Open Access
Add to BibSonomy
Abstract
Off-axis projection is a common practice for reconstructions of Fourier holograms displayed on liquid crystal on silicon (LCoS) spatial light modulators (SLMs), as it spatially separates the image from the undiffracted light. Binary gratings encoded within the holograms enable maximum angular separation. However, as a result, two mirror images of equal intensities are present in the reconstruction. To introduce asymmetry to the intensity distribution and suppress one of those images, we propose a phase mask with a subpixel pattern. Presented results prove the potential of in-built SLM phase-mask layers for optimizing efficiency of the off-axis holographic projection.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Efficient image displays and projectors are one of the most sought after technologies due to their broad commercial applications. Computer-generated holography (CGH) is a solution that has many advantages over the state-of-the-art technologies. With CGH it is possible to project 2D or 3D images at an almost unlimited range of dynamically variable distances [1,2]. Perhaps one of its biggest advantages over other display technologies is the simplicity of the setup, its small bulk size and high energy efficiency [3]. The two most common devices for holographic modulation are digital micromirror devices (DMDs) which are typically used for binary amplitude modulation [4,5], and LCoS spatial light modulators of more flexible applications. LCoS devices have high diffraction efficiency, though it is limited by the restraints of device miniaturization: a so-called zeroth order of diffraction is visible in reconstructions Fourier holograms due to the light reflected from inactive areas between the pixels of an SLM [6,7]. Typically, diffraction gratings are used to achieve off-axis projection and avoid the overlap of the undiffracted light and the useful image, both in holographic projection [8–10] and other applications, such as beam shaping [11,12] or optical information processing [13]. The highest separation angle can be achieved when a binary mask of dense, 2-pixel-wide period is added to the displayed phase information. As the result, however, two images of equal intensities are obtained on the opposite sides of the zeroth order of diffraction, decreasing the maximal diffraction efficiency twofold.
Limiting CGH reconstruction to a single image is a known challenge and many attempts have been proposed to date. State-of-the-art solutions include spatial filtering [14] which leads to bulky setups, correlation systems [15] of high susceptibility to noise, and more complex calculations [16], which all complicate the highly advantageous simplicity of a holographic projector. Wang et al. proposed a solution for static DOE structures based on the implementation of blazed caps on each rectangular pixel of a fabricated DOE in order to shift the resulting image away from the zeroth order of diffraction [17]. In our work, we propose a phase mask added as an SLM layer that serves as a static, blazed, sub-pixel modulation for dynamic display of computer-generated holograms. The concept of such mask can be applied to pixels of non-rectangular shapes as well. The main application of the presented solution is two-dimensional projection using Fourier holograms. However, it can also be adopted in other applications where the holographic image is reconstructed in the Fraunhofer far field of diffraction, such as Fresnel holograms [18–20] or optical information processing, e.g. optical correlators [21,22] designed to for converging beam illumination.
2. Theoretical discussion
A true reconstruction of a Fourier hologram is, up to a constant, identical to a Fourier transform of its complex amplitude. In practice, the intensity distribution is determined by the amplitude and phase present on the surface of the SLM. When the hologram is displayed on a typical LCoS SLM, the periodic pattern of pixels adds additional frequencies to the complex amplitude, which leads to a presence of image duplicates in the reconstructed wavefront [23].
To support this reasoning, we assume a hologram of transmittance $F(u,v)$ illuminated by an incident spherical convergent wave of radius $z$, which encodes a field of a complex amplitude $O(x-x_0,y-y_0)$ corresponding to an object shifted by a vector $[x_0, y_0]$:
where the symbol $\mathcal {F}$ denotes a Fourier transform of the function $O(x-x_0,y-y_0)$ with spatial frequencies $f_x=-u/\lambda z, f_y=-v/\lambda z$. At a distance $z$ we obtain the intensity distribution:The transmittance of the hologram sampled with the constant $d$ (see Fig. 1) within the sample centered at the point $(u=md, v=nd)$ can be described as $t(u-md, v-nd)\times F(md, nd)$, where $t(u,v)$ corresponds to a single elementary cell of the sampling. In general, this function can be a complex one, defined by the following product of amplitude and phase factors: $t(u,v) = |t(u,v)|e^{[i \varphi (u,v)]}$.
