Homodyne detection of holographic memory systems Adam C. Urness* a,b, William L. Wilson a, Mark R. Ayres b a Department of Materials Science and Engineering, Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA; b Akonia Holographics, LLC, Longmont, CO 80021, USA ABSTRACT We present a homodyne detection system implemented for a page-wise holographic memory architecture. Homodyne detection by holographic memory systems enables phase quadrature multiplexing (doubling address space), and lower exposure times (increasing read transfer rates). It also enables phase modulation, which improves signal-to-noise ratio (SNR) to further increase data capacity. We believe this is the first experimental demonstration of homodyne detection for a page-wise holographic memory system suitable for a commercial design. Keywords: Holography, Coherent detection, Data storage 1. INTRODUCTION The theoretical storage density of holographic data storage (HDS) is immense, owing largely to the fact that data is stored throughout the volume rather than merely on one or a few 2D surfaces [1]. The most advanced HDS drive to date relied on the direct detection of amplitude-modulated holographic signals [3] to encode and recover the data. However, encoding and recovering amplitude-modulated holographic signals resulted in low signal to noise (SNR), resulting in a low code rate and therefore reduced disk capacity and read/write transfer rates. To improve code rate, capacity and transfer rate, we propose incorporating a coherent data channel employing homodyne detection and quadrature phase shift keying (QPSK) [2][3]. In addition to many benefits predicted by communications theory, the new channel will fundamentally alter the dynamics of signal-induced noise formation during the storage step of the data channel, increasing storage capacity and SNR. Coherent detection techniques have been used in communications channels for over a century [4]. The superheterodyne receiver, the ubiquitous circuit that brings radio-frequency signals down to manageable electrical bandwidths, was invented in 1921 [5]. Coherent detection enables the reception of signals modulated in phase and frequency, and generally improves the reception of signals modulated in amplitude when compared to direct detection. Coherent detection can amplify the signal directly in the optical domain, thereby not only decreasing the required detector integration time, but also boosting the signal level above the thermal noise of the detector so as to approach the shot noise limit. Coherent detection is performed by combining the signal with a reference signal, called the local oscillator, and then detecting the intensity of the mixed signal. The coherent superposition includes a cross term that varies in intensity according to the phase difference of the two constituents. Generally, the local oscillator may differ in frequency from the signal carrier, and the process is known as heterodyne detection. The intensity of the cross term will vary in time at a rate determined by this difference (beat frequenc, and the frequency, as well as the phase, may thereby be determined. If the local oscillator has the same frequency as the signal carrier, then the process is also known by the more specific term, homodyne detection. Coherent optical detection was not developed for many years after the advent of coherent radio wave detection owing mostly to the lack of coherent optical sources. However, there is currently great interest in optical coherent detection, especially for fiber optic communications systems [6][7][8][9]. In some respects, coherent optical detection is analytically simpler than coherent radio-frequency detection because optical detectors detect intensity (i.e., photon absorption onl whereas radio wave antennae detect field (i.e., mixed photon absorption and emission) [10]. Optical detectors are not sensitive to the sum and doubled frequency components of the squared field, responding only to the information-bearing difference frequency term.
