Suppression of undesired diffraction orders of binary phase holograms Christian Maurer, Andreas Schwaighofer, Alexander Jesacher, Stefan Bernet,* and Monika Ritsch-Marte Division for Biomedical Physics, Innsbruck Medical University, 6020 Innsbruck, Austria *Corresponding author:
[email protected] Received 7 March 2008; revised 5 June 2008; accepted 9 June 2008; posted 24 June 2008 (Doc. ID 93545); published 22 July 2008
A method to remove undesired diffraction orders of computer-generated binary phase holograms is demonstrated. Normally, the reconstruction of binary Fourier holograms, made from just two phase levels, results in an undesired inverted image from the minus first diffraction order, which is superposed with the desired one. This can be avoided by reconstructing the hologram with a diffuse light field with a pseudorandom, but known, phase distribution, which is taken into account for the hologram computation. As a consequence, only the desired image is reconstructed, whereas all residual light is dispersed, propagating as a diffuse background wave. The method may be advantageous to employ ferroelectric spatial light modulators as holographic display devices, which can display only binary phase holograms, but which have the advantage of fast switching rates. © 2008 Optical Society of America OCIS codes: 050.1380, 090.1995, 230.6120.
1. Introduction
Typically, binary phase holograms, corresponding to two-dimensional pixel arrays that consist of only two phase values, diffract not only into the desired first diffraction order, but with equal efficiency also into the disturbing minus first order [1,2]. The appearance of a conjugate diffraction order is particularly disturbing for on-axis Fourier holograms where the reconstructed image field is superposed with a second, inverted (“pseudoscopic”) image of the same intensity. Depending on the size of the phase jump between the two available phase levels, also higher order components appear, which show enlarged copies of the image with a magnification factor corresponding to the diffraction order. As an example, a comparison between reconstructed continuous and binary phase holograms, displayed at the same spatial light modulator, is shown in Fig. 1. Whereas continuous phase holograms [Figs. 1(a) and 1(c)] reconstruct just the desired images, in the case of bin0003-6935/08/223994-05$15.00/0 © 2008 Optical Society of America 3994
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ary phase holograms [Figs. 1(b) and 1(d)], these are superposed by their corresponding inverted images. These effects limit the usefulness of ferroelectric spatial light modulators (SLMs) as display or projection devices for computer-generated holograms (CGHs), since this type of SLM can display only binary phase structures. In other respects, ferroelectric SLMs would be highly suitable as holographic displays, since they allow a very fast switching rate in the 10 kHz regime (or even faster) [3–7]. This can be useful for displaying time-multiplexed holograms [8] for, e.g., reproducing color holograms by switching the displayed phase patterns in synchronization with light sources of different colors, optical manipulation of mesoscopic particles in microscopy [9–15], or the trapping of atoms [16,17]. There are well-known methods that separate the desired image field from the undesired diffraction orders for binary holograms. One approach is to produce off-axis holograms by multiplying the transmission function of the original on-axis DOE with an inclined rÞ, where ~ kp is the grating vecplane term ∝ expði~ kp ·~ tor that defines the direction and the magnitude of the tilt. The undesired negative diffraction orders are
Fig. 1. Reconstructed computer-generated on-axis Fourier holograms using a spatial light modulator as a phase display. (a) and (c) were displayed using 256 phase levels in the interval between 0 and 2π, whereas (b) and (d) are the corresponding binary holograms using only the phase levels 0 and π.
