**5. Dot-matrix holographic recording on As2S3-Al films**

For the recording of relief-phase holograms, organic photoresists are mainly used; they are sensitive enough only in the ultraviolet part of spectra λ < 480 nm. For lasers with the wavelength λ=480-630 nm, it is possible to use chalcogenide film photoresists. Due to the high values of the photo-induced changes of optical properties, the thin films of amorphous chalcogenide semiconductors make a very promising medium for holographic recording. But the use of such films for recording dot-matrix holograms is complicated because of the low sensitivity E=5-30 J/cm2. For example, the time for producing dot-matrix holograms consisting of 106 pixels is measured by tens of hours. That is why a possibility of increasing sensitivity of films As2S3 in the system As2S3-Al was studied. Usually, in order to get a relief hologram, a film of photoresist is put onto the substrate with an absorbing layer. In this case the interference pattern in the photoresist layer is made only by incident waves. The interference pattern in the As2S3-Al system during the holographic recording can be considered as a sum of incident waves and those reflected by the aluminum sub layer. In case of their overlap, the maximum diffraction efficiency (DE) and sensitivity of the medium are obtained. One of the main conditions for coincidence of interference patterns of incident and reflected waves is equality of incidence angles upon the film surface. If this condition is met, any one of the incident beams after reflecting by the aluminum layer goes in the direction of the incidence of another one (**Fig. 19**). Optical schemes having equal incidence angles of beams are mainly used for the dot-matrix and image-matrix holographic recordings. Taking into consideration the possibility of selective etching of As2S3 films, the As2S3-Al system is promising from the point of view of obtaining relief-phase holograms.

Fig. 19. Scheme of interference field in As2S3-Al films. **a)** The incidence angles are equal. **b)** The incidence angles are diffrent.

Interference pattern of reflected waves is less stable in time, since before the interference the waves pass the As2S3 layer, which has some changes of optical properties during exposure. The aim of the experiments was to determine the conditions ensuring the maximum overlap of interference patterns of incident and reflected waves. A relatively easy way of obtaining the minimum reflection coefficient in a definite part of spectra is viewed as an important feature of the system As2S3-Al. Monitoring interference minimum of reflection during the evaporation of film makes it possible to get the system As2S3-Al with a minimum value of initial reflection, i.e. 10-15%. Thus, maximum energy absorption of incident waves is obtained and, as a result, the sensitivity of composition during the recording of dot-matrix holograms increases up to 40%.

As2S3 films were obtained by thermal evaporation in vacuum onto aluminum (~100 nm) coated glass substrates. The thickness of the As2S3 film was closely controlled during the

Digital Holographic Recording in Amorphous Chalcogenide Films 89

absorption make it possible to observe the result of holographic recording in thin films. The interference pictures in As2S3-Al films of different thickness after the recording of dot-matrix holograms are shown in **Fig. 22a-e**. As seen in **Fig. 22c**,**d,** two systems of interference bands have equal periods and are mutually shifted. One system of bands corresponds to the interference of incident waves; the other one corresponds to the waves which are reflected from the aluminum surface. Interference maxima of the incident and reflected waves in the films of thickness 2.4 μm are situated very near to each other (see **Fig. 22c**). This case corresponds to the maximum diffraction efficiency of a holographic pixel recording. The optimal exposure for the recording of holograms in the As2S3-Al system is E=12-15 J/cm2. This half that needed for the As2S3 films of the same thickness on substrates with absorbing layer of iron oxide. There are two factors significantly influencing the sensitivity of the

The first is reduction of exposure dose due to superposition with matching of interference patterns for the incident and Al layer reflected waves. This happens when the film thickness is ~2.4 μm (λ=532 nm). The second factor is the possibility of production of As2S3-Al films with the minimum reflection coefficient in the part of spectra where the optical recording is made. The main problem in practical implementation of the As2S3-Al system in dot-matrix holography is that of depositing an As2S3 layer of high uniformity on

Fig. 22. Image of optical absorption in volume of As2S3-Al films after dot-matrix hologram

The principle of dot-matrix holography is based on decomposition of a hologram image into a two-dimensional array of elementary pixels containing diffraction gratings with parameters that need mathematical calculation. Their size usually lies in the range of 10-100 μm and depends on the technology of optical recording that is used. In each elementary pixel of the hologram there is a diffraction grating with certain period *d*and orientation

recording (λ=532 nm, N=800 mm-1, E=10 J/сm2). The pictures were obtained while performing Z-scanning (λ=632 nm) by the confocal microscope LEICA TCS SP5. The left part of each picture corresponds to the layer Al, the right one to the film-air surface; **a)** without recording, **b)** film 1.2 μm, **c)** film 2.4 μm, **d)** film 3.6 μm, **e)** film 4.8 μm.

