Statistic Model of Homodyne Acousto-Optic Spectrum Analyzer Statistical Model of a Homodyne Acousto-Optic Spectrum Analyzer

Introduction. Interferometric schemes of acousto-optic spectrum analyzers were intended for increasing their dynamic range. The application of these schemes was assumed to provide a twofold increase in the dynamic range expressed in decibels. This article theoretically proves the impossibility of achieving this aim. Aim. To analyze the noise characteristics of a homodyne acousto-optic spectrum analyzer (HAOSA), as well as to estimate its signal-to-noise ratio and dynamic range. Materials and methods. A mathematical model describing the HAOSA work was compiled. This model consid-ers the formation of quadrature components for obtaining an amplitude spectrum of an input signal, as well as takes into account the shot and readout noise. Results. In comparison with an acousto-optical power spectrum analyzer, the application of an interferometric scheme does not provide a twofold increase in the dynamic range. The achieved increase in the dynamic range did not exceed the level of 1.35 dB. Constant illumination led to a significant increase in the self-noise of the spectrum analyzer due to the shot noise, compared to which the thermal noise and the readout noise became insignificant. An expression for estimating the spurious-free dynamic range was obtained, with its value being primarily determined by acousto-optic interaction nonlinearity. Under standard parameters of analyzer blocks, the spurious-free dynamic range is shown to cover a single-signal dynamic range. An expression for estimating the signal-to-noise ratio was derived. Conclusion. The single-signal dynamic range of a homodyne acousto-optic spectrum analyzer is determined primarily by the photodetector saturation charge. When developing such systems, the question of an optimal ratio of these parameters should be solved, taking into account the light source power, the diffraction efficiency of the acousto-optical modulator and the photodetector saturation charge. The developed statistical HAOSA model provides a more accurate estimation of the dynamic range with an error of 1 dB. Statistic Model of Homodyne Acousto-Optic Spectrum Analyzer Statistic Model of Homodyne Acousto-Optic Spectrum Analyzer


