Patent Publication Number: US-2023132851-A1

Title: Probabilistic Constellation Shaping for Point-To-Multipoint Optical Fiber Communication Systems Employing Subcarriers

Description:
The present patent application hereby claims priority to the provisional patent application identified by U.S. Ser. No. 63/275,050 filed on Nov. 3, 2021 the entire content of which is hereby incorporated by reference. 
    
    
     BACKGROUND 
     So-called point-to-multi-point optical communication systems typically include a hub or primary node that communicates with a plurality of leaf nodes that are remote from the hub node. The leaf nodes, however, in some systems may not be equidistant to the hub node. Rather, the leaf nodes may be located at different distances from the hub node. 
     The noise associated with each leaf transmission path is, in many instances, Gaussian in nature, and such noise has been termed additive white Gaussian noise (AWGN) in a linear power limited regime. Optimal capacity for optical signals propagating in an AWGN channel has been achieved with Gaussian probability distributions in which transmission probability of symbols (and their corresponding constellation points) correspond to a Gaussian distribution. Such Gaussian probability distributions are not uniform and are therefore different from the uniform distribution that normally exists on “standard” modulation formats, such as quadrature phase shift keying (“QPSK”) and m-quadrature amplitude modulation (“QAM”, where m is typically an integer greater than 4). 
     For a given optical fiber path distance between the leaf and the hub, and at a desired signal-to-noise (SNR) margin, there is an optimal spectral efficiency SE for which the transmission rate is maximized. Typically, however, such optimal SE cannot be achieved with the standard modulation formats noted above, because the fixed SEs with coarse granularities associated with such modulation formats may either be too high or too low for the link. Thus, the deployed transmission data rate on each of the hub-leaf links is often less than what the link ideally can carry. 
     SUMMARY 
     Consistent with an aspect of the present disclosure, an apparatus is provided that comprises a hub node operable to transmit a plurality of optical subcarriers, a first one of the plurality of optical subcarriers carrying first symbols and second symbols. The first symbols are transmitted with a first probability and the second symbols are transmitted with a second probability less than the first probability. A second one of the plurality of optical subcarriers carries third and fourth symbols, the third symbol being transmitted with a third probability and the fourth symbol us transmitted with a fourth probability less than the third probability. In addition, the apparatus includes a first leaf node operable to receive the first one of the plurality of optical subcarriers. Further, the apparatus includes a second leaf node operable to receive the second one of the plurality of optical subcarriers. The first leaf node is configured to be located a first distance from the hub node and the second leaf node is configured to be located a second distance from the hub node, the first distance being greater than the second distance. 
     It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention, as claimed. 
     The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate one (several) embodiment(s) and together with the description, serve to explain the principles of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    shows an example of a network consistent with an aspect of the present disclosure; 
         FIG.  2    shows a further example of a network consistent with the present disclosure; 
         FIG.  3    shows an example of power spectral density plot consistent with an aspect of the present disclosure; 
         FIGS.  4   a  and  4   b    show further examples of spectral density plots consistent with the present disclosure; 
         FIGS.  5   a  and  5   b    shows example of probabilistic shaping distributions consistent with the present disclosure; 
         FIG.  6    shows an example of a transmitter consistent with the present disclosure; 
         FIG.  7    shows a block diagram of an application specific integrated circuit (ASIC) consistent with the present disclosure; 
         FIG.  8    shows a block diagram of a digital signal processor (DSP) consistent with the present disclosure; 
         FIGS.  9  and  10    show example of circuitry provided in the DSP shown in  FIG.  8   ; 
         FIG.  11    shows an example of a network consistent with a further aspect of the present disclosure; 
         FIG.  12    a power spectral density plot showing subgroups of optical subcarriers consistent with the present disclosure; 
         FIG.  13    shows an example optical communications network consistent with a further aspect of the present disclosure; 
         FIG.  14    shows a block diagram of a hub node included in the network shown in  FIG.  13   ; and 
         FIG.  15    shows optical subcarriers and wavelength selective switch filter passbands consistent with a further aspect of the present disclosure. 
