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Abstract
The spatial phase distortion caused by a rough target causes a decoherence effect which, in turn, produces system sensitivity degradation. The decoherence phenomenon is the primary problem that restricts the application of active optical heterodyne detection, e.g., synthetic aperture radar and long-range coherent laser detection and ranging. By establishing a one-to-one correspondence between the combination of array signals and the system signal-to-noise ratio (SNR), a scheme for spatial phase distortion correction based on the intelligent optimization algorithm is proposed in this paper. The calculation of phase adjustments for each array signal is transformed into an optimization problem for the combination of array signals, experiments are conducted using rough target heterodyne images, and the parallel genetic algorithm (PGA) is used to calculate the phase adjustment of each array element. The results show that the spatial random phase distortion is corrected effectively without prior knowledge, and the PGA achieves an excellent computational performance which, along the efficiency of the proposed technology, has wide-scale implications for the application of active heterodyne detection and optical coherent communication.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical heterodyne detection is an extension of the microwave heterodyne technology to the optical band [1–3]. The coherence of light waves and optical heterodyne detection have significantly improved with the invention and subsequent development of laser technology. In contrast to direct detection, which can only obtain information on the intensity of a signal, heterodyne detection can obtain information on the intensity, phase, and frequency of the signal under test from the intermediate frequency (IF) signal. Theoretical and experimental results show that the signal-to-noise ratio (SNR) of heterodyne detection can reach the quantum limit of the detector [4–6]. Accordingly, thanks to its prominent technical superiority in high sensitivity and suitable selectivity, laser heterodyne detection technology offers a prospect of wide-ranging applications in various aerospace science and technology fields, such as advanced remote sensing, geographical mapping, and space coherent optical communication [7,8].
Heterodyne detection is extremely sensitive to the wavefront matching state between the local oscillation (LO) and signal fields [2,3,6,9,10]. The system has the best SNR when the LO field of the detector surface is perfectly matched to that of the signal. However, in most cases, the roughness of the detection target surface is characterized by a statistical measure in terms of the optical wavelength. In general, the wavelength of the laser is in the order of micrometers, therefore the majority of detected targets are rough for light waves. Rough targets may modulate the spatial phase of the signal field and cause mismatches between the LO and signal beams. These mismatches, caused by the rough target, lead to a decoherence effect that results in system sensitivity degradation. Currently, the decoherence effect is a major challenge limiting the application of active heterodyne detection techniques. Therefore, it is crucial that the SNR is improved through the correction of the spatial phase distortion of the signal field.
Adaptive optic methods are a common and effective approach to compensate for wavefront aberrations in optical heterodyne detection [11–14]. The method first detects the wavefront aberration of the signal light using a wavefront sensor, and subsequently compensates for the wavefront aberration using a system consisting of deformation mirrors. However, flicker phenomena in complex environments can disrupt the light intensity and phase of the signal beam, generating phase discontinuity points. In recent years, sensorless adaptive optic technology based on phase-diverse or phase-retrieval technology has significantly simplified the system [15–19]. However, this technology utilizes a single intensity image for wavefront recovery which leads to phase blurring. Therefore, this method has strong restrictions on its application.
Another method to compensate for wavefront aberrations is the array detector technique, which uses an array of detectors as receivers of the heterodyne signal [20,21]. Each detector in the array corresponds to a part of the spatial random phase aberration that causes a random phase noise to be added to the signal output from each array element. The spatial phase aberration is compensated for by adding a defined amount of phase adjustment to the array element signals. Therefore, obtaining the exact phase adjustment of each array element signal is the central problem of the array detector method. In Ref. [22], the array detector heterodyne system is designed to calculate the phase adjustment of each array element signal given that the heterodyne signal is measurable. Furthermore, a method to correct the phase difference between array heterodyne signals is proposed [23,24]. The basic underlying idea is to take one of the array element signals as a reference, and then correct its phase difference with respect to the other array signals through a greedy or a traversal algorithm. However, because of the decoherence effect, the SNR of the system remains low even when the signal optical return optical power is strong. In this case, the above methods cannot accurately calculate the phase adjustments of the array element signals.