A one-dimensional case with a non-limited aperture of the hologram simplifies the discussion while upholding the universality of conclusions regarding the method. As such, up to a certain constant, the transmittance of a sampled Fourier hologram can be written as follows:
where $\otimes$ is a convolution operator, $comb(u)=\Sigma _n \delta (u-n)$ and $\delta (u)$ denotes a Dirac delta function. Then, omitting the factor $e^{i2\pi z/\lambda }/{i\lambda z} \cdot e^{i\pi x^2/z\lambda }$ which does not impact the intensity distribution, the reconstruction of the field at the distance $z$ leads to: where $D=\lambda z/d$ is the distance between neighboring diffraction orders of the SLM, $T$ and $\Phi$ denote the Fourier transforms of the amplitude factor $|t(u)|$ and the phase factor $e^{i \varphi (u)}$ respectively.In general, the transmittance $t(u,v)$ of a single sampling cell can be a phase-and-amplitude function. Amplitude apodization [24], that is a modification of amplitude transmittance of hologram pixels, changes the shape of modulating function $T$, which in turn alters intensities of image duplicates in different diffraction orders. This method is particularly useful in increasing the uniformity of intensity in off-axis projections. In this work, however, we focus on the phase-only factor which can further influence the modulating function without sacrificing total intensity of the reconstruction. In particular, we propose phase apodization with a linear phase function which enables a shift of the signal envelope without affecting its shape, leading to more uniform off-axis projections as well.
This simple and promising case involves linear phase $\varphi (u)=2\pi (au)$. With the Fourier transform $\mathcal {F}\{e^{i2 \pi (au)}\}=\delta (f_x-a)$ we obtain, in the reconstruction plane, the field of:
In two dimensions, with $\varphi (u,v)=2\pi (au+bv)$, Eq. (5) can be written as:
When $a=b=0$, the obtained images are periodically spaced and additionally modulated by the function $T(x/dD, y/dD)$, which corresponds only to the amplitude factor $|t(u,v)|$ and usually has its central maximum in the point (0,0). According to Eq. (6), output intensity distribution of diffractive orders can be modified by phase apodization. To reconstruct the image around point $(x=x_o+nD,y=y_o+mD)$ with maximum efficiency, the modulating function needs to be shifted appropriately. Taking into account the distribution of parasitic orders of diffraction on the SLM, the most efficient projection can be achieved in between those orders. In practice that means the image is shifted by a multiple and half-period in one or both directions, e.g. for a single direction:
If we limit ourselves to the area between the central and first diffraction order, then $n=m=0$ and the apodization is limited to the lowest phase values. In that case, there should be a $\pi$-deep linear increase of phase in one of the directions along the width $d$ of the cell.
2.1 Apodization design
The ideal phase mask function $\varphi (u)$ increases the phase value linearly within the SLM pixel. However, for preliminary testing of the method, we decided to consider binary approximations of that apodization function as well. In this paper, we propose two phase apodization designs: a binary grating of $0.5\pi$ depth of modulation and a blazed kinoform of maximum depth modulation of $\pi$.
The designed additional phase information shifts the envelope of the off-axis hologram reconstruction, so that the intensity maximum follows the center of the displayed image. For rectangular pixels, the envelope follows a sinc function shape in both X and Y directions. However, in our solution the effect does not rely on the fixed shape of a pixel and as such, varied envelopes would follow the intensity shift caused by the chosen structure.
The densest pattern that can be displayed on an LCoS SLM is a binary mask of a 2-pixel period. Thus, applying a software-based phase mask in a setup without any additional elements can only be realized by undersampling the SLM pixel pattern in at least one direction, creating areas of 2x1 or 2x2 pixels, or more, that correspond to a single pixel of a calculated hologram. Each of these super-pixels can carry additional, easily controlled phase information of a period smaller than itself. However, this solution sacrifices available diffraction angles in one or both directions and limits the overall angular sizes of the reconstructed images.
