Abstract:
A feed-forward equalizer can be used in the host optical receiver to perform at least some of the desired signal processing in the optical domain, e.g., prior to coherently detecting and digitizing the received optical signal(s). In some embodiments, the signal processing implemented in the feed-forward equalizer can at least partially compensate the adverse effects of chromatic dispersion, polarization-mode dispersion, and/or spatial-mode mixing/crosstalk imparted on the received optical signal(s) in the optical transport link. This reduces the signal-processing load of and the signal-processing requirements to the receiver&#39;s electrical DSP.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims priority from U.S. Provisional Patent Application No. 61/608,246, filed on Mar. 8, 2012, and entitled “OPTICAL FEED-FORWARD EQUALIZER, SUCH AS FOR AN OPTICAL RECEIVER.” This provisional patent application is incorporated herein by reference in its entirety. 
     
    
     BACKGROUND 
       [0002]    1. Field of the Invention 
         [0003]    The present invention relates to optical communication equipment and, more specifically but not exclusively, to an optical feed-forward equalizer that can be used in MIMO (multiple input, multiple output) signal processing. 
         [0004]    2. Description of the Related Art 
         [0005]    This section introduces aspects that may help facilitate a better understanding of the invention(s). Accordingly, the statements of this section are to be read in this light and are not to be understood as admissions about what is in the prior art or what is not in the prior art. 
         [0006]    The next-generation of optical communication systems is being designed for relatively high data-transmission rates, e.g., higher than about 100 Gbit/s per channel. At these rates, the effects of chromatic dispersion (CD) and polarization-mode dispersion (PMD) may significantly degrade the transmission performance of optical transport links. A representative prior-art approach to dealing with these signal impairments is to perform appropriate signal processing in the electrical digital domain, e.g., after the corresponding optical signal has been coherently detected and digitized at the receiver. This electrical digital signal processing is typically implemented using a customized ASIC, which can be relatively expensive to design and/or fabricate. In addition, such an ASIC typically requires relatively high power to operate, with the consumed power being approximately proportional to the operative baud rate squared. 
         [0007]    The optical MIMO methods that exploit the inherently high transmission capacity of multipath (e.g., multimode and/or multi-core) optical fibers tend to further increase the complexity of digital signal processing at the receiver, e.g., because the corresponding ASIC may be additionally configured to deal with the effects of spatial-mode mixing and crosstalk in the corresponding multipath optical transport link. By some estimates, the complexity of an ASIC configured to process MIMO signals might be about one hundred times higher than that of an ASIC that does not implement MIMO processing. As a result, practical implementation and operation of an electrical digital signal processor (DSP) configured to handle the effects of CD, PMD, and spatial-mode mixing/crosstalk in an optical MIMO system might be too expensive for commercial applications. 
       SUMMARY OF SOME SPECIFIC EMBODIMENTS 
       [0008]    At least some of the above-indicated problems are addressed by various embodiments of an optical feed-forward equalizer disclosed herein. In one embodiment, the optical feed-forward equalizer is part of the host optical receiver that enables the latter to perform at least some of the desired signal processing in the optical domain, e.g., prior to coherently detecting and digitizing the received optical signal(s). The optical signal processing implemented in the optical feed-forward equalizer can be used, e.g., to at least partially compensate the adverse effects of chromatic dispersion, polarization-mode dispersion, and/or spatial-mode mixing/crosstalk imposed on the received optical signal(s) by the optical transport link. This reduces the signal-processing load of and/or the signal-processing requirements to the receiver&#39;s electrical DSP. 
         [0009]    According to one embodiment, provided is an apparatus comprising an optical feed-forward equalizer configured to be coupled between an input fiber and one or more output fibers. The feed-forward equalizer comprises: an optical splitter configured to optically split light received from the input fiber into a plurality of sub-beams; and a first spatial light modulator (SLM), wherein the optical splitter is configured to pass at least some sub-beams of said plurality of sub-beams to the first SLM such that a sub-beam from the at least some sub-beams impinges on a portion of the first SLM. The first SLM is configured to (i) receive said at least some sub-beams from the optical splitter with relative time delays with respect to one another and (ii) spatially modulate each of the received sub-beams to generate a respective modulated beam. The feed-forward equalizer is configured to couple the modulated beams generated by the first SLM into an optical fiber of the one or more output fibers. 
         [0010]    In some embodiments of the above apparatus, the first SLM is configured to receive said at least some sub-beams from the optical splitter with the relative time delays that have values from a set consisting of integer multiples of a constant time delay. 
