Akonia Holographics Patent | Methods And Devices For Coherent Optical Data Detection And Coherent Data Channel Modulation

Patent: Methods And Devices For Coherent Optical Data Detection And Coherent Data Channel Modulation

Publication Number: 20150070739

Publication Date: 20150312

Applicants: Akonia Holographics

Abstract

Methods and devices for coherent holographic data channel techniques are presented. Coherent techniques for data detection generally include homodyne and heterodyne detection. Techniques for Quadrature homodyne detection (QHD), Enhanced resampling quadrature homodyne detection (ERQHD), N-rature homodyne detection (NHD), and local oscillator fringe demodulation are presented. Coherent detection techniques in turn enable coherent channel modulation techniques such as phase modulation (including binary phase shift keying, or BPSK; phase quadrature holographic multiplexing, or QPSK; and quadrature amplitude modulation, or QAM). Coherent detection may also enable or improve the performance of other channel techniques such as Partial response maximum likelihood (PRML), the various classes of extended PRML, and of Noise predictive maximum likelihood (NPML) detection.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The present application claims priority to U.S. patent application Nos. 61/876,725 filed Sep. 11, 2013 and titled MULTI-TERABYTE HOLOGRAPHIC DATA STORAGE SYSTEMS; 61/941,974 filed Feb. 19, 2014 and titled REFLECTIVE HOLOGRAPHIC STORAGE MEDIUM; and 61/986,083 filed Apr. 29, 2014 and titled N-RATURE HOMODYNE DETECTION. The entire disclosures of the aforementioned US patent applications are incorporated herein by reference.

[0002] In addition, the entire disclosures of U.S. patent application Ser. No. 11/562,533 filed Nov. 22, 2006 and titled METHOD FOR HOLOGRAPHIC DATA RETRIEVAL BY QUADRATURE HOMODYNE DETECTION, which issued as U.S. Pat. No. 7,623,279 on Nov. 24, 2009; and Ser. No. 11/069,007 filed Feb. 28, 2005 and titled PROCESSING DATA PIXELS IN A HOLOGRAPHIC DATA STORAGE SYSTEM, which issued as U.S. Pat. No. 7,848,595 on Dec. 7, 2010, are incorporated herein by reference.

BACKGROUND

[0003] Demand for increased data storage density and data transfer rate continues to grow, and holographic data storage offers advantages in storage density and transfer rate compared to competing methods. However, holographic data storage can be limited by relatively low signal to noise ratio (SNR) and relatively high bit-error rates for data recovered therefrom. Limitations related to SNR and bit-error rates can be exacerbated where data is stored at very high density.

[0004] In order to facilitate reliable recovery of stored data, some storage capacity is generally devoted to error correction and duplication of stored data, and low SNR and high bit-error rate generally increase the need for such error correction and data duplication. Thus the relatively low SNR and high bit-error rates associated with recovery of holographic data diminish data density advantages of holographic data storage. In addition, decreased SNR and increased bit-error effects are magnified by some multiplexing methods used to increase holographic data storage density.

[0005] Accordingly, techniques for increasing SNR and reducing bit-error rate when recovering stored holographic data are needed. Similarly, multiplexing methods compatible with data recovery techniques that enjoy high SNR and low bit-error rate are required to more fully exploit benefits of holographic data storage.

BRIEF DESCRIPTION OF THE DRAWINGS

[0006] FIG. 1 illustrates schematic views a) and b) of a device adapted to perform coherent holographic data channel techniques according to an embodiment.

[0007] FIG. 2 shows quadrature image pair simulations of a PSK-encoded hologram recovered with a local oscillator in a 0.degree. phase state and in a 90.degree. phase state according to an embodiment.

[0008] FIG. 3 shows two cross correlation peak strength maps for the images shown in FIG. 2, according to an embodiment.

[0009] FIG. 4 shows an image produced by a combination of the quadrature image pair illustrated in FIG. 2, according to an embodiment.