**6. Calculation principles for dot-matrix holograms** 

As2S3-Al system during the recording of dot-matrix holograms.

large substrates.

process of evaporation. Monitoring reflectivity with the interference technique at the wavelength of 532 nm during evaporation of the As2S3 layer allowed for obtaining a As2S3- Al pattern with the initial reflection of 10-15%. The process of evaporation was stopped immediately as soon as the interference minimum of reflection was reached.

Laser irradiation of As2S3 films causes modulation of optical refractive index, which brings about modification of interference conditions, that is modification of the reflectivity coefficient of the As2S3-Al system. It was experimentally established that the minimum value of reflection coefficient in the As2S3-Al system depends on the thickness of As2S3 film and reaches Rmin=10% at d=2.4 μm (**Fig. 20a**). Dependence of the pixel hologram DE on film thickness is presented in **Fig. 20b**. The maximum diffraction efficiency DE ~40 % was obtained when the film thickness was d=2.4 μm (**Fig. 21**). With an increase of the film thickness, absorption in the As2S3 layer at the wavelength λ=532 nm becomes high and lessens the influence of Al-reflected waves on the formation of the interference pattern. In the case of thick films (d > 5 μm), holographic recording takes place only due to interference of incident waves.

Fig. 20. **a)** Dependence of minimum reflection Rmin of the system As2S3-Al on the film thickness (λ=532 nm). **b)** Dependence of the maximum diffraction efficiency of dot-matrix holograms on the thickness of films As2S3-Al

Fig. 21. Photos of the test dot-matrix holograms recorded on As2S3-Al films at different magnifications.

During holographic recording, local changes of optical properties of the recording medium, corresponding to the distribution of interference maxima, take place. Changes in local

process of evaporation. Monitoring reflectivity with the interference technique at the wavelength of 532 nm during evaporation of the As2S3 layer allowed for obtaining a As2S3- Al pattern with the initial reflection of 10-15%. The process of evaporation was stopped

Laser irradiation of As2S3 films causes modulation of optical refractive index, which brings about modification of interference conditions, that is modification of the reflectivity coefficient of the As2S3-Al system. It was experimentally established that the minimum value of reflection coefficient in the As2S3-Al system depends on the thickness of As2S3 film and reaches Rmin=10% at d=2.4 μm (**Fig. 20a**). Dependence of the pixel hologram DE on film thickness is presented in **Fig. 20b**. The maximum diffraction efficiency DE ~40 % was obtained when the film thickness was d=2.4 μm (**Fig. 21**). With an increase of the film thickness, absorption in the As2S3 layer at the wavelength λ=532 nm becomes high and lessens the influence of Al-reflected waves on the formation of the interference pattern. In the case of thick films (d > 5 μm), holographic recording takes place only due to interference

Fig. 20. **a)** Dependence of minimum reflection Rmin of the system As2S3-Al on the film thickness (λ=532 nm). **b)** Dependence of the maximum diffraction efficiency of dot-matrix

Fig. 21. Photos of the test dot-matrix holograms recorded on As2S3-Al films at different

During holographic recording, local changes of optical properties of the recording medium, corresponding to the distribution of interference maxima, take place. Changes in local

holograms on the thickness of films As2S3-Al

immediately as soon as the interference minimum of reflection was reached.

of incident waves.

magnifications.

absorption make it possible to observe the result of holographic recording in thin films. The interference pictures in As2S3-Al films of different thickness after the recording of dot-matrix holograms are shown in **Fig. 22a-e**. As seen in **Fig. 22c**,**d,** two systems of interference bands have equal periods and are mutually shifted. One system of bands corresponds to the interference of incident waves; the other one corresponds to the waves which are reflected from the aluminum surface. Interference maxima of the incident and reflected waves in the films of thickness 2.4 μm are situated very near to each other (see **Fig. 22c**). This case corresponds to the maximum diffraction efficiency of a holographic pixel recording. The optimal exposure for the recording of holograms in the As2S3-Al system is E=12-15 J/cm2. This half that needed for the As2S3 films of the same thickness on substrates with absorbing layer of iron oxide. There are two factors significantly influencing the sensitivity of the As2S3-Al system during the recording of dot-matrix holograms.

