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In this paper, we propose and demonstrate the synthetic aperture imaging by using spatial modulation diversity technology with stochastic parallel gradient descent (SPGD) algorithm. Instead of creating diversity images by means of focus adjustments, the technology, proposed in this paper, creates diversity images by modulating the transmittance of individual sub-aperture of multi-aperture system, respectively. Specifically, spatial modulation is realized by switching off the transmittance of each sub-aperture with electrical shutters, alternately. Based on these multi diversity images, SPGD algorithm is used for adaptively optimizing the coefficients of Zernike polynomials to reconstruct the real phase distortions of multi-aperture system and to restore the near-diffraction-limited image of object. Numerical simulation and experimental results show that this technology can be used for joint estimation of both pupil aberrations and an high resolution image of the object, successfully. The technology proposed in this paper can have wide applications in segmented and multi-aperture imaging systems.
© 2015 Optical Society of America
1. Introduction
The resolution of a diffraction-limited imaging system is proportional to its pupil diameter [1]. Modern ground-based and space-based telescopes are nearing the limits on the size and weight of monolithic primary mirrors, hindering light collecting efficiency and imaging resolution. In response, synthetic aperture imaging has grown out of the quest for higher angular resolutions in astronomy [2–4]. For synthetic aperture imaging system, sub-apertures must be aligned to very tight tolerances in order to achieve high resolution. Actuation is usually needed to maintain equivalent optical path distances (OPD) between sub-apertures. So it is necessary to measure the states of sub-apertures for applying appropriate corrections to actuators.
There exist many techniques for measuring states of sub-apertures, such as laser interferometry, Shack-Hartmann wavefront sensors, image-based wavefront sensing techniques and etc. Because frequent recalibration is needed, laser interferometry is not a practical method of wavefront sensing for synthetic aperture imaging. Shack-Hartmann wavefront sensors generally do not work well for extended objects or for high complexity wavefronts. Image-based wavefront sensing techniques, which include phase retrieval and phase diversity, allow for an estimate of pupil phase to be made directly from images produced by the system [5]. A trade-off is that Image-based wavefront sensing techniques are typically computationally intensive. Compared to the phase retrieval algorithms, phase diversity algorithms do not require the object to be known and can estimate the object in addition to the system phase. With very little extra optical hardware, phase diversity can simultaneously recover information about the object and system aberrations such as sub-aperture misalignments [6]. Phase diversity was first proposed as a method of wavefront sensing by Gonsalves and has been studied and developed extensively [7–12]. Conventional phase diversity is achieved by introducing a global focus error on a series of two or more defocused images. As an alternative to conventional focus diversity, M. R. Bolcar and J. R. Fienup reported an sub-aperture piston phase diversity, which utilize sub-aperture piston phase as the diversity function.
In this paper, we propose and demonstrate a new form of phase diversity associated with multi-aperture telescope systems by using spatial modulation on its each sub-aperture in proper order. Specifically, the spatial modulation is realized by switching off the transmittance of each sub-aperture with electrical shutters, alternately. Based on the knowledge of diversity, stochastic parallel gradient descent (SPGD) algorithm adaptively optimizes the coefficients of Zernike polynomials to reconstruct the phase distortion and restore the near-diffraction-limited images of the object. This improved technology does not require multiple separate image planes, defocusing optics or high precision actuators. The cost and complexity of the system can be greatly reduced. The influences of actuator’s precision have also been greatly reduced. To the best of our knowledge, the sub-aperture diversity by using spatial modulation with electrical shutters proposed in this paper has never been reported.
This paper is organized as follows. In the second section, the working principle of spatial modulation diversity with SPGD algorithm are given. The third section reports the corresponding numerical analysis. The fourth section gives the experimental verification. In the fifth section, the conclusions are given.
2. Analysis of the theory
Fig. 1 shows a configuration of synthetic aperture imaging system (two sub-apertures). Each sub-aperture is equipped with an electrical shutter, respectively. The fast steering mirrors are used to maintain equivalent path lengths between two sub-apertures. Compared with conventional phase diversity technology, spatial modulation diversity technology, which is proposed in this paper, use electrical shutters to modulate the complex pupil function of multi-apertures, alternately. In the procedure of spatial modulation, electrical shutters are switched off in proper order and corresponding images are captured by the CCD camera.
