12SiO2o. (Crystal area 1 x 1 cm 2 , d = 2 mm, ~ 0 = 7 kV cm -1 , hi = 0.44 ~m, h2 0.63 ~m).

c)

=

case because o f high angular selectivity o f t h i c k holograms. Really one can show t h a t to correlate t w o wideband two-dimensional pictures by means of a volume m e d i u m it is necessary t o fulfil collinearity o f the t w o o b j e c t waves to b e processed and t h e resulting correlat i o n wave. F o r t h e usual m e t h o d s o f h o l o g r a m w r i t i n g r e c o n s t r u c t i o n this c o n d i t i o n can b e m e t only in G a b o r ' s a n d D e n i s j u k ' s schemes, u n s u i t a b l e for a B i l 2 S i O 2 0 crystal because o f t h e high noise level in t h e f o r m e r case a n d inevitable h o l o g r a m erasure u p o n r e a d o u t in t h e latter one. H o w e v e r t h e p r o b l e m can be solved w i t h the help o f the n o n l i n e a r t e c h n i q u e . In this case t h e crystal is exposed by two independent interference patterns formed b y t h e F o u r i e r t r a n s f o r m s o f the images to be processed and t w o c o r r e s p o n d i n g plane reference waves. T h e

304

Fig. 4. Correlation analysis. (a) K 0 is the wave vector of the object beams; K1, K2 and K_ are wave vectors of two basic holograms and the combinational one. (b, c) Autocorrelation functions of two parallel transparent stripes (b) and a transparent outline of a square (c). (Crystal area 1 x 1 cm 2 , d = 2 mm, o = 7 kV cm -1 , ~.1 = 0.44 ~m, ~.2 = 0.63 ~m, f = 250 ram).

necessary i n d e p e n d e n c e can b e p r o v i d e d b y the o r t h o gonal p o l a r i z a t i o n s or the little difference o f t h e wavelengths in t h e i n t e r f e r e n c e p a t t e r n s . The w h o l e arrangem e n t (fig. 4a) is c h o o s e n to result in collinearity o f the o b j e c t waves u n d e r processing and t h e wave recon-

Volume 31, number 3

OPTICS COMMUNICATIONS

structed from the combinational hologram at X2 4=)k1 . It is simple to show that after the inverse Fourier transformation of this wave one can obtain either convolution or correlation of the initial images corresponding to the combinational hologram K± used in the experiment. As distinct from the case of the focused image holograms we have used in our experiments on nondestructive readout (fig. 3), in the case of Fourier holograms the crystal thickness d limits the linear dimensions of the images under processing up to the value

Dm~~ = Xlf x/;/X2 a ,

(5)

where f i s the focus length of the input Fourier lens. For the typical values: d = 2 mm, f = 250 mm, X1 = 0.5/am, X2 = 1.0 #m, is Dmax = 4.4 ram. Experimental results on the autocorrelation of the simple binary images by means of a Bi12SiO20 crystal are presented in figs. 4b, c.

December 1979

References [1] S.G. Lipson and P. Nisenson, Appl. Optics 13 (1974) 2054. [2] J.P. Huignard and F. Micheron, Appl. Phys. Lett. 29 (1976) 591; M. Peltier and F. Micheron, J. Appl. Phys. 48 (1977) 3683. [3] J.P. Huignard, J.P. Herriau and T. Valentin, Appl. Optics 16 (1977) 2796; J.P. Huignard and J.P. Herriau and T. Valentin, Appl. Optics 16 (1977) 1807. [4] M.P. Petrov, S.I. Stepanov and A.A. Kamshilin, Ferroelectrics 21 (1978) 631. [5] S.V. Miridonov, M.P. Petrov and S.I. Stepanov, Piz'ma v JTP 4 (1978) 976 (in russian). [6] H. Kogelnik, Bell Syst. Tech. J. 48 (1969) 2909. [7] J.P. Herriau, J.P. Huignard and P. Aubourg, Appl. Optics (1978) 1851. [8] A.A. Kamshilin, M.P. Petrov and S.I. Stepanov, Piz'ma v JTP 5 (1979) 374 (in russian). [9] M.P. Petrov, S.I. Stepanov and A.A. Kamshilin, Optics Comm. 29 (1979) 44. [10] J.W. Goodman, Introduction to Fourier Optics (McGrawHill Book Company, 1968).

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