In this paper, it is proposed that the calculation of phase adjustments for each array signal is transformed into an optimization problem for the combination of array signals. Once the correspondence between the combination of the array signal phase adjustments and the fitness function has been established, iterations of the intelligent algorithm are used to determine the optimal phase adjustments for all array signals without prior knowledge. Subsequently, the array detector signals are simulated to verify the effectiveness of the different intelligent optimization algorithms used in the optimization problem for the combination of array signals. Thereafter, rough target heterodyne images are used as the target for experiments, and the parallel genetic algorithm (PGA) is used to calculate the phase adjustment of each array element signal to ensure the iteration speed and computational accuracy. The method proposed in this paper can effectively overcome the problem of decoherence in remote sensing applications, such as synthetic aperture radar and long-range coherent laser detection and ranging. Moreover, the proposed solution can also be applied to overcome the decoherence problem caused by atmospheric turbulence, which, in turn, benefits the application of coherent communication techniques.
2. Decoherence effect in heterodyne detection
Optical heterodyne detection is essentially optical frequency coherent detection that is based on the principle of mixing coherent LO light and incident signal light on the photosensitive surface, as shown in Fig. 1. The principle of heterodyne detection determines that this detection method has the advantages of high sensitivity and suitable selectivity. In general, the heterodyne signal can be expressed as a cosine function:
Fig. 1. Schematic diagram of heterodyne detection. RS represents the received signal; LO represents the local oscillator; BS represents the beam splitter; PD represents the photodetector.
The heterodyne SNR is the ratio of the effective coherent power to the total incoherent power and is an important standard for evaluating the system performance. Assuming that the polarization states of both beams are the same, the SNR can be expressed as [1]:
If the influence of amplitude modulation is not considered, both the LO and the signal have the same amplitude distribution over the active surface of the detector and the SNR can be written as:
The spatial phase distribution generates heterodyne signals at different positions of the detector, adding a uniformly distributed random phase noise [25]. When the detector receives sufficiently independent scattered phase units, then [3]:
Increasing the output signal power or IF amplifier gain coefficient is a common way to improve the SNR of the heterodyne system. However, Eq. (4) implies that the heterodyne signal cannot be obtained because of the speckle phase, regardless of the magnitude of the IF amplifier gain coefficient or the output signal power. This phenomenon is the decoherence effect in heterodyne detection.
If the influence of phase modulation is not considered, the LO and signal have the same phase distribution across the detector area. The SNR can be described as follows:
The amplitude matching state of the LO and signal over the detector active surface determines the SNR of the heterodyne system. According to the theory in Ref. [25,26], the amplitude of the target speckle satisfies the Rayleigh statistics when the independent units are sufficiently large. Therefore, based on the assumption that the amplitude distribution of the LO is uniform, the SNR can be expressed as:
Equation (6) indicates that the signal-to-noise ratio of the system is a fixed value when the amplitude distribution of the signal obeys a specific distribution. In this case, the heterodyne SNR can be improved by increasing the output signal power or IF amplifier gain coefficient.
3. Combinatorial optimization problem in heterodyne detection
As discussed in Section 2, the heterodyne SNR attains a maximum value when the signal beam matches the LO beam. Eliminating the spatial phase mismatch is extremely difficult compared with the amplitude mismatch, that has an adverse effect on the heterodyne detection system. The array detector method provides a novel solution to the problem of spatial phase distortion. Its principle is to ensure that all the array elements signal in-phase superimposed output, such that the adverse effect of spatial phase distortion on the system performance is eliminated. In the array, the spatial phase distortion can be uniformly split into several parts using the array detector. Each array element corresponds to one part of the spatial phase distortion. Generally, the noise signal is Gaussian white noise and does not lead to the accumulation of noise when multiple channels are superimposed. Consequently, the SNR of the system can be significantly improved by adjusting the output of the array signal for in-phase superposition. From theoretical and simulation study, it has been shown that the incremental SNR of an N × N array detector system is proportional to N [5,27]. However, determining the phase adjustment for the individual array elements and compensating the phase difference between the signals from different array elements are the central challenges of the array detector method.
Combinatorial optimization problems are a class of problems that seek extreme values in discrete states. Moreover, the signal output from the detector can be transformed into a discrete digital signal after being sampled by an analog-to-digital converter (ADC). Once the correspondence between the combination of the array signal phase adjustments and the fitness function has been established, the array detector method can be transformed into a combinatorial optimization problem to compute the extreme value of multiple discrete array signal superpositions. At present, intelligent optimization algorithm is a common tool to solve complex combinatorial optimization problems. Hence, the optimal phase adjustment combination of the array signals can be calculated iteratively using an intelligent optimization algorithm.
