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Cross-talk compensation of a spatial light modulator for iterative phase retrieval applications PIERRE GEMAYEL, BRUNO COLICCHIO, ALAIN DIETERLEN,*
AND
PIERRE AMBS
Laboratoire MIPS EA-2332, Université de Haute-Alsace, 12 rue des Frères Lumière—68093 MULHOUSE CEDEX, France *Corresponding author: [email protected] Received 29 October 2015; revised 18 December 2015; accepted 18 December 2015; posted 18 December 2015 (Doc. ID 252897); published 29 January 2016
Beam-propagation-based phase recovery approaches, also known as phase retrieval methods, retrieve the amplitude and the phase of arbitrary complex-valued fields. We present and experimentally demonstrate a simple and robust iterative method using a liquid crystal spatial light modulator located at an object diffraction plane. M random phase masks are applied between the object and the image sensor using the modulator, and then M diffraction patterns are collected in the Fourier plane. An iterative algorithm using these patterns and simulating the propagation of the light between the two planes allow us to recover the object wavefront. The use of this type of dynamic modulator makes the experimental setup simpler and more flexible. We need no a priori knowledge about the object field, and the convergence rate is high. Simulation results show that the method exhibits high immunity to noise and does not suffer any stagnation problem. However, experimental results have shown that the technique is sensitive to the cross talk of the modulator. We propose a method for compensating these modulator defects that are validated by experimental results. © 2016 Optical Society of America OCIS codes: (000.4430) Numerical approximation and analysis; (100.5070) Phase retrieval; (120.5060) Phase modulation; (230.6120) Spatial light modulators. http://dx.doi.org/10.1364/AO.55.000802
1. INTRODUCTION Both the amplitude and phase of a diffracted beam carry a part of the information about the diffracting object’s surface or its inner structure. While the amplitude may easily be derived from the square root of the intensity, the phase distribution, which carries most of the information, cannot be measured directly, and consequently its loss is more difficult to overcome. Three solutions to the phase problem exist: holography [1–4], wavefront sensing like the Shack–Hartmann sensor, and phase retrieval methods [5–7]. The latter are considered an inverse problem using an iterative algorithm alternating between the real domain and the frequency domain. These methods are based on the fact that the variation of the diffraction patterns along the direction of beam propagation may be used to extract the phase information of the object [5], so when the diffraction pattern is oversampled by a factor larger than 2, the phase information might be recovered using an iterative algorithm [8]. In addition, the progress in computer technologies as well in digital imaging devices makes the application of beam-propagationbased phase recovery approaches easier, especially in that these methods are relatively insensitive to noise, robust, have a simple experimental setup, and do not require the application of a 1559-128X/16/040802-09$15/0$15.00 © 2016 Optical Society of America
particular reference wave like holography. For these reasons, phase retrieval methods have known a lot of interest in various domains such as electron microscopy [9], x-ray diffraction imaging [10], material science, and astronomy. In 1972, Gerchberg and Saxton (GS) introduced the first widely accepted iterative method for phase retrieval based on projections [5]. The idea is that we can recover the phase if partial information about the object density as well as about the magnitude of the object Fourier transform are known. In 1982, Fienup proposed a number of iterative algorithms founded on various interpretations of the GS method [6] by using finite support and a nonnegativity constraint in object space instead of the magnitude of the object. The most basic method and the most used one is known as the hybrid input–output algorithm, which is closely related to the steepest-descent method, but it is more robust and converges much faster than the error-reduction algorithm [11], especially for real-valued nonnegative signals. However, it has been shown that recovering a complex object requires a sufficiently accurate hard support [7] or acquiring low resolution images [12]; otherwise these algorithms are not capable of successful reconstructions [13–15]. The loss of real and positive constraints for a complex valued object must be compensated
Research Article by the use of other constraints as done in several experimental strategies in order to overcome stagnation and nonconvergence problems. Faulkner and Rodenburg [16,17] proposed a method consisting on taking more than two diffraction patterns which correspond to different areas of the sample, using a moving aperture. Nugent et al. [18,19] proposed the use of diffraction data obtained with illumination modulated in both intensity and phase. Chen and Chen proposed to record several diffraction intensity maps through variable function orders in the fractional Fourier transform [20], or to record diffraction patterns while modifying the focal length of a lens function displayed on a phase mask [21]. Other methods also exist involving the modulation of the illumination wavelength [22] or recording intensities across two or more parallel planes connected through Fresnel or Fourier transforms [23–27]. In 2007, Zhang et al. suggested a method known as spread-spectrum phase retrieval (SSPR) inspired from the essence of the wavefront sensing technique [28]. The idea is to introduce a strong phase modulation into the object field using a phase mask. This way, any sampling of the Fresnel-recorded diffraction pattern contains a contribution of all points in the object, which will eliminate the effects of sensor noise and convergence stagnation. In the first place a phase plate is used to introduce the phase modulation, so before initiating the algorithm, more than three diffraction patterns are collected, as the plate is shifted transversely. Then, in order to make the experimental setup easier and more flexible, the phase plate was replaced by a liquid crystal spatial light modulator (LC-SLM) with which there is no need to modify the setup during acquisitions [29]. This SSPR approach has been applied in the interferometric evaluation of rotating smooth objects using a fixed phase diffuser plate with the speckle measurements taken at multiple axially displaced planes [30]. Recently, an SLM in a 4f -setup has also been used in conjunction with the multiple-plane SSPR technique [30] for the reconstruction of smooth object wavefront where the SLM displays a transfer function for free-space propagation [31]. One note is that experimental results obtained when an SLM is employed for the SSPR technique are inferior to the results obtained by using a phase plate, even if adjoining pixels were binned into a superpixel, and this is mainly due to the fringing field effect highly present in liquid crystal on silicon (LCOS) displays. This effect refers to the cross talk between neighboring pixels, and it appears when there is a significant phase jump. This effect leads to the broadening of the phase profile in a LC device [32,33]. In fact, phase retardation is not constant over the entire surface of the pixel, especially at the edges, and this can be considered as a spatial low pass filtering of the phase retardation displayed on a SLM with no pixel cross talk [34]. In this paper we present a method for phase retrieval based on SSPR. We collect diffraction patterns in the Fourier domain instead of the Fresnel domain, which is simpler because we just need to position the CCD in the focal plane of the lens without worrying about distances between components. The phase modulation is introduced by a LC-SLM located between the object and the CCD. After presenting simulation and experimental results, we discuss how the cross talk of the modulator can limit the convergence of the method and deteriorate the
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quality of the reconstruction. In the final section we propose a procedure to overcome the cross talk and experimentally demonstrate that the reconstructions are greatly improved. 2. PHASE RETRIEVAL METHOD A. Principles
In this section the principles of the method mentioned above are explained. Figure 1 shows the general principle of iterative methods for phase retrieval. All algorithms should alternate between the spatial and frequency domains while applying progressively different type of constraints. On the other hand, as we have mentioned in the introduction, the constraint we have chosen consists of introducing a random phase modulation in the object plane using the experimental setup illustrated in Fig. 2. A polarized plane wave diffracted by the object passes through the SLM, which modulates it with a known, random phase distribution included in the range of 0 to 2π. The object and the modulator can be of transmission or reflection type. Then the modulated beam passes through a lens that focuses it in its focal plane where is placed a CCD camera in order to collect the spectrum. Thus, the intensity of the Fourier transform of the modulated beam is acquired in the CCD plane. The strong modulation introduced by the SLM allows diffracting all incident wavefronts with a wide angle, which reduces the correlation between recorded intensities and so gets more information from each recording to avoid the stagnation problem. Furthermore, ambiguities in the Fienup algorithm like the object field shift or the twin image [15] disappear
Fig. 1. General principle of iterative phase retrieval methods. Progressively applying constraints while altering between spatial and frequency domains.