The different variants of coherent optical detection include most, but not all, combinations of modulation scheme (e.g., ASK, FSK, or PSK) with either detection method (heterodyne or homodyne). Other technical differences may affect performance, for instance the difference between synchronous heterodyne detection (where the local oscillator is harmonically phase-locked to the signal) and asynchronous heterodyne detection (where it is not). In either case, the local oscillator must be carefully frequency-controlled in order to produce a suitably slow difference frequency. The enormous sensitivity improvements that may be achieved have motivated fiber optic system designers to explore coherent detection despite its considerable complexity and competing technologies. The incremental improvements in performance gained by going from a relatively simple asynchronous heterodyne system to the best performer homodyne PSK are unlikely to warrant the cost and complexity of a fully phase-locked local oscillator laser for fiber optic communications. However, this difficulty is considerably alleviated in holographic data storage. The same laser used to generate the probe beam for holographic reconstruction provides a ready source of coherent photons for the local oscillator, and prompts development of a superior homodyne system rather than heterodyne detection. The task of controlling the phase difference between the signal and the local oscillator can then be achieved by adjusting the optical path length of one or both of the beams rather than phase-locking a laser to generate a local oscillator from scratch. Conversely, page-oriented holographic data storage presents a special challenge because of the large number of parallel channels often over a million that must be detected. In general, the phase of the holographic signal carrier varies across the page so that a single-channel variable phase retarder could not perform the path length adjustment even if a suitable feedback signal were available. The phase lock requirement becomes an adaptive optics problem wherein the wavefront of the local oscillator must be carefully matched to the holographic carrier wavefront over the whole image. However, interferometry path-length matching is difficult enough for two single-mode fields, let alone across the entire field of a high-bandwidth holographic signal beam. However, this process may be performed algorithmically using a simple switchable phase retarder and the method of quadrature homodyne detection, rather than physically using a complex multichannel adaptive optics device. In this paper, we present the method of quadrature homodyne detection and theoretically show SNR improvements in HDS data recovery. Performance of quadrature homodyne detection is first evaluated by applying the algorithm to simulated images and comparing them to direct detection to demonstrate the theoretical SNR improvement. This improvement is verified experimentally by implementing a quadrature homodyne read channel. The SNR improvement demonstrated by the experimental system actually outperforms simulated predictions, owing to improvements in the detection algorithm. 2.1 Resampling Direct Detection Data Channel 2. TECHNICAL BACKGROUND Akonia currently operates a number of non-homodyne prototypes. The direct detection data channel of these prototypes recovers recorded data from the holographic image during a read operation [11][12]. In the Akonia system, the data are composed into a two-dimensional image using a spatial light modulator (SLM), wherein each pixel takes a bright or dark value to indicate either a one or zero of the recorded binary data. This image is recorded as a hologram, and later reconstructed upon a detector array during a data recovery operation. A critical step in the recovery process is the image alignment measurement. This is represented by the quiver function, a map of the difference in coordinates of the imaged SLM pixel array on the detector pixel array compared to nominal. This function encapsulates image shift, rotation, magnification, and higher order distortions caused by all recovery perturbations. Since the detector array is fixed, the channel must be able to recover data images that are shifted tens of pixels from nominal, and/or rotated and magnified by up to several percent. Each holographic pixel image impinges with an arbitrary alignment with respect to the nearest detector pixels, so it is necessary to oversample the recorded image in order to insure adequate resolution of the data pattern. In the Akonia 300 GB prototype, the SLM pixel spacing is nominally 4/3 greater than the detector pixel spacing, and that ratio is also used in all of the QHD simulations that follow. The quiver function is determined by embedding known data patterns ( reserved blocks ) within each holographic image and locating them within the detected images. The reserved blocks are found by cross correlation (matched filter) operations within the oversampled image. In both the direct detection and QHD data channels, the interpolated locations of the cross correlation peaks are used for the resampling algorithm (see [11]). In the QHD channel, however, an extra piece of information, the magnitude of each cross correlation peak, is used for QHD as described below.