The first is reduction of exposure dose due to superposition with matching of interference patterns for the incident and Al layer reflected waves. This happens when the film thickness is ~2.4 μm (λ=532 nm). The second factor is the possibility of production of As2S3-Al films with the minimum reflection coefficient in the part of spectra where the optical recording is made. The main problem in practical implementation of the As2S3-Al system in dot-matrix holography is that of depositing an As2S3 layer of high uniformity on large substrates.

Fig. 22. Image of optical absorption in volume of As2S3-Al films after dot-matrix hologram recording (λ=532 nm, N=800 mm-1, E=10 J/сm2). The pictures were obtained while performing Z-scanning (λ=632 nm) by the confocal microscope LEICA TCS SP5. The left part of each picture corresponds to the layer Al, the right one to the film-air surface; **a)** without recording, **b)** film 1.2 μm, **c)** film 2.4 μm, **d)** film 3.6 μm, **e)** film 4.8 μm.

#### **6. Calculation principles for dot-matrix holograms**

The principle of dot-matrix holography is based on decomposition of a hologram image into a two-dimensional array of elementary pixels containing diffraction gratings with parameters that need mathematical calculation. Their size usually lies in the range of 10-100 μm and depends on the technology of optical recording that is used. In each elementary pixel of the hologram there is a diffraction grating with certain period *d*and orientation

Digital Holographic Recording in Amorphous Chalcogenide Films 91

We will make calculations in order to form image in the +1 order of diffraction. We take the position of a point source with continuous spectrum of light in the plane YZ. If the angle between the Z axis and direction from the light source to the hologram center is

reflection from the hologram surface is taken into account. The direction of the +1 diffraction order from the pixel with the coordinates ( , ) *x y* to the eye of the observer is defined by the

the eye of the observer. Taking into account the above mentioned conditions, the formulas

*d* λ

λ

θ<sup>=</sup> .

formulas in (2), we obtain the following parameters of the diffraction

<sup>0</sup> sin *d*

= − 30 45 , for example, if <sup>0</sup>

<sup>0</sup> *<sup>f</sup> <sup>d</sup>* 1135 650 *mm* <sup>−</sup> == − . At the next step, we can define the parameters (,) *<sup>d</sup>*

0

*d* λ

=− + ; <sup>2</sup> cos -

ϕ

sin (,)

*d L dxy*

=

pixel with the coordinates ( , ) *x y* having the same visual characteristics (color, brightness)

ϕ

=− + ; 2 2

γ

θ

α

*<sup>x</sup> x y arctg <sup>L</sup> <sup>y</sup>*

( sin - )

θ

*L y x*

<sup>=</sup>

θ

cos cos sin

For the central pixel of the hologram with coordinates (0, 0) and the color

spatial frequencies of diffraction gratings for all colors in spectral range 0

This requires solving a system of equations for the variables *d* and

2

2

θ

<sup>=</sup> <sup>0</sup>

α

2 cos - cos sin *tg*

> 2 sin - cos

α

ϕ

β

0

λ β

cos cos sin

 θ

From the equations in (3) we can find the formulas for calculating the parameters of

<sup>=</sup> <sup>+</sup> (,) sin -

The size of security holograms is usually not bigger than 3x3 cm and corresponds to the conditions *z x* >> and *z* >> *y* ; in this case, the following simplified formulas can be used

ϕ

 θ

β

 β *y d* λ

, where the change of the Z-component of incedent light after

*L LL* <sup>=</sup> , where 2 22 *L x* = ++ *<sup>y</sup> <sup>z</sup>* is the distance between the pixel and

= ± ; 2 2

γ

2 22 cos 1 - cos cos

2 22 cos 1 - cos cos

ϕ:

*x L*

θ

2 2

+

<sup>=</sup> ; <sup>2</sup> cos - *<sup>y</sup>*

β

α β

λ

= 30 , the desired interval of

λ

*L*

= . (3)

(4)

(5)

. (2)

0 based on the

= − 440 650 *nm*

ϕof a

= −

α β

(1)

θ,

= −

2 1 cos cos

θ

in (1) will look like the following:

2

α

grating: <sup>0</sup>

will be <sup>1</sup> -1

2

α

diffraction gratings of the pixel:

cos

for calculations:

