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Multispectral random illumination using a liquid crystal spatial light modulator
Journal of the European Optical SocietyRapid Publications volume 14, Article number: 19 (2018)
Abstract
Background
Using phaseonly liquid crystal spatial light modulator (LCSLM) to generate random illumination is always used in phaseretrieval and single pixel imaging as it have high energy efficiency. While the methods were only used to generate monochromatic illumination. This paper aims to use LCSLM generating the multispectral random illumination.
Methods
The method requires only one SLM; multiple laser beams are simultaneously used to illuminate the entire SLM. Random voltages designed with a probability density function (PDF) are applied to modulate the random phases in the multispectral emitted light beam, leading to random amplitude spots in the far field. By optimize the PDF, the random illumination spot can be close to uniform distributed with high efficiency.
Results
The average intensities of the random light spots are almost uniformly distributed within the range of the first diffraction order. And the zeroorder beam is eliminated. The proposed method has a high energy efficiency and refresh rate and does not require the SLM to have an excessive modulation range.
Conclusion
This paper provides a method to generate multispectral random illumination using a LCSLM, which can be used in phaseretrieval and single pixel imaging.
Background
A purephase spatial light modulator (SLM) is a diffractive optical device with a high energy efficiency. The generation of random phase illumination by an SLM has been used in many applications such as phase retrieval [1] and microscopic imaging [2, 3]. In singlepixel imaging [4] and structured illumination imaging [5, 6] applications, an SLM may also be used to generate random amplitude patterns. As the device only modulates the phase, its energy efficiency will be higher than conventional projection devices based on amplitude modulation. In most current applications, SLMs are generally used only for a singlewavelength beam. If an SLM can be used for fullcolor or multispectral beam control, the usage of the above applications can be extended, thereby obtaining more information in the spectral dimension. As the LCSLM technology is mature and cheap, it is the most commonly used SLM device in laboratory and commercial use. Thus, our study is focused in LCSLM. For the sake of simplicity, we will still use SLM to refer to LCSLM in the following paper.
McManamon analyzed the use of an SLM for steering a broadband beam and found that a beam with different wavelengths will be steered to different angles and orders owing to the dispersion caused by the wavelengthdependent response of the SLM [7]. Gruneisen theoretically analyzed broadband beam control and pointed out that the use of an SLM with a very large modulation range could avoid beams pointed to the wrong orders. However, they are still dispersed to different angles [8]. Stockley studied a chiral smectic liquid crystal (CSLC) writable grating that can minimize the wavelength dependence of the phaseshifting elements, thereby steering multispectral beams to the same order. However, such devices are not common and still need an achromatic optical system to offset the angle dispersion [9]. These explorations pointed out the difficulties for SLM multispectral beam control without a very ideal solution.
The use of an SLM for fullcolor holographic displays has been widely studied. In one approach, three SLMs are utilized for the corresponding red, green, and blue (RGB) channels [10]. However, this obviously increases the complexity of the system. Another approach is the timedivision method (TDM), which cannot simultaneously modulate a multispectral beam [11]. In the spacedivision method (SDM), beams with different spectra are projected onto different areas of the SLM [12]. The beams generated using this method cannot utilize all of the units of an SLM; thus, the resolution is lowered.
In summary, it is difficult to utilize an SLM to generate a certain multispectral projection efficiently. However, it may be feasible for multispectral random illumination, which is the focus of this study. To improve the performance of SLMbased phase retrieval, Chen et al. [13] generated the random illumination by a designed phase pattern which can eliminate the zerothorder beam and let the light be more evenly distributed. While, this method is only for the singlewavelength random illuminate. Inspired by this approach, we generate the multispectral random illumination by designing the phase pattern of SLM. According to our investigation, this is the first research generating the multispectral random illumination using an LCSLM without timedivision or spacialdivision methods.
Methods
The configuration of the optical system is shown in Fig. 1. Three (or more) lasers are fused in a beam spliter and expand by a beam expander then modified by a LCSLM. After that the modified light is imaged on the screen through the lens.