2.2 Phase Shift Keying (PSK) Conventional HDS systems represent data with amplitude shift keying (ASK), i.e., by recording SLM pixels in a bright (1) or dark (0) state. ASK is suitable for direct detection systems because brightness can be directly distinguished by photodetectors. An alternative modulation scheme, phase shift keying (PSK), admits several significant advantages over ASK modulation. In the simplest such scheme, 1 and 0 are represented by opposite phases of the impinging optical fields, e.g., 0 o for 1 and 180 o for 0. All pixels are in a bright 'on' state. A straightforward analysis shows that PSK signaling requires only half as many signal photons to achieve the same bit error rate (ber) as ASK, indicating that PSK enjoys a 3 db signal-to-noise ratio (S/N) advantage over ASK [13]. Furthermore, PSK can improve the optical throughput of the SLM if photons need not be discarded striking dark pixels. Medium usage efficiency is improved for the same reason. However, perhaps the biggest advantage of PSK will accrue from the reduction of signal-induced noise. 2.3 Signal-Induced Holographic Noise Holograms for HDS are typically recorded near a Fourier plane of the SLM in order to minimize their areal footprint. For pure ASK modulation, however, fully half of the energy of the signal beam resides in the DC component, and hence half of the energy comes to a focus at a bright spot in the center of the Fourier plane. This and other properties of ASK modulation necessitate the use of a phase mask during recording [14][15]. Not only does the existing Akonia prototype require a phase mask, it also must be mounted to a motion stage to further reduce signal-induced noise between holograms. Elimination of this bulky and expensive component, as well as the lens relay required to image it, would greatly enhance the prospects for a compact, low-cost commercial design. Signal-induced noise can be divided into two broad categories. The linear first-order category includes only those terms in the weak, linear regime for both recording modulation and diffraction. Intra-signal modulation noise is an important example of this it can be shown that the DC component of a signal field results in noise terms that add coherently both within and between holograms, whereas the AC component produces terms that add incoherently [16]. In this respect, PSK modulation will result in a dramatic reduction of noise compared to ASK modulation. The second category consists of higher order effects. An important example of this includes localized saturation of media response at intensity hot spots such as the DC focus. Failure to record the DC component of an ASK signal field results directly in the loss of contrast between bright and dark pixels; for a PSK field it produces very little impairment at all. Intensity variation may also produce higher order effects via localized shrinkage causing optical path length variation and grating distortion. However, we are unable to quantify these effects in the quadrature homodyne read channel testbed, described in section 4, because it does not include the necessary write channel. 2.4 Local Oscillator Generation Figure 1 shows a possible method for introducing a collimated local oscillator beam into the detector path in Akonia s HDS architecture. All elements in the figure are common to Akonia s planned design except for the polarization rotator, the switchable retarder, and the analyzer. Polarization Rotator SLM Polarization Rotator SLM SLM Illumination PBS Detector Local Oscillator PBS Detector Switchable Retarder Reference Beam Medium Signal Beam Switchable Retarder Data Beam Medium Analyzer a) b) Probe Beam
Figure 1. Schematic architecture for (a) coherent recording; and (b) coherent recovery. The local oscillator is derived from the same beam used to illuminate the SLM during recording. The switchable retarder is a simple one-channel device that transmits the local oscillator in one of two states with an optical path length difference of 90 o. This might be accomplished, for example, with an electro-optic (EO) modulator. The polarization rotator is configured to switch the input beam to p polarization during recovery and s polarization during recording. During recovery, the p polarized local oscillator and the s polarized data beam are incident on the analyzer, which is in proximity to the detector. The analyzer is simply a linear polarizer with its axis oriented to pass components of both the data beam and the local oscillator, thus projecting them onto a common polarization axis at a desired mixing ratio. The mixing ratio is selected to preferentially pass most of the weak signal at the cost of blocking local oscillator photons, but the much stronger local oscillator still reaches the detector with higher power than the signal. 