thus spatially separated from the positive ones and can be blocked [18]. However, this operation discards the half planes in k space that carry unwanted information and, thus, reduces the available bandwidth of the DOE and the utilizable field of view of the projected image by one-half. Another method is to use Fresnel-type holograms, instead of Fourier holograms, which means that the original on-axis transfer function is multiplied by a lens term ∝ exp ð−iπj~ rj2 =λf Þ. In this case, the desired image is not reconstructed at infinite distance (as for Fourier holograms), but at a certain distance f behind the hologram, whereas the undesired negative diffraction orders are simultaneously defocused and produce a diffuse background [15,19]. However, this method also reduces the available bandwidth and it does not allow the use of certain advantageous features of Fourier holograms, such as their maximal depth of sharpness or the shift invariance of the output image with respect to the transverse position of the hologram. In this paper we demonstrate an alternative method to eliminate the undesired diffraction orders of binary CGHs by illuminating them with a pseudorandom light field with a known phase distribution. This might be achieved with a lithographically produced phase mask that is directly attached to the SLM display and that shifts the phase of each SLM pixel by a randomly chosen, but known, value in ½0; 2π. Note that for reflective displays the double pass through the mask has to be considered. Alternatively, a given random phase mask, for instance, a slab of ground glass, could be used after measuring its phase distribution interferometrically. This known “offset” phase distribution is stored in the computer and taken into account when calculating the binary CGH of any desired image field, as explained later. First attempts in this direction have been reported in [20]. As we will explain, in the ideal case of two available phase levels that differ by π, the maximal first-order diffraction efficiency that can be achieved is 40.5% of the incoming light, which is identical to the maximal efficiency of a standard binary phase hologram and is the same efficiency that can be reached with the other above-mentioned methods (off-axis or Fresnel holograms) to suppress the undesired diffraction orders. However, the dispersion of the diffuse background wave is maximized by the random phase plate, i.e., the background noise is smaller than in the Fresnel
approach. The contrast between the programmed image and the diffuse background depends on the image structure, i.e., the 40.5% “usable” light is concentrated in the programmed bright image areas, whereas the remaining 59.5% is diffusely distributed over the entire image plane. Therefore, the contrast is particularly high for images with only a few bright pixels, or one-dimensional line structures, like those used for optical trapping patterns. To demonstrate the principle, we simulated the pseudorandom phase mask by a phase pattern that was displayed at a part of the same SLM that was used for displaying the binary CGH. In our case, we could use a nematic SLM, which could display continuous phase patterns (with 256 programmable phase levels in the interval between 0 and 2π), as well as simulating a binary ferroelectric SLM by displaying only the two phase values 0 and π. Section 2 describes the basic principle of the method, followed by its experimental demonstration. 2.
Underlying Concept
The diffraction efficiency of an ideal phase CGH (without absorption and with a 100% fill factor) depends on the number n of phase values in the interval between 0 and 2π that are actually used to construct the underlying grating structure. If we consider a blazed grating (“sawtooth grating”) as an elementary example, n would be the number of different phase levels that are addressed within one grating period. In this case, the diffraction efficiency η becomes [21] η¼
n π 2 sin : π n
ð1Þ
A one-dimensional sawtooth grating with a grating period of 14 pixels (with a linear phase spacing in the interval ½0; 2π) is plotted in Fig. 2(a). According to Eq. (1), the corresponding first-order diffraction efficiency of such a blazed grating is 98.3%. The corresponding diffraction pattern is displayed in the inset of Fig. 2(a), where m denotes the diffraction order and η the corresponding diffraction efficiency. The diffraction efficiency η is calculated by simply computing the squared absolute value of the Fourier transform of the corresponding complex transmission function. In this case, there is a dominant peak at m ¼ 1 with η ≈ 0:98. The next part of the figure [Fig. 2(b)] shows a phase pattern corresponding to a binary grating (with the optimal phase step of π). In this case, Eq. (1) predicts an efficiency of 40.5% for the first-order efficiency. However, the same energy is also scattered to the minus first order (and the residual energy to higher, odd-numbered diffraction orders), such that the diffraction pattern looks symmetrical [see inset of Fig. 2(b)]. Next, Fig. 2(c) shows the diffraction pattern of a one-dimensional random phase mask with phase values that are uniformly distributed in the interval 1 August 2008 / Vol. 47, No. 22 / APPLIED OPTICS
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½0; 2π. In this case, the scattered light is randomly distributed as a speckle pattern [inset of Fig. 2(c)]. In Fig. 2(d), the random phase pattern of Fig. 2(c) has been corrected with a binary phase mask, by adding either 0 or π to the phase of each pixel, such that the best possible approximation to the ideal sawtooth grating of Fig. 2(a) is obtained. This one-dimensional situation can straightforwardly be generalized to our two-dimensional diffuse illumination method, where a first pseudorandom phase distribution (c) is corrected by a binary phase mask to get the closest possible approximation to a desired ideal (i.e., continuous) phase hologram. The corresponding diffraction efficiency of the approximated blazed grating in Fig. 2(d) is displayed again in the inset. The firstorder diffraction efficiency is close to 40.5%, whereas the residual intensity is distributed over the whole image plane as a fine speckle pattern. An intuitive explanation for this is given in the remaining part of the figure: the approximated blazed grating of Fig. 2(d) can be numerically decomposed into the sum of two other phase patterns, namely, an ideal sawtooth grating [Fig. 2(e)] and a random phase mask [Fig. 2(f)]. However, now the modulation amplitude of the random phase pattern in Fig. 2(f) is confined to the interval ½0; π, rather than ½0; 2π as before [Fig. 2(c)]. The reason for this confinement is the “intelligent” choice to add 0 or π to the random field in Fig. 2(c) in order to approximate the desired phase pattern in Fig. 2(e). Therefore, the difference of the created phase pattern from its ideal shape [Fig. 2(e)] cannot exceed π. The overall scattering property of Fig. 2(d) can now be obtained by first diffracting the incident wave at the sawtooth grating [Fig. 2(e)], followed by scattering at the random phase mask [Fig. 2(f)]. The first diffraction process at the “perfect” blazed grating has an efficiency of 98.3% [as in Fig. 2(a)] for the desired field distribution (see inset), which is then scattered by the second random phase mask. However, its reduced phase distribution amplitude of only π reduces its scattering strength. It can be shown [22] that such a phase mask diffusely scatters only 59.5% of the incoming light, whereas the remaining 40.5% is still confined in the zero order [see inset of Fig. 2(f)], which means that this percentage of the
Fig. 2. Explanation for the diffraction properties of different grating types. Details are explained in the text. 3996
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incoming light field is transmitted without distortion. Thus, the overall efficiency of the desired image is the product of the first-order efficiency of the first pattern and the zero-order efficiency of the second one, yielding about 40.5%. This efficiency is also obtainable by an “ordinary” binary phase CGH, with the advantage that now the energy of the undesired diffraction orders is distributed over the whole image field, since the nonzero-order light behind the mask in Fig. 2(f) is homogeneously distributed [see inset of Fig. 2(f)] such that no conjugate image is formed. 3.