     
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     Consistent with the present disclosure, a transmitter is provided in a hub node of a point-to-multi-point optical communication system. The transmitter supplies a plurality or group of optical subcarriers, each subgroup of the group of optical subcarriers in, one example, is associated with a respective leaf node. Since the leaf nodes may be located at different distances from the hub node, the modulation of each optical subcarrier subgroup is optimized for the distance and impairments associated with the optical path over which the subcarrier subgroup is transmitted to its designated leaf. Namely, probabilistic shaping is employed, in one example, to adjust the modulation so that a maximum spectral efficiency is supported for a given SNR associated with the hub-leaf link. 
     In addition, optical subcarriers may be selectively blocked or transmitted so that the bandwidth or spectral width associated with a particular group or subgroup of optical subcarriers is within the passband of a wavelength selective switch (WSS), such that each optical subcarrier in the group or subgroup of optical carriers may be switched to a particular output of the WSS, and, in one example, directed to a particular leaf node. 
     Reference will now be made in detail to the present embodiments of the present disclosure, examples of which are illustrated in the accompanying drawings. In general, the same reference numbers will be used throughout the drawings to refer to the same or like parts. The terms “hub,” “hub node,” and “primary node” are used interchangeably herein. In addition, the terms “leaf,” “leaf node,” and “secondary node” are used interchangeably herein. 
       FIG.  1    illustrates an example of an aggregation network  100  consistent with the present disclosure in which primary node  110  may communicate with multiple secondary nodes  112 - j  to  112 - m , which sometimes may be referred to individually or collectively as secondary node(s)  112 . Secondary nodes  112 , in one example, are remote from primary node  110 . Primary node  110  may transmit a group of optical subcarriers, including subgroups of optical subcarriers described in greater detail below, in a downstream direction onto an optical communication path  111 , which, like each of optical communication paths  113 - j  to  113 - m , may include one or more segments of optical fiber, as well as one or more optical amplifiers, reconfigurable add-drop multiplexers (ROADMs) or other optical fiber communication equipment. In the example shown in  FIG.  1   , splitter  114  may be coupled to an end of optical communication path  111  to receive the optical subcarriers and provide a power split portion of each subcarrier to a corresponding one of secondary nodes  112 - j  to  112 - m  via a respective one of optical communication paths  113 - j  to  113 - m.    
     As further shown in  FIG.  1   , primary node  110  has a data capacity to receive n Gbit/s of data (e.g., a data stream) for transmission to secondary node  112 . Each secondary node  112  may receive and output to a user or customer a portion of the data input to primary node  110 . In this example, secondary nodes  112 - j ,  112 - k ,  112 - l , and  112 - m  output j Gbit/s, k Gbit/s, l Gbit/s, and m Gbit/s of data (data streams), respectively, whereby the sum of the j, k, l, and m may equal n (where j, k, l, m, and n are positive numbers). 
       FIG.  2    show transmission of additional subcarriers in an upstream direction from secondary nodes  112 - j  to  112 - m  to primary node  110 . As further shown in  FIG.  2   , each of secondary nodes  112 - j  to  112 - m  may transmit a corresponding group of subcarriers or one subcarrier to optical combiner  116  via a respective one of optical communication paths  115 - 1  to  115 - m . Optical combiner  116  may, in turn, combine the received optical subcarriers from secondary nodes  112 - j  to  112 - m  onto optical communication path  117 . Optical communication paths  115 - 1  to  115 - m  and  117  may have a similar construction as optical communication paths  111  and  112 - 1  to  112 - m.    
     As further shown in  FIG.  2   , each of secondary nodes  112 - j  to  112 - m  receives a respective data stream having a corresponding data rate of j Gbit/s, k Gbit/s, l Gbit/s, and m Gbit/s. At primary node  110 , data contained in these streams may be output such that the aggregate data supplied by primary node  110  is n Gbit/s, such that, as noted above, n may equal the sum of j, k, l, and m. 
