Volume holograms of co-orthogonal waves for optical channel switching and expansion of light waves in Walsh basis functions

I January 1997

OPTICS COMMUNICATIONS ELSEVIER

Optics Communications

133 (1997) 415-433

Full length article

Volume holograms of co-orthogonal waves for optical channel switching and expansion of light waves in Walsh basis functions V.V. Orlov, A.R. Bulygin XI.Vavibv

State Optical Institute. 12 Birzhevuya Line. St. PetersburR. 199034. Russia

Received 29 March 1995; revised version received 5 February 1996; accepted 13 June 1996

Abstract Study is presented of the volume superposed holograms, recorded with the use of light waves, whose amplitudes can be described in terms of mutually orthogonal functions. In the experiment the possibility to use such holograms for switching of optical channels and for expansion in Walsh functions of the light waves was shown. One channel commutation to 8 output channels with a signal/cross-talk ratio of 15-25 dB and expansion of the light waves, scattered by objects consisting of 7 points, across the set of 8 Walsh functions were realized. On the basis of the evaluated expansion coefficients the real components of the complex amplitudes of the light field from the objects were found out. Mean square error of experimental values with respect to the theoretically predicted values was 4.2-8.8%. The theoretical study of the cross-talk, revealing itself in the case of frustrated orthogonality of the reference waves, was carried out. Signal wave intensity and cross-talk intensity and their relation were evaluated for the case of nearly 100% diffraction efficiency of the holograms and for the case of low diffraction efficiency. Light field expansion in the case of frustrated orthogonality of the reference waves was discussed. Keywork

Volume superposed

holograms;

Optical channels

switching;

1. Introduction It is well known,

that a lens carries

out a Fourier

transformation of the incident light field. Namely, the light wave field is expanded by the lens action in the continuous spectrum of spatial harmonics. A similar property is revealed by the superposed volume holograms, recorded by light waves whose components have complex amplitudes, which can be described in terms of mutually co-orthonotmal functions. Such a hologram makes it possible to carry out the wave field expansion in any predetermined basis of discrete co-orthogonal functions, for example, the 0030-4018/97/S17.00 Copyright PII SOO30-4018(96)00423-3

Light wave expansion;

Cross-talk

basis of discrete Fourier transform functions or Walsh function basis. This property of holograms was found out in Refs. [1,2] by the theoretical investigation in the kinematic approximation. Later [3] it was shown, that under some conditions such a hologram does not contain the intermodulation gratings of dielectric permeability. Namely, the superposed holograms are to be recorded by the pairs of object and reference waves, whose complex amplitudes are proportional to two arbitrary complete sets of the discrete, orthonormal functions - one for the object waves and another for the reference waves. A recorded

0 1997 Elsevier Science B.V. All rights reserved.

416

V.V. Orlou. A.R. Bulygin/

Optics Communications

hologram will contain only the cross-modulation gratings produced by the interference of the reference waves with object waves, and not the intermodulation gratings produced by the interference of reference or object waves with themselves. The intermodulation gratings are eliminated by integration of the intermodulation gratings of all the superposed holograms, because the sum of amplitudes of these gratings from different superposed holograms is equal to zero. Two additional conditions are to be fulfilled: first, the response of the dielectric permeability is to be linear, and, second, in the course of hologram recording the gratings, which were already recorded, should not to distort the object and reference waves, used for sequent grating recording. We shall call the set of the superposed volume holograms, recorded under such conditions, the volume hologram of coorthogonal waves (VHCOW). The properties of VHCOW were theoretically studied in Refs. [3,4] in the approximation of the modal theory of volume hologram (the dynamic theory of diffraction). This theory treats the holograms, recorded by the paraxial waves, whose incidence angles to the hologram surface are small [5,6]. According to our results, for both transmission and reflection, phase and absorption VHCOW reconstruction by a certain reference wave yields only the object wave that was recorded with the given reference wave. The diffraction efficiency of the phase VHCOW can be as high as 100%. Hence, the VHCOW is transforming any wave from one set of mutually co-orthogonal waves to the corresponding wave from another set of such functions. In particular, the case of object waves, described by the Kronecker F-function, corresponds to the expansion in the set of the mutually orthonormal functions. In this case VHCOW reconstructs selectively the object waves with high diffraction efficiency also in the case of large incidence angle of the object wave with respect to the hologram surface [7]. This property of VHCOW provides the possibility to use it both for light wave expansion and for the switch of optical channels or for the tilt of light beams in discrete directions. Similar to VHCOW in their properties are the superposed holograms, recorded by the arbitrary object waves and mutually co-orthogonal reference waves. This sort of superposed holograms, known as

133 (1997) 415-433

the phase-coded multiplexing holograms, was studied in Refs. [8-141 for the case of recording of numerous images and their selective restoration, in Ref. [ 151 for the case of images summation and extraction and in Ref. [16] for the case of images encryption. In Ref. 1171 the phase-coded reference waves were used in recording of the so called pseudo-deep holograms. Unlike VHCOW, the superposed hologram with the arbitrary object field can be selectively reconstructed only in the case of low diffraction efficiency. The theoretical analysis gives the conclusion that, in the case of non-mutually co-orthogonal object waves, such a hologram with high diffraction efficiency reconstructs the superposition of object waves instead of one object wave [ 181. The mentioned properties of VHCOW were found out in the course of their theoretical study. This paper is devoted to the experimental study of the properties of VHCOW, used for optical channel switching and for wave expansion. We have also studied theoretically the cross-talk, resulting from the frustrated mutual co-orthogonality of reference waves in the course of VHCOW recording. We have found out the relation of the intensities of the signal wave to that of the cross-talk wave. We have analyzed the light wave expansion in the case of frustrated mutual coorthogonality of the reference waves. Preliminary reports of our experimental results were given in Refs. [I 9,201.

2. Recording

of the VHCOW

VHCOWs were recorded, in which the reference waves presented the Walsh functions, while the object waves were Kronecker &functions. The Walsh function equals 1 and - 1 and makes the complete set of the discrete orthonormal functions. For each value of n = 1, 2, 3,. . . there exists a set of 2” Walsh functions [21]. We have used the set, consisting of 8 functions. 8 superposed holograms were recorded, using 8 reference and 8 object point sources. Two matrices A^and g, Eq. (I), describe the complex amplitudes of the waves, emitted by the point sources while the supe_rposed holograms were recorded. Here the matrix A stays for the reference waves and l? for the object waves. The matrix A^or B^ element, in the mth row and nth column corre-

V.V. Orlov, A.R. Bulygin/Optics

Communications

sponds to the complex amplitude of the nth oint source when the mth hologram was recorded Qm, n =o, I,..., 7). 111111

I

I

1

I

a= t

i

1

-1

-1 -1

I

1;

-’

t

-1 -I

-11 1

-1I

-11

-1

i 1I

I

1 -1

1

-1 -1

I1 -1 1

-I 1 1

’

1:

-’

-11 1

-1I1

-1 -11 1

-1

00100000 01000000 10000000

,-=ooo

1

,

1

(la)

1 10000

00001000’ 00000100 00000010 00000001

I

(lb)

Each row of the matrix A^and I? corresponds to one of the Walsh functions and s-function correspondingly. In Fig. 1 the scheme of the experimental setup for VHCOW recording is shown. The collimated beam of He-Ne laser radiation was split by mirror BS into the object and reference waves. The reference beam

417

133 (1997) 415-433

was collimated by two cylinder lenses CL1 and CL2, which concentrated it into the thin light stripe in the direction, perpendicular to the figure plane. This concentrated beam came to the 1D bar of 8 microlenses (period 0.79 mm). These lenses produced 8 reference point sources; their localization was determined by one of 8 phase masks, spatially modulating the wave by the Walsh function. Each phase masks had its order k (k = 0, 1, . . . ,7) and shifted the phase of the point sources 0 or 7~ radians according to the values 1 or - 1 of the Walsh function with the same order k. The order of the Walsh function is determined by the number of its value sign changes. These phase masks were made of flat glass plates, one side of which was coated by a photoresist layer. Phase modulation of ?r was done with an accuracy not worse than + r/10. The modulated reference wave was imaged by lens Ll on the plane, behind the photorefractive recording medium Reoksan [22] (the photorefractive polymer plate was 3.16 mm thick). The object used for recording holograms was formed by lens L2. To obtain successively eight object point sources the direction of the narrow beam produced by diaphragm Dl and transformed by lens L2, was varied in the image plane by

Collimator ,____________________, He-Ne laser ______________________

-El Light meter

M3

1.2

Phase mask i

Microlenses

Fig. 1. Optical scheme of the setup. VHCOW: volume hologram of co-orthogonal waves, MI, M2, M3: mirrors, BS: beam splitter, CL1 and CL2: cylindrical lenses, RM: rotary mirror, Dl: diaphragm, D2: mobile diaphragm, Fd: photodiode.