[0010] FIG. 5 is a flow chart illustrating of a method of performing QHD absent enhanced resampling.

[0011] FIG. 6 is a flow chart illustrating a method of performing ERQHD according to an embodiment.

[0012] FIG. 7a is a graph comparing theoretical signal to noise plots for various coherent holographic data channel techniques according to embodiments of the present invention.

[0013] FIG. 7b is a graph comparing actual signal to noise plots for various coherent holographic data channel techniques according to embodiments of the present invention.

[0014] FIG. 8 is a map of .DELTA.{circumflex over (.phi.)} estimate derived from a .DELTA..phi. calibration page according to an embodiment.

[0015] FIG. 9 is a graph showing plots of SNR vs kx for a standard algorithm compared to a fringe demodulated algorithm according to an embodiment.

[0016] FIG. 10 illustrates a PR1 channel represented by a discrete convolution according to an embodiment.

[0017] FIG. 11 illustrates a two dimensional generalization of PR1 according to an embodiment.

[0018] FIG. 12a is a graph showing a plot of single sinc response of a rectangular aperture according to an embodiment.

[0019] FIG. 12b is a graph showing a plot of double sinc response of equalized aperture according to an embodiment.

DETAILED DESCRIPTION

[0020] Homodyne detection comprises a method of blending a coherent reference field (referred to as the local oscillator in the literature) with a signal and detecting the interference pattern between the two. This has an effect of amplifying the signal, eliminating nonlinear effects of coherent noise, and allowing detection of phase as well as amplitude. Formerly, homodyne detection required careful phase control of the local oscillator and the signal, requiring complex adaptive optics and/or phase servo loops. Conversely, embodiments of the present invention include algorithms that allow homodyne detection to be performed simply by combining two or more blended signals with relative phase changes. In some embodiments, homodyne detection according to the present invention boosts signal to noise ratio (SNR) enough to at least double user capacity, as well as providing a host of other benefits.

TERMINOLOGY

[0021] The terms and phrases as indicated in quotation marks (” “) in this section are intended to have the meaning ascribed to them in this Terminology section applied to them throughout this document, including in the claims, unless clearly indicated otherwise in context. Further, as applicable, the stated definitions are to apply, regardless of the word or phrase’s case, to the singular and plural variations of the defined word or phrase.

[0022] The term “or” as used in this specification and the appended claims is not meant to be exclusive; rather the term is inclusive, meaning either or both.

[0023] References in the specification to “one embodiment”, “an embodiment”, “another embodiment, “a preferred embodiment”, “an alternative embodiment”, “one variation”, “a variation” and similar phrases mean that a particular feature, structure, or characteristic described in connection with the embodiment or variation, is included in at least an embodiment or variation of the invention. The phrase “in one embodiment”, “in one variation” or similar phrases, as used in various places in the specification, are not necessarily meant to refer to the same embodiment or the same variation, components, or objects, in which no other element, component, or object resides between those identified as being directly coupled.

[0024] The term “approximately,” as used in this specification and appended claims, refers to plus or minus 10% of the value given.

[0025] The term “about,” as used in this specification and appended claims, refers to plus or minus 20% of the value given.

[0026] The terms “generally” and “substantially,” as used in this specification and appended claims, mean mostly, or for the most part.

[0027] The term “data pattern,” as used in this specification and appended claims, refers to a two-dimensional array of binary values. The binary values are typically, but not necessarily, ones and zeros. A modulated data pattern refers to a data pattern as it appears on an image of an SLM.

[0028] The term “reserved block pattern,” as used in this specification and appended claims, refers to a data pattern corresponding to a reserved block. For example, an 8.times.8 array of ones and zeroes defining the pixel values within a reserved block is referred to as a reserved block pattern.

[0029] The term “reserved block sample grid,” as used in this specification and appended claims, refers to a set of positions corresponding to centers of reserved blocks. For example, a grid with spacing of 64.times.64 SLM pixels with each grid point corresponding to the center of a reserved block.