Fig. 2. Experimental setup used to collect the M diffraction patterns before launching the SSPR algorithm.
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because of the strong modulation which breaks any symmetry in the object field [28]. Moreover, this modulation attenuates the intensity of the Fourier zeroth order due to the spectrum spreading, and thus avoids the overexposure problem. The distance Z between the object and the modulator is not important, but it must be large (compared with the specimen size) so the propagation of the wavefront among these two planes could be described with the Fresnel integral [35]. Otherwise, if Z is short and the object is not strongly diffracting, the angular spectrum algorithm should be used, as it is more general and requires no minimum distance [35]. This aspect of the problem is well described in these two studies [20,21]. On the other hand, the distance between the lens L2 and the CCD must be exactly equal to the focal distance of the lens, so the propagation between the SLM and the CCD can be numerically obtained with a simple two-dimensional Fourier transform. The method is composed of two parts: The first one consists of recovering the object field in the plane of the modulator by means of an iterative algorithm. Then the obtained wavefront is propagated back toward the object plane using a Fresnel integral. At first, M diffraction patterns of the object in the Fourier plane are collected as the phase mask at the modulator is changed M times. Then the phase retrieval algorithm, as illustrated in Fig. 3, is executed. The first iteration begins with a random guess of the field at the modulator plane; then the following steps are performed: (1) Modulate the current estimation of U n with the first random phase (m 1), giving U˜ n1 U n1 expjφm1 , where n is the iteration number and m the random phase number (m 1; 2…M ). (2) Compute the Fourier transform of the resulting field, which will bring us to the CCD plane with the spectrum F U˜ n1 jF U˜ n1 j expjθm1 , where jF U˜ n1 j and θm1 represent the calculated spectrum magnitude and phase, respectively.
(3) Replace the obtained magnitude by the square root of the experimentally measured intensity for the first random phase (m 1: F U˜ n1 I 1∕2 m1 expjθm1 . (4) Propagate back the corrected spectrum to the modulator plane, performing an inverse Fourier transform. (5) Remove the random phase added in step 1. Thus, we have a new estimation of the wavefront U n1 at the SLM, closer to reality, which will be used in the second iteration where the same five steps will be repeated for the second random phase (m 2). If the last random phase (m M ) is reached, then use the first one. This iterative algorithm stops when the mean square error between two successive wavefronts is sufficiently small. For our setup, a Hamamatsu parallel aligned LC-SLM, series ×10468, is used. The modulator is of reflection type, which results in a better fill factor, having a resolution of 792 × 600 pixels. Each pixel can be analogically addressed with 256 gray levels to modulate the phase between 0 and 2π using simple software. According to the literature and our experiments, this way of addressing the modulator leads to a more stable phase. The main advantage of using an LC-SLM and not a phase plate is its ease of use, especially that there is no need for human intervention during the acquisition stage. For better functioning of the SSPR method, one must ensure that the modulator has a linear response and a flat surface; therefore, one must calculate the corresponding look-up table and compensate its surface aberration. Also, the recorded intensities were acquired with a 1280 × 960, 16-bit camera, and then cropped to 512 × 512 pixels for the calculation. The laser beam has a wavelength of 632.8 nm. B. Simulation Results
To prove the accuracy of the method, many simulations have been done for various kinds of objects. Regardless the object type and shape, the algorithm converges quickly for all simulations, even when using a small number of diffraction patterns (M > 3). Figures 4(a) and 4(b) show the simulation of the reconstruction of the amplitude and phase of a complex object using five diffraction patterns and 200 iterations. The retrieved amplitude and phase are very similar to the initial field, even though the phase has been calculated with a difference of a constant factor from the original one. Once this constant shift is determined and added to the recovered phase, the phase difference no longer appears. This simulation was performed assuming that the CCD camera codes intensity levels using 16 bits, so diffraction patterns (amplitude) were divided into 65,536 integer gray levels. In order to quantify the accuracy of the method as well as the speed of the algorithm convergence, one can use two parameters. The first, ε, known as mean square error, measures the difference between two successive reconstructions defined as X jU n1 m; nj − jU n m; nj2 ; (1) ε m;n
Fig. 3. Phase retrieval iterative algorithm: n is the iteration number, M is the total number of phase distributions, I m is the mth recorded Fourier pattern, and F.T. the Fourier transform operator.
where U n1 and U n , respectively, represent the retrieved wavefront for the iterations n 1 and n. The second is the signal-to-noise ratio (SNR): PP jU n j2 ; (2) SNR P P jU in j − jU n j2
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Fig. 4. (a) and (b) show simulation results of the reconstruction of a complex valued-field using the SSPR method, (c) represent the logarithmic evolution of the SNR with respect to iteration number when using five random phases. One can see the three parts of SNR evolution.
where U in and U n , respectively, represent the original wavefront and the recovered wavefront before the modulator plane. When plotting the SNR curve, one notes that the algorithm converges in three parts as shown in Fig. 4(c), which expresses the logarithmic values of the SNR versus iteration number. One distinguishes a first slow step where the algorithm tries to find a correct estimation. This phase may last from tens to thousands iterations depending on the number of recordings. The more diffraction patterns one records, the faster the convergence will be. The second step begins when an improved estimation is found. At this level the algorithm converges to a final state where the SNR value stagnates [28]. C. Experimental Results
An experiment has also been done to verify this approach using the optical setup in Fig. 2 and the components described in the first section. As the phase reconstruction quality depends on the nature of the object [36], different types of object were used. Figure 5 shows the reconstruction of three objects, two centrosymmetric objects and a rectangle. All reconstructions were calculated in the modulator plane, then propagated back to the object plane using an inverse Fresnel transformation. As can be seen, the quality of reconstruction is poor, and experiments have shown that it will be worse for more complex objects. Furthermore, the number of iterations required experimentally for a complete reconstruction greatly exceeds the number of iterations used in the simulation. For the simulation,
Fig. 5. (a), (c), and (e) are real objects we want to reconstruct including two centrosymmetric objects and one rectangle. (b), (d), and (f ) are their respective reconstructions. One can see that for centrosymmetric objects (two and three circular apertures), the reconstruction includes the object and its opposite to 180°. For the rectangle, the reconstruction quality is very poor, and the reconstructed wavefront is highly spread when returning to the object plane.
the wavefront is restored within a dozen iterations, yet experimentally we have to wait more than 4000 iterations before the reconstructed object become distinguishable. On the other hand, we would point out that the algorithm does not converge for all tests. In some cases, the error value remains high and the reconstruction fails. Other problems can also occur, such as the translation of the object’s final position according to the number of recordings. Also, it was found that for centrosymmetric objects (e.g., two or three holes), the final reconstruction may contain the object, and it is opposite to 180°, as shown in Figs. 5(b) and 5(d). D. Results Analysis
The difference between simulation and experimental results can be attributed to several noise sources. Some noises are specific to the optical components and their calibration, when others are specific to the used modulator. The number of iterations needed for convergence, and the quality of reconstruction
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Table 1. SNR of the Simulated Reconstructions When Error Sources Were Intentionally Added in Order to Introduce Degradationa
Lena Circle Rectangle Cross “X”
Ideal Situation
8 bits CCD Camera
SLM Rotation (1°)
CCD Rotation (1°)
79,9 79,8 78,9 77,2
33,3 32,9 32 31
4,31 79,8 3,61 1,7
4,13 8,9 2,3 2,5
a
The first column indicates the object types, and the lines show the introduced error.
