CC BY
20
УДК 681.758
Influence of the Matrix Structure of the Modulator and Detector on the Optical Spectrum
Analyzer Output Signal
Kolobrodov, V. H., Tymchik, G. S.: Mykytenko, V. I., Kolobrodov, M. S., Lutsiuk, M, M,
National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute"
E-mail: tfiermo@ukr.net
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.
Key words: digital optoelectronic spectrum analyzer; matrix light modulator; matrix detector; spatial spectrum of the image
1 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 [1-3]. 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 mid 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.
2 Problem formulation
The purpose of the article is to develop physic-mathematical model of coherent optical spectrum analyzer which has the space-time matrix fight modulator mid 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 (tci,го) 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 Vp and forms the field distribution V\{xity\) 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 Vi (vx, uy) of the function Vi(xi,yi). The matrix detector converts the intensity of the field /3 (.г'з, Ул) into video signal уз). If the COS A is used in the spectral filtering system of the optical signal, i.e. when the spectrum of the signal V) {vx, vy) is multiplied by the transfer function of the filter Hf{vx, vy) 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 vy ).
Fourier lens
input signal is determined by the function (11]
t™u (aq,!/i) = jrcct ^ *
comb (-
m \
#1 ^lmO\
К
w„
/si-ttlmoVn
vm J 4 Xm )\ J
У1 У ’ \
-comb
< rcct I l V «
(y\ - y^no \ (y\ -mmo\l 1 _
\ wm ) \ vm JJj
('У1 )i (1)
)
у i Virno
where ximQ,yimo are coordinates of the center of zero (central) pixel of the modulator with respect to the origin of the coordinate system x 1 y\, which determine the displacement of the modulator matrix center relative to the COSA optical axis;
rect(a) - rect-function, rcct (z) -cornb(z) - comb-function, comb
j 1, when \z\ < 1/2 ( 0, when \z\ > 1/2
(=0 = £^-00 *(*-")-
Fig. 2. A model of the matrix spatial light modulator: (a) is the matrix; (b) is the center pixel
I11 the formula (1) the expressions in square brackets determine the periodic structure of the matrix Xm x Ym.
Let suppose an optical signal (a video signal) is input to the modulator and is normalized to usn(xltyi) = us (:cb yi) fus.max (:tb yi). Then it is converted to the amplitude transmittance coefficient of the modulator ts (oq, j/i).
Then the amplitude transmission coefficient of the modulator can be represented in the form
Fig. 1. Digital coherent optical spectrum analyzer scheme
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 Xm x Ym size and the period of Vm x Wm (Fig. 2). Each pixel has a transparent zone vm x wm. The amplitude coefficient of such a modulator in the absence of ail
tm (qji) = tm0 {xi,yi)te (xuyi). (2)
After diffraction on the matrix structure fight 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 aqjq of the Fourier-lens and is illuminated by a plane wave with amplitude Vp (Fig. 1), then the distribution of the field amplitude £3-1/3 in back focal plane is described by the expression (1,2]
v f f30
V (ЖЗ,Ы = -Щ j J tm. (*1, Vl)*
*е-7 3т(-з Ч+КУ1) dxjdyu (3)
where / is a focal length of the Fourier-1 ens.
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 Vp/jXf, is the spatial spectrum of the modulator amplitude transmission coefficient tm (xi, yi) with spatial frequencies:
(4)
_ £3 _ Уз_
^ " A/’ Uy ~ Af To model the Fourier-lens following characteristics were used: the focal length /, entrance aperture diameter Dp, point spread function (PSF) h0(x,y). For a diffract ion-limited Fourier lens, its PSF is determined by the diameter of the entrance aperture and has the form [1,2]
'h, (x3, y:i) = jj P„{Xfvx, Xfvy)-
■ e-D<^+y^y)dl/xdl/yt (5)
where P0 is a pupil function.
The matrix detector (MD) is used to record the intensity of the light field I{xz,yz) = |V (:гз,Уз)|2 in the focal plane of the Fourier-lens.