         [0011]    In some embodiments of any of the above apparatus, the feed-forward equalizer is configured to operate as an optical finite-impulse-response filter. 
         [0012]    In some embodiments of any of the above apparatus, the input fiber is a multimode fiber. 
         [0013]    In some embodiments of any of the above apparatus, the feed-forward equalizer further comprises an optical spatial-mode de-multiplexer coupled to the input fiber and configured to spatially separate light that populates different spatial modes of the input fiber; and the optical splitter is configured to optically split the spatially separated light to generate the plurality of sub-beams. 
         [0014]    In some embodiments of any of the above apparatus, the optical splitter is configured to cause said at least some sub-beams of the plurality of sub-beams to impinge onto the first SLM as a rectangular array in which the sub-beams are parallel to one another. 
         [0015]    In some embodiments of any of the above apparatus, the first SLM is configured to operate in transmission. 
         [0016]    In some embodiments of any of the above apparatus, the first SLM comprises an array of MEMS mirrors. 
         [0017]    In some embodiments of any of the above apparatus, the optical splitter comprises an optically transparent plate having first and second opposing surfaces, wherein: the first surface has a first portion covered by an anti-reflection coating and a second portion covered by a non-transparent mirror; and the second surface has a first portion covered by a partially transparent mirror. 
         [0018]    In some embodiments of any of the above apparatus, the first portion of the first surface is configured to receive light from the input fiber and couple the received light into an interior of the optically transparent plate; and the optical splitter is configured to generate at least a subset of the plurality of sub-beams using light that exits the interior of the optically transparent plate through the partially transparent mirror. 
         [0019]    In some embodiments of any of the above apparatus, the apparatus is configured such that the relative time delay of a sub-beam in said subset is determined by a respective number of back-and-forth trips in a zigzag pattern that the light of the sub-beam takes in the interior of the optically transparent plate between the first surface and the second surface before leaving the interior through the partially transparent mirror. 
         [0020]    In some embodiments of any of the above apparatus, the optically transparent plate is oriented at a tilt angle with respect to an input plane of the first SLM. 
         [0021]    In some embodiments of any of the above apparatus, the feed-forward equalizer is configured to have the input fiber oriented at a tilt angle with respect to the optically transparent plate. 
         [0022]    In some embodiments of any of the above apparatus, the second surface has a second portion covered by an anti-reflection coating; and the optical splitter is configured to generate at least one of the sub-beams using light that exits the interior of the optically transparent plate through the second portion of the second surface. 
         [0023]    In some embodiments of any of the above apparatus, the partially transparent mirror has non-uniform reflectivity across the first portion of the second surface. 
         [0024]    In some embodiments of any of the above apparatus, the feed-forward equalizer further comprises one or more additional SLMs, wherein: the optical splitter is further configured to apply a respective subset of said plurality of sub-beams to each of the one or more additional SLMs such that different sub-beams of the respective subset impinge on different respective portions of the corresponding additional SLM; each of the one or more additional SLMs is configured to (i) receive the respective subset of said plurality of sub-beams from the optical splitter with relative time delays with respect to one another and (ii) spatially modulate each of the received sub-beams to generate a respective modulated beam; and the feed-forward equalizer is further configured to couple the modulated beams generated by each of the one or more additional SLMs into a respective one of the output fibers different from the optical fiber configured to receive the modulated beams generated by the first SLM. 
         [0025]    In some embodiments of any of the above apparatus, the apparatus further comprises a controller coupled to the first SLM and the one or more additional SLMs to control respective spatial modulation patterns imparted by the SLMs onto the respective received sub-beams. 
         [0026]    In some embodiments of any of the above apparatus, the controller is configured to cause the SLMs to display the respective spatial modulation patterns in a manner that causes the feed-forward equalizer to reverse spatial-mode mixing imparted, by a corresponding optical transport link, on the light received from the input fiber. 
         [0027]    In some embodiments of any of the above apparatus, the apparatus further comprises a controller coupled to the first SLM to control spatial modulation patterns imparted by the first SLM onto the received sub-beams, wherein the controller is configured to cause the first SLM to display the spatial modulation patterns in a manner that causes the feed-forward equalizer to mitigate effects of dispersion imparted, by an optical transport link, on the light received from the input fiber. 
         [0028]    In some embodiments of any of the above apparatus, the feed-forward equalizer is part of an optical receiver; and the optical receiver comprises a coherent optical detector configured to receive light from the feed-forward equalizer through the optical fiber configured to receive the modulated beams generated by the first SLM. 