Based on the imaging principle, intensity distribution of the captured image can be modeled as [1,13]:
where is the kth captured image, is intensity distribution of the object, is noise in the kth image, * denotes a convolution, is the kth intensity point spread function (PSF). The intensity PSF is magnitude squared of coherent impulse response.where is a Fresnel-like transform of the pupil and can be expressed as [1,13]:where are pupil plane coordinates. ,and are elements of ABCD ray-transfer matrix that relates pupil plane to image plane for the kth diversity image., which is the total complex pupil function, is given by a sum over sub-aperture functions.where Q is the number of sub-apertures, is the pupil function of sub-aperture q, and is the known diversity contribution to the phase on sub-aperture q by using electrical shutters, alternately. can be expressed as Eq. (5).is the unknown phase error, which can be represented as the combination of a series of M Zernike polynomials.where Zm(u,v) is the mth order Zernike polynomial, αm is the Zernike expansion coefficient, which is also the control parameter. The Zernike polynomials, Zernike polynomials ordering and normalization constant are defined and shown in Table 1 of [14]. To implement the sub-aperture spatial modulation diversity, a series of images would be taken, each with a different amount of diversity by using electrical shutters, respectively. This known error adds to the unknown phase error of the sub-aperture to give the total complex pupil function. In Fourier domain, we define the following metric.where is a 2-D Fourier domain coordinate, K is the number of diversity images. ,and are 2-D Fourier transforms of , and . According to the treatment of Gonsalves and Fienup, error metric of Eq. (7) can be simplified to the following form.where is the coefficient matrix of Zernike polynomials, is the complex conjugate value of. It can be found that error metric is a function of coefficients . It is now possible to optimize Eq. (8) with respect to the set of phase parameters . According to analysis of Fienup, global minimum of corresponds to phase distribution, which is closed to the real phase distortion of the system. The corresponding Fourier spectrum of the restored near-diffraction-limited image of the object can be represented as follows.The SPGD algorithm is the steepest descent algorithm and is well suited for finding a global minimum or maximum of some objective error functions. According to the analysis of Vorontsov, the maximum convergence speed can be achieved if Zernike polynomials are chosen as a set of influence functions, which is regarded as modal control strategy in [15]. The quality metric E = E(fx, fy, α) is a function of Zernike polynomial expansion coefficients α = {α1, α2,…, αn}. The SPGD algorithm is used to optimize the Zernike polynomial coefficients to minimize the quality metric. Steps for SPGD algorithm can be briefly described as follows. Each iteration cycle works as follows:
- (1)Generate statistically independent random perturbations δα1, δα2,…, δαn, where all δαi are small values that are typically chosen as statistically independent variables having zero mean and equal variances, <δαi> = 0, <δαiδαi> = σ2δij where δij is the Kronecker symbol.
- (2)Apply the control signal with perturbations and get the metric function, E+ = E(α1 + δα1, α2 + δα2,…, αn + δαn), then apply the control signals with the opposite perturbations and get the metric function, E- = E(α1-δα1, α2-δα2,…, αn-δαn). Calculate the difference between two evaluations of the metric function δE = E+-E-.
- (3)Update the control signals, αi = αi + γδαiδE, i = 1, 2,…, n, where γ is the update gain. γ>0 and γ<0 according to the procedure of maximization and minimization, respectively. In this paper, γ<0 is chosen to minimize the error metric.
3. Numerical simulation
It is necessary to evaluate the performance of spatial modulation synthetic aperture imaging technique with SPGD algorithm. In this section, a series of digital numerical simulations are conducted to prove the validity. Simulations are performed with monochromatic light and the wavelength is 532nm. The synthetic imaging system being modeled is a four circular sub-apertures and laid as square configuration. The four aperture configuration is shown in Fig. 2(a). The USAF 1951 resolution test chart, which is shown in Fig. 2(b), is used to evaluate the performance of the imaging system.
Fig. 2 The four-aperture configuration and the USAF 1951 resolution test chart used in the numerical simulation, (a) the four-aperture configuration, (b) the USAF 1951 resolution test chart.
