High-speed phase modulation using the RPC method with a digital micromirror-array device

An improved implementation of the reverse phase contrast (RPC) method for rapid optical transformation of amplitude patterns into spatially similar phase patterns using a high-speed digital micromirror-array device (DMD) is presented. Aside from its fast response, the DMD also provides an electronically adjustable and inherently aligned input iris that simplifies the optimization of the RPC system. In the RPC optimization, we illustrate good agreement between experimentally obtained and theoretically predicted optimal iris size. Finally, we demonstrate the conversion of a binary amplitude grating encoded on the DMD into a binary (0-π) phase grating. ©2006 Optical Society of America OCIS codes: (230.6120) Spatial light modulators; (100.1160) Analog optical image processing; (210.0210) Holographic and volume memories References and links 1. P. C. Mogensen and J. Glückstad, “Phase-only optical encryption,” Opt. Lett. 25, 566-568 (2000). 2. P. C. Mogensen, R. L. Eriksen, and J. Glückstad, “High capacity optical encryption system using ferro-electric spatial light modulators,” J. Opt. A: Pure Appl. Opt. 3, 10-15 (2001). 3. P. C. Mogensen and J. Glückstad, “Phase-only optical decryption of a fixed mask,” Appl. Opt. 40, 1226-1235 (2001). 4. C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. 85, 171-176 (1991). 5. R. John, J. Joseph, and K. Singh, “Holographic digital data storage using phase modulated pixels,” Opt. Lasers Eng. 43, 183-194 (2005). 6. T. D. Wilkinson, W. A. Crossland, and V. Kapsalis, “Binary phase-only 1/f joint transform correlator using a ferroelectric liquid-crystal spatial light modulator,” Opt. Eng. 38, 357-360 (1999). 7. J. Glückstad and P. C. Mogensen, “Reverse phase contrast for the generation of phase-only spatial light modulation,” Opt. Commun. 197, 261-266 (2001). 8. P. C. Mogensen and J. Glückstad, “Reverse phase contrast: an experimental demonstration,” Appl. Opt. 41, 2103-2110 (2002). 9. L. Yoder, W. Duncan, E. M. Koontz, J. So, T. Bartlett, B. Lee, B. Sawyers, D. A. Powell, and P. Rancuret, “DLP Technology: Applications in Optical Networking,” Proc. SPIE, 4457, 54-61 (2001).


Introduction
Reconfigurable two-dimensional (2D) phase patterns that can be imprinted onto a laser beam's otherwise planar wavefront have a number of applications in optical engineering.These include optical encryption-decryption [1][2][3], holographic optical memory or datastorage [4,5], and optical joint-transform correlation (JTC) [6].Well-controlled modulation of phase in 2D space has become conveniently straightforward with the emergence of phaseonly spatial light modulators (PO-SLM).PO-SLMs may be liquid crystal (LC) or microelectromechanical system (MEMS) -based devices that generate phase profiles by spatial variation of the optical path of an incident wavefront.The former does it by a refractive index or birefringence variation while the latter does it by a geometrical path alteration.
Previously, a new technique was proposed for achieving spatial phase modulation using the reverse phase contrast (RPC) method [7].An RPC setup, upon simple optimization, optically converts an amplitude-only input pattern into a spatially identical (and almost purely) phase pattern at the output plane.The RPC method gives a number of advantages as it can make use of fixed amplitude masks (which are easier to fabricate than phase masks) and comparatively low-cost amplitude-only SLMs that create reconfigurable input patterns.Furthermore, in terms of speed, RPC can enable faster modification of spatial phase modulation than what LC-based PO-SLMs can offer by utilizing a high-speed MEMS-based amplitude-only modulator.
The viability of RPC has been experimentally established both with the use of static amplitude transparencies and a tandem of LC-based SLM with a polarizer [8].Though dynamic phase modulation can be achieved with RPC utilizing an LC-SLM-polarizer tandem, the speed is hampered by the relatively slow response time of the LC device.In this work, we implement a fast and robust RPC system with the use of a state-of-the-art digital micromirrorarray device (DMD TM ) [9].Specifically, we use a Texas Instrument DMD consisting of an array of light-deflecting aluminum micromirrors (1024×768 square pixels; 13.68 μm pixel pitch; 88% active area fill), each of which can achieve two positional states, ON or OFF, i.e. electromechanically induced diagonal mirror tilt γ of +12° or -12°, respectively.Achievable switching speeds of the individually addressable bistable micromirrors can be up to ~15 μs.In addition, the DMD also has significantly higher illumination power tolerances supporting several spectral regions from 350-2000 nm and has much less stringent requirement on the polarization of the incident field compared to LC-based SLMs.