4. Combinatorial optimization problem of heterodyne signals
To model the combinatorial optimization problem for the array signals, we assume that the array element signal consists of the sum of the heterodyne signal Sij and the noise Nij, and that the random phase factor φij due to spatial phase distortion obeys a uniform distribution in the range [0,2π]. When the array signal is phase-adjusted, the detector output is as follows:
Equation (7) shows that different combinations of phase adjustments correspond to different signal outputs, or to different system SNR results. Therefore, if the system SNR is set as a fitness function, a one-to-one correspondence between the phase adjustment combination and the fitness function is achieved. The spectral range of the IF heterodyne signal in active detection applications is determined, while no noise accumulation is caused, when multiple channels are superimposed. Therefore, the spectral component values of the useful signals can also reflect the system SNR, and such a design can significantly reduce the number of operations during iterative calculations. Thus, the fitness function is expressed as follows:
Theoretically, the value of the fitness function is maximized when the array signals are accurately superimposed on the output. Accordingly, the problem of array signal combination optimization is to iteratively calculate, through the intelligent optimization algorithm, the combination of the phase adjustments that best fits the phase difference between the array signals.
In this study, the N × N array heterodyne system model was composed of N × N heterodyne signals with amplitudes 1/N2 of and with different phases. The IF heterodyne signal frequency was assumed to be 50 Hz, the sampling rate was 1 kHz, and the SNR of each array element signal was -10 dB. All noise signals are Gaussian white noise. The phases of array signals obey a uniform distribution in the range [0,2π]. Figure 2 shows the simulated outputs of individual array elements of a 3 × 3 array detector and the amplitudes of array signals were 1/9.
Fig. 2. Simulated outputs of the individual array elements with random phases of a 3 × 3 array detector.
The nine in-phase superimposed array heterodyne signal spectrum corresponding to an array detector size of 3 × 3, is shown in Fig. 3(a). The fitness function value is 1.004, and the SNR is -0.43 dB. Owing to the existence of noise, the value of the system fitness function is greater than the sum of the heterodyne amplitudes of the nine array signals. The useful IF heterodyne signal can be extracted from the total output of the array detector using an IF amplifier. An array detector with in-phase output can effectively improve the SNR of the heterodyne system. For comparison, the superimposed spectrum of the nine random phase array heterodyne signals is shown in Fig. 3(b). The fitness function value is 0.0664, and the SNR is -23 dB. This phenomenon may be due to the fact that the cosine heterodyne signals cancel each other out owing to random phases. Therefore, an array detector without phase adjustment may not improve the heterodyne system performance.
Fig. 3. Spectral analysis of the total output of a 3 × 3 array heterodyne system. (a) Nine array signals without phase difference. (b) Nine array signals with random phases.
Intelligent optimization algorithms usually consist of three parts: population generation, iterative operations, and judgment output. The intelligent optimization algorithm randomly initializes the individual within the feasible region and regulates the individual through the fitness function without prior knowledge. However, the computational power of various types of algorithms varies in different problems due to the different iterative strategies. The problem of falling into the local optimal solution owing to slow convergence or premature convergence in the operation of the algorithm will lead to unsatisfactory optimization results.
To compare the computational power of various types of intelligent optimization algorithms in the array signal combination optimization problem, we introduce array signals into different algorithmic models. These include genetic algorithms (GAs) that simulate the evolutionary mechanisms of organisms in nature, particle swarm optimization (PSO) algorithms that simulate the behavior of a flock of birds or a school of fish, differential evolution (DE) algorithms that simulate the competition and cooperation mechanisms between individuals within a group, seeker optimization algorithms (SOAs) that simulate the behavior of human social activities, gravitational search algorithms (GSAs) that simulate the model of gravity in physics, and artificial immune algorithms (AIAs) that simulate the biological immune behavior. The number of individuals in the population was 200, and the maximum number of iterations was 100. The algorithms continuously update the population through iteration and finally output the global optimal results.
Fig. 4 shows the resulting fitness function from the different intelligent optimization algorithms for the 3 × 3 array signals in Fig. 3(b). Each figure corresponds to an intelligent optimization algorithm that performs 100 independent operations on an identical dataset. In general, by establishing a combinatorial optimization model of array heterodyne signals, all intelligent optimization algorithms can effectively improve the SNR of the heterodyne system. Some of the results are even higher than the fitness function value of the in-phase output condition in Fig. 3(a), and the phase differences between the array signals are accurately compensated. However, the results of some algorithms, such as GSA and DE, fluctuate significantly.
Fig. 4. Resulting fitness function from six intelligent optimization algorithms for a 3 × 3 array signals model.