Experimental Realization
The basic outline of the experiment is sketched in Fig. 3. The light source for the following experiments was a continuous wave 5 mW helium–neon laser with a wavelength of 633 nm, emitting a linearly polarized pure TEM00 mode. It illuminated a reflective nematic SLM (Holoeye, HEO 1080P) with a resolution of 1920 pixels × 1080 pixels, each with a size of 8 μm × 8 μm, and a fill factor (corresponding to the usable phase modulating area) of 87%. For the correctly adjusted incident light polarization the SLM acted as a pure phase modulator with a resolution of 256 linearly adjustable phase levels in the interval between 0 and 2π. For the proof-of-principle of the diffuse illumination method, the SLM was used both for generating the pseudorandom diffuse illumination wave and for displaying the binary phase CGH. For this purpose, the SLM displayed two different phase masks side by side, each with a size of 960 pixels × 960 pixels. The laser beam was expanded to a plane wave that homogeneously illuminated the first phase pattern (H 1 ), acting as the pseudorandom diffuser. The reflected
Fig. 3. (Color online) Experimental setup. The expanded beam of a He–Ne laser illuminates one-half of a high-resolution SLM with a size of 1920 pixels × 1080 pixels, acting as a diffuser by displaying a continuous pseudorandom phase pattern. From there, the light is reflected and sharply imaged (by a set of two lenses) at the other half of the SLM display. There a binary CGH is displayed, which takes the pseudorandom illumination into account. The hologram is reconstructed in its Fourier plane, using a Fourier-transforming lens in front of a CCD camera. In another configuration (not displayed), the Fourier transform of the light reflected from the first SLM pattern is projected at the second one, using only one Fourier-transforming lens arranged in the middle of the optical path between the two SLM displays.
light was then guided by two mirrors and a (variable) lens arrangement to the second phase mask (H 2 ), which displayed the CGH (corrected by the illumination phase). Finally, the light reflected from H 2 was projected through a Fourier-transforming lens to a CCD camera, where the reconstructed hologram was recorded. This setup was used to simulate different situations; Fig. 4 shows the results. The lower diagrams sketch the realized experimental configurations. First, a standard binary Fourier CGH reconstruction was performed in Fig. 4(a). In this case, H 1 was unmodulated, just acting as a mirror. Thus, the incident laser beam was reflected without being changed considerably and it illuminated H 2 as a plane wave. Displaying a hologram pattern composed of the two phase levels 0 and π at H 2 , this corresponded to a “normal” binary Fourier hologram readout. As expected, the reconstructed image in Fig. 4(a) shows the superposition of the desired image (i.e., the “X” and “O” characters) in the upper part of the figure with their conjugates in the lower part. The bright spot in the center results from light reflected from the nonactive area of the SLM, due to its limited fill factor, and makes up 13% of the total reflected light. Second, the setup could be used to simulate the situation where a pseudorandom phase mask is “attached” to the CGH. Since it was not possible to physically attach such a diffuser to the SLM, this situation was simulated by sharply imaging the pseudorandom phase mask H 1 with a size-preserving telescopic lens arrangement into the plane of H 2 [as indicated below Fig. 4(b)]. For the simulation of the diffuser, H 1 was chosen as a pseudorandom array with a uniform continuous phase distribution ϕðx; yÞ in the interval between 0 and 2π. To calculate a CGH that can be displayed in the plane of H 2, and that corrects for the illuminating phase distribution, first the “normal” CGH Ψðx; yÞ for the desired image was calculated with a standard Gerchberg–Saxton optimization algorithm [23], just assuming the availability of continuous (i.e., nonbinary) phase levels. This pattern
was then corrected by subtracting the known pseudorandom offset phase distribution ϕðx; yÞ. The resulting phase distribution was “wrapped” to the interval ½0; 2π by taking it modulo-2π and, finally, “binarized” by setting all phase values of mod2πfΨðx; yÞ − ϕðx; yÞg in ½0; π to 0 and in ½π; 2π to π. The resulting reconstructed image is shown in Fig. 4(b). As expected, there is no conjugate image anymore. Instead, its energy is dispersed as a diffuse speckle pattern, which is sufficiently diluted to display the desired image with high contrast. Because of the first pseudorandom mask, the zero-order spot (i.e., the just specularly reflected spot) in the center of the image [Fig. 4(a)] has now vanished. A third alternative simulates the situation where the CGH is illuminated by a diffuse wave that results from a first scattering process at a random phase mask situated