The signal at the MD output is determined by the convolution of the functions
ud (хз,Уз) = 1(хз,Уа) * *Ро(хз,уз), (6)
where Rd(:с'з>Уз) is the spectral sensitivity of the detector at the wavelength A up of the laser radiation and ** is an operator of two-dimensional convolution. For modeling MD we will use following features:
1. matrix format po x qo\
2. pixel size Vo x Wo, rn.kni2;
3. detector matrix structure period Vo x WD, mkm2;
4. spectral sensitivity Ro, V/(lx ■ s);
5. accumulation time f,, sec.
Let’s present the sensitivity of the MD Яп(хз,уз) similarly to the modulator transmission function (1)
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 htl {хз,уз), then the real amplitude of the field VT (.cj, y:}) in the plane £з, уз is determined by the convolution of the functions
У г (^з, Уз) = V (ж3, уз) * *h0 {xz, y3). (8)
Then the video signal at the MD output is determined by convolution (6), which can be represented as
ud (хз,уз) =
= j|V (ж3,уз) * К (х3,уз)|2 * Д^(ж3,'Уз)| =
= -^Гр { [F C*Mh)}* h° (жз, Уз)]2*
^(23,Уз)} = {tma (xipyi)ts (a: 1, yi)} *
* М^з/уз)]2 * Яо(я3,Уз)}> (9)
* R
where Tj} is an operator of a two-dimensional Fourier transform mid Ip = \VP\2 is mi 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 [ I ]
Ге
l,22Ayf.
Up
(10)
Rd (хз,Уз) = Rd jrect *
1 u(X‘Z X3D3 \ , / X3 -x3do\1 1
* kcorab {-^^)rcct /x
= Rnx (хз) Rny (уз) , (7)
where ^"1 do,У1 do are the coordinates of the center of zero (central) pixel of the MD relative to the origin of the coordinate system хзуз, which determine the center of the modulator matrix relative to COSA optical axis.
If the diaphragm number of the lens is f/D = 2 and the wavelength of the laser radiation is A up =
0.63 fmpm, then the diameter of the lens scattering circle 2?’£ = 1, fqiupm will be much smaller than the MD pixel size Vo =7 pupni, 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
tmOn
(u’i) = rect
/ xi — £l)np\
\ vm J
1 (Xi-Xim0\ . ЛТ1
лСсошЬ A——Jroc*
Rdx (£3) = Rd\ rect
'x3 — X30Q \ v vn
)
1
VD
fx3 — X3Dq\
(Чь~)
Then expression (9) will have the form
nD (-тз) = {V (.t3)|2 * ffopfo)} =
= {xi)ts (^)}|2*Ло(а;з)} ■
(12)
(13)
Let’s define the spatial spectrum of the modulator transmission function tm(:Ei) in the presence of the input signal ts (x 1)
fo 1 (yx) = fo, = F (fm (a-l)} =
{f mO (‘fo)fo(^'l)} i‘mO ( * fo (^b,) 1 ( f 4.)
where fo (i/x) = F {tm (,7q)} is a spectrum of the input signal and tmo{i/x) is modulator spatial transmission spectrum in the absence of an input signal, which is determined by the function [1,11]
folder (Цс) — fo {fonOir 1
% 1 mO A
3? 1 mO ^
—comb I * I rcct
* TTl V *7
" F{roM
f ~ 34 mt) A 1 1 =
l Xm )\ f
-J2lT Itaoli,
= Umsincfum/y^e { C() 1 ПО I lj>, // ,- i ] *' A",,; .‘-‘illC I -V;., //_, )
= vmsmc{vmvx)e-ji™1™0''* ■
sin (гг Vfo иx)
} =
. (15)
is 1’”Хт71 sine (vm-rfo } and the phase is - 4 л-ж,m(J-rfo. The maximum value of the amplitude is located in the central maximum, when i = 0, With increasing of diffraction order i, the amplitude of the maxima decreases. Taking into account expressions (14) and (15), the signal at the output of the detector is equal to
uu (жз) =
д2у2 { |foi.0rc (,J*) * fo (^д:)|
_ h f
A2/2 \ .