2.5 Quadrature Image Recovery Using the configuration of Figure 1 (b), two different images of a hologram may be recovered which bear a quadrature relationship to each other. Since the data and the local oscillator are propagating along the same optical axis, the difference wavefront Δϕ x, y between the holographic signal and the local oscillator will contain only slowly varying components, producing slowly varying fringes. The first image of Figure 2, ( x the 0 o phase state, and the second image, ( x I P,, shows the simulated detector intensity when the switchable retarder is in I Q,, corresponds to the 90 o state. The simulation uses a local oscillator with intensity 100 times that of the hologram, which has the effect of amplifying the signal by 20. Because of the 90 o phase change, the entire fringe pattern has also shifted by 90 o so that regions of low contrast in the P image are high contrast in the Q image, and vice-versa. Thus, in principle, the two images contain all the information needed to recover every pixel in the hologram. 50 50 100 100 150 150 200 200 250 250 300 300 50 100 150 200 250 300 50 100 150 200 250 300 Figure 2. P and Q Quadrature image simulations of a PSK-encoded hologram recovered with the local oscillator in each of the two phase states (0 o and 90 o ). Δϕ x, y 2.6 Estimating If the hologram contains reserved block patterns as described in section 2.1, then the quadrature image pair also contains all the information necessary to determine Δϕ x, y. Recalling the image alignment measurement method of section 2.1, a cross correlation operation is performed between a portion of the detected image and a corresponding reserved block pattern. Figure 3 shows normalized cross correlation peak strength maps for the simulated P and Q images of Figure 2. Comparing Figure 2 with Figure 3, it is apparent that the regions that are in high contrast and noninverted show large positive peak strength values (approaching +1). Similarly, inverted regions with high contrast have large negative peak strength values approaching 1. Together, the two peak strength maps represent quadrature projections of Δϕ x, y onto the locally-varying recovery phase basis. It may be estimated by the expression
where tan -1 (a, b) is the four-quadrant arctangent. [ ] 1 ( i, j) = tan X ( i, j), X ( i j) Δϕ ˆ (1) Q P, 1 1 2 0.8 2 0.8 4 0.6 0.4 4 0.6 0.4 6 0.2 6 0.2 8 0 8 0-0.2-0.2 10-0.4 10-0.4 12-0.6 12-0.6-0.8 14 14-1 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Figure 3. Cross correlation peak strength maps, X P (i, j) and X Q (i, j), for the images of Figure 2. -0.8-1 The peak strength maps are sampled only at the locations of the reserved blocks. These maps are then up-sampled to positions between the reserved blocks using a simple bilinear interpolation function. 2.7 Quadrature Image Combination be Following a derivation here omitted for brevity [17], we find the rule for optimally combining the P and Q images to Eˆ S ( x, ~ cos[ Δϕ( x, ] I P( x, + sin[ Δϕ( x, ] IQ ( x, X ( x X ( x P, ~ Q, = I P( x, + 2 2 1/ 2 2 2 X ( x, + X ( x, X ( x, + X ( x, ( ) P Q ~ ( ) P ~ Q 1/ 2 ~ I Q ( x, I ~ and I ~ Q are DC-free versions of the quadrature images. In this form, it where Ê is the estimated signal field, and S P can be seen that the estimate simply combines the two images in proportion to their local cross correlation peak strength. The negative sign of the peak will restore the intensity pattern to positive polarity in the inverted regions, and the denominator will normalize local variations in image intensity. Figure shows the image produced by combination of the images of Figure 2, showing high-contrast non-inverted data throughout. The recombination process reconstructs the optical field at the spatial resolution of the detector. For oversampled detection, the recombined image may be subsequently resampled using a technique similar to that of section 2.1. (2)
50 100 150 200 250 300 50 100 150 200 250 300 2.8 Coherent Noise Reduction Figure 4. Image produced by combination of the quadrature image pair of Figure 2. Lowering the diffraction efficiency of the holograms allows for more to be stored, so designers tend to be driven to operate at a diffraction efficiency that is only marginally higher than the dominant noise source, reference beam scatter. Reference beam scatter noise typically dominates detection shot noise and thermal electronic noise in HDS systems. Scattered reference light mixes coherently with the holographic signal, and moreover is of comparable spatial bandwidth since it issues from the same aperture. Lengthening the exposure time during hologram recovery cannot reduce it. Coherent noise degrades performance more than additive noise of equivalent power. Akonia has estimated that QHD will enjoy approximately 6 db increase in signal-to-noise ratio (S/N) from coherent noise sources [18][17][19]. This improvement results generally from linearizing coherent noise by coherent detection and theoretically adds to the 3 db improvement for utilizing PSK modulation, as noted in section 2.1. Thus, QHD could in principle provide an S/N improvement of 9 db over direct ASK detection. Such an improvement would put HDS, a chronically low S/N technology, into a regime more typically occupied by other data storage products. 