ESTIMATION OF SINUSOIDAL OSCILLATORY MODULATION OF A RADIANT FLUX

V. I. Lukovnikov

(Presented by the scientific and technical seminar of the Department of Electrical Equipment, AEMF)

In [1] the spectrum of a radiant flux modulated by two gratings moving rectilinearly and uniformly toward one another is considered. The present article describes a more general case of oscillatory modulation, frequently encountered in optoelectronic devices [2], when the gratings (raster disks or masks) oscillate about a common axis relative to one another according to a sinusoidal law.

As is known, for oscillatory modulation by two rasters the expression for the modulated flux \(\Phi(t)\) can be written in the form:

\[ \Phi(t)=\iint\limits_{(S)} p(\rho,\varphi)\cdot f_1(t,\varphi)\cdot f_2(t,\varphi)\,dS, \tag{1} \]

where \(p(\rho,\varphi)\) is the density of the radiant flux at a point with polar coordinates \(\rho\varphi\) (the vertical diameter of the disks is taken as the polar axis) on an aperture of area \(S\) of arbitrary shape; \(f_1(t,\varphi)\) and \(f_2(t,\varphi)\) are the transmission functions of the first and second raster disks, respectively. Denoting, in accordance with the figure, \(\tau_i=\varphi_{\mathrm{wi}}/\varphi_{0i}\), \(N_i=2\pi/\varphi_{0i}\), where \(i=1,2\), and, for the law of motion of the disks, \(\varphi_i(t)=\varphi_{mi}\sin(\omega_{\mathrm{osc}\,i}t+\gamma_{\psi i})\), following the method of [1], one can obtain the expression for the modulation function \(h(t)\) of a uniformly distributed flux over a circular aperture of radius \(R\) with its center displaced by a distance \(L_a\) from the axis of the disks in the following form:

\[ \begin{aligned} h(t)={}&\Phi(t)\cdot \Phi_0^{-1} =\tau_1\tau_2+2\pi^{-2}\tau_2\sum_{m=1}^{\infty} m^{-1}\sin(\pi m\tau_1)\cdot\\ &\cdot Q_0(mN_1)\cdot \cos\{mN_1[\varphi_{m1}\sin(\omega_{\mathrm{col}1}t+\gamma_{\varphi1})-(\varphi_{\mathrm{n}1}-\varphi_{\mathrm{na}})]\}+\\ &+2\pi^{-2}\tau_1\sum_{n=1}^{\infty} n^{-1}\sin(\pi n\tau_2)\cdot Q_0(nN_2)\cdot \cos\{nN_2\cdot\\ &\cdot[\varphi_{m2}\sin(\omega_{\mathrm{col}2}t+\gamma_{\varphi2})-(\varphi_{\mathrm{na}}-\varphi_{\mathrm{n}2})]\}+2\pi^{-2}\cdot\\ &\cdot\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}(mn)^{-1}\sin(\pi m\tau_1)\cdot \sin(\pi n\tau_2)\cdot Q_0(mN_1+\\ &+nN_2)\cdot \cos\{mN_1[\varphi_{m1}\sin(\omega_{\mathrm{col}1}t+\gamma_{\varphi1})-(\varphi_{\mathrm{n}1}-\\ &-\varphi_{\mathrm{na}})]-nN_2[\varphi_{m2}\sin(\omega_{\mathrm{col}2}t+\gamma_{\varphi2})-(\varphi_{\mathrm{na}}-\varphi_{\mathrm{n}2})]\}+\\ &+2\pi^{-2}\cdot \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}(mn)^{-1}\sin(\pi m\tau_1)\sin(\pi n\tau_2)\cdot Q_0(mN_1-nN_2)\cdot\\ &\cdot \cos\{mN_1[\varphi_{m1}\sin(\omega_{\mathrm{col}1}t+\gamma_{\varphi1})-(\varphi_{\mathrm{n}1}-\varphi_{\mathrm{na}})]+nN_2\cdot\\ &\cdot[\varphi_{m2}\sin(\omega_{\mathrm{col}2}t+\gamma_{\varphi2})-(\varphi_{\mathrm{na}}-\varphi_{\mathrm{n}2})]\}, \end{aligned} \tag{2} \]

where

\[ \Phi_0=\iint_{(S)} p(\rho,\varphi)\,dS,\qquad Q_0(q)=2\pi\cdot \int_{0}^{1}\cos\{q\cdot \]

\[ \cdot \arcsin(\sqrt{x_0}\sin\varphi_{0a})\}\,x_0^{-0.5}(1-x_0)^{0.5}\,dx_0, \]

with

\[ q=mN_1;\ nN_2;\ mN_1+nN_2;\ mN_1-nN_2. \]