         [0029]    According to another embodiment, provided is a method of processing optical signals, wherein the method has the steps of: (A) splitting, in an optical splitter, light received from an input fiber into a plurality of sub-beams; (B) applying at least some sub-beams of said plurality to a first SLM so that (i) different sub-beams impinge on different respective portions of the first SLM and (ii) the sub-beams applied to the first SLM have relative time delays with respect to one another; (C) spatially modulating, in the first SLM, each of the sub-beams applied to it by the optical splitter to generate a respective modulated beam; and (D) optically coupling the modulated beams generated by the first SLM into a first of one or more output fibers. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0030]    Other aspects, features, and benefits of various embodiments of the disclosure will become more fully apparent, by way of example, from the following detailed description and the accompanying drawings, in which: 
           [0031]      FIG. 1  shows a block diagram of an optical receiver according to an embodiment of the disclosure; 
           [0032]      FIGS. 2A-2B  illustrate a feed-forward equalizer (FFE) circuit that can be used in the optical receiver of  FIG. 1  according to an embodiment of the disclosure; 
           [0033]      FIGS. 3A-3C  illustrate possible modifications to the FFE circuit shown in  FIG. 2A  according to an embodiment of the disclosure; 
           [0034]      FIG. 4  shows a block diagram representing a side view of an FFE circuit that can be used in the optical receiver of  FIG. 1  according to an embodiment of the disclosure; and 
           [0035]      FIG. 5  illustrates a method of MIMO signal processing that can be implemented in an FFE circuit according to yet another embodiment of the disclosure. 
       
    
    
     DETAILED DESCRIPTION 
       [0036]      FIG. 1  shows a block diagram of an optical receiver  100  according to an embodiment of the disclosure. Receiver  100  is configured to receive an optical input signal  102  via a corresponding optical transport link (not explicitly shown in  FIG. 1 ) from a remote optical transmitter. The optical transport link may be implemented using a single-mode fiber, a multimode fiber, a multi-core fiber, and/or a fiber-optic cable having a plurality of optical fibers. Representative examples of optical fibers that can be used to deliver signal  102  to receiver  100  are disclosed, e.g., in U.S. Patent Application Publication Nos. 2010/0329670 and 2010/0329671, both of which are incorporated herein by reference in their entirety. In some embodiments, optical input signal  102  can be a space-division multiplexed (SDM) signal generated as described, e.g., in U.S. patent application Ser. No. 12/986,468 (filed on Jan. 7, 2011) and U.S. Patent Application Publication No. 2011/0243490, both of which are incorporated herein by reference in their entirety. 
         [0037]    Optical input signal  102  is applied to a feed-forward equalizer (FFE) circuit  110  configured to perform optical signal processing, e.g., as further described below in reference to  FIGS. 2-5 . In various embodiments, the optical signal processing implemented in FFE circuit  110  is directed at reducing the adverse effects of certain signal impairments imposed on optical input signal  102  in the optical transport link. Representative examples of such impairments include, but are not limited to CD, PMD, spatial-mode mixing, and other linear signal distortions. A control signal  148  generated by a controller  150  may be used to dynamically change the configuration of FFE circuit  110 , e.g., to track the changing conditions in the optical transport link. Control signal  148  can be generated, e.g., based on the results of signal decoding in an electrical digital signal processor (DSP)  140  that are provided to controller  150  via a DSP/controller interface  144 . 
         [0038]    Based on the optical signal processing implemented therein, FFE circuit  110  generates K optical signals  112   1 - 112   K , where K is a positive integer. Although the embodiment shown in  FIG. 1  corresponds to K&gt;1, an embodiment for which K=1 is also possible (for example, see  FIG. 2A ). 
         [0039]    Each of optical signals  112   1 - 112   K  generated by FFE circuit  110  is coherently detected, as known in the art, by a corresponding coherent detector  130  using a local oscillator (LO) signal  122  supplied by an LO source  120 . The detection results generated by coherent detector  130  are digitized and supplied in electrical digital form to DSP  140 . In a representative embodiment, each electrical digital sample provided by coherent detector  130  to DSP  140  includes digital measures of the in-phase and quadrature components of signal  112 , e.g., as indicated in  FIG. 1  by digital signals labeled I and Q, respectively. For each signaling interval (e.g., symbol period), DSP  140  processes a full set of digital measures I and Q generated by coherent detectors  130   1 - 130   K  to generate an output data stream  142 . Provided that receiver  100  appropriately handles optical input signal  102 , output data stream  142  contains all the data that have been transmitted by the remote transmitter. 