The light field in the focal plane of an SLM can be obtained with the Fourier transform of the emitted light. Thus, I=FFT(U)^{2}, where I is the intensity of beam in the far field, and U is the light field of emitted beam. If the phase of U is randomly distributed, then beams are scattered to spots with a random intensity in the far field. To obtain random spots with a large bandwidth, the size of the SLM should be large. With a larger scattering angle, the spatial correlation of the random phase should be weak. If a particular device is selected, the optimal values of these parameters are fixed.
When an SLM is used, it is important to reduce the intensity of the zerothorder of the farfield beam because it is often very strong, which seriously deteriorates the performance of structured light illumination and phase retrieval. Because of the wavelengthdependent response of the SLM, strong zerothorders will appear in each spectrum if arbitrary voltages are applied to the device. In this study, we are aiming to design a control voltage to eliminate these zerothorder spots.
For the kth beam spectrum, the zerothorder intensity is related to the phase of the modulated beam; that is,
Where I_{0k} denotes the zerothorder intensity of the kth spectrum; A_{k} is a fixed intensity factor; N is the total number of rows (and columns), i.e., the resolution of the SLM is N×N; and ϕ_{k} denotes the phase retardation of the corresponding spectrum. Accounting for the crosstalk effect in liquid crystal SLM, we have
Where f_{k}[] denotes the voltagephase response of the kth spectrum, x(m,n) is the control voltage. h(m,n) is the point spread function (PSF) of the crosstalk [14], which has nonzero values only in a small region around (0,0). ⊗ represents the convolution operator.
To reduce every I_{0k}, we need to solve for the control voltages. An apparent way is to optimize x(m,n) with the objective function \(J = \sum _{k}\ I_{0k}\). However, it is very hard.
In order to generate random illumination, x(m,n) should be independent random variables to broaden the scattering angle. Assuming that they are identically distributed, as N→∞, the expectation of the zerothorder intensity is
The objective function can then be converted to
and
To convert the twodimensional convolution to onedimensional vector multiplication, we reshape h(m,n) into a S×1 vector h^{′}, and x(m,n) into a N^{2}×S matrix x^{′}. Here, S equals to the number of nonzero elements in h(m,n). We assign a new index t to represent the tth row of x^{′}. Further, we denote the PDF of x by p(x). Thus, we have
Since p(x)>0 and the range of f_{k}[] is only a few multiples of 2πs, reducing any of J^{′′}(i) will reduce the J. To simplify the optimization, let h_{max}=max(h^{′}), and choose the corresponding term \(J^{\prime \prime }_{max} = J^{\prime \prime }(i_{max})\) as the objective function. Where, i_{max} is the index of the maximum of h^{′}. We denote \(x^{\prime }_{t}(i_{max})\) as \(x_{t}^{\prime \prime }\), then the objective function becomes
Further, \(x^{\prime \prime }_{t}\) is independent and identically distributed, then
In order to solve this problem with a computer, p(x) is discretized. Finally, by ignoring the N^{2}th power,the objective function becomes
Here, p(x) is a discrete PDF that has a probability of p_{l} at \(x^{\prime \prime }_{l}\). Then, only 2M parameters need to be optimized to minimize \(J^{\prime \prime }_{max}\), thereby suppressing the zerothorder of every spectrum.
Result and discussion
The nonlinearity of the optimization problem is very strong. Traditional optimization methods may easily converge to the local maxima. Thus, we use the genetic algorithm function ga in MATLAB to optimize the parameters in Eq. 14. After that, the function fminsearch is utilized for further optimization. In the simulation, in order to demonstrate the results clearly, we assume that there are three spectra (k=1,2,3): 632 nm (R), 514 nm (G), and 454 nm (B), and the SLM has modulation ranges of 2.4π, 3.4π, and 3.7π, respectively. Namely, the minimum modulation range appears in the red band (the longest wavelength), which is slightly greater than 1λ. However, they are larger in other spectra due to the dispersion of the liquid crystal. The PDF of the crosstalk is a Gaussian kernel with a standard deviation of 0.4 pixels. The voltagephase response curves are similar to the one in literature [5], as shown in Fig. 2. In our simulation experiment, the number of points of the discrete PDF is eight. The optimization function only requires 10s to find a result. The optimized PDF is summarized in Table 1. In the tests, one of the results is shown in Fig. 3. A random voltage pattern with the optimized PDF is shown in Fig. 3a, and the random intensity pattern in the farfield (in RGB color) is shown in Fig. 3b.