In numerical simulation, the diameter of each sub-aperture is 20mm. The distance between the USAF 1951 resolution test chart and the imaging lens is about 30m. The adjacent sub-aperture is closed to each other. After passing through the four aperture, imaging beams are focused by a imaging lens with 300mm focal length to form the image of the test chart. We use the first 9 Zernike polynomials excluding the first 3 Zernike polynomials (piston, x tilt, y tilt) to denote the phase distortion of imaging beams. In practical applications, the phase distortion of imaging beams is usually considered for representing the phase distortions generated by the transmission medium, such as atmospheric turbulence, eyes' crystalline lens and etc. The corresponding phase distribution is shown in Fig. 3(a) and the peak-to-valley (PV) value is about 12rad. Only the piston phase errors, which are shown in Fig. 3(b), are considered for representing the phase distribution of four apertures. By combining the phase distortions mentioned above, we can obtain the whole phase distortions of the synthetic aperture imaging system, which are shown in Fig. 3(c).
Fig. 3 Phase distributions of imaging beams and four sub-apertures telescopes (the unit is radian), (a) phase aberration of imaging beams, (b) piston errors of the four sub-apertures telescopes, (c) the whole phase distortion of the synthetic aperture imaging system.
According to phase distortions of imaging beams and pupil aberration of four aperture telescopes, we numerically simulated the image of the resolution test chart based on incoherent imaging theory [1, 13]. The corresponding blurred image is shown in Fig. 4. It is recognized that because of phase aberrations, the image of the resolution test chart is blurred seriously and the image of the inner resolution test chart cannot be distinguished. In this paper, the spatial modulation diversity technology is proposed to realize the synthetic aperture imaging. The spatial modulation are realized by modulating the transmittance of four apertures with electrical shutters in proper order. The spatial modulated images of the resolution test chart are shown in Figs. 5(a)-5(d), which are formed by four apertures with aperture 1 turned off (other apertures turned on), aperture 2 turned off (other apertures turned on), aperture 3 turned off (other apertures turned on), aperture 4 turned off (other apertures turned on), respectively. Using these modulated images, the technology proposed in this paper, cannot only be used to estimate the phase distortion of the imaging system, but also can be used for reconstructing the near-diffraction-limited image of the resolution test chart.
Fig. 5 Images of the resolution test chart obtained by using spatial modulation technology, (a) image formed by four sub-apertures with aperture 1 turned off, (b) image formed by four sub-apertures with aperture 2 turned off, (c) image formed by four sub-apertures with aperture 3 turned off, (d) image formed by four sub-apertures with aperture 4 turned off.
For spatial modulation diversity technology, SPGD algorithm is performed for a modal wavefront corrector with Zernike function {Zi(u,v)} as a set of influence functions. During the process, we use the first 12 Zernike polynomials for correcting the phase aberration of imaging beams and the piston phase errors of four sub-apertures. What need to be explained is that the first order Zernike polynomial (piston) with different centroid is only used for reconstructing piston phase errors of four sub-apertures. The SPGD controller continuously updates the phase distribution to minimize the metric function, which is shown in Eq. (8). The corresponding results are shown in Figs. 6(a) and 6(b). The relative value of metric function is defined as the ratio of metric function value in the process to the original metric function value. According to Figs. 6(a) and 6(b), it can be found that the whole phase distortion of the imaging system is achieved after 120 iterations with use of modal control. After 120 iterations, the SPGD algorithm converges to its minimum value. It is recognized that the restored phase distortion by using spatial modulation diversity technology is nearly the same as the loaded phase distortion, which is shown in Fig. 3(c). Fig. 7(a) shows the phase error, which is defined as the difference between the restored phase distortion and the loaded phase distortion. The root-mean-square (RMS) value of the phase error is about 0.3rad. Based on the phase distortion shown in Fig. 6(b), we can get the reconstructed near-diffraction-limited image of the resolution test chart, which is shown in Fig. 7(b). It can be found that the sharpness of the reconstructed near-diffraction-limited image has been greatly improved. The reconstructed near-diffraction-limited image is nearly the same as the original test chart, which is shown in Fig. 2(b).
Fig. 6 The results generated by the spatial modulation diversity technology, (a) corresponding evolutions of the relative value of metric function as the SPGD algorithm proceeds, (b) the restored phase distribution (the unit is radian).
Fig. 7 The results generated by the spatial modulation diversity technology, (a) the difference between the restored phase distortion and the loaded phase distortion, the RMS value is about 0.3rad, (b) the reconstructed near-diffraction-limited image of the resolution test chart.