2.9 Phase Quadrature Holographic Multiplexing (PQHM) In addition to the benefits described above, coherent detection techniques such as QHD have the potential to directly increase the storage capacity of HDS devices. Since homodyne detection allows for the recovery of both amplitude and phase, it in principle transmits two channels of information where direct detection provides only one. This principle is exploited in other communications technologies by coding techniques such as quadrature amplitude modulation (QAM) which simultaneously transmits two independent data channels on a single carrier. Because sinusoids separated by 90 in phase are mutually orthogonal, the real and imaginary parts of a complex index grating represent independent degrees of freedom in the holographic address space. 2.10 Phase Quadrature Recording In one scheme, PQHM recording can be accomplished by using elements that are already present for quadrature homodyne detection. Figure 1 (a) illustrates the Akonia architecture during PQHM recording. The polarization rotator is now configured to switch the illuminating beam to s polarization so it will be directed towards the SLM by the PBS. Recording PQHM holograms thus becomes a simple matter of performing two exposures with different data images and reference beams differing only by a 90 o phase factor using the switchable retarder. So long as the optical paths in the device remain stable over both exposures, the two holograms will be recorded in phase quadrature with respect to each other.
2.11 Recovery of PQHM Holograms PQHM holograms may be recovered by using a coherent detection method to isolate the quadrature field components. For QHD, recovery of PQHM holograms will produce quadrature image pairs similar to those of Figure 2, excepting that there will be no low-contrast gray fringes visible. This is because the two holograms are intermixed such that the Q hologram occupies the low-contrast regions of the P hologram, and vice-versa. The data of the quadrature hologram may be recovered using Eˆ ~ ~ x, y = sin Δϕ x, y I x, y + cos Δϕ x, y I x y (3) ( ) [ ( )] ( ) [ ( )] ( ) Q P Q, Separate quiver patterns for the Q image need not be calculated, though the Q reserved block patterns must be equalized by a method here omitted for brevity [17]. 3. SIMULATION RESULTS Performance of quadrature homodyne detection was evaluated by applying the algorithm, detailed in section 2, to simulated images. Tests were performed comparing PSK quadrature homodyne, PSK homodyne, ASK homodyne, and ASK direct detection. The data page was 752 x 752 SLM pixels including a 16 pixel border, and contained 11 x 12 reserved blocks on a 64 pixels pacing grid. The simulated detector page was oversampled by 4:3, producing a 1003 x 1003 image. The simulated point spread function was established by the 1.124 Nyquist filtering aperture, and the images contained a slight (~0.1%) rotation component to make the resampling coefficient usage approximately uniform. All parameters except for the encoding and detection method were kept identical. In the homodyne trials, the local oscillator was set to 100 times the power of the signal image. For quadrature homodyne, two waves of tilt in the x direction were also applied. Figure 5 summarizes the results of the simulation. The x axis represents the ex ante signal-to-noise power ratio, which was varied by adding coherent noise to the noise-free simulated images. The y axis is the raw bit error rate (ber) determined by simple binary threshold detection of the resampled image Figure 5. Raw bit error rate versus optical SNR for the various cases including phase-quadrature multiplexed holograms 4. EXPERIMENTAL RESULTS Performance of quadrature PSK homodyne predicted by the simulation was confirmed experimentally by implementing a holographic read channel that incorporated homodyne detection, shown in Figure 6. The test setup does not contain any media and thus can only verify SNR and transfer rate improvements in the read channel. To simulate coherent noise that is created during holographic recording, a holographic diffuser with the same angular bandwidth is implemented in a separate arm and injected in the simulated media plane of the setup. The half wave plate (HWP) just