. sin(7rXm^) .
umsmc(nm^) . ----г- * fo (Ux)
П 2
sui(nVmyx)
*7fD(x3)}. (17)
Analysis of expression (17) shows that the signal at the spectrum analyzer output is a convolution of the ideal signal spectrum ts (ux) with a discrete spatial transmission spectrum of the modulator im3i, 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 47rximorr-, which
'm
is caused by the displacement of the modulator center from the spectrum analyzer optical axis.
5 The analysis of harmonic signal spectrum
where sin cfo) = sin(jrz)/7rz.
Analysis of the function (15) shows that the diffracted beams from adjacent pixels wall be amplified if the condition of the main maximum is fulfilled Vmyx = i, where i = 0, ±1,... is a maximum number. Then the expression (15) for the г-th maximum has the form [Ij
As an example of calculation of the MD output signal, let’s consider the harmonic input signal, which is modeled by the function
ts (a‘i) = ^ + fottcos(27wsa:i), (18)
t
7/1 (Jx
—
a ^ \ ^m
-JWTXimO— 1 ——
’77/ / ’7П
(16)
where fs(J is an amplitude of the signal and n, is a frequency of the signal.
The spatial spectrum of such signal is determined by the function [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 Vfo. The width of diffraction maximums are determined by the size X7n of the modulator. The amplitude of the field in the diffraction maximum is a complex function whose modulus equals
fo iyx) = (vx) + ^6 (vx - i;s) + ^5{ух- vs).
(19)
Let’s substitute the function (19) into the expression (17) and use the filtering property of the delta function
UD (ж3) =
_ 2P
A2/2
H,nainc(ym^) . ----Г*
Sin(7ryml/;r)
'<*Ю t
n2
+ -f5(vx ~ ъ) + -fS(ux - Va)
*Rd(x 3)
t
_ h f [1
A2/2 \ [2
. Sin (irAml/*)
,vmsrnc(vmvx) , ----—+
2 sm
. -so . r ( M sm \^Xm ("s “ "s)] ,
+ — Umsinc [vm [vx - Vs)\ , . ,------yj" +
sm [7rv;n (v* - zaJJ
+ ^-t)msinc [t'm (>* + iy.4J]
sin [rfw (i/x + i/д)]12 sin [ttV,,, (i/;,; + //s )]
■ (20)
From the expression (20) it turns out that the signal at the MD output consists of an infinite number of diffraction maxi mums and each of them has three maxi mums displaced relatively to each other.
Let’s determine the distance between the maxi-mums and their width. An investigation of the function (20) shows that the positions of the central x3:maT:i,o and lateral maxi mums in the г -th diffracti-
on order can be found from the conditions:
14гКг,таг,1,0 —
t r
I'm : I — 1 \
А/
гЛ /
— T7 i (2f)
km
The solution of this equation describes width of the maximum
5vx
8x3
A /
~\r i => 8x3
2Xf
X.~
(26)
Similar result can be obtained along the coordinate 2/3-
Thus, the diffraction maximum can be considered as a rectangle of size <hr3 x Sy3, 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)
^З.тах>1,0* VSymax,!,^
Xf Л/
vrm ’ v;„ •
(27)
„ Уз1 L X D *
i 'd ' L i ДЯз J ■ i 8y3 r
Уз,тих,1,<- 1
^3
Fig. 3. The position of the diffraction maximum on ttie detector array
As a mathematical model of such maximum, we’ll use illumination function
and
fm {Ух.тах,1,± 1 ^ — Ц ^
=k Vm I ’ ’ ’ ± vs\ = г; =S>
^ A/ ^ у -F ■ (22)
From the expressions (21) and (22), two methods for determining the spatial frequency iq,of the test signal could be suggested:
E {хз,Уз) — Fmaiii.o‘
, { ^3.1гииЛ)0\
'rGC V Ы ) '
, I ^ ^3,772(1.1,1,0 \ /Лй\
■”Ч----------si,------]• (28)
where Emtis 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
1. /'s
^3,7f»w,2,+ l 3'3,т/гая:1г, —1
2A7 ;
(23)
2. v,
■^3iTnax,i,0 — 1
А/
(24)
The greatest amplitude has a diffraction maximum of the first order, when
i = 1. The width of this maximum 8x3 can be found from expression (20), when condition is fulfilled
sin {-Xmvx) sin (7rVr„,//3.)