V.V. Orlov, A.R. Bulygin/Optics

418

rotating the RM mirror. Lens L2 imaged the RM mirror surface into the hologram area. Therefore every point object beam passed through the same Reoksan section illuminated by the reference beams. The reference was normal with respect to the Reoksan surface. The average angle of incidence of the object beams was 7r/6. Eight superposed holograms were sequentially recorded, beginning from the 0th and finishing by the 7th (the number of the hologram corresponds to the order of the phase mask). The intensity ratio of the reference and object waves was 1 : 1. Recording the new superposed hologram results in diminishing the diffraction efficiency of the holograms, recorded earlier. To equalize the diffraction efficiencies of all the superposed holograms, the exposure time of each new hologram was chosen in such a way that this hologram diffraction efficiency equaled 0.85 of the previous hologram efficiency. The diffraction efficiency of the holograms was measured during recording using the lens L3, diaphragm D2 and photodiode FD. The lens L3 imaged the point objects in the plane of the movable diaphragm D2, which transmitted the object source, corresponding to the hologram, recorded in this moment. The intensity of the reconstructed object beam was measured in the small pause, when the original object beam was blocked by the shutter. The exposures of all the superposed holograms were approximately equal. Table 1 summarizes the values of the diffraction efficiencies of the superposed holograms, measured in the process of recording and after compleTable I Diffraction efficiency of superposed holograms (in %). The number of the column is the number of the superposed hologram whose diffraction efficiency was measured. The number of the row is the number of the superposed hologram after whose recording the measurements were done. In the cases marked by (-) diffraction efficiency was not measured #

# 0

0

1 2 3 4 5 6 7

16.0 12.8 9.2 4.6

1

13.5 _ 10.4 _ _ 5.8

2

3

4

Il.5 9.7 _ _

9.2 _ _

8.2 _ _

7.0 _

6.0

6.0

5.6

5.7

5

6

7

Communications

133 (1997) 415-433

tion of superposed hologram recording. The diffraction efficiency of hologram #O, which was the first in the recording sequence, equaled 16% when it was the only recorded. After recording completion, the average value of diffraction efficiency was 5.4% while the minimal and maximal values of efficiency were 4.6% and 6%. Total transparency of the Reoksan matrix was 0.4. The angular separation between reference beams was much larger than the Bragg angle selectivity of the Reoksan sample, in accordance with VHCOW theory. The angular separation between object point sources was lo-* rad whereas the angle selectivity of the cross-reconstructions was 4 X 10-j rad. The cross-modulation gratings of the holograms revealed high values of parameters, characterizing their volume properties by 0 = 180 s=- 1 and Q = 2000 * 1 [23]. Consequently, each volume grating of VHCOW reconstructed only the waves, which were used in the recording stage [24]. Mutual co-orthogonality of the reference waves was partly distorted both due to the phase mask errors and to the difference in the reference beam intensities. Mean square deviation of the intensities of the reference point sources with respect to the average value was 6%. As the diffraction efficiency of the superposed hologram was sufficiently high, the reference waves could be distorted by the intermodulation gratings of the holograms, recorded in the preceding stages. (If the reference waves correspond to Walsh functions. in the kinematic approximation of the diffraction theory, i.e. if diffraction efficiency is low, the intermodulation gratings do not distort the reference waves of the consequent superposed holograms [32].)

3. Switching of optical channels The VHCOW feature to reconstruct selectively the object waves can be used to switch optical channels. In this case each of the output optical channels corresponds to one of the object waves. VHCOW, described in Section 2, was used in the experimental study of the possibility to use it in

switching of optical channels. The same experimen-

5.8 5.0

4.8

tal setup, which was used for hologram recording (Fig. 1) was used in these experiments. The reference beam was used as the input beam for optical switching. The kth output channel was selected by

V.V. Orlou. A.R. Bulygin / Optics Cnmmunicarions Table 2 Relative signal and cross-talk intensities, when the optical channels being switched using two superposed holograms #

#

0 0

108

I

I

placing the kth order phase mask in the path of the reference beam. The light beam, making the image of the kth point object, was treated as signal beam. Other reconstructed beams were treated as cross-talk. The intensity of the reconstructed object beams was measured using the lens L3, aperture D2 and photodiode FD, similarly to the diffraction efficiency measurement. The diameter of diaphragm D2 was 0.3 times the distance between adjacent point object images allowing only the first 6 rings of the reconstructed object diffraction pattern to reach the detector. Tables 2, 3 and 4 summarize in relative units the results of the signal beam and cross-talk beam intensity measurement. Tables 2 and 3 present the intensities of the beams, measured after recording two and four first superposed holograms correspondingly. Table 4 presents the results, obtained after complete recording the superposed holograms. The number of the rows in Tables 2, 3 and 4 corresponds to the number of the phase mask, modulating the reference wave and determines the number of the output channel to be connected with the input channel. The number of the columns corresponds to the number of the output channel, in which the intensity was measured. The intensities of signal waves are presented in the crosses of rows and columns with one and the same number. All other intensities correspond to the

415,

# 0

1 114

f 1997) 415-433

Table 4 Relative signal and cross-talk intensities, when the optical channels arc switched using a VHCOW consisting of eight superposed holograms

#

1

133

0 1 2 3 4 5 6 7

270 5 4 2 2 I 2 2

2

I

3

4

7

4

340 2

I

I I

350 4

IO 350

I I

6

5

I

I

2

2

1 2 I

2

2 I I I

I I I I

327 IO

4 295

2

I

1

I

I

I

I

I I

1

2

4 332 2

2

3

I

2

7

I I I 4 282

intensities of the cross-talk waves. In Tables 2, 3 and 4 the relation of the average signal intensity to the average intensity of cross-talk @NRA) correspondingly equals 111, 145 and 147 or, in other words, 20.4, 21.6 and 21.6 dB respectively. Cross-talk in the case of VHCOW was produced by partial frustration of mutual orthogonality of the reference waves due to the errors in the phase masks and to the different intensity of the point reference sources. According to the results of the theoretic analysis presented in Section 5, the cross-talk is determined by the Gram matrix 6, Eq. (91, of the reference waves. We have found this matrix with the assumption that only the amplitude modulus of the reference wave components be distorted. Phase distortions were unknown and thus were not taken into account. The moduli of the components’ amplitudes were determined from the measurement of the intensities of all the point reference sources (see Section 4, Table 5, where the values of these seven components are enumerated as the components of vector A,). According to the results of Section 5, the value of SNRA is determined by the following relationship: N

Table 3 Relative signal and cross-talk intensities, when the optical channels being switched using four superposed holograms #

1) f Q;, II= I

c IQnm12.(2)

f n= I n=+Ptl In=

I

I

Here Q,, is the element of matrix 0. The value of

#

I

0 0

SNRA=(N-

440

I

3

2

5

3

3

3

2 5

500 I I

4 2

460 3

2

I 8 440

SNRA, determined from Eq. (2), equaled 32.7 dB. The difference between this theoretical value of 32.7 dB and the experimentally measured 21.6 dB most probably can be explained by not taking into account phase distortions of the reference waves.

