Influence of the matrix structure of the modulator and detector on the optical spectrum analyzer output signal

In this article, we investigate the physical and mathematical model of a coherent optical spectrum analyzer (COSA), which uses a matrix light modulator and a matrix detector as input and output devices. This model allows to define distortions in the output signal of the spectrum analyzer and the error in determining the signal spatial frequency. The study of this model showed that form of the signal at the COSA’s output depends on the pixels sizes of modulator and detector matrices, as well as on the aberrations of the Fourier lens entrance pupil diameter. The output signal is a convolution of an ideal input signal spectrum with a discrete spatial transmission spectrum of the modulator, which is followed by convolution with a discrete sensitivity of the matrix detector. This means that the spectrum of the signal under investigation is distorted by the spatial spectrum of the modulator and the matrix structure of the matrix detector. An important feature of the signal is its independence from the phase shift, which is caused by the displacement of the modulator center relative to the optical axis of the spectrum analyzer. The output signal of COSA consists of an infinite number of diffraction maximum, each of which has three maximum, the distance between which is proportional to the spatial frequency of the test signal. The position (frequency) of the maximum is determined by the pixel size, and their width by the size of the modulator. Obtain the formulas for determining the spatial frequency of the test signal, which differ substantially from the traditional formula and depend on the position of the central and lateral maximum in the diffraction maximum. The error in measuring the frequency depends on the size of the detector pixel, focal length of the Fourier lens, and the modulator matrix size. Developed the method for determining the error in measuring the spatial frequency of a harmonic signal. The error is defined as the difference between the true frequency corresponding to the position of the center of the diffraction maximum and the measured frequency corresponding to the position of the pixel center which has the maximum signal.


Introduction
Optical methods of information processing have significant advantages in comparison with electronic systems, primarily due to the instant processing of two-dimensional arrays of information at the speed of light [1][2][3]. Most optical processors use coherent spectrum analyzers, which are designed to convert two-dimensional distribution of the field amplitude into the spatial spectrum of this distribution [4,5]. The efficiency limit of a coherent optical spectrum analyzers (COSA) depends on the spatial resolution and speed of input and output devices [6,7]. The matrix spatial light modulators (SLM) with transmission of pixels, which defined by the test signal, are applied as devices for input of signals in modern COSA to process optical signals in real time and increase the measurement accuracy [8,9]. The output signal of the device is registered with a matrix detector (MD) of light (such as digital camera or webcam) with further computer processing, which significantly extends functional capabilities of the COSA [9,10].
There are many monographs and articles, where the features of using SLM or matrix detector in optoelectronic systems are explored [10,11]. At the same time, there is a lack of scientific and technical information about the joint effect of the matrix structure of SLM and MD on the generalized characteristics of COSA.

Problem formulation
The purpose of the article is to develop physicmathematical model of coherent optical spectrum analyzer which has the space-time matrix light modulator and matrix detector. This model allows to determine the distortions in input signal and inaccuracies in defining spatial frequency of signal which is measured.
3 Physical and mathematical model of digital coherent spectrum analyzer A coherent optical spectrum analyzer classic scheme ( Fig. 1) consists of spatial light modulator, Fourier-lens and matrix detector [1,2]. The light modulator is located in the front focal plane of the Fourier-lens and has an amplitude transmission coefficient ( 1 , 1 ) which is determined by the matrix structure of the modulator and the video signal of the image. The modulator is illuminated by a plane monochromatic wave with the amplitude and forms the field distribution 1 ( 1 , 1 ) behind the modulator. The lens realizes a two-dimensional transformation of the amplitude of this field and forms in the back focal plane a two-dimensional spectrum˜1( , ) of the function 1 ( 1 , 1 ). The matrix detector converts the intensity of the field 3 . If the COSA is used in the spectral filtering system of the optical signal, i.e. when the spectrum of the signal˜1 ( , ) is multiplied by the transfer function of the filter ( , ) with the subsequent Fourier transform, the second SLM is placed in the back focal plane of the lens, the transmission of which is determined by the function ( , ). Let's consider the models of the separate components of a coherent spectrum analyzer.