after the simulated media plane can be adjusted to set the system to ASK direct detection or quadrature PSK homodyne detection, enabling an apples-to-apples comparison of both methods on the same system. The local oscillator (LO) is created and interfered with the object beam using a Mach-Zehnder interferometer configuration. To determine the effect the spatial frequency of the interference pattern has on the quadrature homodyne recovery algorithm, a 4f lens system was implemented on a 3D stage to provide the ability to easily vary the LO wavefront. SLM Diffuser Beam Expander ECDL HWP Pol Optical Noise Injector HWP PBS Simulated Media Plane FT lens w/ Polytopic Camera Analyzer Variable LC Retarder PBS Spatial filter PBS Wavefront Adjuster Local Oscillator Figure 6. Experimental setup for testing homodyne data recovery in a page-wise HDS system. An external cavity diode laser (ECDL) centered at 405 nm was used as the illumination source for both homodyne and direct detection recoveries. Polarizing beam splitters (PBS) were used to create the local oscillator and coherent optical noise. To create the phase delay between image pairs we incorporated a variable liquid crystal (LC) retarder into the LO arm of the system. However, this device s modulation bandwidth is limited to 100 s of Hz s, which is too slow for a commercial product. To increase modulation bandwidth an electro-optic modulator (EOM), piezo electric actuator stack, or angularly tuned parallel plate can be incorporated into the system to create the phase delay. We have performed a preliminary study using some of these devices and found they have both superior bandwidth and performance compared to the LC retarder and will be implemented in the integrated system. In addition to testing quadrature homodyne detection described in section 2, we made further improvements to the quadrature homodyne algorithm. The enhanced quadrature homodyne recovery algorithm enabled recoveries of even higher SNR than the original algorithm. Figure 7 depicts a comparison between simulation predicted SNR improvements of homodyne recovery to experimentally demonstrated improvements using both quadrature homodyne and enhanced quadrature homodyne recovery algorithms.
5 db 9 db! Figure 7. Comparison between a) simulated and b) experimental results on the dependence of page SNR to the optical SNR for direct detection, quadrature homodyne and enhanced quadrature homodyne recovery. The LO to signal ratio was 18 to 1 for both the quadrature and enhanced quadrature recoveries. Figure 7 shows that the experimentally demonstrated SNR improvements of the original quadrature homodyne recovery algorithm match simulated improvements, while the new enhanced algorithm matches instead the theory-based prediction outlined in section 2.8. This is both unexpected and encouraging as it will enable the code rate to be reduced further, effectively increasing disk capacity and transfer rate, and potentially reduce spacing between holograms in angular space, increasing addressable space and capacity. In addition to surpassing the SNR improvements predicted by the simulation, recovery exposure times also showed a potential improvement of 12x. The required exposure time for direct detection, 2 ms, was reduced to.175 ms for both the quadrature and enhanced quadrature recovery algorithms. This improvement could be increased further by raising the LO-to-signal ratio using higher performance optical components that maintain wavefront quality. 5. SUMMARY AND CONCLUSION We have presented a homodyne detection read channel implemented on a page-wise holographic memory architecture. Performance of quadrature homodyne detection is first evaluated by applying the quadrature homodyne algorithm to simulated images and comparing them to direct detection to demonstrate the theoretical SNR improvement. This improvement is verified experimentally by implementing a quadrature homodyne read channel. The SNR improvement demonstrated by the experimental system outperforms simulated predictions, enabling higher capacity and read transfer rates than previously thought. The next step is to implement a homodyne write channel with the homodyne read channel and demonstrate a fully integrated system. This would confirm that phase mask-free recording is viable and that the improvements demonstrated here can be achieved in a fully integrated system. This would also enable phase quadrature holographic multiplexing, doubling addressable space and potentially doubling capacity for a commercial product.
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