         [0040]    One skilled in the art will understand that one function of FFE circuit  110  and DSP  140  is to implement signal processing that inverts the transfer function corresponding to the optical transport link between the remote transmitter and receiver  100 . For polarization-division multiplexed (PDM) and/or space-division multiplexed (SDM) signals the transfer function can be mathematically expressed as a matrix, with different matrix elements of the matrix representing individual transfer functions applied to the corresponding different components of the multiplexed signal. As already indicated above, link conditions may change over time, thereby causing the transfer function to change as well, usually on a millisecond time scale or slower. In one embodiment, FFE circuit  110  and DSP  140  are configured to adaptively follow link-condition variations. For example, DSP  140  can employ blind adaptation algorithms to learn the link conditions and to cause the signal processing implemented in FFE circuit  110  and DSP  140  to adapt to the link conditions. Alternatively or in addition, from time to time, controller  150  might request that the remote transmitter send to receiver  100  a training or pilot sequence for DSP  140  to estimate the present transfer function. The signal processing implemented in DSP  140  might also compensate for certain nonlinear impediments, such as the phase shifts induced by self-modal and cross-modal fiber nonlinearity. 
         [0041]      FIGS. 2A-2B  illustrate an FFE circuit  200  that can be configured to operate as FFE circuit  110  ( FIG. 1 ) according to an embodiment of the disclosure. More specifically,  FIG. 2A  shows a block diagram representing a side view of FFE circuit  200 .  FIG. 2B  shows a flowchart  290  that represents the flow of the signal processing implemented in FFE circuit  200 . Note that FFE circuit  200  can be considered to be an embodiment of FFE circuit  110  in which K=1 (also see  FIG. 1 ). 
         [0042]    Referring to  FIG. 2A , FFE circuit  200  has an optical input fiber  202  and an optical output fiber  212 . When FFE circuit  200  is used in an embodiment of receiver  100 , optical input fiber  202  receives optical input signal  102 , and optical output fiber  212  outputs optical signal  112 , as indicated in  FIG. 2A . In one embodiment, each of fibers  202  and  212  can be a single-mode fiber. 
         [0043]    Optical input fiber  202  feeds light into a collimator  204  configured to transform that light into a collimated beam  208  and direct that beam to a beam splitter  210 . In one embodiment, beam splitter  210  comprises a relatively thick optically transparent (e.g., glass) plate  212  whose two opposing surfaces  214  and  216  have four different coating films designated by numerical labels  222 ,  224 ,  226 , and  228 . More specifically, film  222 , which covers a portion of surface  214 , is an anti-reflection coating film that minimizes light reflections and causes substantially all light from beam  208  to couple into the interior of plate  212 . Film  228 , which covers a portion of surface  216 , is configured to function as a partially transparent mirror that causes one portion of the light impinging on it from the interior of plate  212  to be transmitted to the exterior of the plate and another portion of the light to be reflected back into the interior of the plate, e.g., as indicated in  FIG. 2A . Film  224 , which covers a portion of surface  214 , is configured to function as a fully reflecting mirror that causes substantially all light impinging on it from the interior of plate  212  to be reflected back into the interior of the plate, e.g., as indicated in  FIG. 2A . Film  226 , which covers a portion of surface  216 , is an anti-reflection coating film that minimizes light reflections and causes substantially all light impinging on it from the interior of plate  212  to be transmitted to the exterior of the plate, e.g., as indicated in  FIG. 2A . 
         [0044]    The number of internal reflections in plate  212  depends on the tilt angle of optical input fiber  202  with respect to the surface of the plate and also on the relative size of the surface portions covered by films  226  and  228 . In the configuration shown in  FIG. 2A , these parameters are chosen to cause beam splitter  210  to split beam  208  into five sub-beams labeled  230   0 - 230   4 . One of ordinary skill in the art will appreciate that other coating-film configurations resulting in other respective total numbers of the sub-beams generated by beam splitter  210  are also possible. 
         [0045]    In one embodiment, the reflectivity of film  228  may be non-uniform, e.g., gradually changing along the X direction. For example, the reflection profile of film  228  along the X direction may be such that the reflectivity of the film decreases toward the boundary with film  226  in a manner that causes sub-beams  230   0 - 230   4  to have substantially equal intensities. 