Homodyne Detection

Apparatus and Operational Methods

[0030] FIG. 1, views a) and b), illustrates an exemplary embodiment of a device configured for implementing coherent channel techniques in accordance with embodiments of the present invention. FIG. 1, view a) indicates the beam paths during a recording operation, while FIG. 1, view b), shows them during a recovery operation. Most elements are common to both beam paths.

Recording Operation

[0031] In the embodiment illustrated in FIG. 1, views a) and b), an architecture is shown that may be used to practice the invention in conjunction with angle-polytopic multiplexing, phase conjugate recovery, and dynamic aperture multiplexing. In the figure, the SLM implements binary PSK modulation with the aid of the half wave plate (HWP) adjacent to it. In FIG. 1, view a), the variable retarder is located in the path of the collimated beam used to illuminate the SLM. In another embodiment, the variable retarder is located in the reference beam arm instead of the signal beam arm. In any case, the variable retarder used for writing quadrature-multiplexed holograms may be the same variable retarder used for QHD during a recovery operation; and the local oscillator beam used during recovery may derive from the same optical path as the SLM illumination beam used for recording. Utilizing the apparatus of FIG. 1, view a), the process of sequentially recording a quadrature-multiplexed hologram pair proceeds as follows: [0032] 1) Select the desired medium (r, .theta.) location and reference beam angle. [0033] 2) Configure switchable HWP 1 to transmit p-polarized light so as to illuminate the SLM. [0034] 3) Configure switchable HWP 2 to transmit s-polarized light to the recording medium. [0035] 4) Set the variable retarder to a first phase state and compose the I data page image on the SLM. [0036] 5) Open and close the shutter (not shown) to expose the I hologram. [0037] 6) Set the variable retarder to a second phase state and compose the Q data page image on the SLM. [0038] 7) Open and close the shutter to expose the Q hologram.

[0039] So long as the wavelength, the apparatus, and the recording medium are stable during the interval including both exposures, two holograms thusly written will maintain a quadrature relationship for each of their respective fringe components. To this end, it is advantageous if the exposures are completed in the smallest practical interval of time. However, the stability requirements are not significantly different than those needed for holography generally. In some embodiments, the variable retarder is a liquid crystal-based device. In some embodiments, the variable retarder is an electro-optical device. In some embodiments, the variable retarder is a mirror mounted to a piezoelectric actuator (in such an embodiment, the variable retarder is reflective, not transmissive as shown in FIG. 1, views a) and b).

[0040] In embodiments recording higher-order PSK or QAM signals, more than two exposures may be performed.

Recorded Signal DC Blocking

[0041] In some embodiments, a filtering operation is performed to suppress residual DC component of the signal beam during recording. This may be beneficial if the SLM produces a significant DC component due to manufacturing or alignment tolerances. DC blocking may be effected by placing a stop in the DC location of a Fourier plane, for example a small opaque dot in the center of the polytopic aperture. DC blocking may alternatively be effected with an angle filter, e.g., with a multilayer thin-film coating that selectively reflects very low incidence angles (near DC) while transmitting other angles.

Recorded Phase Changing

[0042] In some embodiments, different first and second phase states are used while quadrature hologram pairs are recorded within a book. For example, the sequence of first phase states may correspond to 0.degree., 10.degree., 20.degree., … while the second phase state sequence is to 90.degree., 100.degree., 110.degree… . . In this manner, a phase difference of 90.degree. is maintained within each pair, but the absolute phase of the signal beam within the book changes so as to prevent the coherent build-up of pixel-to-pixel noise gratings within the medium.