The spatial light modulator (SLM) has a matrix structure. An amplitude transmission coefficient of it's pixels corresponds to an amplitude of the input (test) optical signal. Therefore, such devices allow to enter optical signals, which are varying in time and space, into the processor. The matrix structure of the SLM is × size and the period of × (Fig. 2). Each pixel has a transparent zone × . The amplitude coefficient of such a modulator in the absence of an input signal is determined by the function [11] where 1 0 , 1 0 are coordinates of the center of zero (central) pixel of the modulator with respect to the origin of the coordinate system 1 1 , which determine the displacement of the modulator matrix center relative to the COSA optical axis; In the formula (1) the expressions in square brackets determine the periodic structure of the matrix × . Let suppose an optical signal (a video signal) is input to the modulator and is normalized to . Then it is converted to the amplitude transmittance coefficient of the modulator ( 1 , 1 ).
Then the amplitude transmission coefficient of the modulator can be represented in the form After diffraction on the matrix structure light enters the entrance aperture of the lens.
The Fourier-lens is designed to form the spatial spectrum of the input optical signal. If SLM is located in the front focal plane 1 1 of the Fourier-lens and is illuminated by a plane wave with amplitude ( Fig. 1), then the distribution of the field amplitude 3 3 in back focal plane is described by the expression [1,2] where f is a focal length of the Fourier-lens.
Analysis of expression (3) shows that the complex amplitude of the light field in the back focal plane of the Fourier-lens, up to a constant factor / , is the spatial spectrum of the modulator amplitude transmission coefficient ( 1 , 1 ) with spatial frequencies: To model the Fourier-lens following characteristics were used: the focal length , entrance aperture diameter , point spread function (PSF) ℎ ( , ). For a diffraction-limited Fourier lens, its PSF is determined by the diameter of the entrance aperture and has the form [1,2] where is a pupil function. The matrix detector (MD) is used to record the intensity of the light field ( 3 , 3 ) = | ( 3 , 3 )| 2 in the focal plane of the Fourier-lens.
The signal at the MD output is determined by the convolution of the functions where is the spectral sensitivity of the detector at the wavelength of the laser radiation and ** is an operator of two-dimensional convolution.

accumulation time , sec.
Let's present the sensitivity of the MD ( 3 , 3 ) similarly to the modulator transmission function (1) where 1 0 , 1 0 are the coordinates of the center of zero (central) pixel of the MD relative to the origin of the coordinate system 3 3 , which determine the center of the modulator matrix relative to COSA optical axis. 4 The output signal of the matrix detector The distribution of the field amplitude in the back focal plane of the Fourier-lens, which is determined by integral (3), is valid for an ideal lens with an infinite entrance aperture. If the lens has PSF ℎ ( 3 , 3 ), then the real amplitude of the field ( 3 , 3 ) in the plane 3 , 3 is determined by the convolution of the functions Then the video signal at the MD output is determined by convolution (6), which can be represented as where {} is an operator of a two-dimensional Fourier transform and = | | 2 is an intensity of the laser beam which illuminates the modulator. After substituting functions (1), (5) and (7) in (9), we can determine the general equation for calculating the video signal at the output of the spectrum analyzer. For a preliminary analysis of the function (9) there are series of approximations: 1. For the diffraction-limited optical system of the Fourier-lens, the radius of the spread circle is equal to the radius of the Airy circle [1] If the diaphragm number of the lens is / = 2 and the wavelength of the laser radiation is = 0.63 m, then the diameter of the lens scattering circle 2 = 1, 5 m will be much smaller than the MD pixel size =7 m. In this case, the PSF (5) can be viewed as a point (delta function).
2. In order to simplify the mathematical transformations, we consider the one-dimensional case. Therefore, the functions (1) and (7) have the form Then expression (9) will have the form Let's define the spatial spectrum of the modulator transmission function where˜( ) = { ( 1 )} is a spectrum of the input signal and˜0 ( ) is modulator spatial transmission spectrum in the absence of an input signal, which is determined by the function [1,11] where sin c( ) = sin( )/ . Analysis of the function (15) shows that the diffracted beams from adjacent pixels will be amplified if the condition of the main maximum is fulfilled = , where = 0, ±1, . . . is a maximum number. Then the expression (15) for the -th maximum has the form [1] It follows from (15) and (16) that the spatial spectrum of such modulator is an infinite number of diffraction maximums which positions (frequencies) are determined by the pixel size . The width of diffraction maximums are determined by the size of the modulator. The amplitude of the field in the diffraction maximum is a complex function whose modulus equals is sinc (︁ )︁ and the phase is -4 1 0 . The maximum value of the amplitude is located in the central maximum, when = 0. With increasing of diffraction order , the amplitude of the maxima decreases. Taking into account expressions (14) and (15), the signal at the output of the detector is equal to Analysis of expression (17) shows that the signal at the spectrum analyzer output is a convolution of the ideal signal spectrum˜( ) with a discrete spatial transmission spectrum of the modulator˜0 , which is followed by convolution with discrete sensitivity of the matrix detector. This means that the spectrum of the test signal is distorted by the spatial spectrum of the modulator and by the matrix structure of the matrix detector. An important feature of the signal is its independence from the phase shift 4 1 0 , which is caused by the displacement of the modulator center from the spectrum analyzer optical axis. 5 The analysis of harmonic signal spectrum As an example of calculation of the MD output signal, let's consider the harmonic input signal, which is modeled by the function where is an amplitude of the signal and is a frequency of the signal.