«Олш,1,0 (жз) = £(жз) * Rn (жз) =
^3 Д^3,тддг,1^0 А
5x3 )
* Ri
(:
= Е-тахЛ,0 rect [ -
ж3 ~ А
{rcct ( —----— I *
\ VD J
1 , / Ж3 - хзо(1 А
_сот„ (——j
(хз- Хзт \
rcct {-хГ~ )
(29)
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 Vu « vu.
a
C
пДп
Fig. 4. The shape of the signal from the diffraction maximum as a function of its size and position on the MD: (a), (b) - Sx3 < VDi (c), (d) - 5x3 = 1, bVD
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
VpN + 0,5Vb
А/
(30)
where N 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
;/:i.С
VpN
А/
(31)
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 жз.таа;.1,о on the detector array (matrix), as well as on the detector Vo pixel sizes and the maximum width 6x3,
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 агз.то:Ел,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 tsa.
Let’s consider a technique of definition inaccuracies when measuring harmonic signal spatial frequency 5i/„n. We define it as
— 'Ум '’r - (32)
where i^o is a true frequency, which corresponds to the position of the center of the diffraction maximum and vxm is a measured frequency, which corresponds to the position of the center of the pixel, which has maximum signal.
In the x3y3 plane, expression (32) according to (4) has the form (Fig. 5a)
bx3m = £30 (33)
According to (29), the output signal of the pixel of the MD uo.max (£3) is a convolution between the area of the diffraction maximum and the area of the pixel (Fig. 5, a).
From the graph of the function uo,max (■%) , shown in Fig. 5, f>, we find the maximum displacement between the centers of the diffraction maximum and the pixel at which the signal has a maximum value: Лс'з = {VD - 6x3) /2.
In the region of spatial frequencies, the quantity Sx3 determines the measurement error of the spatial frequency
Fig. 5. Determination of the measurement error of spatial frequency: (a) is a position of the pixel and diffraction maximum in the plane x3y3; (b) is an amplitude of the pixel signal as a function of the position of the diffraction maximum.
6 Conclusions
1. Modern optical information processing systems, including coherent optical spectrum analyzers, use matrix space-time light modulators as an optical signal (image) input devices. The output signal is registered by matrix detector. However, in this case there are large distortions of the input signal caused by the matrix structures of the modulator and the detector. At the same time, there is lack of scientific and technical literature about distortions of input signals in such processors.
2. Physical and mathematical model of COSA, which uses a matrix modulator and detector, was developed. It allowed to get a general expression for the signal at the spectrum analyzer output. The study of it showed that
2.1. The type of signal at the MD output depends on the pixels size of the modulator mid detector matrices, as well as on the aberrations mid the diameter of the entrance aperture of the Fourier-lens. All this
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 COS A in the future.
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Вплив модулятора i фотоприймача на вих1дний сигнал оптичного спектро-анал!затора
Колобродое В. Г., Тимчик Г. С., Никитенко В. L, Колобродое М. СЛуцюк М. М.