Reverse phase contrast: setup and theory
A compact setup, shown in Fig. 1, for the conversion of a DMD-encoded amplitude-only pattern into a geometrically identical phase pattern via the RPC method is constructed.It employs a collimated monochromatic laser beam (λ ≈ 1064 nm, Ytterbium fiber laser, IPG Laser GmbH) to read out the amplitude-only object generated by the computer controlled DMD.As seen in Fig. 1(a), the DMD chip is oriented in a slightly unorthodox manner such that the micromirror tilt axes are perpendicular to the optical table and the chip normal is along the optical axis of the 4f setup.The expanded, collimated beam is incident at an angle of ~24° with the chip normal [Fig.1(b)] such that the strongest Fraunhofer diffraction order when all micromirrors are ON is parallel to the optical axis and the only order collected by the remaining optics.All our intensity data are normalized to the intensity of this order.
A given input object encoded on the DMD is described by a real-valued amplitude transmittance where and the circ-function, which is unity within a circular region of radius r Δ , and zero otherwise, defines a chosen input aperture truncating the amplitudemodulated signal ) , ( y x α . Unlike in previous RPC implementations [8], the input iris is now generated by the input-encoding device, i.e. the DMD, itself (by simply setting the micromirrors outside the defined circle to OFF-state).The input iris is therefore both imaged in focus and electronically adjustable with the input signal ) , ( y x α .RPC, being a Fourier filtering process, introduces a spatial filter at the midplane between the two 4f setup lenses [Fig.1(b)].For a non-absorbing Fourier filter (experimentally implemented here) that gives a phase shift of π at a circular, on-axis centered region of radius can be considered as a synthetic reference wave (SRW) and the term α given by ( ) describes the spatial average of the input amplitude pattern.

Results and discussion
In optimizing the performance of the RPC system, one needs to correctly achieve a suitable matching between the size of the input aperture and the size of the π phase-shifting region of the Fourier filter.A good point to start is by satisfying the so-called dark-background condition [8].This condition means that the optimal input aperture radius . The results clearly show a good agreement between the experiment and the model, which both give op r Δ ~ 1.33 mm as shown in Fig. 2(c).The uniformity of the achieved dark background is indicative of the flatness of the SRW ) (r g ′ , which approaches a constant value K at the center Keeping the input iris radius op r Δ ~ 1.33, we tested a 50% duty-cycle binary amplitude grating as input pattern, the image of which (obtained with the same 4f setup but without the Fourier filter) is shown in the upper inset of Fig. 3, into our DMD-based RPC system.With correct on-axis alignment of the phase-only Fourier filter, CCD camera 1 detects a lowcontrast output intensity shown in the lower inset of Fig. 3.This can be explained by our model defined in Eq. ( 2) considering that for the particular grating pattern where α ~ 1/2 [see Eq. ( 3)], the output intensity is given by where ON ℜ and OFF ℜ denote regions where the ON and OFF micromirrors are imaged.
Since, in the inner part of the circular iris' image, ) (r g ′ may be approximated by a constant K = 1/2, we expect a four-fold intensity reduction and equalization between ON ℜ and OFF ℜ regions, which is more accurate at the center and is indeed what we obtain experimentally in Fig. 3.It is also worth to note the highly contrasted image of the DMD-defined circular input aperture that is not achievable with the LC-SLM based RPC system requiring an extra polarizer, which usually has a limited extinction ratio [8]. Along with the intensity equalization, it is also apparent from Eq. ( 5) that we have transformed our amplitude-only binary grating into a periodic distribution of (approximately) +1/2 and -1/2 amplitude values, thus a 0-π binary phase grating.To experimentally prove so, we have obtained the optical Fourier transform of the RPC output pattern whose residual halo  is truncated by an iris of radius op r Δ .The result, seen in Fig. 4, is very close to the theoretically expected far-field diffraction pattern of a 50% fill 0-π binary phase grating.Interferometric detection of the RPC output phase pattern is also feasible [8].
It is worth to note that other binary (0-φ Δ ) phase patterns, with π φ <

Δ
, can also be generated by the RPC method.With the inherent loss due to the use of an amplitude input, synthesis of a desired phase pattern must follow a correct procedure [a recipe is provided in Ref. 7] in order to optimize the light throughput at the output.