The spatial spectrum of such signal is determined by the function [1] Let's substitute the function (19) into the expression (17) and use the filtering property of the delta function From the expression (20) it turns out that the signal at the MD output consists of an infinite number of diffraction maximums and each of them has three maximums displaced relatively to each other.
Let's determine the distance between the maximums and their width. An investigation of the function (20) shows that the positions of the central 3, The greatest amplitude has a diffraction maximum of the first order, when = 1. The width of this maximum 3 can be found from expression (20), when condition is fulfilled The solution of this equation describes width of the maximum Similar result can be obtained along the coordinate 3 . Thus, the diffraction maximum can be considered as a rectangle of size 3 × 3 , which is projected onto the matrix structure (7) of the detector.
The center of this rectangle has coordinates for the first-order central maximum (Fig. 3) 3, As a mathematical model of such maximum, we'll use illumination function where ,1,0 is an illumination at the center of the diffraction maximum.
Taking into account the functions (7) and (28), the signal at the MD output for the one-dimensional case is equal to Fig. 4 shows the signal at the MD output, which is formed by the diffraction maximum for different sizes of the maximum and the pixel, when ≈ . In cases a and c, the calculation of the spatial frequency for the central maximum is carried out according to formula (21), which has the form , where is a number of pixels from the optical axis till to the diffraction maximum. In this case, the normalized amplitude of the maximum signal is 1.0 and 0.44, respectively.
In cases b and d, the spatial frequency is calculated by formula (21), which has the form , , ,0 = .
In this case, the normalized amplitude of the maximum signal is 0.5 and 0.33, respectively. Analysis of function (29) shows that the MD output signal, which corresponds to the diffraction maximum, depends on its position 3, ,1,0 on the detector array (matrix), as well as on the detector pixel sizes and the maximum width 3 .
After recovering the image [11], or corresponding processing of signals from individual pixels, it is possible to determine the position of the center of the maximum 3, ,1,0 , with sufficient accuracy, and determine the signal frequency from formulas (23) or (24). Using (20), we can also calculate the amplitude of the signal . Let's consider a technique of definition inaccuracies when measuring harmonic signal spatial frequency . We define it as where 0 is a true frequency, which corresponds to the position of the center of the diffraction maximum and is a measured frequency, which corresponds to the position of the center of the pixel, which has maximum signal.
In the 3 3 plane, expression (32) according to (4) has the form (Fig. 5a) leads to significant distortion of the spectrum of the test signal; 2.2. The signal at the spectrum analyzer output is a convolution of an ideal signal spectrum with a discrete spatial transmission spectrum of the modulator, which is followed by convolution with a discrete sensitivity of the matrix detector. This means that the spectrum of the test signal is distorted by the spatial spectrum of the modulator and the matrix structure of the detector; 2.3. The important feature of the signal is its independence from the phase shift, which is caused by the displacement of the modulator center from the spectrum analyzer optical axis; 2.4. The signal at the MD output consists of an infinite number of diffraction maximums. Each of them has three maximums and the distances between them are proportional to the spatial frequency of the test signal; 2.5. The formulas for determining the spatial frequency (23) or (24) differ substantially from the traditional formula (4) and depend on the position of the central and lateral maximums in the first-order diffraction maximum; 2.6. The inaccuracy in frequency measuring is determined by formula (34) and depends on the pixel size, the focal length of the Fourier-lens and the size of the modulator matrix.
3. It is expedient to study the influence of the Fourier-lens point spread function on the general characteristics of the digital COSA in the future.