420 Table 5 Theoretical

V.V. Orlov. AR. Bulygin/Optics

complex

of the object light waves A = (a,, a*, . . , a,) and real parts of the experimental complex amplitudes, d,): uRe and u,,,, are the rms deviations of real and imaginary parts of experimental complex amplitudes from their

Object #

Point source # 2

1

2

133 (1997) 415-433

amplitudes

ReA’= Re(d,, a; ,..., theoretical values

1

Communications

0 4 0

0 5 0

-8 96 80 0 9 0

-9 98 95 0 2 0 -5 98 105

A, ReA’, A2

ReA’? A3

ReA’, A, ReA’, A5

ReA; -96 -96

‘46

ReAb

3

4

5

100 93 100 94 100 87 -100 -94 -100 -97 -100 -103

0 3 97 93 97 91 0 2 97 92 97 71

0 3 0 0 98 99 0 0 0 0 -98 -98

Our experiment has shown that the VHCOW reconstructs the object waves with high selectivity; it can be used for switching optical channels or tilting the light beam to discrete directions. The scheme of optical channel switching on the base of VHCOW for the case of four input and output channels is shown in Fig. 2. The signal light wave, modulated in time, is sent from any of the input channels to any of the output channels with the help of the correspondent modulation of the phases of its components by 0 or T rad. The maximal frequency of signal modulation is limited by the spectral selectivity of VHCOW. This method of switching optical channels needs very high stability of the optical lengths of all

Beam splitter

Modulator \

I

Condenser

6

7 0 4 0

0 2 0 -4 94 98 0 3 0 -9 94 85

-3 91 90 0 1 0 2 -91 -88

optical paths between the beam splitters and holograms.

4. Light wave expansion

in Walsh functions

VHCOW, expanding the light wave across some specific set of functions, is to be recorded with the use of reference waves, whose complex amplitudes are described in terms of such functions. The object waves correspond to the Kronecker &functions. Such a VHCOW can expand the wave, scattered by the object, consisting of several points, coinciding with some of the reference point sources, used at the stage

VHCOW

/ 1

” 2

Fig. 2. Optical schematic

up,.._= 4.2 cr Irn= 13 u u ;;I:;” g u ;I 1;; u ae = 4.4 D Irn = 11 u Rc = 4.5 crfin = 10 u Rc = 8.8 u Irn = 22

of the optical channel switch 4 X 4.

0 ” t P :

V.V. Orloo, A.R. Bulygin/Optics

of recording. If the VHCOW is illuminated in such a way, the waves, whose complex amplitudes are proportional to the expansion coefficients [ 11, are reconstructed. So, it is necessary to determine the complex amplitudes of the waves, restored by the hologram, with the purpose to find out the expansion coefficients. We propose a method to solve this problem based on the following. The VHCOW, recorded using N point reference sources, is used for expansion of the wave, emitted by the object, consisting of no more than N - 1 points. One point source is used in this case as the reference source for determination of the reconstructed wave phase. Intensities of the reconstructed waves are measured three times. In the first measurement the residual reference source is not used, and VHCOW is reconstructed only by the object waves (Fig. 3a). Measured intensities of the reconstructed waves are equal to the squares of the expansion coefficients, I,, = Ic,,12,

Communicutions 133 (1997) 415-433

421

VHCOW rl

Detector array

la, cf (a)

(b)

(3)

where C, = I C, ( exp(i cp,) is the nth expansion coefficient, n = 1, 2,. . . , N. In the second measurement the hologram is reconstructed both by the reference wave and by the wave from the object (Fig. 3b). Measured intensities of reconstructed waves contain the data on the cosines of the phase shift (v~ - (P,,“) between the phase of the wave, corresponding to the nth expansion coefficient cp, and phase of the wave, reconstructed by the reference object qbn: I,, = I C, I ’ + 2 I C, I I b, I cos(cp,- qbn)+ I b, I ‘,

(4) where b, = I b, I exp(icp,,,) is the complex amplitude of the wave, reconstructed by this reference wave, n= 1,2,..., N. We shall call this wave the reference wave of the nth expansion coefficient. The third measurement is carried out similarly to the second one, but the reference source wave phase is shifted by 7r/2 (Fig. 3~). Consequently, the data on me sines of the phase shifts is measured: I,,~ = I C, I 2 + 2 I C, I I b, I sin( cp,- qbn)+ I b, I 2.

(5)

a.

a, aI

a,

a//

d

Cc)

Fig. 3. The method of finding of the coefficients of expansion of the discrete object wave field in orthogonal functions. VHCOW is a volume hologram of co-orthogonal waves, (h) are the waves emitted by the object points, (*) is a wave emitted by the reference source, (++I+) are the waves occurring due to the interference of waves reconstructed by the object and those reconstructed by the reference source. (a) Obtaining the expansion coefficient moduli; the reference source is ‘turned off. (b) Obtaining the phase difference cosines; the reference source is ‘turned on’. (c) Obtaining the phase difference sine; the wave field phase of the reference source is shifted by rr/2.

So Eq. (3) makes it possible to determine the moduli of the expansion coefficients I C, I and Eqs. (41 and (5) their phases cp,. The moduli I b, I and phases qb,, of the complex amplitudes of the reference waves of the expansion coefficients are in general different for different n,

V.V. Orlov, A.R. Bulygin/Optics

422

i.e, for different expansion coefficients. They depend on the specific chosen set of orthogonal functions and on the choice of the reference point source. For the set of Walsh co-orthogonal functions it is possible to choose the universal reference point source. Moduli and phases of the waves, reconstructed by this universal source, will be one and the same for all n. For example, in the case of Walsh set expansion (matrix & Eq. (la)> the source, which at the stage of VHCOW recording corresponded to the Walsh function values, represented in the first column of matrix i, is the simplest one to use. Noteworthy is that the moduli I b, I are easily found by measurement of the intensities of the waves, reconstructed by VHCOW illumination only by the reference source. So the true valuesof Ib,)arefoundforall n=l,2,...,Nand the influence of the recording and reconstruction errors onto the values of 1b, I in Eqs. (4) and (5) is excluded. This method was used in the expansion in 8 Walsh functions of 6 objects, consisting of 7 points. The objects were formed, using the phase masks of 0th and 7th order, mounted in turn in the reference beam. Some of the seven reference sources were screened. So the objects were realized with the waves, differing in their module and phase. Table 5 summarizes the complex amplitudes of the object waves in relative units. These amplitudes are given in real numbers (positive and negative). It is convenient to treat these amplitudes as the values of 7 components (I, (n= 1,2,..., 7) of six vectors with amplitudes A,=(a,, a2 ,..., a,) (m= 1, 2 ,..., 6). Here the value of each component is equal to the complex amplitude of the wave, emitted by the corresponding point source; each vector stays for the wave of corresponding object. The intensities of the reference waves of the expansion coefficients 1b,, I *, measured while the VHCOW was illuminated by the reference source, are summarized in Table 6. One can see from this table, that the turn from the zero expansion coefficient reference wave to that of the seventh