     In another example, subcarriers may be transmitted in both an upstream and downstream direction over the same optical communication path. In particular, selected subcarriers may be transmitted in the downstream direction from primary node  110  to secondary nodes  112 , and other subcarriers may be transmitted in the upstream direction from secondary nodes  112  to primary node  110 . 
     In some implementations, network  100  may include additional primary and/or secondary nodes and optical communication paths, fewer primary and/or secondary nodes and optical communication paths, or may have a configuration different from that described above. For example, network  100  may have a mesh configuration or a point-to-point configuration. 
       FIG.  3    illustrates a power spectral density plot of a group G of optical subcarriers, which, in this example, includes optical subcarriers  1  to  32 . Optical subcarrier group G may be output from hub node  110  and may include subgroups of optical subcarriers G 1  to G 2 . In the example, shown in  FIG.  3   , each subcarrier subgroup G 1  to G 2  includes an equal number of optical subcarriers, namely, eight. It is understood, however, that each optical subcarrier subgroup may include more or fewer optical subcarriers. Moreover, although four subgroups are shown in  FIG.  3   . It is further understood that more or fewer subgroups may be output from hub node  110 . 
     Subcarriers  1 - 32 , in one example, are Nyquist subcarriers, which are a group of optical signals, each carrying data, wherein (i) the spectrum of each such optical signal within the group is sufficiently non-overlapping such that the optical signals remain distinguishable from each other in the frequency domain, and (ii) such group of optical signals is generated by modulation of light from a single laser. In general, each subcarrier may have an optical spectral bandwidth that is at least equal to the Nyquist frequency, as determined by the baud rate of such subcarrier. 
     Each optical subcarrier  1 - 32  has a respective one of a plurality of optical frequencies f 1  to f 32  distributed about a center frequency f 0 , which is associated with the frequency of light output from a laser (discussed in greater detail below) that is modulated to provide the group of subcarriers G. In the example shown in  FIG.  3   , sixteen optical subcarriers  1 - 16  have frequencies less than frequency f 0  and sixteen optical subcarriers  17 - 32  have optical frequencies greater than f 0 . It is understood, however, that more or fewer optical subcarrier frequencies may be provided above or below frequency f 0 . 
     Consistent with an aspect of the present disclosure, under certain circumstances transmission of each optical subcarrier may be undesirable. In one example, as described in greater detail below, the bandwidth or spectrum associated with frequencies f 1  to f 32  may exceed that of a filter through which optical subcarriers  1 - 32  are to pass. Accordingly, certain subcarriers may need to be blocked or not transmitted while the transmitted optical subcarriers have frequencies within the bandwidth of the filter. In another example, the data capacity of a leaf node may be insufficient to receive the data associated with a particular number of subcarriers. Rather, the leaf node may have a limited capacity to receive and process optical subcarriers, such that a relative low number of optical subcarriers may be designated for such node. Accordingly, in this example, certain subcarriers may be block while a limited number of optical subcarriers may be transmitted to the leaf node. 
       FIG.  4   a    shows a power spectral density plot whereby certain optical subcarriers, for example, optical subcarriers  1 ,  2 ,  31 , and  32  are blocked or deactivated, whereas remaining subcarriers  3 - 30  are transmitted from hub  110 . In  FIG.  4   b   , optical subcarriers  1 - 4  and  29 - 32  are blocked, while remaining subcarriers  5 - 28  are transmitted from hub  110 . 