in its Fourier plane (or in its far field). For this purpose, the optics between H 1 and H 2 are designed such that the Fourier transform of H 1 is reconstructed at H 2 . The pseudorandom phase pattern H 1 thus produces a diffuse illumination wave in the plane of H 2. It consists of a diffuse speckle pattern with a phase distribution that can be calculated by numerically computing the two-dimensional Fourier transform of H 1 (i.e., of Ffexp½iϕðx; yÞg), and considering the correct scaling factor that depends on the focal length of the Fourier-transforming lens. The correction of the binary CGH displayed at H 2 can then be performed with this calculated phase offset as described before in Fig. 4(b). The position of the hologram H 2 must be adjusted properly, such that the actually generated diffuse wave field in the plane H 2 matches the one expected from the numerical computation of the Fourier transform of H 1. The experimentally reconstructed CGH in Fig. 4(c) shows that this method works with approximately the same performance as in Fig. 4(b). Particularly, there is again a suppression of other diffraction orders and of the central bright spot. The reconstruction efficiency in the cases of Figs. 4(b) and 4(c) is approximately equal. This demonstrates that it is possible to consider the phase distribution of an illumination wave scattered by a distant diffuser in the calculation of a corrected CGH. In the case where the pseudorandom scattering mask H 1 is located exactly in the Fourier plane of H 2, a lateral shift of H 1 just results in a corresponding lateral shift of the reconstructed image in the camera plane [24]. 4.
Fig. 4. (Color online) Reconstructed holograms (upper line) in different reconstruction geometries (below). (a) shows a classic binary CGH with plane wave illumination. In (b) the pseudorandom phase mask H 1 is sharply imaged onto the corrected binary CGH H 2 . In (c) the pseudorandom phase mask H 1 acts as a diffuse illumination source by locating it in the Fourier plane of the binary CGH H 2 .
Conclusion
A method was demonstrated to reconstruct on-axis binary CGHs without superposition of the unwanted conjugated order. A practical motivation for this approach is the possibility to extend the use of binary phase displays, such as ferroelectric SLMs, for holographic displays. To date these have the advantage of being the fastest available high-resolution SLMs, with switching rates of almost 100 kHz. In the future, they could be used to reconstruct color holograms by time-multiplexed displaying of different CGHs 1 August 2008 / Vol. 47, No. 22 / APPLIED OPTICS
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illuminated with synchronously switched diodes of different colors. They could also be used in optical tweezers or optical atom traps, or as fast lithography tools to, for example, write sophisticated hologram patterns into photopolymers. The most convenient implementation of the method would probably be to attach a specially fabricated pseudorandom phase mask with a known phase profile close to the liquid crystal layer by using, e.g., a microstructured coverglass or a diffusely reflecting rear mirror. Alternatively, the coverglass of a SLM could be replaced by a ground-glass slide acting as a random phase mask. Its offset spatial phase distribution could afterward be measured interferometrically so that it could be used as a pseudorandom phase mask. For a random phase mask that is directly attached to the SLM, the offset phase of each pixel is much better controlled than in our actual experiment and, therefore, we expect also a better signal-to-noise ratio. Nevertheless, as we have shown, it is also possible to illuminate the SLM with a diffuse light field scattered by a distant diffuser with a known phase profile. In any case, provided that the offset phase distribution in the CGH plane is known, it can be considered in all further hologram calculations. The theoretically achievable diffraction efficiency of the method is 40.5%, and equals the efficiency of a standard binary CGH. The remaining 59.5% of the diffracted light is dispersed as a uniform background over the whole image plane and creates a nonavoidable uniform background. However, the method maximizes the dispersion of the unused light such that, in this respect, it is in any case better than the Fresnel method explained above, which has already been demonstrated to be practicable for certain applications, such as optical tweezers [15] or image filtering [19]. Therefore, the method allows producing CGHs for on-axis reconstruction (i.e., without a superposed grating), which has the advantages that the full bandwidth of the display is used to encode the image information and that the whole field of view in the image plane is accessible. This work was supported by the Austrian Science Fund (FWF) project P19582-N20.
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