*d d*

ϕ

cos

 α

 θ

sin

= − ; 2

ϕ

λ

for the observer as in the case of the central pixel.

sin

ϕ

*d* λ

0

ϕ

*d*

= − ;

θ= = , where <sup>0</sup>

θ

*d* λ

<sup>0</sup> ( , ) ( , 0) ( , 0) sin

Usually, in practical cases 0 0

α

then <sup>1</sup> *k* = − (0, sin ,cos )

unit vector <sup>2</sup> (- ,- , ) *x z <sup>y</sup> <sup>k</sup>*

*x d* λ

= ± ; 2 1 cos cos

β

angle ϕ relative to horizontal axis in the plane of hologram (**Fig 23a.b**). Size and shape of pixels may also change. The shape influences the fill factor of the whole hologram. For example, the fill factor of rectangular pixels can reach 100% and for round ones it is only 80%. In the former case, consequently, the diffraction efficiency of the hologram will be about 20% higher than in the latter. It is possible to modulate diffracted light intensity by changing the size of pixels or exposure time during optical recording.

The calculation task is to define the condition that a particular pixel of a hologram would direct a given spectral part of diffracted light towards the observer. The observer can see diffracted light from each element of a hologram only at a certain angle; and the total perception of diffraction of all elements makes a visual effect that corresponds to the initial graphic design. It is obvious that the main factors for reconstruction of the whole hologram are orientation and period of diffraction grating in each pixel, as well as position of an observer and the light source during the reconstruction. In these conditions, the holographic image will correspond to the initial graphic design. If the positions of an observer and the light source are defined, period *d* and orientation angle ϕ (Fig. 23a) of the grating completely determine the conditions when an incident white light can be diffracted with the given spectral region towards the observer. In this case, the color of the hologram element corresponds to the color of a dot in the initial graphic image.

Fig. 23. **a)** Parameters of diffraction grating. **b)** An image element formation by diffraction pixel.

The coordinate system XYZ is chosen in such a way that the hologram is lying in the plane XY, and the Z axis directed to the observer from the center of the hologram (Fig. 23b). Assume that the observer's eye is on the Z axis at the distance of about 30 cm. The light source illuminating the hologram has a continuous spectrum in the visible region, and the direction of the incident radiation is given by the unit vector <sup>1</sup> *k* . We must define the parameters (,) *d* ϕ of the diffractive pixel with the coordinates ( , ) *x y* , which the observer can see in the color with the wavelength λ<sup>0</sup> . If the light with the wavelength λ falls on a flat diffraction grating along the unit vector 1 1 11 *k* = (cos , cos , cos ) α β γ the direction of diffraction, which is defined by the unit vector 2 2 22 *k* = (cos , cos , cos ) α β γ , can be determined by the formulae (Yaotang et al., 1998):

pixels may also change. The shape influences the fill factor of the whole hologram. For example, the fill factor of rectangular pixels can reach 100% and for round ones it is only 80%. In the former case, consequently, the diffraction efficiency of the hologram will be about 20% higher than in the latter. It is possible to modulate diffracted light intensity by

The calculation task is to define the condition that a particular pixel of a hologram would direct a given spectral part of diffracted light towards the observer. The observer can see diffracted light from each element of a hologram only at a certain angle; and the total perception of diffraction of all elements makes a visual effect that corresponds to the initial graphic design. It is obvious that the main factors for reconstruction of the whole hologram are orientation and period of diffraction grating in each pixel, as well as position of an observer and the light source during the reconstruction. In these conditions, the holographic image will correspond to the initial graphic design. If the positions of an observer and the

completely determine the conditions when an incident white light can be diffracted with the given spectral region towards the observer. In this case, the color of the hologram element

Fig. 23. **a)** Parameters of diffraction grating. **b)** An image element formation by diffraction

The coordinate system XYZ is chosen in such a way that the hologram is lying in the plane XY, and the Z axis directed to the observer from the center of the hologram (Fig. 23b). Assume that the observer's eye is on the Z axis at the distance of about 30 cm. The light source illuminating the hologram has a continuous spectrum in the visible region, and the

of the diffractive pixel with the coordinates ( , ) *x y* , which the observer can

<sup>0</sup> . If the light with the wavelength

 β

α

γthe direction of

> β

, can be

α

direction of the incident radiation is given by the unit vector <sup>1</sup> *k*

diffraction grating along the unit vector 1 1 11 *k* = (cos , cos , cos )