For a particular case, which is technically simple to implement, when

\[ \tau_1=\tau_2=0.5;\qquad N_1=N_2=N;\qquad \varphi_{m1}=\varphi_m;\qquad \omega_{\mathrm{col}1}=\omega_{\mathrm{col}};\qquad \gamma_{\varphi1}=\gamma_{\varphi};\qquad \varphi_{m2}=\varphi_{\mathrm{n}2}=0, \]

from relation (2) we obtain that the spectrum of the modulating function consists of a constant component:

\[ \begin{aligned} h_0={}&0.25+\pi^{-1}\sum_{m=1}^{\infty}(-1)^{m-1}(2m-1)^{-1}Q_0[(2m-1)N]\cdot \{\cos[N\cdot\\ &\cdot(2m-1)(\varphi_{\mathrm{n}1}-\varphi_{\mathrm{na}})]\cdot I_0[N(2m-1)\varphi_m]+\cos[N(2m-1)\varphi_{\mathrm{na}}]\}+\\ &+2\pi^{-2}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}(-1)^{m+n-2}[(2m-1)(2n-1)]^{-1}\cdot I_0[N(2m-1)\varphi_m]\cdot\\ &\cdot\{Q_0[2N(m+n-1)]\cos[N(2m-1)\varphi_{\mathrm{n}1}-2N(m+n-1)\varphi_{\mathrm{na}}]+\\ &+Q_0[2N(m-n)]\cdot \cos[N(2m-1)\varphi_{\mathrm{n}1}-2N(m-n)\varphi_{\mathrm{na}}]\} \end{aligned} \tag{3} \]

of odd harmonics

\[ h_{2i-1}\sin[(2i-1)(\omega_{\mathrm{col}}t+\gamma_{\varphi})] \]

with amplitude:

\[ \begin{aligned} h_{2i-1}={}&2\pi^{-1}\sum_{m=1}^{\infty}(-1)^{m-1}(2m-1)^{-1}Q_0[N(2m-1)]\cdot \sin[N(2m-1)\cdot\\ &\cdot(\varphi_{\mathrm{n}1}-\varphi_{\mathrm{na}})]\cdot I_{2i-1}[(2m-1)N\varphi_m]+4\pi^{-2}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}(-1)^{m+n-2}\cdot\\ &\cdot[(2m-1)(2n-1)]^{-1}\{Q_0[2N(m+n-1)]\cdot \sin[N(2m-1)\cdot \end{aligned} \]

\[ \cdot \varphi_{n1}\left|-2N(m+n-1)\varphi_{pa}\right|+Q_0[2N(m-n)]\cdot \sin\left|N\cdot(2m-1)\varphi_{n1}-2N(m-n)\varphi_{pa}\right|\, I_{2i-1}\left[(2m-1)N\varphi_m\right], \tag{4} \]

and also even harmonics

\[ h_{2i}\cos\left[2i(\omega_{\mathrm{col}}t-\gamma_\varphi)\right] \]

with amplitude:

\[ \begin{aligned} h_{2i}={}&2\pi^{-1}\sum_{m=1}^{\infty}(-1)^{m-1}(2m-1)^{-1}Q_0[(2m-1)N]\cdot \cos\left[(2m-1)N\cdot(\varphi_{n1}-\varphi_{pa})\right]\cdot I_{2i}\left[(2m-1)N\varphi_m\right] \\ &+4\pi^{-2}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}(-1)^m n^{-2}\cdot \left[(2m-1)(2n-1)\right]^{-1} \{Q_0[2N(m+n-1)]\cdot \cos\left[N(2m-1)\cdot\varphi_{n1}-2N(m+n-1)\varphi_{pa}\right] \\ &\qquad\qquad\qquad\qquad\qquad +Q_0[2N(m-n)]\cdot \cos\left[(2m-1)\cdot N\varphi_{n1}-2N(m-n)\varphi_{pa}\right]\}\cdot I_{2i}\left[(2m-1)N\varphi_m\right], \end{aligned} \tag{5} \]

where

\[ I_{2i-1}\left[(2m-1)N\varphi_m\right]\quad \text{and}\quad I_{2i}\left[(2m-1)N\varphi_m\right] \]

are Bessel functions of the first kind.