of the object, can be greatly improved by properly controlling all the optical components. We must ensure that the incident beam is homogeneous and polarized in the right direction, and the modulator is correctly aligned and parallel with the beam splitter surface. Also, the distance between the lens and the CCD must be set precisely to be equal to the focal length of the lens and thus collect the exact Fourier transform. The sensor noises introduced during the acquisition stage, the degrees of saturation of diffraction patterns, as well as the quality of the camera can also deteriorate the quality of the reconstruction and must be compensated. Note that component calibration can be made experimentally as well as numerically. In order to bring out the importance of the error compensation, we performed different simulations where we intentionally introduced different errors sources with varying amplitudes. The amounts of degradation the reconstructed wavefront with respect to the initial object are shown in Table 1. This part of our study aims to prove the importance of the calibration phase for the experimental setup. By simply using a CCD camera with a dynamic range coded on 8 bits instead of 16 bits degrades the quality of reconstruction by a factor of two, regardless of the object type. Also, the rotation of 1° of the modulator or the CCD sensor engenders a significant decrease in SNR values; hence, it is important to place these two components in the optical axis parallel to the object plane or compensate their rotation numerically before launching the iterative algorithm. Despite their importance, these factors can be compensated either experimentally or numerically. However, LCOS phase modulators are known to have a strong cross talk between pixels, and this is recognized as the fringing field effect. This effect has the greatest influence on the quality of reconstruction of objects, so in the next section, we will present a procedure to model it for a given modulator.
Fig. 6. Imaging the surface of the Hamamatsu modulator when addressing horizontal and vertical gratings with two gray levels (0 and 128) and a period equal to 1. For more clarity, the profiles of the obtained phase distributions are also shown.
entire surface. This effect is related to the LC thickness [33] and acts as a low pass filter on the phase distribution, which results in a blurring effect of the desired sharp edge between the pixels as shown in Figs. 6 and 7. In order to see the effect of the cross talk on our own modulator, we imaged its surface using a lens to obtain a magnification greater than 10. Figure 6 shows the experimental setup, as well as the image of horizontal and vertical gratings addressed on the SLM with a period of 1. Ideally the white area should be equal to the black area, but as we can see the transition from white to black, and vice versa, is gradual and not sharp. Thus, the value of the phase retardation in a given pixel is not the same
3. MODULATOR CROSS TALK A. Identification Phase
As can be seen, in order to apply the iterative method explained above, the only information that we must precisely know is the random phase profiles addressed on the modulator. However, as the pixel sizes of the SLM are micrometric, the voltage applied to each pixel in order to achieve different phase levels has a significant effect on its neighbors [37], and thus the phase retardation obtained from each pixel will not be uniform over its
Fig. 7. (a) Phase distribution addressed on the SLM and (b) real phase displayed by an SLM suffering from strong cross talk, simulated by convolution of the ideal distribution with a Gaussian point spread function with a radius of 0.5.
Research Article over its entire surface. This can be easily confirmed when we plot the profiles of the two obtained images. Note that this effect is more significant for horizontal edges than vertical edges [38]. A solution that would seem logical for this problem is to use super pixels; this is when several individual pixels are binned to form one bigger pixel. This can alleviate the effect of cross talk, but nothing can guarantee the convergence of the algorithm, and in many cases the use of super pixels increases the convergence time of the algorithm. In addition, the higher the order of super pixels, the less the Fourier spectrum is spread, making the acquisition more sensitive. Add to this the fact that the super pixels do not work when it comes to small objects. These latter are projected on one or two super pixels in the modulator plan, where the phase does not vary quickly. A simulation was performed to investigate the effect of cross talk on the reconstruction. As we have already pointed out, the cross talk acts like a spatial low pass filter. Thus, it can be represented as the convolution of the ideal phase map with a transfer function specific to the used SLM [33,34]. In the simulation, the cross talk of the modulator is modeled by a Gaussian transfer function, which means that each of the random phases is convolved with a Gaussian in order to get as close as possible to the actual state of the random phase really displayed on the modulator as shown in Fig. 7. Moreover, Fig. 8 shows the simulation of the reconstruction of an object with two circles with a modulator suffering from Gaussian cross talk. As can be seen, the quality of the reconstruction is poor, and the amplitude is not uniform within circles, despite the 800 iterations performed. Also, instead of getting two circles, the reconstruction contains at least three circles easily visible. We also notice that the position of the reconstructed object is not precise; it translates into the plan, and the final position depends on the number of random phases used. As the results of these simulations being very similar to experimental results, one assumes that any compensation of the cross talk of the modulator must provide a better quality of reconstruction, remove the object duplication, and finally find the right position of the object regardless of the number of used random phases. For this purpose, and based on what has been stated above, we will modify the M random phase distributions used in the algorithm to make them identical to those actually displayed on the modulator during the acquisition stage. In other words, we will estimate the effect of cross talk, represented by a transfer function, and then convolve this transfer function with the M random phase distributions. This gives us an estimation of the
Fig. 8. Simulation of the reconstruction of (a) two circles with an SLM suffering from Gaussian cross talk using (b) M 5 random phases and (c) M 7 random phases.
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real M phase distributions actually displayed on the modulator suffering from cross talk. These new phase distributions are then used in the algorithm instead of the ideal phase map. Several studies have shown that the consequences of cross talk are not the same in the horizontal direction as the vertical direction [34,39]. For this reason, it is considered that its transfer function is not Gaussian but elliptical, having the following form: γ ar x r y ;γ x; y e
−
y2 x2 2r 2x 2r 2y
;
(3)
where r x and r y are the following radii in the x and y directions, respectively, and γ is a shape parameter. In order to find the parameter values that can compensate the effect of cross talk, we imagined the method illustrated in Fig. 9. The idea is to model, using simulation software, a simple object where we perfectly know its dimensions, like a simple circle. Then we try to experimentally reconstruct this object with the iterative method by numerically convolving the M random phases with a certain elliptical transfer function. For the first reconstruction, r x , r y , and γ have random values (between 0 and 1). Once the reconstruction is completed, we compare the recovered object with the initial one modeled at the beginning (by calculating the SNR or by performing a correlation), and we save the degree of resemblance. For each remaining reconstruction, the values of the three parameters are scanned in a range between 0 and 1 with a 0.1 step. At the end of all reconstructions only the values of r x , r y , and γ that give the best resemblance between the constructed object and the original one have to be retained. This process may take a long calculation time; however, it is sufficient to do it once for a given area of the modulator and for a given M random phases, then use the obtained values for other reconstructions.
Fig. 9. and γ.
Iterative procedure for finding the optimal values of r x , r y ,
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B. Compensation Phase
As we now have the parameter of the transfer function, we can apply the iterative method taking into account the effect of cross-talk of the modulator as shown in Fig. 10. The major difference with what has been described before is that during the algorithmic stage we now modulate the object wavefront with M random phase distributions convoluted with a PSF proper to the modulator cross-talk. In order to validate our approach on experimental data, the parameters describing the cross talk effect on our modulator are calculated before reconstructing the two objects shown in Fig. 11. Note that the experimental setup of Section 2 remains unchanged. Results of reconstructions are shown in Figs. 12 and 13. One can see that applying the cross-talk correction improves the quality of reconstructions and the clarity of the edges while using fewer iterations. It also reduces the distortion of the wavefront when propagated back to the object plane and removes the opposite of the object to 180°, as well as its position remaining constant even when we modify the number of random phase distributions.