λ

diffraction, which is defined by the unit vector 2 2 22 *k* = (cos , cos , cos )

changing the size of pixels or exposure time during optical recording.

light source are defined, period *d* and orientation angle

corresponds to the color of a dot in the initial graphic image.

relative to horizontal axis in the plane of hologram (**Fig 23a.b**). Size and shape of

ϕ

(Fig. 23a) of the grating

. We must define the

γ

falls on a flat

λ

angle ϕ

pixel.

parameters (,) *d*

ϕ

see in the color with the wavelength

determined by the formulae (Yaotang et al., 1998):

$$\cos\alpha\_2 = \cos\alpha\_1 \pm \frac{\lambda}{d\_x}; \; \cos\beta\_2 = \cos\beta\_1 \pm \frac{\lambda}{d\_y}; \; \cos\gamma\_2 = \sqrt{1 \cdot \cos^2\alpha\_2 - \cos^2\beta\_2} \tag{1}$$

We will make calculations in order to form image in the +1 order of diffraction. We take the position of a point source with continuous spectrum of light in the plane YZ. If the angle between the Z axis and direction from the light source to the hologram center isθ , then <sup>1</sup> *k* = − (0, sin ,cos ) θ θ , where the change of the Z-component of incedent light after reflection from the hologram surface is taken into account. The direction of the +1 diffraction order from the pixel with the coordinates ( , ) *x y* to the eye of the observer is defined by the

unit vector <sup>2</sup> (- ,- , ) *x z <sup>y</sup> <sup>k</sup> L LL* <sup>=</sup> , where 2 22 *L x* = ++ *<sup>y</sup> <sup>z</sup>* is the distance between the pixel and the eye of the observer. Taking into account the above mentioned conditions, the formulas in (1) will look like the following:

$$\cos\alpha\_2 = -\frac{\lambda\sin\varphi}{d};\ \cos\beta\_2 = -\sin\theta + \frac{\lambda\cos\varphi}{d};\ \cos\gamma\_2 = \sqrt{1-\cos^2\alpha\_2-\cos^2\beta\_2}\ . \tag{2}$$

For the central pixel of the hologram with coordinates (0, 0) and the color λ0 based on the formulas in (2), we obtain the following parameters of the diffraction grating: <sup>0</sup> <sup>0</sup> ( , ) ( , 0) ( , 0) sin *d d* λ ϕ θ = = , where <sup>0</sup> <sup>0</sup> sin *d* λ θ<sup>=</sup> .

Usually, in practical cases 0 0 θ = − 30 45 , for example, if <sup>0</sup> θ = 30 , the desired interval of spatial frequencies of diffraction gratings for all colors in spectral range 0 λ = − 440 650 *nm* will be <sup>1</sup> -1 <sup>0</sup> *<sup>f</sup> <sup>d</sup>* 1135 650 *mm* <sup>−</sup> == − . At the next step, we can define the parameters (,) *<sup>d</sup>* ϕ of a pixel with the coordinates ( , ) *x y* having the same visual characteristics (color, brightness) for the observer as in the case of the central pixel.

This requires solving a system of equations for the variables *d* and ϕ:

$$\cos\alpha\_2 = -\frac{\dot{\lambda}\_0 \sin\varphi}{d}; \cos\beta\_2 = -\sin\theta + \frac{\dot{\lambda}\_0 \cos\varphi}{d}; \cos\alpha\_2 = -\frac{x}{L}; \cos\beta\_2 = -\frac{y}{L} \tag{3}$$

From the equations in (3) we can find the formulas for calculating the parameters of diffraction gratings of the pixel:

$$\arg \mathfrak{g} = -\frac{\cos \alpha\_2}{\cos \beta\_2 + \sin \theta} \quad \Rightarrow \quad \mathfrak{q}(\mathbf{x}, \mathbf{y}) = \operatorname{arcctg} \left| \frac{\mathbf{x}}{L \sin \theta \cdot \mathbf{y}} \right| \tag{4}$$

$$d = -\frac{\mathcal{A}\_0 \sin \varphi}{\cos \alpha\_2} \implies d(\mathbf{x}, y) = \frac{d\_0 \ L \sin \theta}{\sqrt{\left(L \sin \theta \cdot y\right)^2 + \mathbf{x}^2}} \tag{5}$$