In the case where the center of the aperture coincides with the origin of coordinates (the center of the modulating disks), \(L_a=0\), relations (3), (4), (5) are transformed into the form:

\[ h_0=0.25+2\pi^{-2}\sum_{m=1}^{\infty}(2m-1)^{-2}I_0\left[(2m-1)N\varphi_m\right]\cdot \cos\left[(2m-1)N\varphi_{n1}\right], \tag{6} \]

\[ h_{2i-1}=4\pi^{-2}\sum_{m=1}^{\infty}(2m-1)^{-2} \sin\left[(2m-1)N\varphi_{n1}\right]\cdot I_{2i-1}\left[(2m-1)N\varphi_m\right], \tag{7} \]

\[ h_{2i}=4\pi^{-2}\sum_{m=1}^{\infty}(2m-1)^{-2} \cos\left[(2m-1)N\varphi_{n1}\right]\cdot I_{2i}\left[(2m-1)N\varphi_m\right]. \tag{8} \]

Calculation by relations (3)--(8) presents no fundamental difficulties; let us note only that the integral \(Q_0'(q)=\frac{1}{2\pi}Q_0(q)\) is tabulated and, according to [3], can be computed in the form of the sum

\[ Q_0'(q)\simeq \sum_{k=1}^{5} A_k'\cos\left[q\cdot \arcsin\left(\sqrt{x_k}\sin\varphi_{oa}\right)\right]. \tag{9} \]

Restricting the series to the first four terms gives an error of about \(0.1\%\). The table gives the values of \(A_k'\), \(x_k\), \(\sqrt{x_k}\) for the case under consideration.

Table 1

The calculations carried out for the spectra \(h(t)\) for various \(\varphi_m^0=\varphi_m/\varphi_{01}\) and \(\varphi_{n1}^0=\varphi_{n1}/\varphi_{01}\) made it possible to obtain a number of conclusions:

1. The “assortment” of harmonics in the spectrum is the greater, the larger \(\varphi_m^0\) is and the closer \(\varphi_{h1}^0\) is to \(0.1250\).

2. The number of the harmonic whose amplitude in the spectrum is the largest increases with increasing \(\varphi_m^0\).

3. The shape \(h(t)\) is the closer to sinusoidal, the smaller \(\varphi_m^0\) is and the closer \(\varphi_{h1}^0\) is to \(0.2500\).

These conclusions are valid for the case in which the aperture coincides with the origin of coordinates (relations (6), (7), and (8)). If, however, the center of the aperture is located at the periphery of the modulating disks (relations (3), (4), and (5)), then items 1 and 2 of the conclusions remain in force and, in addition, the following appear:

1. If the angular size of the light spot on the disks is small, \(\varphi_{0a}^0 \leqslant 0.5\), then for \(\varphi_{ha}^0 \leqslant 0.25 \div 0.5\,\varphi_{0a}^0\) modulation does not occur, since the spot is “lost.”

2. In the case \(0.25 \div 0.5\,\varphi_{0a}^0 \leqslant \varphi_{ha}^0 \leqslant 0.25 + 0.5\,\varphi_{0a}^0\), for a small spot \(\varphi_{0a}^0 \leqslant 0.5\), when \(\varphi_{h1}^0 = 0\), the amplitude of the harmonics changes with variation of \(\varphi_{ha}^0\) almost according to a linear law.

3. When \(\varphi_{h1}^0 \ne 0\), characteristic extrema appear in the curve \(h(t)\), which are most clearly observed at \(\varphi_{h1}^0 = 0.25\).

References

1. Yu. V. Lyubimov, V. V. Dronov. Optical-Mechanical Industry, 3, 18, 1968.

2. P. I. Goskov, V. A. Kocheturov, V. P. Ivanchenkov, V. I. Lukovnikov. USSR Author’s Certificate No. 241029, OIPOTZ, 13, 94, 1969.

3. V. I. Krylov, V. V. Lugin, A. A. Yanovich. Tables for the Numerical Integration of Functions with Power Singularities \(\displaystyle \int_0^1 x^3(1-x)^\alpha f(x)\, \(dx\). Academy of Sciences of the BSSR, Minsk, 1963.