Accuracy and dispersion of the results are crucial evaluation factors. Table 1 shows the statistical results of the fitness function values shown in Fig. 4. The mean and median values reflect the accuracy of the results. The mean values of the fitness function from GA, PSO, SOA, and AIA are greater than 0.95, and those from the other two algorithms are less than 0.9. The standard deviation (S.D.) reflects the dispersion of the results. The results obtained by the GSA and DE algorithms have a large dispersion, and some results appear to be premature convergences. The dispersion of the results of the GA algorithm is minimal, and the stability of the GA algorithm is better than that of the other algorithms in the 3 × 3 array signal model.
Table 1. Statistical results of fitness functions for a 3 × 3 array signals model.
Computational cost is another important evaluation factor. The statistical results of the computation time in Table 1 are the average of 100 independent operations. In the experiments, all the algorithms were implemented in MATLAB R2018b on a PC with a 2.2 GHz CPU and 16 GB RAM. Even if the GA algorithm obtains the best performance in the 3 × 3 array signal model in terms of accuracy and dispersion, the computation time is not optimal. The optimization performance of the PSO algorithm is similar to that of the GA, but the time consumption is larger. The GSA derives in the lowest computational cost among the six algorithms owing to a simple iterative strategy. Simultaneously, the time consumption of the DE is large, although the computing power is not poor.
The number of feasible combinations of the phase adjustments increases exponentially with an increase in the number of array elements, and even results in a data explosion. This phenomenon significantly affects the computing power of intelligent optimization algorithms. Fig. 5 and 6 show the results of the fitness function from the different intelligent optimization algorithms for a 5 × 5 and 7 × 7 array systems, respectively. Each figure corresponds to an intelligent optimization algorithm that performs 100 independent operations on an identical dataset. Similar to the 3 × 3 array detector system, the SNR of each array element signal was also set to -10 dB.
Fig. 5. Resulting fitness function from six intelligent optimization algorithms for a 5 × 5 array signals model.
Fig. 6. Resulting fitness function from six intelligent optimization algorithms for a 7 × 7 array signals model.
Tables 2 and 3 show the statistical results of the fitness function values in Figs. 5 and 6, respectively. Compared with the data in Table 1, the mean value of the fitness functions calculated using each algorithm decreases, and the time consumption increases significantly. The GA algorithm obtains the best adaptability in the problem of array signal combinatorial optimization, and the increase in the number of array elements has negligible effect on the accuracy of the GA algorithm.
Table 2. Statistical results of fitness functions for a 5 × 5 array signals model.
Table 3. Statistical results of fitness functions for a 7 × 7 array signals model.
The parallel genetic algorithm (PGA) is an ideal solution for improving the iteration speed and computational accuracy. In this algorithm, the initial population is first divided into several subpopulations according to the number of processors available. Calculations are performed over each subpopulation independently, applying a GA algorithm per processor, thus taking advantage of processor parallelization to accelerate the calculation process. After a certain number of iterations, several individuals will be exchanged among subpopulations to introduce individuals from other populations and enrich the population diversity. On the other hand, owing to the isolation between subpopulations, the solution accuracy is improved and the possibility of premature convergence is reduced. Table 4 shows the statistical results of the fitness functions for a 10 × 10 array system. In the experiment, the number of independent processors was six, and the SNR of each array element signal is also set to -10 dB. GA, PGA, PSO, parallel PSO (PPSO), and parallel GSA (PGSA) algorithms are used for comparison. The PGSA provides the lowest time consumption but the worst optimization performance. This algorithm is based on the physical model of universal gravitation, such that individuals with small masses gradually approach those with large masses. This simple structure enables the algorithm to achieve excellent convergence speed. However, it also easily falls into a local optimal solution.
Table 4. Statistical results of fitness functions for a 10×10 array signals model.
The PGA obtains the best calculation performance and second-lowest computational cost among the five algorithms. In PGA, the isolation between subpopulations ensures the personality of the population, thus ensuring the diversity of each subpopulation and avoiding the occurrence of precocity. Simultaneously, after a certain evolution, the diffusion of excellent individuals among populations ensures the accuracy of the calculation. In terms of time consumption, under the same total population size, the average calculation time for 100 independent operations of the GA algorithm is 2662.3 s, while the average calculation time for 100 independent operations of the PGA algorithm is 194.70 s. It is evident that the PGA algorithm can simultaneously improve the iteration speed and computational accuracy. In our method, the phase difference between array signals can be corrected by PGSA when the array signal scale is small, while it is best to use the PGA algorithm to correct the phase difference between array signals when the array signal scale is large.
