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Abstract
Complex-amplitude modulation of light fields with a digital micromirror device (DMD) has been widely used in holographic image projection. DMD is a binary-amplitude modulator, and its use for complex field modulation in a 4f configuration requires low-pass filtering. However, the reconstructed fields suffer from low resolution due to the limited bandwidth for the existing methods such as the Lee and superpixel methods. Here, we report a direct binary search (DBS) method to design high-resolution complex-amplitude holograms. The method is able to increase the spatial bandwidth up to twice that of the superpixel method. Numerical simulations and experiments are presented to demonstrate the method, which show that the errors are reduced by about 60% and 40% respectively for the test fields compared to the superpixel method. Furthermore, the measured efficiency of laser light can be improved by a maximum of 60%.
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1. Introduction
Holography was first presented by Gabor in 1948 for reducing aberrations of electron microscope images [1]. Early holograms must be recorded on physical medium, and with the aid of electronic technology, the computer generated hologram (CGH) opens a new door for holography [2,3]. In the production process of CGHs the holograms are no longer physically constrained by illumination coherence, environmental turbulence, and the imperfect recording medium. An arbitrary complex-amplitude light field can be synthesized from the CGH projection, and this ability for complex-amplitude modulation of light gives rise to holographic display [4,5], optical tweezers [6], quantum information processing [7], imaging system [8], optical communication [9] and so on.
The pre-calculated holograms can be imprinted onto traditional optical diffractive elements which are static [10,11]. With the advances in the spatial light modulator (SLM) technology, CGHs are now widely used in dynamic applications [12–14]. The SLM is mainly divided into two types, liquid crystal SLM (LC-SLM) and DMD. The stability and high refreshing rates of DMD make it more popular in many scientific fields [15–17]. The DMD pixels are tiny mirrors that can be tilted to two angles, and these two angles correspond to an ON and OFF configuration. The DMD is a binary amplitude modulation device, but the complex-amplitude hologram is ideally complex valued function. So the complex-amplitude hologram should be encoded into a binary type CGH before it is displayed on the DMD. Although there is the time-multiplexing technique to project the grayscale holograms [18,19], the flicker of the pixels and the low refreshing rates limit its applications where the high stability of light intensity or high refreshing rates are required, such as the ultracold atom experiments [20]. There are many methods to perform the complex-to-binary conversion [2,21–25]. Recently a superpixel method was proposed to realize the complex modulation [26] , and the method can achieve higher fidelity when compared with the traditional Lee method [22,27]. However, there exists the highest allowed spatial frequency to ensure the validity of the method, so the target field must be rescaled to the corresponding resolution before encoding, which makes the high-resolution target field lose sharp features and details [26]. This disadvantage limits its application in the manipulation of quantum systems at the single-atom level [28].
In this paper, we propose a DBS method to modulate the amplitude and phase of light. Compared to the superpixel method, the proposed method can double the upper bound of the allowed spatial frequency. In other words, our method still works when we increase the radius of spatial filter, and it is particularly suitable for the high-resolution holographic projection. We review the superpixel method and introduce the DBS algorithm in section 2. In section 3, the simulations are presented, and experimental results are also provided in Section 4. Finally, we make a conclusion in section 5.
2. Principle
2.1 Method
The superpixel and DBS methods share the same optical setup as illustrated in Fig. 1(a). Two lenses are placed slightly off-axis with respect to each other so that phase difference emerges in the target plane. There is an aperture between the two lenses, and the position of the aperture is chosen $(x,y)=(-a,na)$ with respect to zeroth order diffraction, where $a=\frac {-\lambda f}{dn^{2}}$, $\lambda$ is the wavelength of the light, $f$ is the focal length of the first lens, $d$ is the pixel pitch of the DMD, and $n$ is the superpixel size. With the assignment of the aperture, the neighboring DMD pixels have phase difference in the target plane as shown in Fig. 1(b), where $n = 4$. In the superpixel method the $4\times 4$ DMD pixels are combined into a single superpixel, which can modulate the phase and the amplitude of the field in the target plane. In order to guarantee that $4\times 4$ DMD pixels are effectively averaged, the maximum value of aperture size is limited. However, the upper limit of the size reduces the achievable resolution of the field in the target plane [26]. Comparatively, with the DBS method the upper limit of the aperture size can be improved.