Table 6 Intensities I b,,I2 of the expansion coefficients waves; n is the number of the wave

of the reference

n

0

1

2

3

4

5

6

7

I b, I 2

33

33

29

26

28

23

21

17

Communicurions

I33 (1997) 415-433

results in diminishing the reference waves’ intensity by a factor of 2. According to the proposed method the moduli of the expansion coefficients, I C, I, were found from the results of the first measurement according to Eq. (3). The cosines of the phases of the expansion coefficients were evaluated from the results of the second measurement. Calculation of cos cp, resulted in 7 cases with a total number of 48 (eight expansion coefficients for six objects) with the value lcos qn I > 1; in this case the value of lcos (D,I = 1 was chosen in consequent calculations. The main reason for the occurrence of lcos qn 1 > 1 was the diminishing of the diffraction efficiency of VHCOW due to the washing out of the superposed holograms in the course of their reconstruction: after the second measurement the VHCOW diffraction efficiencies were down to 0.80-0.85 of their primary values (see the last row of Table 1); we did not carry out a third measurement of the intensities. So we did not know the values of sin q,, of the complex coefficients of the expansion in Walsh functions, C,=ReC,+iIm = lC,Icos

C, cp,+ilC,Isin

cp,,

thus we could evaluate only their real parts, ReC, = IC,lcos ‘p,. The Walsh expansion coefficients C, and complex amplitudes of the light waves up are mutually related by the direct and inverted Walsh conversions:

(6a) (6b) where Wal(n, p) is the value of the pth measure of the nth function. According to our experimental conditions, the complex amplitudes of the light waves from the objects were real. The values of Walsh functions are also real. Hence, according to Eq. (61, the expansion coefficients are also real: C, = Re C,. Hence, the values of Re C,, which we have evaluated in the experiment, were sufficient for the evaluation of the real parts of the complex amplitudes of the light waves from the objects. The imaginary parts

V.V. Orlou. A.R. Bulygin / Optics Communications

of the complex amplitudes and of the expansion coefficients are equal to zero: Ima, = ImC, = 0. The conversion (6) is unitary, thus the following equation should hold:

(7)

p=o

n=O

The left hand side of E$. (7) was found as the sum of the squares of moduli of the corresponding vector components A, = (a,, u2,. . . , q). The right hand side of this equation was made equal to the left by multiplying the measured values of ( C, 1 in the corresponding norming factor. The physical essence of this procedure is the assumption of 100% diffraction efficiency of VHCOW. Table 5 summarizes the real parts of the complex amplitudes of the light waves from six objects. These values are given in Table 5 as the real parts of the components of vectors A’,,, = (u’, , d2,. . . , u’,), m = 1. 2,. . . .6. The table presents also the mean square errors uRe of the experimentally measured real parts of the complex amplitudes with respect to their theoretical values. One can see, that the experimental values are close to the theory. For the range of these real parts, - 1OO- 100, their mean square error is in the range cRe = 4.2-8.8. According to the experimental conditions, the imaginary parts of the complex amplitudes Imu, should be equal to zero; hence the intensity corresponding to the imaginary components given by I,,=

i

/Im a,,(‘=

i n=O

p=l

[C,,12-

i

lReC,12

n=O

should be equal to zero. However, for all objects the experimental values were I,, # 0. So we can evaluate the mean square error cl,,, of these measurements: Cl”, = I I,,/8

I “z.

The values of ulrn, enumerated in Table 5, are 2-3 times larger than the corresponding values of rTRe. The reason for this difference may be the following. The complex amplitudes in fact might have a small imaginary part: the phase shift, introduced by the phase masks, might fall into the range 72+ 7r/ 10. A small error in E of the phase mask influenced first of all the imaginary part of the complex amplitude,

I33 (1997) 415-433

a21

because for this part the error influence is proportional to sin(7r + E) = E, while that for the real part is cos(n + E) = 1 - e2/2. In addition to the discussed sources of errors. which were not caused by the hologram itself, there was also the defect of VHCOW itself, which resulted in diminished accuracy of the expansion coefficients’ moduli evaluation - the inhomogeneity of the VHCOW diffraction efficiency. Its essence is the following. Reconstruction of the VHCOW by any point source representing the object, resulted in different intensity of the reconstructed waves of the expansion coefficients. The relationships between the intensities of the reconstructed waves were approximately the same as the values enumerated in Table 6, obtained when the VHCOW was reconstructed by the point source, working as reference source. Hence, the intensities of the waves of the expansion coefficients diminished by approximately 2 times, on the way from the wave of the zero expansion coefficient to the wave of the seventh expansion coefficient. At the same time, for the waves of any point source, expanded in Walsh functions, the moduli of all the expansion coefficients have one and the same value. The results of the experiment confirm the theoretic prediction of the ability of VHCOW to expand the light field in a given set of discrete orthonormal functions and of the possibility to find both the modules and phases of the expansion coefficients from the intensities of the waves, reconstructed by VHCOW. According to the analysis of the experimental errors the accuracy of determination of the complex amplitudes of the light waves can be significantly improved in comparison with our experiment.

5. Cross-talk of VHCOW The reference and object waves, used for VHCOW recording, can have amplitude and phase distortions, frustrating thus their orthonormal character. These frustrations result in intermodulation gratings and cross-talk, when instead of one-object waves, the object waves of various superposed holograms are reconstructed. Cross-talk can result from both the diffraction on the intermodulation and cross-modulation gratings. The cross-talk resulting

V.V. Orlov, AR. Bulygin/Optics

424

from the diffraction on the intermodulation gratings under frustrated mutual orthogonality of the object waves was discussed in Ref. 1251,and the cross-talk caused by the diffraction on cross-modulation gratings in Ref. [18]. In Refs. [10,12,13] the cross-talk, caused by the diffraction on cross-modulation gratings for the case of frustrated orthogonality of the reference waves and for holograms with low diffraction efficiency, were studied. The VHCOW can also reveal, in addition to cross-talk, the noise, caused by diffraction, not meeting the Bragg conditions. This noise, depending on the angular selectivity of the hologram, was studied in Refs. 112,141. In this work we studied the cross-talk, caused by frustrated mutual orthogonality of the reference waves and their diffraction on the cross-modulation gratings. Unlike previous studies, the holograms with both low and high diffraction efficiency are in the scope of our study using the dynamic diffraction approximation. Let the VHCOW be recorded, consisting of N superposed holograms, whose reference and object waves consist of N plane components each. The complex amplitudes of the waves are determined by the matrices A”= [urn,],

in, n= I,...,

B^= [urn,].

m= 1,..., N,

N,

(84

n=N+l,...,

2N, (8b)

where the matrix A^describes the reference and B^ the object waves. The element of the matrices with index mn corresponds to the complex amplitude of the nth component during the recording of the mth superposed hologram. During the recording of the mth superposed hologram the amplitudes of the reference and object waves are described by the row-vectors A, = (u,,,,, . . . , u,~) and B, = I.4m,2N) of the matrices A and B^.Let us (Um,N+i”“’ make up the Gram matrix of the vectors of the reference wave amplitudes: e” =

k,l.

Q,,=&,A,,,

m, n= I,...,

N, (9

where the bar denotes the complex conjugate and K,,, A, the scalar multiplication of the vectors. Let the rate of orthonormity frustration of the reference

Communications

133 (1997) 41.5-433

waves be not large and them to be nearly co-orthogonal -& ; ]Q,,,I
m=l,...,

N,

(10)

n+m

and nearly normed,

Qnn-Qm Qjj

<1

,

n2m

,

n, m, j= I,...,N.