Recovery Operation

[0043] Configuration for a recovery operation in one embodiment is shown in FIG. 1, view b). Using this apparatus, the sequence of operations for performing QHD or NHD recovery proceeds as follows: [0044] 1) Select the desired medium (r, .theta.) location and reference beam angle. [0045] 2) Configure switchable HWP 1 to transmit s-polarized light so as to illuminate the detector, forming the local oscillator beam. [0046] 3) Configure switchable HWP 2 to transmit p-polarized diffracted light to the detector. [0047] 4) Configure the conjugating mirror to retro-reflect the probe beam, thus forming a phase-conjugated replica of the recording reference beam. [0048] 5) Set the variable retarder to a first phase state. [0049] 6) Open and close the shutter (not shown) to detect the I.sub.A detector image. [0050] 7) Set the variable retarder to a second phase state. [0051] 8) Open and close the shutter to detect the I.sub.B detector image.

[0052] For n-rature detection, steps (5) and (6) are repeated for the C detector image, etc… . . The detector images so collected are stored in memory and subsequently processed according to the desired embodiment of the invention.

Local Oscillator Mixing

[0053] In the illustrated embodiment, the signal and local oscillator are incident upon the analyzer in orthogonal linear polarizations. The analyzer is a linear polarizer with its axis oriented so as to transmit components of both the signal and the local oscillator, thereby projecting them onto a common polarization axis. Orientation of the analyzer polarization axis may be used to control the mixing ratio of the signal and local oscillator. In some embodiments, the orientation is set to favor the signal beam, e.g., a 90% signal to 10% local oscillator mixing ratio.

[0054] In other embodiments, recording and recovery is performed in architectures substantially different than that of FIG. 1, views a) and b).

Quadrature Homodyne Detection (QHD)

[0055] This section outlines a method of QHD in the absence of enhanced resampling, as described in [1].

Quadrature Image Recovery

[0056] Using the configuration illustrated in FIG. 1, view 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 .DELTA..phi.(x, y) between the holographic signal carrier and the local oscillator will contain only slowly varying components, producing slowly varying fringes.

[0057] FIG. 2 illustrates I.sub.A and I.sub.B quadrature image simulations of a PSK-encoded hologram recovered with a local oscillator in each of two phase states (0.degree. and 90.degree.). The first image of FIG. 2, I.sub.A, shows the simulated detector intensity when the switchable retarder is in the 0.degree. phase state, and the second image, I.sub.B, corresponds to the 90.degree. state.

[0058] 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.degree. phase change, the entire fringe pattern has also shifted by 90.degree. so that regions of low contrast in the I.sub.A image are high contrast in the I.sub.B image, and vice-versa. Thus, the two images contain all the information needed to recover the data.

[0059] Estimating .DELTA..phi.(x, y)

[0060] If the hologram contains reserved block patterns as described in [2], then the quadrature image pair also contains all the information necessary to determine .DELTA..phi.(x, y) Recalling the image alignment measurement method of [2], a cross correlation operation is performed between a portion of the detected image and a corresponding reserved block pattern. FIG. 3 shows normalized cross correlation peak strength maps P.sub.A and P.sub.B for the simulated I.sub.A and I.sub.B images of FIG. 2, respectively. Comparing FIG. 2 with FIG. 3, it is apparent that the regions that are in high contrast and non-inverted show large positive peak strength values (approaching +1). Similarly, inverted regions with high contrast have large negative peak strength values approaching -1. The peak strength maps thus constitute contrast information, measuring the contrast (including polarity) of the SLM pixel images regionally. Together, the two peak strength maps represent quadrature projections of .DELTA..phi.(x, y) onto the locally-varying recovery phase basis.* It may be estimated by the expression*

.DELTA.{circumflex over (.phi.)}(x,y)=tan.sup.-1[P.sub.B(x,y),P.sub.A(x,y)] (1)

where tan.sup.-1 is the four-quadrant arctangent.

[0061] 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. In one embodiment this is performed using a simple bilinear interpolation function.