Fig. 1. (a) The setup of the superpixel and DBS methods for holographic image projection. (b) The phase map of a superpixel in the target plane. (c) The principle of the DBS algorithm.
Different from the superpixel scheme, the DBS scheme replaces the lookup table with an iterative process. The principle of the DBS algorithm is presented in Fig. 1(c). Firstly, We use a random pattern as the initial binary hologram $H_{0}$. Then the reconstructed field $E_{out}$ in the target plane is calculated using the propagation formula,
where $m\in 1\sim M$ and $n\in 1\sim N$ are the pixel indices of the hologram. $\textrm{FFT}\{ \cdot \}$ and $\textrm{IFFT}\{ \cdot \}$ denote two-dimensional (2D) fast Fourier transform and 2D inverse fast Fourier transform, respectively. $\Theta (m,n)$ is the phase gradient (equivalent to placing the spatial filter off-axis), expressed by $\rho (k,l)$ is the function of the spatial filter, which is expressed asThe convergence behavior of the DBS algorithm is shown in Fig. 2(a), during which we choose the aperture size $r=\frac {\lambda f}{8d}$, and the high-resolution test field which consists of a picture of black cat in its intensity and a picture of white cat in phase profile, as shown in Fig. 2(c). The error reaches 3% in the first iteration and it is reduced to around 2% in the fourth iteration. For the following discussion we adopt 4 iterations in the DBS algorithm.
Fig. 2. (a) The error $\delta$ as a function of iteration number. (b) The error $\delta$ as a function of DLSS. (c) Reconstructed fields using different DLSS.
The error $\delta$ is also greatly affected by the aperture size in the DBS method. In order to optimize the value, we introduce the diffraction limited spot size (DLSS) in target plane to characterize aperture size. The DLSS can be expressed in units of DMD pixels:
We calculate the variation of error with DLSS as illustrated in Fig. 2(b), and some of the reconstructed images are shown in Fig. 2(c). The error reaches its lowest value when DLSS is about 4, and in its vicinity the error changes a little. When the DLSS is greater or less than the critical value the error increases. The reason is that large radius destroys the interference between pixels, while small radius lowers the highest allowed spatial frequency, both of which decrease the modulation fidelity.
Due to the finite aperture size one DMD pixel contributes to the reconstruction plane in all directions because its diffraction pattern overlaps with that of all the surrounding pixels. In the superpixel method, however, only the diffraction of $4\times 4$ pixels is considered, and the influence from others is neglected. In comparison, the adaptive process of the DBS method can balance the diffraction patterns of all pixels. Although it evaluates each pixel individually, the error $\delta$ can be reduced globally.
2.2 Bandwidth of the 4f system
In the 4f system, the spatial filter acts as a low pass filter with a finite ability to resolve detail. In other words, it determines the spatial bandwidth, or the resolution. This finite resolution leads to the blur of the light field when it propagates through the system. So the size of the spatial filter is crucial for the quality of high-resolution fields reconstructed by the superpixel and DBS methods. In the superpixel method, the radius $r$ is specially chosen $r\le \frac {\lambda f}{16d}$ such that the highest allowed spatial frequency is lower than $\frac {\pi }{8d}$ $\rm {rad}\cdot \rm {m}^{-1}$, and the corresponding optimal DLSS is 8 in the modulation of of high-resolution fields [26]. However, when the target field is required to be sampled at the spacing of 4 DMD pixels, the Nyquist frequency is $\frac {\pi }{4d}$ $\rm {rad}\cdot \rm {m}^{-1}$, which is larger than the highest allowed frequency. So it means that the superpixel method remains valid with resolution reduction. In the DBS method the DLSS is 4, and thus the radius of the filter can be improved to $r=\frac {\lambda f}{8d}$. The corresponding highest allowed spatial frequency of the DBS method is $\frac {\pi }{4d}$ $\rm {rad}\cdot \rm {m}^{-1}$, which equals to the Nyquist frequency. Hence, compared with the superpixel method we double the spatial bandwidth and enhance the resolution of the system with the DBS method.