For optimal effect, a subpixel pattern needs to be applied to the SLM surface as an additional, statically manufactured layer. As the light crosses each layer of a reflective SLM twice, those transmission masks should be fabricated with a $0.25\pi$ and $0.5\pi$ maximum depth of modulation on a single pass, respectively (Fig. 2). While we expect to achieve better results using the blazed mask, the binary structure has the advantage of fabrication simplicity.
Fig. 2. Cross-sections of proposed transmission phase masks superimposed on SLM surface: (a) a binary mask, (b) a kinoform mask.
3. Simulations
The used SLM is a HOLOEYE Pluto device, a Full HD modulator with 8 $\mu$m pixel pitch and 93% fill factor. To reproduce the parameters of the device, simulations were carried out by 32x oversampling of an array. A single pixel of the SLM was simulated as an array of 31x31 px, with a 1 px wide spacing in-between, which resulted in 93.8% fill factor, close to the nominal value for the device. In that configuration, each pixel of the simulation array corresponds to 0.25 $\mu$m portion of a physical SLM pixel. The simulations were carried out using a modified convolution approach [25] for an area of 1000x1000 physical pixels.
Using an iterative Fourier transform algorithm [26], we encoded Fourier holograms of a rectangle with a thin dark border, a circle and a letter A (Fig. 3). The rectangle image was chosen for the ease of comparison of the intensity distribution in a non-apodized off-axis projection and one with the phase apodization applied, while the other two correspond to more typical holograms displayed on SLMs. Phase apodization patterns (binary or kinoform, alternatively) were applied to the holograms in those simulations on the whole calculation array, including the inactive part of the SLM surface. We calculated the fast Fourier transform of the resulting phase distributions to obtain image reconstructions.
In particular, holograms were reconstructed for a set of increasing phase depths of apodization, ranging from $0.25\pi$ to $0.75\pi$ for a binary function and from $0.5\pi$ to $1.5\pi$ for a kinoform one, to investigate the influence of maximum phase value on duplicate suppression (Fig. 4). Simulation results suggest that the chosen values of $\varphi = 0.5 \pi$ for a binary grating and $\varphi = \pi$ for a kinoform one successfully suppress the twin image. While there is a small margin for error, phase mismatch reintroduces the presence of image duplicates. This shows that SLM phase masks presented in Fig. 2 should be calculated and fabricated with a specific designed wavelength in mind. In reflective SLMs, where light passes through each layer two times, the depth of modulation should be designed as half of the optimal values, that is $\varphi = 0.25 \pi$ and $\varphi = 0.5\pi$ for binary and kinoform gratings, respectively. While phase masks cannot match the whole spectrum simultaneously, the proposed solution can be used in monochromatic applications or, alternatively, in a configuration where RGB image is realized by spatial multiplexing of the SLM surface into areas displaying each of the constituent colour holograms separately [27,28]. A phase mask could be divided into analogous fragments of adjusted depth of phase modulation.
Fig. 4. Amplitude distribution in hologram reconstructions for varying phase modulation depth of the apodization with its horizontal cross-sections: binary (top) and kinoform (bottom) apodization.
Next, the results for no apodization and both types of phase functions were compared. For the binary mask of $\varphi = 0.5\pi$ phase depth, the envelope of the signal in the image reconstruction was successfully shifted towards the area of off-axis projection (marked with a green line in Fig. 5). As expected, the kinoform function leads to even more optimal results, especially with regards to image duplicates of higher orders, i.e., $-2^{nd}$ order. Binary apodization, while successfully shifting the signal envelope, does not fully suppress the central areas of all the additional images. Numerical analysis (Table 1) shows the intensity integrals in the areas of image duplicates in arbitrary units, as well as the asymmetry ratio calculated as the intensity ratio of the main image to the twin image present in the off-axis projection. In the apodized cases, the intensity is redirected into the area of a main image, as the center of the signal envelope matches its position. The expected values depend on the shape of an object and can be calculated using Eq. (6). For the chosen rectangular object we obtained the theoretical assymetry ratio of 9.6. After applying the kinoform apodization mask, the intensity within the main image is more than ten times higher than the intensity in the twin image area, which is consistent with the calculations. Even higher values of the asymmetry ratio can be obtained for images limited to the central area of the off-axis projection, as proved in the experimental part of presented research.