Conclusion
We have shown the implementation of the reverse phase contrast (RPC) method using a digital mircromirror-array device (DMD).The DMD enables a robust and possibly the fastest amplitude-only 2D spatial modulation that the RPC system converts into a spatially identical phase modulation.We have described the advantages gained from a DMD-based RPC system, particularly the enhanced optimization of electronically tuning the DMD-defined input aperture that matches the RPC Fourier filter.Successful conversion of a binary amplitude grating encoded by the DMD into a binary 0-π phase grating has been achieved.For various applications [1][2][3][4][5][6], arbitrary binary amplitude patterns achieving 50% fill can be converted to corresponding phase patterns by RPC [7,8] and may be encoded by the DMD at promisingly high-speed refresh rates.

Fig. 1 .
Fig. 1.(a) Photograph of the reverse phase contrast (RPC) 4f setup for converting an amplitude-only pattern displayed on a digital micromirror-array device (DMD) into a spatially similar phase pattern at the output plane.(b) Schematic diagram of the whole setup.The expanded and collimated laser beam is made incident to the DMD chip at an angle of ~ 24°, twice the micromirror tilt angle ⎪γ⎪, such that the beam coming out normal to the chip (at ONstate) is the strongest Fraunhofer diffraction order and the only order that passes through the optical train.CCD camera 1 detects the intensity at the output plane.CCD camera 2 captures the optical Fourier transform of the iris-truncated output pattern.Identical lenses with 100-mm focal length are used.A phase-only filter (made from an optical flat with a tiny circular pit) is used to create a phase shift of π over an on-axis circular region (diameter, 2R ~ 39 μm) in the common Fourier plane of the two 4f setup lenses.BS, beam splitter.
the focal length of the lens immediately in front of the DMD.
the boundary of the iris' image when a constant input signal 1 and monitored the output intensity at CCD camera 1 for different radii r Δ .Here, r Δ is electronically (rather than mechanically) tuned and is easily measurable based on the addressed DMD pixels.Selected output images are shown in the insets of Figs.2(a)-2(f) and in the appended animation file (GIF).In these figures, we compare the line scans of the experimentally obtained output intensity profiles with the corresponding

Fig. 2 .
Fig. 2. (GIF, ~140 kB) Comparison of the theoretical and experimental intensity profiles at the output plane of the 4f setup that images a circular iris of diameter, 2Δr = (a) 3.65 mm, (b) 3.15 mm, (c) 2.65 mm, (d) 2.15 mm, (e) 1.65 mm, and (f) 1.15 mm with the phase-only filter centered at the Fourier plane common to the two lenses.Each experimentally obtained intensity profile is a diagonal line-scan through the center of the CCD-captured image (inset).

Fig. 3 .
Fig. 3. Intensity profiles measured along a diagonal (perpendicular to grating bars) for the DMD-encoded binary amplitude grating (red) and for the corresponding phase pattern (blue) produced via RPC when the filter is centered at the Fourier plane.The CCD-captured 2D images for the binary amplitude grating and the RPC output are shown in the upper and bottom insets, respectively.

Fig. 4 .
Fig.4.Measured far-field diffraction profiles of the circular iris' image at the output of the 4f setup without the phase-only filter (circles) and the binary phase pattern produced via RPC (with binary amplitude-only input from the DMD) when the filter is centered at the Fourier plane (triangles).The latter shows the suppressed 0 th and even diffraction orders and the dominant +1 and -1 orders each with strength (four times the actual) approximately equal to the theoretical value of ~0.41 for a 50% duty-cycle, 0-π binary phase pattern.