Fig. 12. Experimental reconstruction of a two circles object [Fig. 11(a)] using the phase retrieval iterative algorithm without compensation of cross talk in (a) SLM plan and (b) object plan. Application of the compensation to the same object in (c) SLM plan and (d) object plan. Intensity profiles of the two final reconstructions are also shown in both SLM and object planes.
On the other hand, one must point out that several combinations of the three parameters r x , r y , and γ permit the compensation of the cross talk and give a good reconstruction of the object, even if the quality of these reconstructions varies from one triplet (r x , r y , γ) to another. Actually, many “correct” values of r x , r y , and γ are obtained because the used strategy calculates mean values for these three parameters, which are all true for the M random phase masks. Thus, it remains to retain the triplet that provides the best SNR. Also, in order to verify whether this method works for the entire surface of the modulator, the same experimental procedure was repeated for four different zones of the modulator. These zones were selected in such a way as to include the entire surface of the SLM. We have found that the triplet γ 0.5, r x 0.6, and r y 0.6 is a common triplet to all the zones of the modulator. This means that regardless of the addressed area of the SLM, these values of r x , r y , and γ permit obtaining a good reconstruction of the object. Beware; this triplet is not the optimal solution for a given area. It is a mean solution or a compromise which will work less well than the optimal one, but it will work for the whole surface of the SLM.
Fig. 10. Phase retrieval iterative algorithm including the compensation of modulator cross talk. ⊗ represents the convolution operator.
Fig. 11. Experimental object to reconstruct: (a) two circles and (b) a rectangle.
Fig. 13. Experimental reconstruction of a rectangle [Fig. 11(b)] using phase retrieval iterative algorithm without compensation of cross talk in (a) SLM plan and (b) object plan. Application of the compensation to the same object in (c) SLM plan and (d) object plan. Intensity profiles are also shown in both SLM and object planes.
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plate with the same thickness, which gives us a total thickness of 2 × E. Phase reconstruction results obtained with the modified SSPR iterative method and with compensation of the modulator cross talk are shown in Fig. 14. As we can notice, the recovered phase distribution have random values except for the rectangle zone, where it becomes constant and where we can easily see the two phase levels for both reconstructions. When we plot the phase profile, after the tilt compensation, we are able to precisely evaluate phase values as shown in Fig. 15. The phase difference between the uncovered and the covered zone of the rectangle for the first case is 1.10 rad whiles it is 2.15 rad for the second case. So, when we have used two plates, the obtained phase value is nearly double the obtained value for one plate. This proves that the calculated phase value is correct.
4. CONCLUSION AND DISCUSSION
Fig. 14. Recovered phase using the iterative SSPR phase retrieval method when covering a zone of the object with one (left) and two (right) plates. Cross-talk compensation is applied.
Finally, and to fully validate this proposed iterative method; it remains to experimentally verify the value of the recovered phase. For this we will twice reconstruct a rectangle object. For the first reconstruction we will cover a part of the object surface with a transparent plate of thickness E, and for the second reconstruction we will cover the same zone by another
Fig. 15. Profile of the recovered phase for one (upper profile) and two (lower profile) plates.
In this paper, we have presented an iterative phase retrieval method commonly known as SSPR which allows recovering random complex objects. This method is based on the strong modulation of the object beam with M random phases distributions introduced by a LC-SLM, then the acquisition of M spectra in Fourier domain. This procedure reduces the correlation between recorded intensities and decreases the intensity of the zeroth order in diffraction patterns. After the presentation of the poor experimental results, we have cited and discussed different sources of noise in order to quantify their effects on the quality of reconstructions. Many noises are introduced by the misalignment of optical components in the setup, which requires experimental or numerical correction. Other noises are due to the nonideal modulation properties of the LCD, especially the cross talk between the pixels. The voltage applied on a pixel can influence all neighboring pixels and introduce an error on their phase shift. We have presented a method that allows taking into account the cross talk in order to compensate it during the algorithmic stage. This improves the reconstruction quality and reduces the number of iterations and the number of acquisitions required for the convergence of the algorithm. On the other hand, one must remember that the calculated values of the three parameters r x , r y , and γ describe a mean transfer function, which represents the cross talk over the entire surface of the SLM and to all the phase masks used. Based on what has been shown, we can say that LC-SLMs can be useful for iterative phase retrieval methods. They have a high resolution and can facilitate the experimental conditions because of their ease of use and control, unlike phase plates. For these components, phase profiles are altered using computer software so there is no more need for human intervention during the acquisition stage. However, some disadvantages of LCSLM remains significant and must be taken into consideration for a successful application of the iterative method. Funding. Contrat de Plan Etat Région (CPER) 20072013; Ministère de l'Education Nationale, de l'Enseignement Supérieur et de la Recherche (MENESR).
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Acknowledgment. We would like to thank M. Fucai Zhang from the London Center for Nanotechnology and M. Giancarlo Pedrini from the University of Stuttgart for giving useful answers to some important questions concerning this study. REFERENCES 1. D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948). 2. S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432, 885–888 (2004). 3. I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier transform X-ray holography,” Science 256, 1009–1012 (1992). 4. T. Kreis, “Holographic interferometry: principles and methods,” in Simulation and Experiment in Laser Metrology: Proceedings of the International Symposium on Laser Applications in Precision Measurements, Balatonfüred, Hungary, June 6, 1996. 5. R. W. Gerchberg and O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972). 6. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978). 7. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987). 8. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998). 9. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419–1421 (2003). 10. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999). 11. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). 12. J. R. Fienup and A. M. Kowalczyk, “Phase retrieval for a complexvalued object by using a low-resolution image,” J. Opt. Soc. Am. A 7, 450–458 (1990). 13. A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984). 14. J. R. Fienup, “Phase retrieval using boundary conditions,” J. Opt. Soc. Am. A 3, 284–288 (1986). 15. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986). 16. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004). 17. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004). 18. K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003). 19. K. A. Nugent, A. G. Peele, H. M. Quiney, and H. N. Chapman, “Diffraction with wavefront curvature: a path to unique phase recovery,” Acta Crystallogr. Sect. A 61, 373–381 (2005).