Fig. 5. Simulation results. Amplitude distribution (a) without a mask; (b) with a binary grating phase mask; (c) with a kinoform phase mask. Green line: main image area. Red line: twin image area.
Table 1. Intensity integrals [a.u.] in areas of neighboring image duplicates and their asymmetry ratio.
The presented successful selection of one of the off-axis images is an important achievement for the holographic displays, as it optimizes the accessible intensity in the main image. At the same time, the uniformity of intensity in the image area improves. Its ratio depends on the slope of the envelope within the area. With no apodization, both twin images are displayed at the steeper edges of the envelope and fade at the edges of the area of interest. Using the proposed phase apodization improves the visibility of the borders of a reconstructed image (Fig. 6), as the target image is placed around the maximum of the sinc envelope function where its slopes are mild. For a kinoform mask, resultant intensity values on both edges of a symmetrical image should be equal, which is consistent with the obtained simulation results.
Fig. 6. Intensity distribution in the area of an off-axis image: (a) with no apodization; (b) with binary and (c) kinoform phase apodization applied (Gamma = 1.3).
4. Experimental confirmation
For the experimental confirmation, a following setup was prepared (Fig. 7): the HOLOEYE Pluto SLM was illuminated with a converging monochromatic wavefront passing through a beam splitter. The wavefront reflected from the modulator propagated towards a screen through said beam splitter. A focused image was formed on the screen.
Fig. 7. Setup used in the experiment. C – collimator, L – lens converging the beam on a screen, P - polarizer, BS – beam splitter, S – observation screen.
The principle of the proposed phase-function solution can be easily, and accurately, proven in an experiment using software-only encoding of the phase apodization. For that purpose, an undersampled hologram was displayed on the HOLOEYE Pluto device, where a superpixel consisted of an area of 2x1 physical pixels. It is the configuration with the least sacrifice of the hologram resolution that still enables software encoding of binary phase apodization. LCoS SLM devices are not capable of displaying gradients of non-discrete values and as such, only a binary grating of discrete values could be implemented with this method.
The obtained hologram reconstructions (Fig. 8) prove that a simple binary function has significant impact on the envelope of the signal. We obtained intensity up to 12 times stronger in the main image than in its twin image (Table 2) – the exact value depending on the displayed image – as well as increase in the maximum intensity of the main image. The proposed function of a $\varphi = 0.5\pi$ phase depth not only suppresses the twin image, but it also increases the uniformity of intensity of the main desired image (Fig. 9).
Fig. 8. Experimental results. Intensity of the undersampled hologram reconstruction with illumination $\lambda$ = 532 nm (a) without phase apodization; (b) with a software-based binary mask of $0.5\pi$ depth of phase modulation.
Fig. 9. Experimental results. Intensity distribution in the area of an off-axis image: (a) with no mask; (b) with a binary mask of $0.5\pi$ depth of phase modulation (Gamma = 1.3).
Table 2. Software-based phase apodization for $\lambda$ = 532 nm. Intensity integrals [a.u.] in areas of twin images and their asymmetry ratio.
In order to prevent the loss of resolution, a physical, subpixel mask needs to be fabricated. For experimental proof, the apodizing mask has to be placed in such a way that it can be then adjusted horizontally, vertically and in rotation to ensure that the periodicity of the structure matches the pixels of the modulator. The outer layers of an LCoS SLM cause a separation too high for the satisfying effect by placing the mask directly on the surface of the device. As an alternative, it is possible to image the surface of the SLM with a 4f system and place the phase mask at the imaging distance, though in that setup phase masks of doubled depth of modulation as compared to Fig. 2 should be used. The fabricated structures need to show good fidelity to the design (see Fig. 4), which have not been achieved yet with technologies available to the authors. Positioning of the mask is also crucial for proper apodization of the device, with a shift as small as $\frac {1}{16}$th of the SLM pixel size (0.5 $\mu$m) causing the reappearance of the suppressed image (Fig. 10), and the difficulty of adjustment further increases with a smaller pixel pitch of modern devices. For those reasons, it would be recommended that the structure is manufactured as an integral SLM layer during device fabrication in order to avoid any potential relative movement of the elements.