Introduction.
In comparison with conventional acousto-optic spectrum analyzers based on spatial integration [1,2], homodyne acousto-optic spectrum analyzers (HAOSA) present interest in terms of having a potentially twofold dynamic range [3][4][5]. HAOSA are composed of ( Fig. 1) a source of monochromatic radiation 1, a collimating lens 2, a twochannel acousto-optical modulator (AOM) 3, the channels of which receive the analyzed signal ( ) s t and the reference signal ( ) , r t a spherical lens 4, and a photodetector (PD) 5.
A spectrum analyzer with an instantaneous PD filter is described in [3], which present little practical interest, since the registration of spatial array distribution requires a line of photodiodes, the output signal of which requires its own processing path. With the advent of matrix PDs with accumulation and numerous cells, a HAOSA scheme with a CCD PD was proposed [6], and a theoretical estimate of the attainable dynamic range was given. At the same time, the authors of [3][4][5] omitted the fact that, in order to obtain the amplitude spectrum of an input signal ( ) s t , it is necessary to additionally form the quadrature component of the spectrum. Moreover, the empirical values of the dynamic range differed from the theoretical estimate by 4 dB only, thus being far from a twofold increase [6].
Statistical HAOSA model. This article proposes a statistical HAOSA model with a PD with accumulation, which takes into account the formation of quadrature components of the spectrum. On the basis of this model, the estimation of an attainable dynamic range is given.
The HAOSA single-signal dynamic range is defined as max min 10 lg , s s P DR P where max s P and min s P are the maximum and minimum powers of the input harmonic signal, respectively, at which the operability of the device is ensured. The upper limit of the range is determined by two independent phenomena: the nonlinearity of the acousto-optic interaction and the saturation of the PD upon charge accumulation under the influence of radiation. The lower limit is determined by the minimum SNR (signal-to-noise ratio) required by the device. The main source of HAOSA noise is the PD, which is in turn characterized by two noise types: dependent and independent of the signal level. The latter include thermal noise, which manifests itself as both a dark charge in PD cells and the noise of the electrical circuit forming the current at the PD output.
The presence of photon noise leads to the fact that the charge accumulated by the PD cell under the influence of external radiation is a random variable obeying Poisson statistics. The average number of photoelectrons e n is determined as are the instantaneous spectra of the analyzed and reference signals, respectively; * is a symbol of complex pairing. Quadrature components can be formed using one of the methods proposed in [9].
In (3), the third term is alternating in sign. For the correct formation of the spectrum of the input signal, the uniform charge on the PD, which corresponds to the energy spectrum of the reference signal ( ) This distribution is not equal to zero and should be taken into account when determining the average distribution in the presence of a signal.
Following similar reasoning, it can be shown that, in the presence of an input signal, the signal at the HAOSA output obeys the Rice distribution: σ is the approximate value of the dispersion of the distributions of the quadrature components; ν is the parameter; ( ) 0 J ⋅ is a modified Bessel function of the first kind of zero order.
The dispersion of the distributions of quadrature components is estimated as where Wr Q and Ws Q are the charge components determined by the energies of the reference and analyzed signals, respectively.
The parameter ν is equal to: is the value of the useful, informational part of the charge in the cell. The average value of the signal at the HAOSA output, determined for the Rice distribution: J ⋅ is the modified Bessel function of the first kind of the first order. The dispersion of the HAOSA output signal is Then the SNR at the HAOSA output is It is not possible to solve (5) relative to s Q ; therefore, the lower boundary of the dynamic range min output SNR can be determined as the limit taking into account the smallness of ν: Following mathematical transformations, a square equation relative to 2 ν is obtained, the positive root of which and (4) Single-signal dynamic range. The ratio of (7) to (6) defines a single-signal dynamic range limited by the capabilities of the PD: Let us evaluate DR on max Q is presented in Fig. 2.
Signal-to-noise ratio. Calculations according to (5) give an estimate of the SNR of the HAOSA output signal for various values of s Q (Fig. 3). To assess the radiation intensity I, forming in where  is Planck's constant; ω is the frequency of the light wave; e is the electron charge.
Then the radiation intensity in the reference channel, which will provide a charge max where λ is the wavelength of light; B θ is the Bragg angle for the reference channel AOM; 2 M is the diffraction quality of the AOM material; sw P is the sound wave power; l is the length of the acousto- Multiplying the left and right sides of (9) by the longitudinal sectional area of the acoustic beam, let us turn to the calculation of powers and determine the radiation level required for exposure to the reference and signal channels:  (Fig. 4). When assessing the laser radiation power las P , it is necessary to take into account the losses associated with the inefficiency of beam focusing, light scattering in the optical path, etc. Assuming the laser utilization factor equal to 1%, las 100 mW. P ≈ In this mode, the single-signal dynamic range of the HAOSA is determined exclusively by the capabilities of the PD and is calculated according to (8).
Two-signal dynamic range. Let us evaluate the level of intermodulation distortions of the third order. In a HAOSA, the nonlinearity determining the twosignal dynamic range ( In this expression, the first term on the left determines the linear component of the signal, and the second is the third-order intermodulation products. To determine the lower boundary of the two-signal dynamic range in (12), it is necessary to leave the first term on the left and substitute s Q on the right with min s Q , which is defined according to (6).

DR
does not depend on max , Q and is determined by the extent and duration of accumulation of the signal spectrum in the PD. It was previously mentioned that the necessary charge can be formed by both achieving the required level of the input signal and using the power of the laser and efficiently. In other words, by increasing inc , s P it is possible to lower the level of the input signal, while maintaining the charge. In this case, a more linear character of the acousto-optical interaction corresponds to lower values of inc s P (see Fig. 4). Fig. 5 shows the dependence of be provided by both a semiconductor [13,14] and a gas laser [13,15]. A longer accumulation period corresponds to a larger charge under the constant power at the AOM input, which also extends the two-signal dynamic range. Thus, the duration of accumulation can also be used to expand the two-signal dynamic range.
Conclusion. When substituting the parameters specified in [6], the developed statistical HAOSA model with a PD with accumulation provides a more accurate estimate of the dynamic range with an error of 1 dB compared to 4 dB obtained by the authors. The model also demonstrates the impossibility of a twofold increase (in decibels) in the dynamic range for interferometric circuits compared to power spectrum analyzers. This is determined by the necessity to provide a relatively high level of reference illumination in the spectral plane, which significantly increases the HAOSA intrinsic noise level.  Radioelectronics. 2020, vol. 23, no. 1, pp. 52-62