     As noted above, leaf nodes  112  are often not equidistant to hub node  110 . For example, some leaf nodes  112  may be significantly closer to hub node  110  than other leaf nodes  112 . Moreover, the distance between each leaf node  112  and hub node  110  may have an associated optimum SE that is different than that corresponding to common modulation formats, such as QPSK and m-QAM. Accordingly, consistent with the present disclosure probabilistic shaping is employed to optimize the SE for a given link. In probabilistic shaping, the signal space is encoded such that the distribution of the projection of the n-D constellation on each of the real and the imaginary dimensions of the constellation follows a desired probability distribution, which may be Gaussian. Probabilistic shaping may be realized by encoding the input information data bits such that, when mapped to a specific 2-D constellation, the probability of occurrence of each of the constellation points follows a desired probability distribution. In other words, unlike standard modulation formats, in which symbols associated with each constellation point are transmitted with equal probability, in probabilistic shaping, certain symbols associated with particular constellation points are transmitted more frequently, i.e., have a higher likelihood or probability of transmission, compared to other symbols corresponding to other constellation points. It has been shown that probabilistic constellation shaping may be able to recover the shaping gain that is lost when standard uniform modulation formats are deployed. 
     A given spectral efficiency (SE) may be associated with a specific probability distribution for a corresponding constellation. Thus, different SEs may be obtained by changing the probability distribution. This is equivalent to designing a single circuit to accommodate many different modulation formats to approximate the Shannon capacity limit for a given link. Thus, in addition to improved SNR gain, probabilistic constellation shaping provides a mechanism to finely tune the SE to maximize the transmission data rate over a communication link at a fixed desired SNR margin. 
       FIG.  5   a    illustrates a three-dimensional histogram plot of a Gaussian probability distribution  3910  (P=2) in which symbols may be transmitted in accordance with an 8 QAM modulation format. The probabilities conform to a Gaussian distribution. Here, symbol transmission probability is shown on the z axis. Inner constellation points in an IQ plane (defined by I and Q axis) are represented by bars  3912 , outer constellation points are represented by bars  3914 , and outermost constellation points  3916  are represented by bars  3916 . As further shown in  FIG.  5   a   , symbols corresponding to inner constellation points are transmitted with a higher probability than symbols corresponding to outer constellation points, and, further, symbols corresponding to the outermost constellation points are transmitted less frequently (lower probability) than symbols corresponding to the outer and inner constellation points. 
       FIG.  5   b    shows a three-dimensional histogram plot of a super Gaussian distribution  3920  (P&gt;2, for example) in which symbols may be transmitted in accordance with an 8 QAM modulation format. The probabilities conform to a super Gaussian distribution. Here, inner constellation points in an IQ plane (defined by I and Q axis shown in  FIG.  5   a   ) are represented by bars  3922 , outer constellation points are represented by bars  3924 , and outermost constellation points bars  3926 . As further shown in  FIG.  5   b   , symbols corresponding to inner constellation points are transmitted with a higher probability than symbols corresponding to outer constellation points, and, further, symbols corresponding to the outermost constellation points are transmitted less frequently (lower probability) than symbols corresponding to the outer and inner constellation points shown in  FIG.  5     a.    
     As noted above, in the distribution shown in  FIG.  5   a    outer symbols (points)  3916  are transmitted with a greater probability than outer symbols  3926  in  FIG.  5 B . Accordingly, the distribution shown in  FIG.  5   a    has an associated SE that is higher than that in  FIG.  5   b   . However, the distribution shown in  FIG.  5   a    may have a higher SNR at longer transmission distances than the SNR associated with the distribution shown in  FIG.  5   b   . Accordingly, higher SE probabilistic shaping distributions, such as that shown in  FIG.  5   a   , may be employed for transmission of optical subcarriers to leaf nodes  112  that are located a relatively short distance away from the hub  110 . On the other hand, lower SE probabilistic shaping distributions, such as that shown in  FIG.  5   b   , may be employed for transmission of optical subcarrier to leaf nodes  112  that are located a relatively long distance away from hub  11 . Such lower SEs are less susceptible to SNR and are therefore more suitable for longer distance transmission. 