У стати дос.'Иджусться заиршюнована физика-мате матична модель когерентного оптичного спектро-ана.шзатора (КОС), тцо використовуе в якосн пристроУв введения та виведення сигнал1в матричного модулятора свггла i приймача пипромi нюван н я. Ця модель дозволяе назначит спогворення у шшдному сигнал! спектроана-jiiзатора i похибки у визначенш просторовоУ частоти досл1джуваного сигналу. Дослщження шеУ модел1 показало, що вид сигналу на виход! КОС залежить в1д розмь pin тксел!в матрицв модулятора i приймача, а також в^д аберац1й г щамегра вх1дноУ 31нищ Фур’е-об’ектива, Сигнал представ ляс згорхку 1деалъного спектра входного сигналу з дискретним просторовим спектром пропускания модулятора з нодалыною згорткою з дискретною чутлинцгпо матричного приймача пи 11 anvi н -ома II нм. Це означае, що спектр дос.щджуваного сигналу спотворю-еться просторовим спектром модулятора i матричною структурою приймача випромшювання. Важливою осо-бливю'хю сиг налу е його незалежшсть вщ фазового зсуву, я кий обумовлений змпценням центру модулятора щодо оптичноУ oei спектроанал1затора. Вихгдний сигнал КОС складаеться з нескжченного числа дифра-ктцйних максим ум in, кожен з яких мае три максимуми, В1дстань м1ж якими пропортцйна просторовш частот! дослщжуваного сигналу. Положения (частота) максиму лив визначаеться розм1ром nine ели, a Ух ширина -розм1ром модулятора. Отримано формули для визна-чення просторовоУ частоти дос.щджуваного сиг налу, як! суттево вЬцйзниютьси шд традшцйноУ формули i зале-жать в/д положения центрального i бокових максиму mlr в дифракщйному максимум!. Похибка втирювання частоти залежитв в!д po3Mipy ткоеля приймача, фокусноУ В1дстан1 Фур’е-об’ектива i розм1ру матршц модулятора. Розроблено методику визначення похибки вим1рюван-ня просторовоУ частоти гармоншного сигналу. Похибка визнат1аеться як р1зииця Mi ж ктинною частотою, що вщ-пошдае положению центра дифракцшного максимуму, i втпряноУ частотою, що вудпов/дак положению центра шкеели, який мае максимальний сигнал.
К л пег слова: цифровий оптико-електронний спе-ктроанал1затор; матричний модулятор свггла; матри-чний приймач випромi нюван н я; просторовий спектр зо-браження
Влияние модулятора и фотоприемника на выходной сигнал оптического спектроанализатора
Колобродов В. Г., Тимчик Г. С., Микитенко В. И., Колобродив Н. С., Луцюк Н. М.
D статье исследуется предложенная физико-матем ати чес ка я модель коге рентного оптического спектроанализагора (КОС), использующего в качестве устройств ввода и вывода сигналов матричный модулятор света и приемник излучения* Однако при этом возникают большие искажения входного сигнала матричными структурами модулятора и приемника* Предложенная модель позволяет определить искажения в выходном сигнале спектроанализатора и погрешности в определении пространственной частоты исследуемого сигнала. Рассмотрены модели основных составляющих КОС: матричного пространственного модулятора света, Фурье-объектива и матричного приемника излучения. Исследование этой модели КОС показало, что вид сигнала на выходе спектроанализатора зависит от размеров пикселов матриц модулятора и приемника, а также от аберраций и диаметра входного зрачка Фурье-объектива* Сигнал представляет собой свертку идеального спектра входного сигнала с дискретным просггрансггвенным спектром пропускания модулятора с по следу' ютцей сверткой с дискретной чувствите.льностыо матричного приемника излучения. Это означает, что спектр исследуемого сигнала искажается пространственным спектром модулятора и матричной структурой
приемника излучения. Важной особенностью сигнала является его независимость от фазового сдвига, который обусловлен смещением центра мо^лячюра относите льно оптическ ой оси one ктроанал изатора. Выходной сигнал КОС состоит из бесконечного числа дифракционных максимумов, каждый из которых имеет три максимума, расстояние между которыми пропорционально пространственной частоте исследуемого сигнала. Положение (частота) максимумов определяется размером пиксела, а их ширина - размером модулятора. Получены формулы для определения пространственной частоты исследуемого сигнала, которые существенно отличаются от традиционной формулы и зависят от положения централъного и боковых максимумов в дифракционном максимуме. Погрешность измерения частоты зависит от размера пиксела приемника, фокусного расстояния Фурье-объектива и размера матрицы модулятора. Разработана методика определения погрешности измерения пространственной частоты гармонического сигнала. Погрешность определяется как разность между истинной частотой, соответствующей положению центра дифракционного максимума, и измеренной частотой, соответствующей положению центра пиксела, который имеет максимальный сигнал.
Клю ч евы е ело еа: ци ф рово й oi гг и ко- э л е ктр о н н ы й спектроанализатор; матричный модулятор света; матричный приемник излучения; пространственный спектр изображения