Research Article 20. W. Chen and X. Chen, “Quantitative phase retrieval of a complexvalued object using variable function orders in the fractional Fourier domain,” Opt. Express 18, 13536–13541 (2010). 21. W. Chen and X. Chen, “Quantitative phase retrieval of complexvalued specimens based on noninterferometric imaging,” Appl. Opt. 50, 2008–2015 (2011). 22. P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval using multiple illumination wavelengths,” Opt. Lett. 33, 309–311 (2008). 23. R. Rolleston and N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. 25, 178–183 (1986). 24. B.-Z. Dong, Y. Zhang, B.-Y. Gu, and G.-Z. Yang, “Numerical investigation of phase retrieval in a fractional Fourier transform,” J. Opt. Soc. Am. A 14, 2709–2714 (1997). 25. V. Y. Ivanov, M. A. Vorontsov, and V. P. Sivokon, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A 9, 1515–1524 (1992). 26. G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30, 833–835 (2005). 27. P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006). 28. F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007). 29. C. Kohler, F. Zhang, and W. Osten, “Characterization of a spatial light modulator and its application in phase retrieval,” Appl. Opt. 48, 4003–4008 (2009). 30. P. F. Almoro, G. Pedrini, A. Anand, W. Osten, and S. G. Hanson, “Interferometric evaluation of angular displacements using phase retrieval,” Opt. Lett. 33, 2041–2043 (2008). 31. M. Agour, P. Almoro, and C. Falldorf, “Investigation of smooth wave fronts using SLM-based phase retrieval and a phase diffuser,” J. Eur. Opt. Soc. 7, 12046 (2012). 32. E. Hällstig, T. Martin, L. Sjöqvist, and M. Lindgren, “Polarization properties of a nematic liquid-crystal spatial light modulator for phase modulation,” J. Opt. Soc. Am. A 22, 177–184 (2005). 33. B. Apter, U. Efron, and E. Bahat-Treidel, “On the fringing-field effect in liquid-crystal beam-steering devices,” Appl. Opt. 43, 11–19 (2004). 34. M. Persson, D. Engström, and M. Goksör, “Reducing the effect of pixel crosstalk in phase only spatial light modulators,” Opt. Express 20, 22334–22343 (2012). 35. J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005). 36. C. Lingel, M. Hasler, T. Haist, G. Pedrini, and W. Osten, “A benchmark system for the evaluation of selected phase retrieval methods,” Proc. SPIE 9132, 91320R (2014). 37. R. James, F. A. Fernández, S. E. Day, M. Komarcevic, and W. A. Crossland, “Modeling of the diffraction efficiency and polarization sensitivity for a liquid crystal 2D spatial light modulator for reconfigurable beam steering,” J. Opt. Soc. Am. A 24, 2464–2473 (2007). 38. C. Lingel, T. Haist, and W. Osten, “Examination and optimizing of a liquid crystal display used as spatial light modulator concerning the fringing field effect,” Proc. SPIE 8490, 84900H (2012). 39. A. Marquez, C. Iemmi, I. Moreno, J. Campos, and M. J. Yzuel, “Anamorphic and spatial frequency dependent phase modulation on liquid crystal displays: optimization of the modulation diffraction efficiency,” Opt. Express 13, 2111–2119 (2005).
Research Article
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Cross-talk compensation of a spatial light modulator for iterative phase retrieval applications PIERRE GEMAYEL, BRUNO COLICCHIO, ALAIN DIETERLEN,*
AND
PIERRE AMBS
Laboratoire MIPS EA-2332, Université de Haute-Alsace, 12 rue des Frères Lumière—68093 MULHOUSE CEDEX, France *Corresponding author: [email protected] Received 29 October 2015; revised 18 December 2015; accepted 18 December 2015; posted 18 December 2015 (Doc. ID 252897); published 29 January 2016
Beam-propagation-based phase recovery approaches, also known as phase retrieval methods, retrieve the amplitude and the phase of arbitrary complex-valued fields. We present and experimentally demonstrate a simple and robust iterative method using a liquid crystal spatial light modulator located at an object diffraction plane. M random phase masks are applied between the object and the image sensor using the modulator, and then M diffraction patterns are collected in the Fourier plane. An iterative algorithm using these patterns and simulating the propagation of the light between the two planes allow us to recover the object wavefront. The use of this type of dynamic modulator makes the experimental setup simpler and more flexible. We need no a priori knowledge about the object field, and the convergence rate is high. Simulation results show that the method exhibits high immunity to noise and does not suffer any stagnation problem. However, experimental results have shown that the technique is sensitive to the cross talk of the modulator. We propose a method for compensating these modulator defects that are validated by experimental results. © 2016 Optical Society of America OCIS codes: (000.4430) Numerical approximation and analysis; (100.5070) Phase retrieval; (120.5060) Phase modulation; (230.6120) Spatial light modulators. http://dx.doi.org/10.1364/AO.55.000802
1. INTRODUCTION Both the amplitude and phase of a diffracted beam carry a part of the information about the diffracting object’s surface or its inner structure. While the amplitude may easily be derived from the square root of the intensity, the phase distribution, which carries most of the information, cannot be measured directly, and consequently its loss is more difficult to overcome. Three solutions to the phase problem exist: holography [1–4], wavefront sensing like the Shack–Hartmann sensor, and phase retrieval methods [5–7]. The latter are considered an inverse problem using an iterative algorithm alternating between the real domain and the frequency domain. These methods are based on the fact that the variation of the diffraction patterns along the direction of beam propagation may be used to extract the phase information of the object [5], so when the diffraction pattern is oversampled by a factor larger than 2, the phase information might be recovered using an iterative algorithm [8]. In addition, the progress in computer technologies as well in digital imaging devices makes the application of beam-propagationbased phase recovery approaches easier, especially in that these methods are relatively insensitive to noise, robust, have a simple experimental setup, and do not require the application of a 1559-128X/16/040802-09$15/0$15.00 © 2016 Optical Society of America
particular reference wave like holography. For these reasons, phase retrieval methods have known a lot of interest in various domains such as electron microscopy [9], x-ray diffraction imaging [10], material science, and astronomy. In 1972, Gerchberg and Saxton (GS) introduced the first widely accepted iterative method for phase retrieval based on projections [5]. The idea is that we can recover the phase if partial information about the object density as well as about the magnitude of the object Fourier transform are known. In 1982, Fienup proposed a number of iterative algorithms founded on various interpretations of the GS method [6] by using finite support and a nonnegativity constraint in object space instead of the magnitude of the object. The most basic method and the most used one is known as the hybrid input–output algorithm, which is closely related to the steepest-descent method, but it is more robust and converges much faster than the error-reduction algorithm [11], especially for real-valued nonnegative signals. However, it has been shown that recovering a complex object requires a sufficiently accurate hard support [7] or acquiring low resolution images [12]; otherwise these algorithms are not capable of successful reconstructions [13–15]. The loss of real and positive constraints for a complex valued object must be compensated
Research Article by the use of other constraints as done in several experimental strategies in order to overcome stagnation and nonconvergence problems. Faulkner and Rodenburg [16,17] proposed a method consisting on taking more than two diffraction patterns which correspond to different areas of the sample, using a moving aperture. Nugent et al. [18,19] proposed the use of diffraction data obtained with illumination modulated in both intensity and phase. Chen and Chen proposed to record several diffraction intensity maps through variable function orders in the fractional Fourier transform [20], or to record diffraction patterns while modifying the focal length of a lens function displayed on a phase mask [21]. Other methods also exist involving the modulation of the illumination wavelength [22] or recording intensities across two or more parallel planes connected through Fresnel or Fourier transforms [23–27]. In 2007, Zhang et al. suggested a method known as spread-spectrum phase retrieval (SSPR) inspired from the essence of the wavefront sensing technique [28]. The idea is to introduce a strong phase modulation into the object field using a phase mask. This way, any sampling of the Fresnel-recorded diffraction pattern contains a contribution of all points in the object, which will eliminate the effects of sensor noise and convergence stagnation. In the first place a phase plate is used to introduce the phase modulation, so before initiating the algorithm, more than three diffraction patterns are collected, as the plate is shifted transversely. Then, in order to make the experimental setup easier and more flexible, the phase plate was