Let us assume, that the object waves are orthonormal: s,,,B, = S,,G,

n, m= I,...,

N,

where a,,,,, is the Croenecker function and G the intensity of the object wave. Let us introduce the coordinate set OXYZ, the axes OX and OY which lie in the plane of the recording medium and the axis OZ normal to it. Let us assume that the wave vectors of the plane components of the waves, K, = (k,,, kRY, k,J are tilted by small angles with respect to the axis OZ and that the waves are polarized in one and the same direction. The amplitude of the wave field, recording the mth superposed hologram, is described by the equation q,,,(R) = nsl~,, exp(iK,R), where R = (x, y, z). Let the c 2N processed recording medium change the dielectric permeability proportionally to the light energy fluency density during the exposure. In this case the dielectric permeability distribution across the hologram volume will be ~(R)=~~+icyt~t~,,

cN I m=lI

Xexp(iK,R)

I 2,

E urn, n=l

(12)

where Ed is the dielectric permeability of the recording medium before the exposure, a0 the variation of its imaginary part in the regions with zero exposure, caused by the processing of the medium, K = K’ + iK” the COmpleX Coefficient Of proportionality, and I the duration of exposure of each superposed hologram. According to the modal theory of the volume

V.V. Orlov. A.R. Buly#~/

Optics Communications

holograms 1561, let us search for the wave field during the hologram reconstruction as T’(R)

= E c,( z) exp(iK,R), (‘3) n= I where the functions c,(z) characterize the variation along the axis z of the intensities of the plane components as well as the difference of their wave vector projections onto the axis z from k,,. Let us treat c,(z) as the components of the amplitude vector c(z) = (c,(z), c,(z), . . . , c,.(z)>. The solution of the set of linear differential equations, describing the propagation of the waves inside the hologram, results in the following relationship between the amplitude vectors 1261(Appendix A): c( z) = exp(iD&)c(O), (‘4) where c(O) is the amplitude vector on the surface of the hologram, c(z) is the amplitude vector at a distance z from the hologram surface, D = $k,t~&, to the wave number of the light in vacuum, and E = [Eni] the so called hologram matrix, whose elements are proportional to the amplitudes of the gratings of the dielectric permeability of the hologram. In the discussed case of the _superposed holograms the elements of the matrix E look like Enj=

i Umnzmj, m=l lSnSN,n+lljS2N for n#jand N+l
E,j = 0,

for n#jand

E,,=L+L,,

for

l
(15)

j
N+lln,

j12N’

(16) 1,2 ,..., 2N,

(17) where L = Ci=, Xi!, 1u,, I 2 and L, = icrO/KtEO. The amplitudes of the intermodulation gratings, Eq. (161, are supposed to be equal to 0, due to Eqs. (IO) and (11). Therefore these amplitude values are much lower than those of the cross-modulation gratings, Eq. (15). If the intermodulation gratings are absent due to the insensitivity of the recording medium to their lower spatial frequencies, the reference waves can be represented by any linearly independent waves, not restricted by the condi$ons (10) and ( 11). The eigenvectors of the matrix E form the basis in n=

425

133 (1997) 415-433

the 2N-dimensional space of the amplitude vectors c(z). The wave, whose amplitude vector c(z) is the eigenvector of the matrix E, is called the ‘mode’ of the hologram. It propagates via the superposed hologram as through the homogeneous medium, whose dielectric permeability depends on the value corresponding to the mentioned eigenvector eigenvalue of the matrix i?. We have found the eigenvectors H,, + ) and H,,_ ) and the eigenvalues h,(+ ) and h,, _ ), n= 1,2,..., N of the matrix ,??(Appendix B):

+G-‘&,,,+ =

f

p,,(A,,

,,...

, + G-‘fium.,,)

iG-‘KB,).

(‘8)

m=l h n(*)=L+Lof&,

(‘9)

where p,, are the components of the vector p, = (P,l9 pn2t*..7 pnN). The vector p, and the scalar T,, are the eigenvector and eigenvalue of the matrix G& G&P,=T,P,,

(20)

where Gf$ is the product of matrix 6 and scalar G. Eqs. (18)~(20) determine the modes of the transparent, amplitude-phase VHCOW, recorded by the distorted reference waves. Let us discuss the reconstruction of the phase, u0 = K” = 0, VHCOW by the same reference waves as used at the stage of hologram recording. Let us expand the reference waves on the surface of VHCOW to the superposition of the modes. The amplitude vector of the reference wave of the mth superposed hologram m = 1, 2,. . . , N will look like A,(O) = $ ,E ‘,j(Hx+) ,= I = ;

25

L j=,

+H,,-,)

dmjpjn(A,,

0 ,...,

0).

(2’)

n=,

The coefficients dmj in Eq. (21) can be found from the elements of the matrix id,,, j], meeting the requirement [d,j][Pjnl=[6tnn19

(22)

V.V. Orlou. A.R. Bulygin/ Optics Communications 133 (1997) 415-433

426

where the left hand side of Eq. (22) is the product of two matrices. Accounting for l!q. (21) in (14), we can find c,,,(z), i.e. the amplitude vector of the waves, produced in the course of the VHCOW reconstruction by the mth reference wave. Describing c,,,(z) in terms of the sum of the amplitude vectors of the reference c,,(z) and object c,,( z> waves, c,(z) = (c,,(z), c,,(z)), we get C,,(Z)

=

exp(iDLz)

2 j=l

2 d,j cos(D\r;JZ)Pj,A,>

z)

=

i -!Gexp(iDLz) k j=l

t

dmjfi

cos( D/cz)A,,,,

(25)

z) = i exp(iDLz){Q,,/Gsin(D\lGe,,z)B,,

=exp(iDLz)

(28b)

= i exp(iDLz)Dz

(28c)

(24)

(26) where Q,, is the intensity of the mth reference wave. In this case the cross-talk is absent. The VHCOW in this case reveals variable diffraction efficiency T&Z) = sin’(D/az) when reconstructed by different reference waves. The maximal diffraction efficiency reaches 100% for the case when D/~~z=1r/2+1rn, n= 1,2 ,.... In the second case let us assume that the diffraction efficiency of VHCOW is small; this is valid for thecaseof Dfiz<1r/2, n=1,2 ,..., N.Letus replace cos and sin in Eqs. (23) and (24) by their series representations, limiting ourselves to the first three terms. Now let us remind the well known feature of the eigenvectors and eigenvalues of matrices [27], GQ,, = Cy’ , d,jTjpj,,. In this case we have, for the reference wave, C,,(Z)

cms( z) = i exp( i DLz) DzQ,,B,.

A, - iDz2G2

i Q,,B,. n=l n#m

The ratio of the signal wave intensity to that of the cross-talk is

Let us analyze the cross-talk and diffraction efficiency of the VHCOW in three cases. First, let us suggest that the reference waves are mutually orthogonal, but not normed: Q,, = &,,,Q,,,,. In this case P,, = a,,,,,, 7, = GQ,, and Eqs. (23) and (24) look like

c,J

P-84

c,,(z)

n=l

Xsin( DJ;Jz)pl,B,.

c,,( z) = exp(iDLz)

c,,( z) = c,,( z) + c*,( z) 9

n=l (23)

c,,(

Here the second term in parentheses describes the attenuation of the mth reference wave and reconstruction of the reference waves of other superposed holograms. The amplitude vector of the object wave, C,,,,,(Z), can be presented in terms of the sum of amplitude vectors of the signal waves, c,,(z), and of the amplitude vector of the cross-talk wave, c,,(z):

k Q,,A, n= 1

SNR=

v

(27)

.

(29)

One can see from Eq. (29) that in the case of small diffraction efficiency of the VHCOW, the SNR is determined only by the Gram matrix of the reference waves and does not depend on the diffraction efficiency of the VHCOW, T,J(Z)= D2GQ,,,,z2. The situation is different in the case of frustrated mutual orthogonality of the object waves. In this case the SNR reveals an unlimited growth with the diffraction efficiency diminishing [ 183. Let us now discuss the case of the normed, but not orthogonal reference waves Q,, = Q, m = 1,2,..., N. Let us also assume that, in accordance with Eq. (lo), the rate of mutual orthogonality frustration is not large. One of the following relationships [27] will be valid for each of the eigenvalues 7j of the matrix GQ: 5 IGQ,,,,l~ n= I

IGQ-T~II

m = 1, 2,. . . , N.

n#t?l

(30) Replacing ,?r, by Q-~ = GQ + Qgj together with Eqs. (10) and (30), the following relationship can be written: N

. i

Qhl

I

;

c tQ,, I

n”#*

1-C 1.