         [0046]    Sub-beams  230   0 - 230   4  generated by beam splitter  210  impinge onto different respective areas of a spatial light modulator (SLM)  240 . At an input plane  238  of SLM  240 , sub-beams  230   1 - 230   4  have a relative time delay with respect to sub-beam  230   0  of τ, 2τ, 3τ, and 4τ, respectively. One of ordinary skill in the art will understand that τ is a constant determined by the thickness of plate  212 , the tilt angle(s) of the plate with respect to fiber  202  and SLM  240 , and the refractive index of the plate&#39;s material. 
         [0047]    SLM  240  is a configurable device that individually modulates each of sub-beams  230   0 - 230   4 , e.g., by applying to each of the sub-beams a respective spatial phase-modulation pattern or a respective spatial phase- and intensity-modulation pattern. For example, U.S. patent application Ser. No. 13/200,072 (filed Sep. 16, 2011) discloses suitable checkerboard phase modulation patterns, which can cause the corresponding optical beam to appear both phase- and amplitude-modulated when averaged, e.g., in the far field, over the pixels of the checkerboard. Such patterns can be used in SLM  240  for individually modulating each of sub-beams  230   0 - 230   4 . For additional details on these modulation patterns, the reader is referred to the above-mentioned U.S. patent application Ser. No. 13/200,072, which is incorporated herein by reference in its entirety. 
         [0048]    When FFE circuit  200  is used in an embodiment of receiver  100 , the pixel configurations of SLM  240  can be controlled by controller  150  via control signal  148 , as indicated in  FIG. 2A . In various embodiments, SLM  240  can be (i) a liquid-crystal-on-silicon (LCOS) SLM configured to operate in reflection or in transmission or (ii) a MEMS minor array configured to operate in reflection. Appropriate polarization-control elements (e.g., one or more quarter-wave plates, not explicitly shown in  FIG. 2A ) can be used in FFE circuit  200 , as known in the art, to ensure proper operation of SLM  240 . 
         [0049]    By applying a respective appropriate modulation pattern to each of sub-beams  230   0 - 230   4 , SLM  240  transforms these sub-beams into beams  242   0 - 242   4 , respectively, and directs the latter beams to a lens  250 . Lens  250  is positioned so that the proximate terminus of optical output fiber  212  is located approximately at the focal point of the lens. As a result, lens  250  functions to spatially recombine beams  242   0 - 242   4  and couple them into optical output fiber  212 . The resulting coupled light forms optical signal  112 . 
         [0050]    In reference to both  FIGS. 2A and 2B , the operation of FFE circuit  200  can be understood as follows. A delay line  260  in flowchart  290  ( FIG. 2B ) that comprises a series of delay elements  262   1 - 262   4  schematically represents the optical-signal reflections within plate  212  ( FIG. 2A ). Each delay element  262  represents one of the back-and-forth trips in a zigzag pattern between surface  216  and surface  214  and can nominally be assigned a delay time of τ. The signal that propagates through delay line  260  is tapped five times. Tap  264   0  ( FIG. 2B ) represents the first partial reflection/transmission of signal  208  by film  228 , which produces sub-beam  230   0  ( FIG. 2A ). Tap  264   1  ( FIG. 2B ) represents the second partial reflection/transmission of the optical signal after the first back-and-forth trip through plate  212  ( FIG. 2A ). The partial transmission through film  228  after the first back-and-forth trip produces sub-beam  230   1  ( FIG. 2A ). Tap  264   2  ( FIG. 2B ) represents the third partial reflection/transmission of the optical signal after the second back-and-forth trip through plate  212 . The partial transmission through film  228  after the second back-and-forth trip produces sub-beam  230   2  ( FIG. 2A ). Tap  264   3  ( FIG. 2B ) represents the fourth partial reflection/transmission of the optical signal after the third back-and-forth trip through plate  212 . The partial transmission through film  228  after the third back-and-forth trip produces sub-beam  230   3  ( FIG. 2A ). Tap  264   4  ( FIG. 2B ) represents the final transmission (without reflection) of the optical signal through film  226  after the final back-and-forth trip through plate  212  ( FIG. 2A ). This transmission produces sub-beam  230   4  ( FIG. 2B ). 
         [0051]    Multipliers  270   0 - 270   4  in flowchart  290  represent different respective portions of SLM  240 . Weighting coefficients b 0 -b 4  applied by multipliers  270   0 - 270   4  to taps  264   0 - 264   4 , respectively, represent the individual modulation patterns applied by the respective portions of SLM  240  to beams  230   0 - 230   4 . Weighting coefficients b 0 -b 4  may have complex values. 
         [0052]    Weighted signals  272   0 - 272   4  generated by multipliers  270   0 - 270   4  represent beams  242   0 - 242   4 , respectively, generated by SLM  240 . 