replaced by a liquid crystal spatial light modulator (LC-SLM) with which there is no need to modify the setup during acquisitions [29]. This SSPR approach has been applied in the interferometric evaluation of rotating smooth objects using a fixed phase diffuser plate with the speckle measurements taken at multiple axially displaced planes [30]. Recently, an SLM in a 4f -setup has also been used in conjunction with the multiple-plane SSPR technique [30] for the reconstruction of smooth object wavefront where the SLM displays a transfer function for free-space propagation [31]. One note is that experimental results obtained when an SLM is employed for the SSPR technique are inferior to the results obtained by using a phase plate, even if adjoining pixels were binned into a superpixel, and this is mainly due to the fringing field effect highly present in liquid crystal on silicon (LCOS) displays. This effect refers to the cross talk between neighboring pixels, and it appears when there is a significant phase jump. This effect leads to the broadening of the phase profile in a LC device [32,33]. In fact, phase retardation is not constant over the entire surface of the pixel, especially at the edges, and this can be considered as a spatial low pass filtering of the phase retardation displayed on a SLM with no pixel cross talk [34]. In this paper we present a method for phase retrieval based on SSPR. We collect diffraction patterns in the Fourier domain instead of the Fresnel domain, which is simpler because we just need to position the CCD in the focal plane of the lens without worrying about distances between components. The phase modulation is introduced by a LC-SLM located between the object and the CCD. After presenting simulation and experimental results, we discuss how the cross talk of the modulator can limit the convergence of the method and deteriorate the
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quality of the reconstruction. In the final section we propose a procedure to overcome the cross talk and experimentally demonstrate that the reconstructions are greatly improved. 2. PHASE RETRIEVAL METHOD A. Principles
In this section the principles of the method mentioned above are explained. Figure 1 shows the general principle of iterative methods for phase retrieval. All algorithms should alternate between the spatial and frequency domains while applying progressively different type of constraints. On the other hand, as we have mentioned in the introduction, the constraint we have chosen consists of introducing a random phase modulation in the object plane using the experimental setup illustrated in Fig. 2. A polarized plane wave diffracted by the object passes through the SLM, which modulates it with a known, random phase distribution included in the range of 0 to 2π. The object and the modulator can be of transmission or reflection type. Then the modulated beam passes through a lens that focuses it in its focal plane where is placed a CCD camera in order to collect the spectrum. Thus, the intensity of the Fourier transform of the modulated beam is acquired in the CCD plane. The strong modulation introduced by the SLM allows diffracting all incident wavefronts with a wide angle, which reduces the correlation between recorded intensities and so gets more information from each recording to avoid the stagnation problem. Furthermore, ambiguities in the Fienup algorithm like the object field shift or the twin image [15] disappear
Fig. 1. General principle of iterative phase retrieval methods. Progressively applying constraints while altering between spatial and frequency domains.
Fig. 2. Experimental setup used to collect the M diffraction patterns before launching the SSPR algorithm.
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because of the strong modulation which breaks any symmetry in the object field [28]. Moreover, this modulation attenuates the intensity of the Fourier zeroth order due to the spectrum spreading, and thus avoids the overexposure problem. The distance Z between the object and the modulator is not important, but it must be large (compared with the specimen size) so the propagation of the wavefront among these two planes could be described with the Fresnel integral [35]. Otherwise, if Z is short and the object is not strongly diffracting, the angular spectrum algorithm should be used, as it is more general and requires no minimum distance [35]. This aspect of the problem is well described in these two studies [20,21]. On the other hand, the distance between the lens L2 and the CCD must be exactly equal to the focal distance of the lens, so the propagation between the SLM and the CCD can be numerically obtained with a simple two-dimensional Fourier transform. The method is composed of two parts: The first one consists of recovering the object field in the plane of the modulator by means of an iterative algorithm. Then the obtained wavefront is propagated back toward the object plane using a Fresnel integral. At first, M diffraction patterns of the object in the Fourier plane are collected as the phase mask at the modulator is changed M times. Then the phase retrieval algorithm, as illustrated in Fig. 3, is executed. The first iteration begins with a random guess of the field at the modulator plane; then the following steps are performed: (1) Modulate the current estimation of U n with the first random phase (m 1), giving U˜ n1 U n1 expjφm1 , where n is the iteration number and m the random phase number (m 1; 2…M ). (2) Compute the Fourier transform of the resulting field, which will bring us to the CCD plane with the spectrum F U˜ n1 jF U˜ n1 j expjθm1 , where jF U˜ n1 j and θm1 represent the calculated spectrum magnitude and phase, respectively.
(3) Replace the obtained magnitude by the square root of the experimentally measured intensity for the first random phase (m 1: F U˜ n1 I 1∕2 m1 expjθm1 . (4) Propagate back the corrected spectrum to the modulator plane, performing an inverse Fourier transform. (5) Remove the random phase added in step 1. Thus, we have a new estimation of the wavefront U n1 at the SLM, closer to reality, which will be used in the second iteration where the same five steps will be repeated for the second random phase (m 2). If the last random phase (m M ) is reached, then use the first one. This iterative algorithm stops when the mean square error between two successive wavefronts is sufficiently small. For our setup, a Hamamatsu parallel aligned LC-SLM, series ×10468, is used. The modulator is of reflection type, which results in a better fill factor, having a resolution of 792 × 600 pixels. Each pixel can be analogically addressed with 256 gray levels to modulate the phase between 0 and 2π using simple software. According to the literature and our experiments, this way of addressing the modulator leads to a more stable phase. The main advantage of using an LC-SLM and not a phase plate is its ease of use, especially that there is no need for human intervention during the acquisition stage. For better functioning of the SSPR method, one must ensure that the modulator has a linear response and a flat surface; therefore, one must calculate the corresponding look-up table and compensate its surface aberration. Also, the recorded intensities were acquired with a 1280 × 960, 16-bit camera, and then cropped to 512 × 512 pixels for the calculation. The laser beam has a wavelength of 632.8 nm. B. Simulation Results
To prove the accuracy of the method, many simulations have been done for various kinds of objects. Regardless the object type and shape, the algorithm converges quickly for all simulations, even when using a small number of diffraction patterns (M > 3). Figures 4(a) and 4(b) show the simulation of the reconstruction of the amplitude and phase of a complex object using five diffraction patterns and 200 iterations. The retrieved amplitude and phase are very similar to the initial field, even though the phase has been calculated with a difference of a constant factor from the original one. Once this constant shift is determined and added to the recovered phase, the phase difference no longer appears. This simulation was performed assuming that the CCD camera codes intensity levels using 16 bits, so diffraction patterns (amplitude) were divided into 65,536 integer gray levels. In order to quantify the accuracy of the method as well as the speed of the algorithm convergence, one can use two parameters. The first, ε, known as mean square error, measures the difference between two successive reconstructions defined as X jU n1 m; nj − jU n m; nj2 ; (1) ε m;n
Fig. 3. Phase retrieval iterative algorithm: n is the iteration number, M is the total number of phase distributions, I m is the mth recorded Fourier pattern, and F.T. the Fourier transform operator.
where U n1 and U n , respectively, represent the retrieved wavefront for the iterations n 1 and n. The second is the signal-to-noise ratio (SNR): PP jU n j2 ; (2) SNR P P jU in j − jU n j2
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Fig. 4. (a) and (b) show simulation results of the reconstruction of a complex valued-field using the SSPR method, (c) represent the logarithmic evolution of the SNR with respect to iteration number when using five random phases. One can see the three parts of SNR evolution.