(31)

V.V. Orbv, A.R. Bulygin/Optics

Communicafions

Let us expand fi (Eqs. (23) and (24)) in power series of gj/Q and limit the series to the terms of second power. Let us assume that the thickness of the VHCOW is z = z,, where z, is determined by the equation D&@z, = n/2. In this case the amplitude vectors of the reference wave, c,,( z, ), of the signal wave, c,,(z,), and of the cross-talk wave, c,,( z, ), fulfill the equations (Appendix C)

d

;I

133 (1997) 4/5-433

‘127

maximal for z = zl. The ratio 1Q,,,,l l/Q2 can be treated as the measure of mutual orthogonality of the nth and mth reference waves. The rate of their orthogonality is better, the lower this ratio is. According to Eq. (35) the diminishing of the mutual orthogonality of the reference waves results in diminishing of the maximal diffraction efficiency of VHCOW, v( z,) = Z,,(z,)/Q. The ratio of signal wave intensity, c,,(z,), to that of the cross-talk. c,,( z,), will be --

1

\

= exp(i

X

DLz,)

Q2

SNR=4

Am+

I? IE

QjnQnmAj

n=l

j=l

n+m,j j+m

II *

(32) c,,,~(z,) = i exp(iDLz,)

n+m

QnjQjm

Bn.

(34) One can see from Eq. (32) that for z = z, the reference wave can be presented as the superposition of the reference waves of all the superposed holograms. The signal wave intensity is then described (with an accuracy of the second power of Q,,/Q terms) as

One can see that the derivative dl,,( z)/d z = 0 for z = z,, and thus the signal wave intensity will be

rr2+4 -___

; lQnml* I m= I l?l+ll

l6

(36) I

Eqs. (29) and (36) result in the following. For one and the same distortion of the reference waves, for the maximal diffraction efficiency of VHCOW, the SNR is nearly 4 times greater than in the case of low diffraction efficiency. With the growth of the number of superposed holograms N and of the number of reference and object wave components, the requirements to the mutual orthogonality of the reference waves grow: for the preservation of SNR with increasing N one has to provide, proportional to N, decrease of I Q,, I ‘/Q*. Let us discuss the case of VHCOW, recorded with the use of distorted waves and reconstructed by the nondistorted reference waves. Let the nondistorted reference waves be described by the amplitude vectors T,,,=F(q,,, q,,,?, . . . . qmN), M= 1,2,...,N, where Cy=, q,jtjnj= a,,,,, and F > 0. Let us describe the distorted reference waves as a superposition of the nondistorted reference waves, N tn= 1,2 ,.... N, r,J,, ?I=I where_ the coefficients rmn = F-‘A,T, fill the matrix R = [ rmn], which we shall call the matrix of the reference wave distortions. The amplitude vectors T,, are described in terms of the amplitude vectors A,,, using the matrix i- ’ = [ (,,,I, inverted with respect to the matrix l?:

A,

T,,=

=

c

; m=l

rhmAIn?

n=

1.2 . . . . . N.

(37)

V.V. Orbv, AR. Bulygin/

428

Optics Communications

In the case of VHCOW reconstruction by the nondistotted reference wave T,, the reconstructed object wave cr “0(z), due to the linearity of the diffraction process, can be described by the superposition of the waves according to Eq. (37), reconstructed by the distorted reference waves:

I33 (1997) 415-433

taking into account Eq. (41) and averaging across the ensemble we get = F4.

(42)

(43)

n # m.

c*.,( z) = 5

Glcm,( z).

(38)

m=l

Here c,,(z) is determined by the relationship (24). If the diffraction efficiency of the VHCOW is small, c,,(z) is determined by Eq. (28). Writing Q,, in Eq. (28) in terms of r,,, we get cr,,( z) = i exp(iDLz)DF*z

rnnBn + 2 m=I tltffl

i

F,,,,,B,,, . I

(39) Here the first term in parentheses corresponds to the signal wave and the second to the wave of cross-talk. The signal wave to cross-talk intensity ratio in this case looks like SNR =

(IQ,,,,1*)=

2F4 -N

(44) Relationships (42) and (44) result in ( 1Q,, I *>/ ( Q& > - 1/N. Hence, the rate of mutual orthogonality of the reference waves grows proportionally to the number of their components under one and the same rate of distortion of these components. The intensity of the cross-talk wave is proportional to m~~(lQ,,12)=2F4~ m+n

(45)

I rnn I2

(40) 5 lrmn12’ !?I= I BZ#lZ

Reference wave distortions depend on the distortions of their plane components. Let us present Q,, and mm in terms of the reference wave components. With this purpose we can write the distorted reference waves as A,=

If the reference waves are determined by Walsh functions, I qmjGnj I* = l/N* and

[@‘+A,,)

exp(ie,,,i)q,+

(F+A,,)

m= l,...,N,

(SNR) = 2(( e*) + (4*)/F*)

(41)

where A,,,,, and E,,,, are the distortions of the moduli and phases of the complex amplitudes of the reference wave components. Let us assume that A,,,,, and E are independent arbitrary values, whose average v%e equals zero, (A,,) = (E,,) = 0, and dispersions, (A;,) = (A*> and (I&) = (E*>, are one and the same for all components. Let us also assume that (A* >/F* -SC 1 and ( E *> +Z 1. Rewriting Eq. (9)

.

So the value of (SNR) does not depend on the number of the superposed holograms N. This feature of the signal wave to cross-talk intensity ratio was established in Ref. [lo]. For the value of r,,, we get

(Ir,,l*)=

Xexp(iE,2)q,2,...,(F+A,n,) Xexp(ie,N)q,N],

For large N it does not depend on N. Taking into account Eqs. (42) and (45) in Eq. (29) we get, for large N, the following equation for the ratio of the average signal intensity to that of the cross-talk: 1

(lr,,I*>=

1,

(E*)+F

(47)

(A*> N

C

l4mj4jn1*+

1 j= I

n #m.

(48) If the VHCOW is reconstructed by the nondistorted reference waves, determined by Walsh functions (for large N), 1 (49) (sNR)= (~*)+(A*)/F*’

V.V. Orlov. AR.

Bulygin / Optics Communications 133 (1997) 415-433

Relationships (46) and (49) result in the following. VHCOW reconstruction by nondistorted reference waves reveals an (SNR) twice as large as in the case of reconstruction by distorted reference waves. According to Eqs. (42), (43) and (47), (48), this conclusion is valid for any system of mutually orthogonal waves. Let us discuss the situation, when the VHCOW, recorded with the use of nondistorted reference waves, is reconstructed by the distorted reference wave. In this case the cross-talk can be evaluated using the feature of VHCOW to expand the distorted reference wave across a set of nondistorted reference waves. The values of I,, are these expansion coefficients and hence, the signal to cross-talk intensity ratio is described by the relationships (40) and (49) for any diffraction efficiency of VHCOW. In Ref. [ 121 the equation equivalent to Eq. (49) for the case of a reconstructing reference wave, having only phase distortions i.e. for (A*> = 0, was derived. Let us discuss the influence of the mutual orthogonality of the reference waves on the accuracy of the expansion coefficients determination in the case when VHCOW is used for this purpose. The wave of the object, illuminating the VHCOW, is described by its amplitude vector X. Let us present X as the superposition of the amplitude vectors of the nondistorted reference waves: X =

5 a,T,,. n=

(50)

I

Here (Y,,are the expansion coefficients to be found. According to Eq. (39) the arbitrary term of Eq. (50) reconstructs the wave: V,( :) = i exp( DLz)DF*z

\

m=l m#n

I (51)

where the first term describes the wave of the nth expansion coefficient, while the second one describes its distortion. The third term describes the distortions, introduced into the waves of other expansion coefficient waves and depending on the nth

429

expansion coefficients. The amplitude vector of the overall wave field, reconstructed by the VHCOW, looks like N

w(z)

N

= C V,(z) n=

= i exp(iDLz)DF’z

I

C f,B,, In=

I

where f, = Cf= ,Tmncrn.Hence the distortions of the reference waves lead to the transformation of the amplitude vector A,=(cY,, (~~,...,a~) of the expansion coefficient waves, described by the matrix 6= [?,,I: F=cA,.