       FIG.  6    illustrates transmitter  202  of primary node  110  in greater detail. Transmitter  202  includes, for example, an application specific integrated circuit (ASIC)  902  that receives data inputs D- 0  to D- 31 . It is understood that ASIC  902  may have more or fewer data inputs. In addition, transmitter  202  includes a D/A and optics block  901 . ASIC  902  supplies a plurality of outputs to D/A and optics block  901  including digital-to-analog conversion (DAC) circuits  904 - 1  to  904 - 4 , which convert digital signal received from ASIC  902  into corresponding analog signals. D/A and optics block  901  also includes driver circuits  906 - 1  to  906 - 2  that receive the analog signals from DACs  904 - 1  to  904 - 4  and adjust the voltages or other characteristics of the analog signal to provide drive signals to a corresponding one of modulators  910 - 1  to  910 - 4 . 
     D/A and optics block  901  further includes modulators  910 - 1  to  910 - 4 , each of which may be, for example, a Mach-Zehnder modulator (MZM) that modulates the phase and/or amplitude of the light output from laser  908 . As further shown in  FIG.  6   , light output from laser  908 , also included in block  901 , is split such that a first portion of the light is supplied to a first MZM pairing, including MZMs  910 - 1  and  910 - 2 , and a second portion of the light is supplied to a second MZM pairing, including MZMs  910 - 3  and  910 - 4 . The first portion of the light is split further into third and fourth portions, such that the third portion is modulated by MZM  910 - 1  to provide an in-phase (I) component of an X (or TE) polarization component of a modulated optical signal, and the fourth portion is modulated by MZM  910 - 2  and fed to phase shifter  912 - 1  to shift the phase of such light by 90 degrees in order to provide a quadrature (Q) component of the X polarization component of the modulated optical signal. Similarly, the second portion of the light is further split into fifth and sixth portions, such that the fifth portion is modulated by MZM  910 - 3  to provide an I component of a Y (or TM) polarization component of the modulated optical signal, and the sixth portion is modulated by MZM  910 - 4  and fed to phase shifter  912 - 2  to shift the phase of such light by 90 degrees to provide a Q component of the Y polarization component of the modulated optical signal. 
     The optical outputs of MZMs  910 - 1  and  910 - 2  are combined to provide an X polarized optical signal including I and Q components and are fed to a polarization beam combiner (PBC)  914  provided in block  901 . In addition, the outputs of MZMs  910 - 3  and  910 - 4  are combined to provide an optical signal that is fed to polarization rotator  913 , further provided in block  901 , that rotates the polarization of such optical signal to provide a modulated optical signal having a Y (or TM) polarization. The Y polarized modulated optical signal also is provided to PBC  914 , which combines the X and Y polarized modulated optical signals to provide a polarization multiplexed (“dual-pol”) modulated optical signal onto optical fiber  916 , for example, which may be included as a segment of optical fiber in optical communication path  111 . 
     The polarization multiplexed optical signal output from D/A and optics block  401  includes, in one example, optical subcarriers  1 - 32  noted above, such that each subcarrier has X and Y polarization components and I and Q components. Moreover, each subcarrier  1 - 32  may be associated with or corresponds to a respective one of inputs D- 0  to D- 31 . In another example, fewer than 32 optical subcarriers may be output from the hub transmitter, such as in  FIGS.  4   a    and  4   b.    
       FIG.  7    shows a block diagram of ASIC  902 . ASIC  902  includes a multiplexer  702  which directs or allocates the data provided to inputs to a corresponding one of multiplexer outputs M 0  to M 31 . For example, data provided to input D- 0  may be supplied to output M 31 . In another example, the data at input D- 0  is supplied to more than one of multiplexer outputs M 0  to M 31 . If fewer inputs supply data to multiplexer  702 , the data provided at such inputs may be allocated to a corresponding one of outputs M 0  to M 31 . In another example, data present on one or more of inputs D- 0  to D- 31  is allocated to one or more of outputs M 0  to M 31 . 