where U in and U n , respectively, represent the original wavefront and the recovered wavefront before the modulator plane. When plotting the SNR curve, one notes that the algorithm converges in three parts as shown in Fig. 4(c), which expresses the logarithmic values of the SNR versus iteration number. One distinguishes a first slow step where the algorithm tries to find a correct estimation. This phase may last from tens to thousands iterations depending on the number of recordings. The more diffraction patterns one records, the faster the convergence will be. The second step begins when an improved estimation is found. At this level the algorithm converges to a final state where the SNR value stagnates [28]. C. Experimental Results
An experiment has also been done to verify this approach using the optical setup in Fig. 2 and the components described in the first section. As the phase reconstruction quality depends on the nature of the object [36], different types of object were used. Figure 5 shows the reconstruction of three objects, two centrosymmetric objects and a rectangle. All reconstructions were calculated in the modulator plane, then propagated back to the object plane using an inverse Fresnel transformation. As can be seen, the quality of reconstruction is poor, and experiments have shown that it will be worse for more complex objects. Furthermore, the number of iterations required experimentally for a complete reconstruction greatly exceeds the number of iterations used in the simulation. For the simulation,
Fig. 5. (a), (c), and (e) are real objects we want to reconstruct including two centrosymmetric objects and one rectangle. (b), (d), and (f ) are their respective reconstructions. One can see that for centrosymmetric objects (two and three circular apertures), the reconstruction includes the object and its opposite to 180°. For the rectangle, the reconstruction quality is very poor, and the reconstructed wavefront is highly spread when returning to the object plane.
the wavefront is restored within a dozen iterations, yet experimentally we have to wait more than 4000 iterations before the reconstructed object become distinguishable. On the other hand, we would point out that the algorithm does not converge for all tests. In some cases, the error value remains high and the reconstruction fails. Other problems can also occur, such as the translation of the object’s final position according to the number of recordings. Also, it was found that for centrosymmetric objects (e.g., two or three holes), the final reconstruction may contain the object, and it is opposite to 180°, as shown in Figs. 5(b) and 5(d). D. Results Analysis
The difference between simulation and experimental results can be attributed to several noise sources. Some noises are specific to the optical components and their calibration, when others are specific to the used modulator. The number of iterations needed for convergence, and the quality of reconstruction
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Table 1. SNR of the Simulated Reconstructions When Error Sources Were Intentionally Added in Order to Introduce Degradationa
Lena Circle Rectangle Cross “X”
Ideal Situation
8 bits CCD Camera
SLM Rotation (1°)
CCD Rotation (1°)
79,9 79,8 78,9 77,2
33,3 32,9 32 31
4,31 79,8 3,61 1,7
4,13 8,9 2,3 2,5
a
The first column indicates the object types, and the lines show the introduced error.
of the object, can be greatly improved by properly controlling all the optical components. We must ensure that the incident beam is homogeneous and polarized in the right direction, and the modulator is correctly aligned and parallel with the beam splitter surface. Also, the distance between the lens and the CCD must be set precisely to be equal to the focal length of the lens and thus collect the exact Fourier transform. The sensor noises introduced during the acquisition stage, the degrees of saturation of diffraction patterns, as well as the quality of the camera can also deteriorate the quality of the reconstruction and must be compensated. Note that component calibration can be made experimentally as well as numerically. In order to bring out the importance of the error compensation, we performed different simulations where we intentionally introduced different errors sources with varying amplitudes. The amounts of degradation the reconstructed wavefront with respect to the initial object are shown in Table 1. This part of our study aims to prove the importance of the calibration phase for the experimental setup. By simply using a CCD camera with a dynamic range coded on 8 bits instead of 16 bits degrades the quality of reconstruction by a factor of two, regardless of the object type. Also, the rotation of 1° of the modulator or the CCD sensor engenders a significant decrease in SNR values; hence, it is important to place these two components in the optical axis parallel to the object plane or compensate their rotation numerically before launching the iterative algorithm. Despite their importance, these factors can be compensated either experimentally or numerically. However, LCOS phase modulators are known to have a strong cross talk between pixels, and this is recognized as the fringing field effect. This effect has the greatest influence on the quality of reconstruction of objects, so in the next section, we will present a procedure to model it for a given modulator.
Fig. 6. Imaging the surface of the Hamamatsu modulator when addressing horizontal and vertical gratings with two gray levels (0 and 128) and a period equal to 1. For more clarity, the profiles of the obtained phase distributions are also shown.
entire surface. This effect is related to the LC thickness [33] and acts as a low pass filter on the phase distribution, which results in a blurring effect of the desired sharp edge between the pixels as shown in Figs. 6 and 7. In order to see the effect of the cross talk on our own modulator, we imaged its surface using a lens to obtain a magnification greater than 10. Figure 6 shows the experimental setup, as well as the image of horizontal and vertical gratings addressed on the SLM with a period of 1. Ideally the white area should be equal to the black area, but as we can see the transition from white to black, and vice versa, is gradual and not sharp. Thus, the value of the phase retardation in a given pixel is not the same
3. MODULATOR CROSS TALK A. Identification Phase
As can be seen, in order to apply the iterative method explained above, the only information that we must precisely know is the random phase profiles addressed on the modulator. However, as the pixel sizes of the SLM are micrometric, the voltage applied to each pixel in order to achieve different phase levels has a significant effect on its neighbors [37], and thus the phase retardation obtained from each pixel will not be uniform over its
Fig. 7. (a) Phase distribution addressed on the SLM and (b) real phase displayed by an SLM suffering from strong cross talk, simulated by convolution of the ideal distribution with a Gaussian point spread function with a radius of 0.5.
Research Article over its entire surface. This can be easily confirmed when we plot the profiles of the two obtained images. Note that this effect is more significant for horizontal edges than vertical edges [38]. A solution that would seem logical for this problem is to use super pixels; this is when several individual pixels are binned to form one bigger pixel. This can alleviate the effect of cross talk, but nothing can guarantee the convergence of the algorithm, and in many cases the use of super pixels increases the convergence time of the algorithm. In addition, the higher the order of super pixels, the less the Fourier spectrum is spread, making the acquisition more sensitive. Add to this the fact that the super pixels do not work when it comes to small objects. These latter are projected on one or two super pixels in the modulator plan, where the phase does not vary quickly. A simulation was performed to investigate the effect of cross talk on the reconstruction. As we have already pointed out, the cross talk acts like a spatial low pass filter. Thus, it can be represented as the convolution of the ideal phase map with a transfer function specific to the used SLM [33,34]. In the simulation, the cross talk of the modulator is modeled by a Gaussian transfer function, which means that each of the random phases is convolved with a Gaussian in order to get as close as possible to the actual state of the random phase really displayed on the modulator as shown in Fig. 7. Moreover, Fig. 8 shows the simulation of the reconstruction of an object with two circles with a modulator suffering from Gaussian cross talk. As can be seen, the quality of the reconstruction is poor, and the amplitude is not uniform within circles, despite the 800 iterations performed. Also, instead of getting two circles, the reconstruction contains at least three circles easily visible. We also notice that the position of the reconstructed object is not precise; it translates into the plan, and the final position depends on the number of random phases used. As the results of these simulations being very similar to experimental results, one assumes that any compensation of the cross talk of the modulator must provide a better quality of reconstruction, remove the object duplication, and finally find the right position of the object regardless of the number of used random phases. For this purpose, and based on what has been stated above, we will modify the M random phase distributions used in the algorithm to make them identical to those actually displayed on the modulator during the acquisition stage. In other words, we will estimate the effect of cross talk, represented by a transfer function, and then convolve this transfer function with the M random phase distributions. This gives us an estimation of the
Fig. 8. Simulation of the reconstruction of (a) two circles with an SLM suffering from Gaussian cross talk using (b) M 5 random phases and (c) M 7 random phases.