Here F=(f,, f2 ,..., f,) is the amplitude vector of the distorted expansion coefficient waves. If the distortions of the reference waves preserve the linear independence of the distorted reference waves, the matrix 6’ exists, inverted with respect to the matrix c*;.In this case the expansion coefficients can be found with the equation A=?‘F.

(52)

The amplitude vector Y of the reference source wave, used for the wave expansion, can be written as Y=

f n=

p,T,. I

The reference wave of the nth expansion coefficient, produced by the VHCOW reconstruction by the wave Y, is described by the amplitude vector y,(z)

= i exp(iDLz)DF*z

Eq. (53) means that in the case of reference wave distortions, i.e. for F,,, Z S,,, the reference waves of the expansion coefficients are distorted both in amplitude modulus and in phase. The distortions of the amplitude moduli are measured and their action is eliminated, using the approach discussed in Section 4. Phase distortions of the reference waves lead to errors in determination of the phases of the waves of the expansion coefficients. These errors can be excluded if one has the possibility to find out the phase distortions of the reference waves of the expansion coefficients. The distortions of the moduli and phases of the reference waves of the expansion coefficients

V.V. Orlov, A.R. Bulygin/Optics

430

can be found when one knows the matrix R^ of distortions of the reference waves. Our analysis of the light waves expansion was carried out for the case of low diffraction efficiency of the VHCOW. In the case of high diffraction efficiency, when the reconstructed waves are described by Eqs. (33) and (341, the results of this analysis will be the same. The only difference is that in this case the matrix describing the distortions of the waves of the expansion coefficients and of their reference waves is another:

F=

f(t+l?-‘T)As

(with an accuracy of the first power of the Q,,/Q term in Eqs. (33) and (34)). Here $-IT is the transposed matrix R^-’ . Hence, the VHCOW, recorded by the distorted reference waves, can be used for the expansion of light waves across the given set of orthonormal functions. These distortions, however, have to be low and their magnitude has to be known. Note that in this study we assumed that the VHCOW lacks intermodulation gratings.

6. Conclusion

The main result of the experimental part of this work is the confirmation of the preceding theoretical studies on the possibility of the VHCOW to transform any wave field from one set of mutually orthogonal wave fields to the corresponding field in another set of mutually orthogonal fields. According to the theory such a transformation can be carried out with a diffraction efficiency of 100%. In our experiment the wave fields, described by 8 Walsh functions, were transformed into the wave fields, described by 8 Kroenecker &functions, with a diffraction efficiency of 5.4%. We have proposed a method for determining the moduli and phases of the expansion coefficients when VHCOW is used for the wave field expansion along the set of orthonormal functions. We have studied theoretically the influence of

Communications

133 (1997) 415-433

frustration of the mutual orthogonality of the reference waves, used in the course of VHCOW recording, on its properties. The signal wave to cross-talk wave intensity ratio (SNR) was found for the cases of low and high diffraction efficiencies of VHCOW. It was shown that in the case of high diffraction efficiency SNR exceeds 4 times the value corresponding to the case of low diffraction efficiency and that in the latter case the SNR does not depend on the diffraction efficiency. It was also shown that in the case when the distortions of the reference waves are described by arbitrary values, whose average value equals zero, the SNR does not depend on the number of superposed holograms in VHCOW. It was shown that a VHCOW recorded with the use of distorted reference waves can be used for the expansion of light waves, when these distortions are not too large and their magnitude is known. Our studies revealed the possibility to use VHCOW for some practical purposes. The possibility of VHCOW to reconstruct in a selective way the recorded wave fields can be used for switching optical channels or for discrete tilt of a light beam. The VHCOW feature to expand the wave field along a set of orthonormal fttnctions can be implemented in wavefront sensors used, for example, for control of optical systems or for studies, in the course of which one has to know the phase of the wave field, scattered by the studied object. In this case the object studied can be formed by a set of discrete points or have a periodical structure. So the interesting question ‘do crystals, whose structure can be treated as a VHCOW, exist?’ can be analyzed. If yes, such crystals can be used as wavefront sensors in the X-ray range of the spectrum for tomography purposes.

Acknowledgements The authors are sincerely grateful to Yu.N. Denisuk for his attention to the work and consultations. The authors are thankful to the reviewer for his proposals on improvement of the style and English language of the paper. The work was supported by the Russian Foundation of Fundamental Investigations (grant 95-02-03887-A) and by the International Science Foundation (grant MU-4000).

V.V. Orloo. A.R. Bulygin /Optics

Appendix

Communications

A

Substitution of Eqs. (12) and (13) to the wave equation. A’P’( R) + k&( R)‘P’(

R) = 0,

(A’)

with the consequent necessary transformation, gives “c/A

2ik I); a_

z)

4

+ ikiuoc,,(

z)

exp(iK,R) 1

umn"mjcp( m=I

n=l

j=l

z,

p=l

X exp[i( K, - Kj + K,,)R]

= 0.

(‘42)

For deriving Eq. (A2) we have used the usual approximation of the slowly changing amplitudes and nullified the second derivative of c,(z) with respect to z. We have also cancelled out two terms, because the light wave number in the medium equals k = ko&,. Let us now add the factor exp(-iK,R) to Eq. (A2) and integrate across the arbitrary plane, perpendicular to the axis OZ for z.2 0. Now we get the following system of 2N linear differential equations for slowly changing amplitudes:

ac,t2) + iki(+,c,(

2ik I: az + k*Kr

; m=l

z)

F E E u,,,&,~c,,( n=l

j=l

z) = 0, (A3)

where I = I , . . . ,2N. In Eq. (A3) the summation along the indexes n, j, p is carried out under the condition

(A41 [K,,-Kj+K,,]~nv=[Kll.~,i, where [Kl,,, means the K vector projection to the plane XOY. inthe case, when the angular separation of the wave vectors of plane components K, far exceeds the angular selectivity of the hologram, the requirement (A4) will be fulfilled just when (j=nand

p=i)

or

(j=p,j+nand

n=l), (A5)

under which the diffractive waves fulfill the Bragg condition. Account for the diffraction waves, not meeting the Bragg condition, results in distorting and

I

noise waves. The distorting waves have a wave vector K for which, according to Eq. (A4). [K 1,o, = [KILW but the projections to the axis OZ are not equal: k, # k,,. For the noise waves, [K],,, f for any 1 = 1,. . . ,2N. The mode theory [KJ.,,, does not account for distorting and noise waves and describes the diffraction in an adequate manner when their intensity is low. Under some conditions the intensity of distorting and noise waves is small notwithstanding the relationship between the angular selectivity of hologram and angular separation of plain components of waves. These conditions are described in Ref. [28] (Eqs. (29) and (30) of this paper). These conditions relate to the case when the intensity of any plain component of the waves. recorded in the hologram, is much less than the summary intensity of other plain components. Integration of Eq. (A3) under the conditions (A5) and limiting ourselves by the paraxial approximation of diffraction theory, when k,. = k, we get the following set of linear differential equations: Jc,( z)

az = iD (L + L,)c,( z) + E [

p=I

J3

133 (I9971 415-433

j=l j’l

i m=l

1 (A61

u,,U,~C~( z)