Quadrature Image Combination

[0062] Following a derivation here omitted for brevity [3], we find the rule for optimally combining the I.sub.A and I.sub.B images to be

E ^ S ( x , y ) = cos [ .DELTA. .phi. ( x , y ) ] I ~ A ( x , y ) + sin [ .DELTA. .phi. ( x , y ) ] I ~ B ( x , y ) = P A ( x , y ) ( P A 2 ( x , y ) + P B 2 ( x , y ) ) 1 / 2 I ~ A ( x , y ) + = P B ( x , y ) ( P A 2 ( x , y ) + P B 2 ( x , y ) ) 1 / 2 I ~ B ( x , y ) ( 2 ) ##EQU00001##

where E.sub.S is the estimated signal field, and .sub.A and .sub.B are (potentially modified) versions of the quadrature images (see Detector Image Modification, below). In this form, it can be seen that the estimate simply combines the two images in proportion to their local contrast, as measured by cross correlation peak strength. The negative sign of the peak will restore the correct polarity in the inverted regions, and the denominator will normalize local variations in image intensity. FIG. 4 shows an image produced by combination of the quadrature image pair of FIG. 2, showing high-contrast non-inverted data throughout.

Enhanced Resampling Quadrature Homodyne Detection Embodiments

[0063] Embodiments of enhanced resampling quadrature homodyne detection (ERQHD) are species of quadrature homodyne detection (QHD). Embodiments of ERQHD include a quadrature image recombination operation combined with a pixel resampling process. FIG. 5 illustrates operations in a generic embodiment of QHD. As shown in FIG. 5, each block represents a data array containing data associated with the a process operation, along with the size/resolution of the data array. Arrays marked with size/resolution “[det]” in FIG. 5 indicate that the data therein correspond to detector pixels, and the data array typically has size equal to that of the detector array, e.g., 1710 rows by 1696 columns of pixels in an exemplary embodiment. Data arrays marked with size/resolution “[SLM]” indicate that the data therein correspond to SLM pixels, typically with data array size equal to the SLM size, e.g., 1200 rows by 1200 columns of pixels in an exemplary embodiment. Typically, a detector has more pixels than a corresponding SLM since the detector must oversample the SLM in order to resolve a modulated data pattern.

[0064] Data arrays marked with size/resolution “[rb]” indicate that the data therein correspond to reserved blocks, which are known data patterns embedded within the holographic data page format. These data arrays would have, e.g., 18 rows by 19 columns of entries corresponding to the 18.times.19 reserved block sample grid of an exemplary data page.

[0065] FIG. 5 is a flow chart illustrating a method of QHD absent enhanced resampling. The QHD method illustrated in FIG. 5 includes operations of upsampling the A and B Quiver Peaks from the [rb] resolution to the [det] resolution, as well as an operation of upsampling the Quiver Alignment from the [rb] resolution to the [SLM] resolution. The upsampled A and B Peaks are subsequently used to produce the Quadrature Combined Image at the [det] resolution. FIG. 6, by contrast, illustrates an exemplary embodiment of a method of ERQHD in which the A and B Quiver Peaks are instead upsampled to the [SLM] resolution, and the operation of producing the Quadrature Combined Image is omitted.

[0066] Instantiation of a QHD process in a form of ERQHD, including, but not limited to embodiments such as illustrated in FIG. 6, may confer several advantages over a QHD process in the absence of enhanced resampling illustrated in FIG. 5:

[0067] 1) Upsampling from reserved blocks ([rb]) to SLM ([SLM]) resolution is computationally simpler than upsampling to detector ([det]) resolution because the reserved blocks may be positioned on a rectilinear grid within the SLM image. Upsampling may thus be performed by relatively simple processes such as inserting an integral number of values in each dimension, e.g., using a bi-linear interpolation algorithm. Upsampling from the reserved block to the detector resolution, by contrast, involves upsampling reserved block information that does NOT generally lie on a rectilinear grid due to real-world image distortions, and has an upsampling ratio that is not only non-integer, but one that varies throughout the image. Thus the process of upsampling to SLM resolution may be both simpler and more accurate than upsampling to detector resolution.