Comparisons between the conventional focus adjustment phase diversity and the spatial modulation diversity have also been numerically simulated. For ease of comparison, the parameters, which are used in conventional focus adjustment phase diversity technology, are the same as that used in spatial modulation diversity technology. To perform the conventional focus adjustment phase diversity, three images differing by a known global phase aberration are used, one image with no diversity phase aberration and the other two images with equal but opposite amounts of defocus phase aberration. The defocus phase aberrations are shown in Figs. 8(a) and 8(b), respectively. The SPGD algorithm is also used to optimize the coefficients of the first 12 Zernike polynomials for reconstructing the phase aberration of imaging beams and to optimize the coefficient of the first order Zernike polynomial with different centroid for restoring the piston phase errors of four sub-apertures. The restored phase aberration and the corresponding phase error are shown in Figs. 9(a) and 9(b). It can be found that the restored phase aberration in Fig. 9(a) looks like the loaded phase aberration in Fig. 3(c). The restored phase error is defined as the difference between the restored phase distortion and the loaded phase distortion. The RMS value of the phase error is about 0.85rad, which is larger than that generated by spatial modulation diversity technology. In addition, numerical simulation results show that the conventional focus adjustment phase diversity technology is sensitive to the precision of focus adjustments. In order to obtain high phase restoration precision, we should know the focus adjustment phase distribution exactly. On the contrary, the spatial modulation diversity technology is realized by switching off the transmittance of each sub-aperture with electrical shutters, alternately. The influences of modulation precisions have been greatly reduced.
Fig. 8 Defocus phase aberration used in the conventional focus adjustment phase diversity for focus adjustment (the unit is radian.), (a) with positive value, (b) with negative value.
Fig. 9 The results generated by the conventional focus adjustment phase diversity, (a) the restored phase distribution, (b) the difference between the restored phase distribution and the load phase distortion, the RMS value is about 0.85rad.
4. Experimental analysis
We apply this technology to the problem of measuring the phase distortion of imaging system and reconstructing the near-diffraction-limited image to prove the validity. The experimental setup is shown schematically in Fig. 10. The phase screen (binary optics phase plate, with 30mm diameter.) is used to load phase distortion to the imaging beams. The diaphragm with two identical apertures is used to simulate the two sub-aperture telescopes. In order to obtain high imaging resolution in horizontal direction, two apertures are laid closed to each other and the diameter of each aperture is chosen as 15mm. So the phase screen can covers two whole apertures. After passing through phase screen and two-aperture diaphragm, the imaging beams of the target (resolution test chart with another distribution, shown below) are focused by Fourier imaging lens (with 50mm diameter and 300mm focal length) to form the image on the detector plane of CCD camera. The configuration of the target is also shown in Fig. 10. The LED light source (Thorlabs’ product, M530L3, with 530nm wavelength) is used to illuminate the resolution test chart. The target contains a frosted glass, which is used to smooth the intensity distribution of LED light. In the experiment, the distance between target and imaging lens is about 6m. CCD camera (PIKE F-032B) with 640 × 480 7.4 × 7.4um2 pixels is used to diagnose the images of the object. By monitoring and processing images from CCD camera, the technology is capable of adjusting phase distribution adaptively to restore the phase distortion of the imaging system and reconstruct the near-diffraction-limited image of the resolution test chart.
Fig. 10 Proof-of-principle experimental setup for proving the spatial modulation diversity technology.
The phase distribution loaded on two apertures by using the phase screen is shown in Fig. 11. The images of the resolution test target without and with being blurred by phase screen are shown in Figs. 12(a) and 12(b), respectively. It is found that because of the phase aberration, image of the resolution test chart is blurred seriously and cannot be distinguished. In the experiment, spatial modulation is realized by warding off the sub-aperture, in proper order. The spatial modulated images of the resolution test chart, which are captured by CCD camera, are shown in Figs. 13(a) and 13(b), respectively. Fig. 13(a) shows the image formed by the right aperture (Left aperture is warded off.). Fig. 13(b) shows the image formed by the left aperture (Right aperture is warded off.). Based on these modulated images Figs. 12(b), 13(a) and 13(b), the technology proposed in this paper is used for joint estimation of the phase distortion of the imaging system and the image of the resolution test chart.
Fig. 12 Images of the resolution test chart formed by two aperture diaphragm, (a) image of the resolution test chart without being blurred by phase distortion of the phase screen, (b) image of the resolution test chart blurred by phase distortion of the phase screen.