.

where I= 1,. . . ,2N. (The parameters D, L and L, are described in the body of this paper.) The solution of the set (A6) looks like Eq. (14) [29]. Note, that the mode theory of volume holograms, which we have used, is not only used for the description of static holograms. It was used as well in dynamic holography [30], in particular, in the theory of phase conjugation via stimulated Brillouin scattering 1281 and in the theory of hologram recording in amplifying media [31]. Appendix B Let us search for the eigenvectors of the matrix k as a superposition of amplitude vectors of the reference and object waves:

H, = ? (P,,A,. IT*=I

&?lR”)~

(Bl)

V.V. Orlov, A.R. Bulygin/Optics

432

where p,, and pb,,, are the coefficients to be found. The following equation,

Communications

133 (1997) 415-433

Taking into account (C2) in Eq. (24) results in the terms N

l%,,= h,H,, taking into account Eq. (B 1) leads to a system of 2 N linear equations. The first N equations result in the relationship W)

GP:*=[k(~+~o)lPnm and the next N equations in the relationship

N

C dmjgjpjn and C d,jg,fPj,,

j=

1

j= I

The properties of eigenvectors and eigenvalues of the matrices [27] and Eq. (22) results, in turn, in N

C dmjgjPjn=(1 -

'mn)Qnmv

(C34

j= 1 N

IL e,jPnj=[h,-(‘+‘,)]Ph,y j=

(B3)

1

C dmjgj2Pjn j= I

where m, n = 1, 2,. . . , N. Eq. (B2) results in Pb,

=

h,-(L+kl) G

W)

Pnm*

Substitution of Eq. (B4) in (B3) leads to

=

2 IQj,IzS,"+

QnjQjm(l

-

'mn).

j#m,n

j+m

G~Q,jP,j=[h,-(‘+‘,)]‘P,,. j= 1

i j= I

j= 1

(C3b)

where m,n=l,2 ,..., N. The same relationship is valid for pi,. Hence, p,, and p’,, are the components of eigenvectors of the matrix Cd = [GQ,,,,], with corresponding eigenvalues

On this basis we can derive Eqs. (33) and (34). For Eq. (23) the conditions (Cl) result in

rn= [h,-(L+&-J]*,

Hence the use of Eqs. (C3) gives us Eq. (32).

(B%

cos(D+,)

= -;[

2

- ;( ;ii].

where n= 1, 2,..., N. Eq. (B5) results in Eq. (19) for the eigenvalues h, of hologram matrix t?. References Appendix

C

The term fi.sin(Dfiz) conditions I+$;;

z=z,=r/2D\rc;Q,

in Eq. (24) under the

(

2 ,

Pa)

)I

(Cl b)

gives, with an accuracy of up to the second power of

g/Q, fisin(

Dfiz,)

[l] Yu.N. Denisyuk and I.N. Davydova, Opt. Spektrosc. 60 ( 1986) 365 [Opt. Spectrosc. 60 ( 1986) 2231. [2] Yu.N. Denisyuk, I.N. Davydova and L.N. Baikova, Opt. Spektrosc. 63 (1987) 1351 [Opt. Spectrosc. 63 (1987) 8011. [3] V.V. Orlov, Pis’ma Zh. Tekn. Fiz. 16 (1990) 9 [Sov. Techn. Phys. Lett. 16 (1990) 461. [4] V.V. Orlov, Zh. Tekn. Fiz. 62 (1992) 117 [Sov. Phys. Techn. Phys. 37 (1992) 8731. [51 V.G. Sidorovich, Zh. Tekn. Fiz. 46 (1976) 1306 [Sov. Phys. Techn. Phys. 21 (1976) 7421. [6] A.A. Leshchev and V.G. Sidorovich, Opt. Spektrosc. 44 (1978) 302 [Opt. Spectrosc. 44 (1978) 1751. [7] N.S. Shlyapochnikova, Zh. Tekn. Fiz. 64 (1994) 67 [Techn. Phys. 39 ( 1994) 7831. [8] C. Denz, G. Pauliat, G. Roosen and T. Tschudj, Optics Comm. 86 (1991) 171. (91 Y. Taketomi, J.E. Ford, H. Sasaki, J. Ma, Y. Fainman and S.H. Lee, Optics Lett. 16 (1991) 1774. [lo] C. Denz, G. Pauliat, G. Roosen and T. Tschudj, Appl. Optics 31 (1992) 5700.

V.V. Orlov. A.R. Bulygin / Optics Communicarions 133 (1997) 415-433 [I I] C. Alves, G. Pauliat and G. Roosen, Optics Lett. I9 (1994) 1894. [I21 M.C. Bashaw, J.F. Heanue, A. Aharoni, J.F. Walkup and L. Hesselink, J. Opt. Sot. Am. B 11 (1994) 1820. [ 131 K. Curtis and D. Psaltis, J. Opt. Sot. Am. A I I (1994) 2547. [I41 J.F. Heanue, M.C. Bashaw and L. Heeselink, J. Opt. Sot. Am. A 12 (1995) 1671. [ 151 J.F. Heanue, M.C. Bashaw and L. Heeselink, Optics Len. I9 (1994) 1079. [I61 J.F. Heanue, M.C. Bashaw and L. Heeselink, Appl. Optics 34 (1995) 6012. [I71 Yu.N. Denisyuk, J. Opt. Sot. Am. A 9 (1992) 1147. [I81 V.V. Orlov, Pis’ma Zh. Tekn. Fiz. 21 (1995) 64 [Techn. Phys. Lett. 21 (1995) 5671. [I91 V.V. Orlov and A.R. Bulygin, Pis’ma Zh. Tekn. Fiz. 19 (1993) 3 [Techn. Phys. Lett. 19 (1993) 11. [20] V.V. Orlov and A.R. Bulygin, in: Advances in Optical Information Processing, VI, ed. D.R. Pape, Proc. SPIE 2240 ( 1994) 227. [21] A.K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, NJ, 1988) p. 155. [221 G.I. Lashkov and V.I. Sukhanov, Opt. Spektrosc. 44 (1978) I008 [Opt. Spectrosc. 44 (1978) 5901.

1231B.

433

Benlarbi and L. Solymar, Int. J. Electron. 52 (1982) 95. and T.K. Gaylor, J. Opt. Sot. Am. 68 [241 R. Magnusson ( 1978) 809. V.V. Skunov and T.V. Yakovleva. in: 1251 B.Ya. Zel’dovich, Problemu opticheskoi golografii, ed. Yu.N. Denisyuk (Nauka, Leningrad, 198 I) p. 80 (in Russian). PhD Thesis, S.I. Vavilov State Optical [261 V.G. Sidorovich, Institute (I 977). (271 Handbook of Applicable Mathematics, Vol. I. Algebra. ed. W. Lederman (John Wiley, Chichester, 1980). t281 V.G. Sidorovich, Zh. Tekn. Fiz. 46 (1976) 2168 [Sov. Phys. Techn. Phys. 21 (1976) 12701. [291 R. Bellman, Introduction to Matrix Analysis (McGran-Hill, New York-Toronto-London, 1960) ch. IO. [301A.V. Groznyi, A.M. Dukhovnyi, A.A. Leshchev and V.G. Sidorovich, in: Opticheskaya Golografiya, ed. Yu.N. Denisyuk (Nauka, Leningrad, 1979) p. 92 (in Russian). 1311A.A. Leshchev and V.G. Sidorovich, Opt. Spektrosc. 57 (1984) 765 [Opt. Spectrosc. 57 (1984) 4671. [321Yu.N. Denisuk, private communication.