         [0053]    A series of adders  280   1 - 280   4  in flowchart  290  represent lens  250 . As already explained above, lens  250  in FFE circuit  200  functions to combine beams  242   0 - 242   4  into a single optical signal. Similarly, adders  280   1 - 280   4  in flowchart  290  combine weighted signals  272   0 - 272   4  to generate a single output signal  282 . Output signal  282  in flowchart  290  represents optical output signal  112  in FFE circuit  200 . 
         [0054]    In an alternative embodiment, a single adder configured to appropriately combine weighted signals  272   0 - 272   4  can be used instead of the series of adders comprising adders  280   1 - 280   4  to functionally represent lens  250 . 
         [0055]    One of ordinary skill in the art will recognize that flowchart  290  corresponds to signal processing associated with a finite impulse response (FIR) filter. It therefore follows that FFE circuit  200  implements an optical variant of an FIR filter. One of ordinary skill in the art will further recognize that, when appropriately configured, an FIR filter, such as that implemented by FFE circuit  200 , can be configured to mitigate the detrimental effects of various linear signal distortions, such as those caused by the effects of CD and PMD. 
         [0056]      FIGS. 3A-3C  illustrate possible modifications to FFE circuit  200  ( FIG. 2A ) according to an embodiment of the disclosure. These modifications enable the modified FFE circuit  200  to also mitigate the effects of spatial-mode mixing/crosstalk imposed onto optical input signal  102  by a multimode transmission fiber. 
         [0057]      FIG. 3A  shows a block diagram of an input module  301  that can be used to replace optical input fiber  202  and collimator  204  in FFE circuit  200 . The proper orientation of input module  301  with respect to other elements of FFE circuit  200  is indicated by the coordinate-axis system X′YZ, which is shown in both  FIGS. 2A and 3A . 
         [0058]    Input module  301  includes a multimode fiber  302 . When input module  301  is used in an embodiment of FFE circuit  200 , multimode fiber  302  is configured to receive optical input signal  102 . 
         [0059]    Input module  301  further includes a spatial-mode (SM) de-multiplexer  310  that is coupled between multimode fiber  302  and six single-mode fibers  312   1 - 312   6 . SM de-multiplexer operates to separate the optical signals that populate different spatial modes of multimode fiber  302  and couple the separated optical signals into fibers  312   1 - 312   6 , respectively. Representative optical devices that can be configured to operate as SM de-multiplexer  310  in input module  301  are disclosed, e.g., in the above-cited U.S. patent application Ser. Nos. 13/200,072 and 12/986,468 and the above-cited U.S. Patent Application Publication Nos. 2010/0329670, 2010/0329671, and 2011/0243490. 
         [0060]    Each of optical fibers  312   1 - 312   6  feeds light into a corresponding one of collimators  304   1 - 304   6 . An individual collimator  304  is generally similar to collimator  204  ( FIG. 2A ) and operates to generate a corresponding collimated beam  308 . Collimated beams  308   1 - 308   6  generated by collimators  304   1 - 304   6 , respectively, are directed to beam splitter  210 . 
         [0061]      FIG. 3B  shows light spots  322   1 - 322   6  generated by collimated beams  308   1 - 308   6 , respectively, on the surface of film  222  in beam splitter  210 . Upon receiving collimated beams  308   1 - 308   6 , beam splitter splits each of these beams, e.g., as described above in reference to  FIG. 2A  and beam  208 . The result of this splitting is a 6×5 array of parallel optical sub-beams, each of which is similar to one of sub-beams  230   0 - 230   4  shown in  FIG. 2A . 
         [0062]      FIG. 3C  shows thirty light spots  330  generated at input plane  238  of SLM  240  by the 6×5 rectangular array of the optical sub-beams generated by beam splitter  210  from collimated beams  308   1 - 308   6 . Each of the thirty sub-beams is individually modulated by SLM  240 , which transforms each sub-beam into a beam that is analogous to one of beams  242   0 - 242   4 . Lens  250  then spatially recombines these spatially modulated beams and couples them into optical output fiber  212 . The resulting coupled light forms optical signal  112 , as already indicated in  FIG. 2A . 
         [0063]    In various alternative embodiments, the FFE circuit illustrated by  FIGS. 3A-3C  can similarly be designed to generate a differently sized array of sub-beams analogous to sub-beams  230   0 - 230   4 . For example, the FFE circuit can employ, in place of SM de-multiplexer  310 , an SM de-multiplexer designed to be coupled to a different (# 6 ) number of optical fibers  312 . Alternatively or in addition, beam splitter  210  can be configured to generate a different (# 5 ) number of sub-beams from each received beam, e.g., by changing the thickness of plate  212 , the geometry of the film coatings, and the tilt angles. 