The size of security holograms is usually not bigger than 3x3 cm and corresponds to the conditions *z x* >> and *z* >> *y* ; in this case, the following simplified formulas can be used for calculations:

Digital Holographic Recording in Amorphous Chalcogenide Films 93

The developed and assembled optical devices for dot-matrix holographic and image-matrix recording have been successfully used for scientific purposes as well as for producing holograms for the protection and identification of industrial products and documents. Its compact dimensions, reliability and low cost price may be interesting for the needs of small

The possibilities of hologram recording on As-S-Se chalcogenide films have been studied. The obtained results show that the above mentioned chalcogenides may be successfully used in applied dot-matrix and image-matrix holography as an excellent alternative to organic photoresists for producing high-quality security holograms with high diffraction efficiency up to 65%. The increase in film sensitivity with increase of the exposure power density has been discovered. It makes the application of pulse

This research was partly supported by the ESF project "Starpdisciplinārās zinātniskās grupas izveidošana jaunu fluorescentu materiālu un metožu izstrādei un ieviešanai" Nr.

Pizzanelly D. (2004). The development of direct-write digital holography.

Teteris J. (2002). Holographic recording in amorphous chalcogenide semiconductor thin

Teteris J. (2003). Holographic recording in amorphous chalcogenide thin films. Current

Chih-Kung L., Wen-Jong W., Sheng-Lie Y., (2000) Optical configuration and color

Yaotang.L, Tianji W., Shining Y., Shining Y., Shichao Z. (1998). Theoretical and experimental

Frumar M., Cernosek Z., Jedelsky J., Frumarova B., Wagner T. (2001). Photoinduced changes

Ozols A.,Reinfelde M., Nordman O. (2001). Photoinduced Anisotrophy and holographic

Reinfelde M., Teteris J., Kuzmina I. (2003). Amorphous As-S-Se films for holographic

Kostyukevych S. Moskalenko N. (2001). Using non-organic resist based on As-S-Se

representation range of a variable-pitch dot matrix holographic printer. Applied

of structure and properties of amorphous binary and ternary chalcogenides, JOAM

chalcogenide glasses for combined optical/digital security devices, Semiconductor

**7. Conclusion** 

recording attractive.

**9. References** 

**8. Acknowledgments** 

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$$d(\mathbf{x}, y) = \frac{d\_0}{1 - \frac{y}{z \sin \theta}}, \text{ or the equivalent } f(\mathbf{x}, y) = \frac{1}{d(\mathbf{x}, y)} = f\_0 \left( 1 - \frac{y}{z \sin \theta} \right), \text{ where } f\_0 = d\_0^{-1} \text{ (6)}$$

and

$$\varphi(\mathbf{x}, \mathbf{y}) = \frac{\mathbf{x}}{z \sin \theta \cdot \mathbf{y}} \,\mathrm{\,\,\,\mathrm{\,} \,\mathrm{\,\,} \,\mathrm{\,\,}\,\tag{7}$$

with the distance from hologram to observer, as mentioned above, 30 *z cm* ≈ . In the extreme case, when *z* → ∞ (or 2 2 *x y* + → 0 ) equations (6 and 7) become quite simple:

$$d(\mathbf{x}, y) = d\_0 \quad \text{and} \quad \varphi(\mathbf{x}, y) = 0 \tag{8}$$

Equations (8) can be used for holograms of up to 1.5x1.5 cm in size, with holograms of larger sizes demonstrating a noticeable color distortion and irregularity of the brightness of pixels in the hologram area.

In the photos presented in Fig.24, one can see two holograms of the same size recorded on the basis of the same original image, a uniform background of a certain color. In case of Fig.24a, calculation was made according to the formulas in (8), and in case of Fig.24b, taking into account the formula in (7). It is clearly seen that in the first case, the brightness of pixels in the direction of observation strongly depends on the position of pixels in the hologram.

Fig. 24. Photos of a recovered image from a dot-matrix hologram recorded on As40S20Se40 photoresist. a) Orientation angle of diffraction grating of all pixels ϕ *= 0*  b) Orientation angle of diffraction grating depending on the position of pixels ϕ *= K x*

Calculation of pixel parameters for stereograms or kinematic effects is similar to the considered case. It is only necessary to take into account a new position of observation of the hologram that can be achieved by replacing the old variables (,) *x y* in formulas (4)-(7) by the new ones (,) *x xy y* −Δ −Δ , where ( , ) Δ Δ *x y* indicate displacement of the observer in the observation plane.