Fig. 10. Simulated amplitude distribution in hologram reconstructions for varying misalignment of binary phase masks with their horizontal cross-sections: (a) no misalignment, (b) misalignment of $\frac {1}{16}$th and (c) $\frac {1}{8}$th of the SLM pixel size (Gamma = 1.2).
5. Conclusions
In our work, we presented the successful means for introducing intensity asymmetry to the off-axis projection of Fourier holograms on LCoS spatial light modulators. In the experiment, we achieved more than tenfold increase of intensity ratio between the selected image and its twin image.
To our knowledge, it is the first time a solution was proposed for a phase mask for dynamic display of holograms. The proposed masks shift the intensity envelope of the hologram reconstruction, independent on its shape: the solution would be equally applicable for non-rectangular pixels of any SLM. An important strength of the method is the lack of resolution loss in any of the directions, thanks to the subpixel phase variations in the apodizing mask. The simplicity of the holographic projection setup is retained as well, as only the laser source of an adjusted polarisation, the spatial light modulator and the thin apodization mask are needed for achieving the improved hologram reconstruction. This makes our solution especially beneficial for applications such as near-eye displays where more bulky state-of-the-art solutions cannot be applied. We believe that the shown research holds a potential for in-built SLM layers in the devices intended for holographic display applications.
Funding
Narodowe Centrum Nauki (2021/41/N/ST7/01520).
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.
References
1. C. Ying, H. Pang, C. Fan, and W. Zhou, “New method for the design of a phase-only computer hologram for multiplane reconstruction,” Opt. Eng. 50(5), 055802 (2011). [CrossRef]
2. A. Velez-Zea, “Iterative multiplane hologram generation with mixed constraint,” Appl. Opt. 60(2), 224–231 (2021). [CrossRef]
3. M. Makowski, I. Ducin, K. Kakarenko, J. Suszek, M. Sypek, and A. Kołodziejczyk, “Simple holographic projection in color,” Opt. Express 20(22), 25130–25136 (2012). [CrossRef]
4. D. Akbulut, T. J. Huisman, E. G. van Putten, W. L. Vos, and A. P. Mosk, “Focusing light through random photonic media by binary amplitude modulation,” Opt. Express 19(5), 4017–4029 (2011). [CrossRef]
5. D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” MOEMS display and imaging systems 4985, 14–25 (2003). [CrossRef]
6. D. Palima and V. R. Daria, “Effect of spurious diffraction orders in arbitrary multifoci patterns produced via phase-only holograms,” Appl. Opt. 45(26), 6689–6693 (2006). [CrossRef]
7. V. Arrizón, E. Carreón, and M. Testorf, “Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations,” Opt. Commun. 160(4-6), 207–213 (1999). [CrossRef]
8. A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Broadband suppression of the zero diffraction order of an SLM using its extended phase modulation range,” Opt. Express 22(14), 17590–17599 (2014). [CrossRef]
9. H. Zhang, J. Xie, J. Liu, and Y. Wang, “Elimination of a zero-order beam induced by a pixelated spatial light modulator for holographic projection,” Appl. Opt. 48(30), 5834–5841 (2009). [CrossRef]
10. M. Rubin, G. Dardikman, S. K. Mirsky, N. A. Turko, and N. T. Shaked, “Six-pack off-axis holography,” Opt. Lett. 42(22), 4611–4614 (2017). [CrossRef]