         [0064]      FIG. 4  shows a block diagram representing a side view of an FFE circuit  400  that can be used as FFE circuit  110  ( FIG. 1 ) according to another embodiment of the disclosure. Note that FFE circuit  400  corresponds to K=6. 
         [0065]    FFE circuit  400  includes input module  301  (also see  FIG. 3A ). The projection shown in  FIG. 4  corresponds to a view along the Y-coordinate axis in  FIG. 3A . Due to this projection being shown in  FIG. 4 , only optical fiber  312   1  is visible, with the view of optical fibers  312   2 - 312   6  being blocked in  FIG. 4  by optical fiber  312   1 , and only collimator  304   1  is visible, with the view of collimators  304   2 - 304   6  being blocked in  FIG. 4  by collimator  304   1 . 
         [0066]    FFE circuit  400  further includes six output fibers  412   1 - 412   6 . When FFE circuit  400  is used in an embodiment of receiver  100 , optical input fiber  302  of input module  301  receives optical input signal  102 , and optical output fibers  412   1 - 412   6  yield optical signals  112   1 - 112   6 , respectively, as indicated in  FIG. 4 . 
         [0067]    The collimated beams generated by collimators  304   1 - 304   6  in FFE circuit  400  are directed to a beam splitter  410 , which is configured to split each of the received beams into five corresponding sub-beams  430 . In one embodiment, beam splitter  410  is generally analogous to beam splitter  210  ( FIG. 2A ). 
         [0068]    In addition to beam splitter  410 , FFE circuit  400  includes beam splitters  460   1 - 460   3  configured to operate as indicated in  FIG. 4 . More specifically, beam splitters  460   1 - 460   3  operate to further split each of sub-beams  430  generated by beam splitter  410  and distribute the resulting sub-beams among SLMs  440   1 - 440   6 , as indicated in  FIG. 4 . In one embodiment, each of beam splitters  460   1 - 460   2  is a cube having two internal planar interfaces arranged diagonally in an X shape. Beam splitter  460   3  can be a conventional 3-dB beam-splitting cube. 
         [0069]    Each of SLMs  440   1 - 440   6  used in FFE circuit  400  is generally analogous to SLM  240  ( FIG. 2A ). As such, each of SLMs  440   1 - 440   6  is configurable to display spatial modulation patterns that act to mitigate the detrimental effects of various above-indicated linear signal distortions. 
         [0070]    Each of lenses  450   1 - 450   6  used in FFE circuit  400  is generally analogous to lens  250  ( FIG. 2A ). As such, each lens  450  functions to spatially recombine the optical beams received from the corresponding one of SLMs  440   1 - 440   6  and couple them into the corresponding one of optical output fibers  412   1 - 412   6 . The resulting coupled light forms a respective one of optical output signals  112   1 - 112   6 . 
         [0071]    One of ordinary skill in the art will appreciate that the design concept illustrated by  FIG. 4  can be used to design an alternative embodiment of FFE circuit  400 , in which (i) input module  301  is replaced by a similar input module, but having any desired number N of fibers  312  and collimators  304  and (ii) the number of beam splitters  460 , SLMs  440 , and lenses  450  is appropriately changed to enable the FFE circuit to have any desired number K of output fibers  412 . 
         [0072]      FIG. 5  illustrates a method  500  of MIMO signal processing that can be implemented in an embodiment of FFE circuit  400  having N fibers  312 /collimators  304  in input module  301  and K output fibers  412  (also see  FIG. 4 ). 
         [0073]    Each of input blocks IN 1 -IN N  in  FIG. 5  represents an optical beam generated by a corresponding one of the N collimators analogous to collimators  304  in  FIG. 4 . Each of output blocks OUT 1 -OUT K  in  FIG. 5  represents a corresponding one of the K output signals coupled into the output fibers analogous to output fibers  412  in  FIG. 4 . Each of the processing blocks labeled h nk  (where n=1, 2, . . . , N and k=1, 2, . . . , K) represents a corresponding matrix element of the inverse transfer function corresponding to the optical transport link between the remote transmitter and receiver  100 . The value of each matrix element h nk  is set by the configuration of the respective surface portion of the respective one of SLMs  440  (also see  FIG. 4 ). For example, the value of each of matrix elements h n1  (where n=1, 2, . . . , N) is set by the configuration of the respective portion of SLM  440   1 . The value of each of matrix elements h n2  (where n=1, 2, . . . , N) is set by the configuration of the respective portion of SLM  440   2 . The value of each of matrix elements h n3  (where n=1, 2, . . . , N) is set by the configuration of the respective portion of SLM  440   3 , and so on. As already indicated above, by using appropriate h nk  values, the signal processing of method  500  can substantially undo the effects of spatial-mode mixing/crosstalk in a multimode fiber, thereby causing the optical signals received at each of output blocks OUT 1 -OUT K  to represent a corresponding one of the optical signals originally coupled into the corresponding spatial mode of the multimode fiber at the transmitter end of the optical transport link. Further adjustment of the h nk  values can be used to reduce the detrimental effects of CD and/or PMD caused by the optical transport link. 