5. Experiment results
To evaluate the effectiveness of PGA for the combinatorial optimization problem of array heterodyne signals, a high-speed camera was utilized to record heterodyne signals. The active heterodyne detection system included a high-speed camera as illustrated in Fig. 7(a). The system was composed of a camera, a laser, two acoustic-optic modulators (AOMs), several splitters and reflectors. The wavelength of the continuous-wave laser (Verdi-II, Coherence, Co,. Ltd.) was 532 nm. The sampling rate of the high-speed camera (MEMRECAM HX-7, NAC, Co,. Ltd.) was 100 kfps. The heterodyne frequency was determined from the frequency shift difference between the AOM1 and AOM2. The heterodyne bandwidth was approximately 20 kHz given by the line width of the laser source.
Fig. 7. Scheme of heterodyne detection system. (a) Configuration of the system based on a high-speed camera. AOM, acoustic-optic modulator. (b) Schematic of the principle of obtaining 3 × 3 array heterodyne signals from a series of heterodyne images.
In the experiment, print paper was used as the detection target to introduce the spatial phase distortion of the signal light field. The optical power of the LO and signal beams received by the camera were adjusted to 0.07 μW and 0.35 μW, respectively. Fig. 7(b) presents a schematic of the procedure to obtain 3 × 3 array heterodyne signals from a series of heterodyne images. In synthesis, the heterodyne images recorded by the camera are first evenly divided in an arrange of 3 × 3 heterodyne images, and then the time-domain heterodyne signals are calculated by summing all the light intensity values of each heterodyne image in different array elements.
Fig. 8(a) shows examples of heterodyne images of print paper. Each picture contained 60 × 60 pixels, with a pixel size of 11 μm. The spectral analysis results are shown to the right of the heterodyne images. In some applications of active heterodyne detection, the frequency of the heterodyne signal is unknown, but its range can be determined. Hence, the fitness function in this section is set to the summation of the frequency components in the range of 10 kHz to 30 kHz. Then, the phase adjustment of each array element was calculated using the PGA.
Fig. 8. Examples of heterodyne images and their spectral analysis results. (a) Examples of heterodyne images of print paper. (b)–(d) examples of heterodyne images after phase adjustment through PGA for 3 × 3, 5 × 5, and 10 × 10 array detector systems, respectively. Rightmost column: spectral analysis of each group of heterodyne images.
Fig. 8(b)–8(d) are examples of heterodyne images and their spectral analysis results after phase adjustment processing through PGA for 3 × 3, 5 × 5, and 10 × 10 array detector systems, respectively. After phase adjustment processing, the periodic variation of the heterodyne signal can be seen from a continuous set of heterodyne images, in which the spatial random phase distortion is effectively corrected. On the right side of the heterodyne images in Fig. 8(b)–8(d) are the spectral analysis results. The heterodyne signals increased rapidly as the size of the detector element decreased. Conversely, no noise accumulation occurs when multiple heterodyne signals are superimposed. By establishing a one-to-one correspondence between the combination of the array signals and the fitness function, the SNR of the heterodyne can be effectively improved by employing PGA, even without prior knowledge. It provides a novel scheme to solve the decoherence problem of heterodyne signals in the application of active detection.
It could be observed that, despite the rapid development of high-speed camera technology, using high-speed cameras as signal receiving devices for heterodyne detection systems is still not practical. Instead, it is best to use a PIN photodiode array to apply the proposed technique in practice. Furthermore, one can use multi-computers for distributed calculation or transplant the PGA to a high-speed signal processor, such as a microcontroller unit and digital signal processor, to satisfy real-time processing requirements.
6. Conclusion
This paper presents a parallel array signal processing technology for spatial phase-distortion correction in heterodyne detection. By establishing a one-to-one correspondence between the combination of the array signals and the SNR of the system, six different iterative intelligent algorithms were used to determine the optimal phase adjustments for all array signals without prior knowledge. Finally, rough target heterodyne images were used as the target for the experiments, and the PGA was used to calculate the phase adjustment of each array element. Experimental results demonstrate that the proposed parallel array signal processing technology corrects spatial phase distortion and achieves excellent computational performance, overcoming the decoherence of heterodyne signals, and thereby potentiating the application of active heterodyne detection and optical coherent communication. However, the time consumption caused by iterative calculation is still inevitable. In recent years, deep-learning-based intelligent optimization algorithms have gradually become mainstream. Therefore, we will conduct further research based on deep learning to apply this technique in practice.
Funding
The S&T Program of Hebei under Grant (216Z1706G); National Key Research and Development Program of China (2019YFC1407904); National Natural Science Foundation of China (61575170).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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