3. Simulations
In order to analyze the performance of our method, high-resolution fields are numerically reconstructed with different methods. The influence of the DLSS on the image quality is presented as well. The first test field contains two ’cat’ pictures in its intensity and phase, as illustrated in Fig. 3(a) and 3(b). The binary holograms with different methods and DLSS are shown in Fig. 3(c–f). The corresponding reconstructed fields are shown in Fig. 3(g–n). We adopt DLSS = 4 in Fig. 3(g–j) while it is 8 in Fig. 3(k–n). Figure 3(g), 3(i), 3(k) and 3(m) present the reconstructed intensity and phase patterns by the DBS method. The best result of the DBS method is shown in Fig. 3(g) and 3(i), where we are able to see clear fur and the edges are preserved favorably. Using this method, the error is about 2.2%. Similarly, the resulting fields by applying the superpixel scheme are also calculated as shown in Fig. 3(h), 3(j), 3(l) and 3(n). Among them the best reconstructed field is Fig. 3(l) and Fig. 3(n) with an error of 5.3%. However, the intensity and phase profiles of the obtained field are blurred badly. To further quantify the ability for the detail reconstruction, the intensities of the enlarged area are normalized to total intensity, as shown in Fig. 3(g) and 3(l), and we calculate the root-mean-square error (RMSE) of the area. It is $1.4\times 10^{-5}$ for the DBS method and $1.7\times 10^{-5}$ for the superpixel method. The comparison shows that the DBS method can reconstruct the field with higher resolution.
Fig. 3. Simulated results of the ’cat’ pictures under different DLSS. (a,b) Intensity and phase profiles of the target field. (c,e) Binary holograms according to the DBS method with DLSS = 4 and DLSS = 8. (d,f) The two binary holograms are the same when using the superpixel method. (g,i,k,m) Reconstructed fields by the DBS method. (h,j,l,n) Reconstructed fields by the superpixel method.
In addition, the light efficiency of the two methods is also computed as the fraction of energy incident on the DMD directed into the reconstruction:
where the $MN$ is the total power of the incident light. For the superpixel scheme the optimal value of DLSS is 8 and the corresponding aperture size is smaller than that in the DBS scheme. Based on this, more light can be collected with the proposed method. In Fig. 3(l) and 3(n) the light efficiency of the superpixel method is 1.2%, and we find it’s 2.1% for the DBS method in Fig. 3(g) and 3(i). The light efficiency is improved by 75% compared with the superpixel method.Next, another test field with sharply varied phase profile is presented in Fig. 4. As shown in Fig. 4(l), the intensity profile in the enlarged region is severely contaminated by the superpixel method because of the variation. The RMSE of the enlarged region is $6.2\times 10^{-5}$ and the total error is $\delta =6.0\%$. However, the contamination is largely suppressed in Fig. 4(g) with the RMSE of $3.8\times 10^{-5}$ using our method, and the total error is $\delta =2.3\%$. The results demonstrate the ability to suppress the undesired ripples in the reconstructed intensity distribution through our method. The light efficiency of this test field is not changed much.
Fig. 4. Simulated results of the ’chimpanzee’ and ’mandrill’ object images under different DLSS. (a,b) Intensity and phase profiles of the target field. (c,e) Binary holograms using the DBS method with DLSS = 4 and DLSS = 8. (d,f) The two binary holograms are the same according to the superpixel method. (g,i,k,m) Reconstructed fields by the DBS method. (h,j,l,n) Reconstructed fields by the superpixel method.