         [0074]    Certain embodiments of the apparatus and methods disclosed herein may benefit from the various aspects of the apparatus and methods disclosed in provisional U.S. patent application Ser. No. 61/608,139 (filed on Mar. 8, 2012, as attorney reference 811579-US-PSP) by Roland Ryf, Rene-Jean Essiambre, and Nicolas K. Fontaine, entitled “Multimode Optical Communication Apparatus and Methods.” This provisional patent application is incorporated herein by reference in its entirety. 
         [0075]    While this invention has been described with reference to illustrative embodiments, this description is not intended to be construed in a limiting sense. Various modifications of the described embodiments, as well as other embodiments of the invention, which are apparent to persons skilled in the art to which the invention pertains are deemed to lie within the principle and scope of the invention as expressed in the following claims. 
         [0076]    For the purposes of this specification, a MEMS device is a device having two or more parts adapted to move relative to one another, where the motion is based on any suitable interaction or combination of interactions, such as mechanical, thermal, electrical, magnetic, optical, and/or chemical interactions. MEMS devices are fabricated using micro- or smaller fabrication techniques (including nano-fabrication techniques) that may include, but are not necessarily limited to: (1) self-assembly techniques employing, e.g., self-assembling monolayers, chemical coatings having high affinity to a desired chemical substance, and production and saturation of dangling chemical bonds and (2) wafer/material processing techniques employing, e.g., lithography, chemical vapor deposition, patterning and selective etching of materials, and treating, shaping, plating, and texturing of surfaces. The scale/size of certain elements in a MEMS device may be such as to permit manifestation of quantum effects. Examples of MEMS devices include, without limitation, NEMS (nano-electromechanical systems) devices, MOEMS (micro-opto-electromechanical systems) devices, micromachines, microsystems, and devices produced using microsystems technology or microsystems integration. 
         [0077]    Unless explicitly stated otherwise, each numerical value and range should be interpreted as being approximate as if the word “about” or “approximately” preceded the value of the value or range. 
         [0078]    It will be further understood that various changes in the details, materials, and arrangements of the parts which have been described and illustrated in order to explain the nature of this invention may be made by those skilled in the art without departing from the scope of the invention as expressed in the following claims. 
         [0079]    Although the elements in the following method claims, if any, are recited in a particular sequence with corresponding labeling, unless the claim recitations otherwise imply a particular sequence for implementing some or all of those elements, those elements are not necessarily intended to be limited to being implemented in that particular sequence. 
         [0080]    Reference herein to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment can be included in at least one embodiment of the invention. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments necessarily mutually exclusive of other embodiments. The same applies to the term “implementation.” 
         [0081]    Also for purposes of this description, the terms “couple,” “coupling,” “coupled,” “connect,” “connecting,” or “connected” refer to any manner known in the art or later developed in which energy is allowed to be transferred between two or more elements, and the interposition of one or more additional elements is contemplated, although not required. Conversely, the terms “directly coupled,” “directly connected,” etc., imply the absence of such additional elements. 
         [0082]    The description and drawings merely illustrate the principles of the invention. It will thus be appreciated that those of ordinary skill in the art will be able to devise various arrangements that, although not explicitly described or shown herein, embody the principles of the invention and are included within its spirit and scope. Furthermore, all examples recited herein are principally intended expressly to be only for pedagogical purposes to aid the reader in understanding the principles of the invention and the concepts contributed by the inventor(s) to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions. Moreover, all statements herein reciting principles, aspects, and embodiments of the invention, as well as specific examples thereof, are intended to encompass equivalents thereof. 
         [0083]    It should be appreciated by those of ordinary skill in the art that any block diagrams herein represent conceptual views of illustrative circuitry embodying the principles of the invention.