11. A. B. Stilgoe, A. V. Kashchuk, D. Preece, and H. Rubinsztein-Dunlop, “An interpretation and guide to single-pass beam shaping methods using SLMs and DMDs,” J. Opt. 18(6), 065609 (2016). [CrossRef]
12. E. Y. Zlokazov, “Methods and algorithms for computer synthesis of holographic elements to obtain a complex impulse response of optical information processing systems based on modern spatial light modulators,” Quantum Electron. 50(7), 643 (2020). [CrossRef]
13. R. A. Suarez, A. A. Neves, and M. R. Gesualdi, “Optical trapping with non-diffracting Airy beams array using a holographic optical tweezers,” Opt. Laser Technol. 135, 106678 (2021). [CrossRef]
14. M. Agour, C. Falldorf, and C. Von Kopylow, “Digital pre-filtering approach to improve optically reconstructed wavefields in opto-electronic holography,” J. Opt. 12(5), 055401 (2010). [CrossRef]
15. D. G. Crowe, J. Shamir, and T. W. Ryan, “Sidelobe reduction in optical signal processing,” Appl. Opt. 32(2), 179–183 (1993). [CrossRef]
16. M. Gopakumar, J. Kim, S. Choi, Y. Peng, and G. Wetzstein, “Unfiltered holography: optimizing high diffraction orders without optical filtering for compact holographic displays,” Opt. Lett. 46(23), 5822–5825 (2021). [CrossRef]
17. H. Wang, Y. Liu, Q. Ruan, H. Liu, R. J. Ng, Y. S. Tan, H. Wang, Y. Li, C. Qiu, and J. K. Yang, “Off-axis holography with uniform illumination via 3D printed diffractive optical elements,” Adv. Opt. Mater. 7(12), 1900068 (2019). [CrossRef]
18. S. Reichelt and N. Leister, “Computational hologram synthesis and representation on spatial light modulators for real-time 3D holographic imaging,” J. Phys.: Conf. Series 415(1), 012038 (2013). [CrossRef]
19. C. Chang, Y. Qi, J. Wu, J. Xia, and S. Nie, “Speckle reduced lensless holographic projection from phase-only computer-generated hologram,” Opt. Express 25(6), 6568–6580 (2017). [CrossRef]
20. C. Haupt, A. Kolodziejczyk, and H. J. Tiziani, “Resolution and intensity distribution of output images reconstructed by sampled computer-generated holograms,” Appl. Opt. 34(17), 3077–3086 (1995). [CrossRef]
21. J. Nicolás, J. Campos, C. Iemmi, I. Moreno, and M. J. Yzuel, “Convergent optical correlator alignment based on frequency filtering,” Appl. Opt. 41(8), 1505–1514 (2002). [CrossRef]
22. O. Lópe-Coronado, I. Moreno, J. Campos, and M. J. Yzuel, “Full in-situ characterization of spatial light modulators in an optical correlator. Filter adaptation to operating curves,” J. Modern Opt. 45(12), 2461–2472 (1998). [CrossRef]
23. J. W. Goodman, Introduction to Fourier Optics, (Roberts and Company, 2005).
24. J. Starobrat, S. Fiderkiewicz, A. Kołodziejczyk, M. Sypek, R. Beck, K. Pavłov, M. Słowikowski, A. Kowalczyk, J. Suszek, and M. Makowski, “Suppression of spurious image duplicates in Fourier holograms by pixel apodization of a spatial light modulator,” Opt. Express 29(24), 40259–40273 (2021). [CrossRef]
25. M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995). [CrossRef]
26. M. Makowski, M. Sypek, A. Kolodziejczyk, and G. Mikula, “Three-plane phase-only computer hologram generated with iterative Fresnel algorithm,” Opt. Eng. 44(12), 125805 (2005). [CrossRef]
27. M. Makowski, M. Sypek, I. Ducin, A. Fajst, A. Siemion, J. Suszek, and A. Kolodziejczyk, “Experimental evaluation of a full-color compact lensless holographic display,” Opt. Express 17(23), 20840–20846 (2009). [CrossRef]
28. P. He, J. Liu, T. Zhao, Y. Han, and Y. Wang, “Compact large size colour 3D dynamic holographic display using liquid crystal display panel,” Opt. Commun. 432, 54–58 (2019). [CrossRef]