4. Experiments
Experiments are also carried out to verify the proposed method. The optical setup is shown in Fig. 5. A laser beam with a wavelength of $\lambda$ = 785 nm is used as a light source, and the DMD (DLP 3000, Texas Instruments) has $608\times 684$ pixels with pitch size of $d$ = 7.64$\mu$m. The two lenses have equal focal lengths of $f$ = 400 mm, and the images are recorded by a CCD camera (DCU224M-GL, Thorlabs) located in the back focal plane of the second lens. The phase distribution of the reconstructed field is extracted by the interferometer setup.
Fig. 5. Optical setup for holographic display. Light emerging from the DMD interferes with the reference beam in the position of the CCD camera, and the phase profile of the reconstructed field is extracted by the interference fringes. The intensity profile is directly measured by the CCD camera when blocking out the reference beam.
In the experiments, for the DBS scheme we choose the parameter DLSS = 4, and the corresponding aperture size is $r$ = 5.2 mm, while it’s 2.6 mm for the superpixel scheme. The optical reconstruction with different methods is presented in Fig. 6. Figure 6(a) and 6(c) show the obtained field of ’cat’ pictures using the DBS approach, and the error with respect to the target field is 4.0%. Meanwhile, it’s 6.5% for the one encoded with the superpixel method as shown in Fig. 6(b) and 6(d). The difference can be clearly observed by comparing the two groups of reconstruction. The images obtained by the DBS method give higher contrast and preserve more edge information. However, for the superpixel scheme the details of the intensity and phase profiles are blurred and undiscerned, as presented in Fig. 6(b) and 6(d). The RMSE of the zoomed area in Fig. 6(a) is $1.5\times 10^{-5}$ using the DBS method, whereas we find it is $2.0\times 10^{-5}$ in Fig. 6(b) using the superpixel method. In Fig. 6(e)–6(f), the RMSE of the zoomed part is $4.8\times 10^{-5}$ for the DBS method and $6.0\times 10^{-5}$ for the superpixel method. Although the high frequency components of images are lost due to the actual aperture size in the experiments, the RMSE may be reduced compared to the simulations if enough low frequency components are well preserved. Nevertheless, it can be observed that the problem of ripples caused by steep phase gradients is improved by our method. Besides, the whole image reconstructed by the DBS method is of higher visual quality and contains less noise compared to the superpixel method, and the total errors in the two methods are 4.9% and 7.8% respectively. The deviations of the experimental values from the simulated results mainly arise from the process of extracting phase, during which some high frequency information of the phase profile is lost [29]. To avoid aberration of the system, we first measure the reference phase by constructing a plane wave, and then subtract it from the measured phase to get Fig. 6(c), 6(d), 6(g) and 6(h).
In addition, we measure the laser power of the ’cat’ images in the reconstruction plane with the same input power for the two methods. As a result the light efficiency of the DBS scheme increases by 60%. From the above discussion our method has better performance than the superpixel method in the modulation of high-resolution fields.
5. Conclusion
In this paper, we have proposed a DBS method for high-fidelity spatial light modulation with DMD and a 4f double-lens setup. In the method, the binary value of each pixel of DMD is independently determined according to the reconstruction error. Although the operation is uncorrelated to each other, the global error is reduced gradually. Based on the principle of DBS we can double the bandwidth of the 4f system compared with the superpixel method. The computational simulations indicate our method has a nearly 60% lower error, and meanwhile we observe about 40% reduction of the error in the experiments. Besides, the total measured efficiency of light usage for the test fields can be improved up to be 160% of the traditional value. All of the above results validate that our method outperforms the superpixel method for high-resolution holographic image projection. In the future, our method can be improved in its computational time and extended to generate complex optical potentials for the ultracold atom experiments [28].
Funding
Fundamental Research Funds for the Central Universities; National Natural Science Foundation of China ); Anhui Initiative in Quantum Information Technologies; Chinese Academy of Sciences. (11874341).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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