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real M phase distributions actually displayed on the modulator suffering from cross talk. These new phase distributions are then used in the algorithm instead of the ideal phase map. Several studies have shown that the consequences of cross talk are not the same in the horizontal direction as the vertical direction [34,39]. For this reason, it is considered that its transfer function is not Gaussian but elliptical, having the following form: γ ar x r y ;γ x; y e
−
y2 x2 2r 2x 2r 2y
;
(3)
where r x and r y are the following radii in the x and y directions, respectively, and γ is a shape parameter. In order to find the parameter values that can compensate the effect of cross talk, we imagined the method illustrated in Fig. 9. The idea is to model, using simulation software, a simple object where we perfectly know its dimensions, like a simple circle. Then we try to experimentally reconstruct this object with the iterative method by numerically convolving the M random phases with a certain elliptical transfer function. For the first reconstruction, r x , r y , and γ have random values (between 0 and 1). Once the reconstruction is completed, we compare the recovered object with the initial one modeled at the beginning (by calculating the SNR or by performing a correlation), and we save the degree of resemblance. For each remaining reconstruction, the values of the three parameters are scanned in a range between 0 and 1 with a 0.1 step. At the end of all reconstructions only the values of r x , r y , and γ that give the best resemblance between the constructed object and the original one have to be retained. This process may take a long calculation time; however, it is sufficient to do it once for a given area of the modulator and for a given M random phases, then use the obtained values for other reconstructions.
Fig. 9. and γ.
Iterative procedure for finding the optimal values of r x , r y ,
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B. Compensation Phase
As we now have the parameter of the transfer function, we can apply the iterative method taking into account the effect of cross-talk of the modulator as shown in Fig. 10. The major difference with what has been described before is that during the algorithmic stage we now modulate the object wavefront with M random phase distributions convoluted with a PSF proper to the modulator cross-talk. In order to validate our approach on experimental data, the parameters describing the cross talk effect on our modulator are calculated before reconstructing the two objects shown in Fig. 11. Note that the experimental setup of Section 2 remains unchanged. Results of reconstructions are shown in Figs. 12 and 13. One can see that applying the cross-talk correction improves the quality of reconstructions and the clarity of the edges while using fewer iterations. It also reduces the distortion of the wavefront when propagated back to the object plane and removes the opposite of the object to 180°, as well as its position remaining constant even when we modify the number of random phase distributions.
Fig. 12. Experimental reconstruction of a two circles object [Fig. 11(a)] using the phase retrieval iterative algorithm without compensation of cross talk in (a) SLM plan and (b) object plan. Application of the compensation to the same object in (c) SLM plan and (d) object plan. Intensity profiles of the two final reconstructions are also shown in both SLM and object planes.
On the other hand, one must point out that several combinations of the three parameters r x , r y , and γ permit the compensation of the cross talk and give a good reconstruction of the object, even if the quality of these reconstructions varies from one triplet (r x , r y , γ) to another. Actually, many “correct” values of r x , r y , and γ are obtained because the used strategy calculates mean values for these three parameters, which are all true for the M random phase masks. Thus, it remains to retain the triplet that provides the best SNR. Also, in order to verify whether this method works for the entire surface of the modulator, the same experimental procedure was repeated for four different zones of the modulator. These zones were selected in such a way as to include the entire surface of the SLM. We have found that the triplet γ 0.5, r x 0.6, and r y 0.6 is a common triplet to all the zones of the modulator. This means that regardless of the addressed area of the SLM, these values of r x , r y , and γ permit obtaining a good reconstruction of the object. Beware; this triplet is not the optimal solution for a given area. It is a mean solution or a compromise which will work less well than the optimal one, but it will work for the whole surface of the SLM.
Fig. 10. Phase retrieval iterative algorithm including the compensation of modulator cross talk. ⊗ represents the convolution operator.
Fig. 11. Experimental object to reconstruct: (a) two circles and (b) a rectangle.
Fig. 13. Experimental reconstruction of a rectangle [Fig. 11(b)] using phase retrieval iterative algorithm without compensation of cross talk in (a) SLM plan and (b) object plan. Application of the compensation to the same object in (c) SLM plan and (d) object plan. Intensity profiles are also shown in both SLM and object planes.
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plate with the same thickness, which gives us a total thickness of 2 × E. Phase reconstruction results obtained with the modified SSPR iterative method and with compensation of the modulator cross talk are shown in Fig. 14. As we can notice, the recovered phase distribution have random values except for the rectangle zone, where it becomes constant and where we can easily see the two phase levels for both reconstructions. When we plot the phase profile, after the tilt compensation, we are able to precisely evaluate phase values as shown in Fig. 15. The phase difference between the uncovered and the covered zone of the rectangle for the first case is 1.10 rad whiles it is 2.15 rad for the second case. So, when we have used two plates, the obtained phase value is nearly double the obtained value for one plate. This proves that the calculated phase value is correct.
4. CONCLUSION AND DISCUSSION
Fig. 14. Recovered phase using the iterative SSPR phase retrieval method when covering a zone of the object with one (left) and two (right) plates. Cross-talk compensation is applied.
Finally, and to fully validate this proposed iterative method; it remains to experimentally verify the value of the recovered phase. For this we will twice reconstruct a rectangle object. For the first reconstruction we will cover a part of the object surface with a transparent plate of thickness E, and for the second reconstruction we will cover the same zone by another
Fig. 15. Profile of the recovered phase for one (upper profile) and two (lower profile) plates.
In this paper, we have presented an iterative phase retrieval method commonly known as SSPR which allows recovering random complex objects. This method is based on the strong modulation of the object beam with M random phases distributions introduced by a LC-SLM, then the acquisition of M spectra in Fourier domain. This procedure reduces the correlation between recorded intensities and decreases the intensity of the zeroth order in diffraction patterns. After the presentation of the poor experimental results, we have cited and discussed different sources of noise in order to quantify their effects on the quality of reconstructions. Many noises are introduced by the misalignment of optical components in the setup, which requires experimental or numerical correction. Other noises are due to the nonideal modulation properties of the LCD, especially the cross talk between the pixels. The voltage applied on a pixel can influence all neighboring pixels and introduce an error on their phase shift. We have presented a method that allows taking into account the cross talk in order to compensate it during the algorithmic stage. This improves the reconstruction quality and reduces the number of iterations and the number of acquisitions required for the convergence of the algorithm. On the other hand, one must remember that the calculated values of the three parameters r x , r y , and γ describe a mean transfer function, which represents the cross talk over the entire surface of the SLM and to all the phase masks used. Based on what has been shown, we can say that LC-SLMs can be useful for iterative phase retrieval methods. They have a high resolution and can facilitate the experimental conditions because of their ease of use and control, unlike phase plates. For these components, phase profiles are altered using computer software so there is no more need for human intervention during the acquisition stage. However, some disadvantages of LCSLM remains significant and must be taken into consideration for a successful application of the iterative method. Funding. Contrat de Plan Etat Région (CPER) 20072013; Ministère de l'Education Nationale, de l'Enseignement Supérieur et de la Recherche (MENESR).
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