We calculated 1000 random patterns in the far field to validate the performance of our random illumination. In the simulation, we set the SLM parameters as mentioned above with the following additional parameters: device resolution: 128×128, electrode interval: 10 μm and fill factor: 100% (in fact, the fill factor only affects the PDF h). Simulation experiments were carried out using an Intel Core i32130 3.4GHz central processing unit (CPU) using MATLAB version 2013a. It takes 1.26 s to generate 1000 voltage patterns, i.e., the refresh rate could be higher than 800 Hz, which is far greater than the frame rate of most commercial SLMs. In comparison, the pattern generated without optimization is also presented. An average intensity image of the random spots is shown in Fig. 4.
As shown in Fig. 4a, there exists an extremely bright zerothorder spot; therefore, it must be saturated in the visualization. However, in Fig. 4b, the average intensity distribution of the random spots is uniform without any large fluctuations. Upon closer inspection, a brighter spot could be found in the zerothorder, which is mainly attributed to the residuals of the optimization. The zerothorder intensities are 1.5, 2, and 1.2 times larger than their neighborhoods in the R, G, and B bands, respectively, but their powers are only 0.009, 0.011, and 0.004% of the entire beam in each spectrum. In contrast, in the nonoptimized case, the zerothorder intensities are 6000 times lager then their neighborhoods, and they take up 7.2, 6.7, and 3.8% of the entire energy, respectively. Hence, our algorithm successfully suppressed the zerothorder spot. If we increase the modulation range of the SLM or use a PDF with a larger number of degrees of freedom, the zerothorder intensities will be reduced more significantly.
In Fig. 4b, if the average intensity of the spot is greater than 1/e^{2} of the maximum value (except the zerothorder), we define this area as a valid area (VA). In this area, if an 8bit camera is used, the average gray value of the random spot will be greater than 35, which is sufficient for measurement. For the R, G, and B bands, the angles of the VAs are approximately 5.3°, 4.3°, and 3.8°, respectively. Although they are narrow, the VAs can be increased through a telescope system. In addition, the energy utilization efficiency is greater than 75% in the VA of each spectrum. If we choose the minimum VA as the VA of entire system, i.e., 3.8%, the efficiencies of each spectrum are 57, 70, and 76%, which are also far higher than the efficiency of the random spots generated by amplitude modulation (< 45%).
By analyzing the autocorrelation of the spot intensity, the widths of the correlation peaks can be obtained as 0.028°, 0.023°, and 0.020° for R, G, and B, respectively, which determine the average size of a random spot. A highresolution SLM can be used to obtain a finer spot. Moreover, we find that the spots of the three spectra are nearly independent, and the crosscorrelations of RG, GB, and BR are 0.007, 0.004, and 0.005, respectively, which means that the random spots are independent.
Experimental results show that we can use a single SLM to generate multispectral random illumination which is close to the wight uniform distribution with high energy efficiency. The method in this paper can be used for multispectral phaseretrieval, multispectral microscopy imaging, single pixel imaging and other fields.
Conclusions
In summary, we have developed a method to generate multispectral random illumination by a piece of an SLM whose modulation range is a slightly greater than 1λ. This method has a high energy efficiency and computing efficiency. Moreover, it does not require a complex system, and the complexities of the system and algorithm will not significantly increase when a new spectrum is added. In our method, every spectrum illuminates the entire area of the SLM; thus, the random illumination has a very high resolution. We also suppressed the zerothorder spot in the far field to improve the illumination quality. This method can be used for multispectral microscopy, phase retrieval, structured illumination imaging, singlepixel imaging, and other applications.
Abbreviations
 LCSLM:

Liquid crystal SLM
 PDF:

Probability density function
 PSF:

Point spread function
 SLM:

Spatial light modulator
 VA:

Valid area
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XC deduce the formulas and conduct the simulations and wrote the manuscript. ZS and WH supervised the project and contributed the main conceptual ideas. All authors read and approved the final manuscript.
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Correspondence to Xiao Chen.
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XC is now a doctoral student in Automatic Target Recognition(ATR) laboratories, National University of Defense Technology. His research interests are imaging and phase retrieval. ZS and WH are associate professor and professor in ATR, NUDT. There research interests are signal processing and imaging.
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Keywords
 Spatial light modulators
 Multispectrum
 Random illumination
 Genetic algorithm