Fig. 13 Images of the resolution test chart obtained by using spatial modulation technology, (a) image formed by two sub-apertures with left sub-aperture turned off, (b) image formed by two sub-apertures with right sub-aperture turned off.
In the experiment, the SPGD algorithm is also performed for a modal wavefront corrector with Zernike function as a set of influence functions. During the process, the SPGD algorithm controller continuously updates the coefficients of the first 15 Zernike polynomials excluding the first order Zernike polynomial (piston) to fit the phase distortion for minimizing the metric function. It can be found that after 70 iterations, SPGD algorithm converges to its minimum value and the corresponding restored phase distribution is shown in Fig. 14(b). The processing time is related with the execution velocity of the hardware and iteration steps. On our personal computer (Parameters, CPU, Intel Xeon E3-1225 @3.2GHz and RAM, 4GB), the time consumed by one iteration step is about 0.5s. In this paper, the velocity is mainly restricted by the hardware of our computer. If we use FPGA, DSP or GPU, the velocity can be improved greatly and the imaging system can be a quasi real time system.
Fig. 14 Experimental results generated by spatial modulation diversity technology, (a) corresponding evolutions of the relative value of metric function as the SPGD algorithm proceeds, (b) phase distribution of the imaging system (the unit is raidan).
The phase error, which is defined as the difference between the loaded phase distortion shown in Fig. 11 and the restored phase distortion shown in Fig. 14(b), is shown in Fig. 15(a). The PV value and RMS value of the phase error are 0.78λ and 0.2λ, respectively. λ is the wavelength of LED light and is 530nm. After careful analysis, we find that the main component of the phase error is tilt errors. The reason for this phenomenon is that in the phase restoration, positions of the multi diversity images have not been considered. By using calibration, the tilted errors can be greatly reduced. The tilt error only have influences on the position of images, but have no influences on the quality of the images. The phase error excluding the tilt errors is shown in Fig. 15(b). The PV value and RMS value of the phase error excluding tilt error are 0.36λ and 0.13λ. These results show that the precision of the phase restoration is high.
Fig. 15 The phase errors between the restored phase distortion and loaded phase distortion (the unit is radian), (a) the phase error without excluding the tilted errors, the PV value is about 0.78λ, the RMS value is about 0.2λ, (b) the phase error with excluding the tilted errors, the PV value is about 0.36λ, the RMS value is about 0.13λ.
Based on the restored phase distortion shown in Fig. 14(b), we can get the reconstructed near-diffraction-limited image of the resolution test chart, which is shown in Fig. 16. It can be found that the sharpness of the reconstructed near-diffraction-limited image has been greatly improved and more details are obtained. The reconstructed near-diffraction-limited image is nearly the same as the original test chart which is shown in Fig. 12(a). In order to evaluate the performance, we use the following Roberts sharpness operator to estimate the quality of images shown in Figs. 12(a) and 16.
where is the input image, and are the number of pixels of the image. We calculate the Roberts sharpness operator of the reconstructed near-diffraction-limited image and the image without being blurred by phase distortion of the phase screen, respectively. Results show that the Roberts sharpness operator of Fig. 16 is about twice as much as that of Fig. 12(a). These results indicate that the technology proposed in this paper have not only restore the phase distortion of the phase screen, but also restore the phase distortion of the imaging systems.5. Conclusion
The synthetic aperture imaging by using spatial modulation diversity with SPGD algorithm has been proposed and demonstrated. The modulation is realized by switching off the transmittance of the sub-aperture with electrical shutters, alternately. The numerical simulation with four apertures and experiment analysis with two apertures are used to prove the validity of the technique. Results show that spatial modulation phase diversity can be used for joint estimation of both pupil aberrations and an image of the object successfully. Numerical simulation results show that the restored phase distribution is nearly the same as the phase distortion of the imaging system. The RMS value of the phase error between input phase distortion and restored phase distortion is about 0.3rad. Experimental results indicate that excluding the tilt errors, the RMS value of the phase error between the loaded phase distortion and restored phase distortion is about 0.13λ. The technology proposed in this paper cannot only restore the phase distortion of the phase screen, but also can restore the phase distortion of the imaging systems. The technology will have wide applications in segmented and multi-aperture imaging systems.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (No. 61205144), Research Project of National University of Defense Technology (No. JC13-07-01) and the key Laboratory of High Power Laser and Physics, CAS.
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