This article considers algorithm implemented in an optoelectronic circuit. The main particularity of this algorithm is implementation in the working circuit of a part consisting of laser on the level of hardware, grey-level optical modulator of light and a diffraction pattern camera in the Fourier plane. This design allows using a laser with any initial distribution of intensity and adjusting a phase function of calculated DOE exactly for this distribution in order to decrease an error in forming a diffraction pattern of the output distribution. For problems of synthesizing diffraction optical component (DOE) iterational (iterative) methods were developed and are used widely [1-11]. Their main advantage is that iterational algorithms prove to be more precise in comparison with other algorithms for DOE phase computation[6-10]. On the other hand, focusators computated with their help have irregular microrelief, which raises requirements for production technology of the components computated. Besides, the DOE computation using iterative algorithms requires significant expenses.
= (| ( , )|2 − | ( , )|2), minimizing functionality of amplitude deviation in a recovered image from a set value [1-3]: where | ( , )|и| ( , )| - is a set and calculated wave amplitude in the plane of spatial spectrum.
Let us take a coordinate descent algorithm as a basis for functionality minimizing algorithm (
Let us use coefficients obtained with two-dimension unary re-expression [4] as coordinates for the coordinate descent: ( , ) = ∑
∑ ( , ) ( , ) = ∑
∑ ( , ) 1 1 −1 −1 =0 =0 1−1 1−1 =0 =0 { −2 { 2 ( ( + + )} , )} .
Let us compile algorithm scheme:
1) computation of expansion factors for a given initial approximation of a recovered image with expansion formula (
image recovery using known expansion coordinates with an inverse transform (
−1 ∑ =0
∑ =−01 ( , ) ( , , , ), where ( , , , ) - is the null-space for forward transformation; ( , ) - size imaging matrix
× ; ( , )-transformed size imaging matrix 1 × 1 than dimensions of the initial data, which in its turn facilitates computational speed.
Recover y of an initial image is carried out by means of inverse transformation: ( , ) = ∑ 1−1 =0
∑ =10−1 ( , ) ( , , , ), where ( , , , ) - is the null-space for the forward transformation; inverse Fourier transforms:
One of possible forward and inverse transformations representations for images by size × may have form of forward and
Computer Modeling / S.K. Misievich, R.V. Skidanov
3. functionality recalculating (
At realization of this algorithm in an optoelectronic circuit, Fourier transform performance, which make a significant part of
computations is transferred to hardware component [5]. Realization of these calculations is carried out by means of spatial light
modulator, which modifies amplitude of illuminating beam. After this, the light passes through a collecting lens, forming a
distribution in focussing plane, which is recorded by the camera and serves as a basis for functionality recalculation (
Let us conduct a simulation experiment for a DOE phase function optimization using a worked out iterational algorithm.
Let us use a result received with the iterational algorithm as initial approximation of a phase. We conduct results of numerical experiments for images that describe amplitude and phase distributions with dimensions of 256x256 pixels . For acceleration of the algorithm work we employ radially-symmetrical phase and amplitude distributions as investigated.
A ring given at figure 11 is used as a reference distribution.
As a method for inaccuracy estimation, a mean square deviation was used:
√1 ∑( , )∈ [ ( , )− `( , )]2 = ,
1 ∑( , )∈ `( , ) where ( , ) - is the distribution of the intensity formed; `( , ) – reference distribution of intensity; – area of inaccuracy estimation; - space of the area .
For the calculated initial approximation, inaccuracy was 0.81. After implementation of iterational algorithm, the inaccuracy reduced to 0.48. Results of the algorithm performance a re given in figure 2.
(
b) He-Ne – helium-neon solid-state laser, OF – optical filters, CL1, CL2, CL3 – convex lens, SP – splitter, D – diaphragm, GOM – grey-level spatial optical modulator CRL OPTO, CAM – camera VSTT-252, M – rotating mirror, PC – personal computer
Fig. 5. Optical scheme used in the experiment.
Figure 6 shows a photography of the optoelectronic system in action.
Implementation of such scheme in an algorithm for computating DOE phase function is that the phase function corrected at each algorithm's iteration is brought off to the modulator, which functions as an actual DOE, then the camera records the diffraction pattern in the lens focussing plane. Further, the distribution obtained is processed by the computer. Discrepancy error between the reference and the obtained distribution is calculated and an operation of the phase correction occurs.
As an initial approximation of the intensity formed we shall employ DOE shown on figure 3a. After its forming on the optical scheme, the distribution shown on fig. 7 was received.
Such distribution possesses a significant peak related to re-reflection effect, which occurs in the modulator. Further, fig.8 shows the result of a programme deduction from the distribution, which describes this peak.
The calculated deviation of this intensity distribution from the reference one makes approx. 95%.
The result of the algorithm's work, which is implementedd in the optoelectronic circuit is given in fig. 9. The distribution corresponding it, as obtained by means of diminution from the central peak is given in fig. 10. The DOE phase function computated has the form represente din fig. 11.
The distribution obtain as a result of the algorithm performance has a more a more distinct ringed structure, which corresponds more to the set reference distribution. The calculated deviation of this image from the reference one makes 83%.
An essential disadvantage of such computation method for the DOE phase function is a low speed of the algorithm's work, which correlated to output of a frequently alternating phase distribution onto the modulator.
In work process, an algorithm for calculating the DOE phase function was elaborated, which is used for forming a radiallysymmetrical distribution in the Fourier plane of the convex lens. Such algorithm was implementedd on the computer for conducting a work simulation and convergence research, then transposed to the optoelectronic circuit.
The results of conducting a live experiment showed possibility of implementing this algorithm for calculating the diffraction optical components. The main advantage of the algorithm is its universality in relation to the intensity distribution of the illuminating beam . When calculating the DOE phase function, there is no need for taking into account laser intensity distribution, as it is counted on the hardware level, as a result of the algorithm's work the phase function is corrected exactly for the given distribution. Although, a low speed of the algorithm's work in the optoelectronic circuit hinders its implementation.
Such algorithm may be used in cases, when it is necessary to calculate a component that gives a more distinct image at the output, using a laser with a specific intensity distribution.
The work was funded by the Russian Federation 3.3025.2017/8.9 Ministry of Education and Science of state-assigned task No.