[0068] 2) The memory size to store SLM resolution data arrays is typically smaller than that to store detector resolution data arrays.

[0069] 3) Reserved block ([rb]) to SLM ([SLM]) upsampling is required in both cases in order to generate the Upsampled Alignment from the Quiver Alignment, thus in the ERQHD case the upsampling algorithm and perhaps the hardware itself may be shared for both purposes.

[0070] Note that according to the n-rature embodiment of the present invention, the number of detector images used may be increased from two to three or more, e.g., using detector images I.sub.A, I.sub.B, I.sub.C, … .

ERQHD Resampling

[0071] The resampling stage for ERQHD must be modified to operate directly on the I.sub.A and I.sub.B … images from the detector rather than the Quadrature Combined Image. Resampling of the Quadrature Combined Image for QHD or NHD may be performed in the same manner as resampling in a direct detection channel according to [2][4]. In this approach, the position of each SLM pixel image upon the detector is established by locating the positions of the reserved blocks within the page image. Resampling is then performed by choosing a set of detector pixel values I near to the SLM pixel image (e.g., the nearest 4.times.4 window of detector pixels), and applying a set of resampling coefficients w, i.e.,

{circumflex over (d)}=Iw (3)

where {circumflex over (d)} is the estimated data value d, of the SLM pixel image, e.g., d.epsilon.{-1,+1} for BPSK data. Note that while d.epsilon.{-1,+1}, d may assume non-integer values such as, e.g., 0.90, for subsequent soft error correction. w may be chosen to minimize the squared error between {circumflex over (d)} and the actual data, d, over many detection cases. Furthermore, differing w coefficient sets may be optimized and applied for differing alignment cases, e.g., 256 different w coefficient sets could be used corresponding to differing 2D fractional pixel alignment cases of the 4.times.4 window of detector pixels with respect to the SLM pixel image.

[0072] For ERQHD, the detector pixel values I in the Quadrature Combined Image may be replaced by the corresponding pixel values in the I.sub.A and I.sub.B detector images. For ERQHD n-rature detection, there may be additional images, e.g., C, etc., as well. Furthermore, the detected pixel values in these images may be modified, such as by subtracting a global image mean, etc., in accordance with a Detector Image Modification rule described in Detector Image Modification, below. Let the set of (modified) detector pixel values near the SLM pixel image be designated by , {tilde over (B)}, {tilde over (C)}, etc… . . Then an ERQHD resampling procedure might be used,

{circumflex over (d)}=[c.sub.A +c.sub.B{tilde over (B)}+c.sub.C{tilde over (C)}+ … ]w (4)

where c.sub.A, c.sub.B, … represent the combination coefficients for the respective detector pixel value sets. In one embodiment, these combination coefficients may be determined by the cosine projection of the data page upon the local oscillator used to detect it as measured by the reserved block correlation peak strengths, e.g.,

c A = P A P A 2 + P B 2 + , c B = P B P A 2 + P B 2 + , ( 5 ) ##EQU00002##

where P.sub.A, P.sub.B, … are the Upsampled A Peaks, B Peaks, etc., … for the corresponding SLM pixel image as determined from the reserved block correlation operations on the corresponding I.sub.A, I.sub.B, … detector images. In another embodiment, the normalizing denominators in the cosine projections may be omitted, e.g.,

C.sub.A=P.sub.A, C.sub.B=P.sub.B, (6)

[0073] In still other embodiments, the combination coefficients may be determined from the cosine projections of the reserved blocks from a different data page, e.g., when performing phase quadrature holographic multiplexing (see the description of Phase Quadrature Holographic Multiplexing below). For example, the data value of the corresponding SLM pixel in the I (inphase) data page could be estimated by applying equations (4) and (5) as presented, and then the data value of the corresponding SLM pixel image in the Q (quadrature) image could be estimated by using different combination coefficients, e.g.,

c A = cos ( cos – 1 ( P A P A 2 + P B 2 + ) – .phi. Q ) , c B = cos ( cos – 1 ( P B P A 2 + P B 2 + ) – .phi. Q ) , ( 7 ) ##EQU00003##

where .phi..sub.Q is the known phase difference between the I and Q images, e.g., 90.degree.. In this manner separate correlation and upsampling operations do not need to be performed for the reserved block patterns in the Q image; instead the entire image combination and resampling process may be accomplished using the reserved block patterns of the I image. An I image is sometimes referred to a P image, such that an I and Q image pair can alternatively be referred to as a P and Q image pair. An I image of an I and Q image pair should not be confused with I.sub.A, I.sub.B, I.sub.C, etc., which refer to intensities of image A, image B, image C, etc., respectively.

N-Rature Homodyne Detection

[0074] In another embodiment of the present invention, n-rature homodyne detection (NHD) may be performed in lieu of quadrature homodyne detection (QHD). In QHD, two detector images, I.sub.A and I.sub.B, are acquired of each hologram. A phase difference, typically but not necessarily 90.degree., is introduced into either the signal beam or the local oscillator beam so that different projections (typically, but not necessarily, orthogonal) of the complex signal will appear in each detector image. In NHD, this process is generalized to use n detector images, I.sub.A, I.sub.B, I.sub.C, etc., and a phase difference typically of 360.degree./n is introduced for each subsequent image. Though NHD requires additional holographic exposures, it enjoys other benefits, most notably the rejection of common intensity noise, as described below.

In traditional QHD,* the detected images may be described by*

I.sub.A=I.sub.LO+I.sub.S+2|E.sub.LO.parallel.E.sub.S|cos(.DELTA..phi.)

I.sub.B=I.sub.LO+I.sub.S+2|E.sub.LO.parallel.E.sub.S|cos(.DELTA..phi.+9.- degree.) (8)

where I.sub.S is the signal intensity, I.sub.LO is the local oscillator intensity, and .DELTA..phi. is the phase difference between the two. |E.sub.S| and |E.sub.LO| are the magnitudes of the optical fields, i.e., |E.sub.S|= {square root over (I.sub.S)},|E.sub.LO|= {square root over (I.sub.LO)}. The quantities .phi..sub.A.apprxeq..DELTA..phi. and .phi..sub.B.apprxeq..DELTA..phi.+90.degree. are measured using reserved block correlations, and then the signal magnitude |E.sub.S|* may be estimated by*

E ^ S = 1 2 E LO [ I A cos ( .phi. A ) + I B cos ( .phi. B ) ] = 1 2 E LO { [ I LO + I S + 2 E LO E S cos ( .DELTA. .phi. ) ] cos ( .phi. A ) + [ I LO + I S + 2 E LO E S cos ( .DELTA. .phi. + 90 .degree. ) ] cos ( .phi. B ) } .apprxeq. E S ( cos 2 ( .DELTA. .phi. ) + cos 2 ( .DELTA. .phi. + 90 .degree. ) ) + 1 2 E LO ( I LO + I S ) ( cos ( .DELTA. .phi. ) + cos ( .DELTA. .phi. + 90 .degree. ) ) ( 9 ) ##EQU00004##

The final term in the last expression represents common intensity noise. Since the factor (cos.sup.2(.DELTA..phi.)+cos.sup.2(.DELTA..phi.+90.degree.))=1 for all .DELTA..phi. in the final expression, common intensity noise is an additive noise source in the estimate of |E.sub.S|. Since, typically, I.sub.LO<<I.sub.S and I.sub.LO.apprxeq.constant for homodyne detection, common intensity noise is usually small compared to the signal. Common intensity noise may also be mitigated by Detector Image Modification as described subsequently. Nevertheless, the elimination of common intensity noise by n-rature detection has been shown to produce a significant SNR boost in laboratory tests.

[0075] For n=3, n-rature detection,* the detector images may be written as*

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