     As further shown in  FIG.  7   , groups of multiplexer outputs M 0  to M 31  are provided to a respective one of scheduler circuits Sch  0  to Sch  3 . In the example shown in  FIG.  7   , multiplexer outputs M 0  to M 7  are provided to scheduler circuit Sch  0 ; multiplexer outputs M 8  to M 15  are provided to scheduler circuit Sch  1 ; multiplexer outputs M 16  to M 23  are provided to scheduler circuit Sch  3 ; and multiplexer outputs M 24  to M 31  are provided to scheduler circuit Sch  3 . Scheduler circuits are operable to supply outputs S 0  to S 31  including the data included in inputs D- 0  to D- 31  to corresponding probabilistic shaping (PS) encoders PS 0  to PS 31 . Scheduler circuits are further operable to distribute the data to one or more PS encoders PS 0  to PS 31 , such as those described in U.S. Patent Application Publication No. 2022-0014300 the entire contents of which are incorporated herein by reference. Other known PS encoders may also be employed. 
     Each PS encoder PS 0  to PS 31  is associated with a respective optical subcarrier that is ultimately output from the modulators described above. If fewer than 32 optical subcarriers are transmitted, only the PS encoders PS 0  to PS 31  associated with a transmitted subcarrier may be activated, in one example. Each of PS encoders PS 0  to PS 31 , if activated, outputs a respective one of encoded outputs to a multiplexer  703 , which, in turn, has outputs SC 0  to SC 31 , each of which corresponding to a respective one of optical subcarriers  1 - 32 . In one example, encoded data output from one or more of PS encoders PS 0  to PS 31  is allocated by multiplexer  703  to one or more outputs SC 0  to SC 31 . 
     As further shown in  FIG.  7   , outputs SC 0  to SC 31  are next provided to DSP  704 , which is shown in greater detail in  FIG.  8   . 
     DSP  704  may include FEC encoders  1002 - 0  to  1002 - 31 , each of which may receive a respective one of a plurality of the outputs from switches SW 0  to SW 19 . FEC encoders  1002 - 0  to  1002 - 31  carry out forward error correction coding on a corresponding one of the switch outputs, such as, by adding parity bits to the received data. FEC encoders  1002 - 0  to  1002 - 31  may also provide timing skew between the subcarriers to correct for skew induced by link between nodes  110  and  112 - j  to  112 - m  described above. In addition, FEC encoders  1002 - 0  to  1002 - 31  may interleave the received data. 
     Each of FEC encoders  1002 - 0  to  1002 - 31  provides an output to a corresponding one of a plurality of bits-to-symbol circuits,  1004 - 0  to  1004 - 31  (collectively referred to herein as “ 1004 ”). Each of bits-to-symbol circuits  1004  may map the encoded bits to symbols on a complex plane. For example, bits-to-symbol circuits  1004  may map encoded bits described above to a distribution of symbols consistent with probabilistic shaping. Each of bits-to-symbol circuits  1004  provides first symbols, having the complex representation XI+j*XQ, associated with a respective one of the switch outputs, such as D- 0 , to DSP portion  1003 . Data indicative of such first symbols is carried by the X polarization component of each subcarrier  1  to  32  or each transmitted subcarrier. 
     Each of bits-to-symbol circuits  1004  further may provide second symbols having the complex representation YI+j*YQ. Data indicative of such second symbols, however, is carried by the Y polarization component of each of subcarriers  1  to  32 . 
     Such mapping, as carried by about circuit  1004 - 0  to  1004 - 31  define, in one example, based on the encoded data, a distribution, as described above having an associated SE optimized for a particular hub-leaf link. Accordingly, consistent with an aspect of the present disclosure one or more of the optical subcarriers may carry data at the same or a different data or baud rate than one or more of the other optical subcarriers in accordance with the distribution associated with that subcarrier. 
     In addition, circuits  1004  may further be operable to tailor the distribution, based on the encoded data, to result in an optimized SE and baud and/or data rate for a particular hub-leaf link. 
     As further shown in  FIG.  8   , each of the first symbols output from each of bits-to-symbol circuits  1004  is supplied to a respective one of first overlap and save buffers  1005 - 0  to  1005 - 31  (collectively referred to herein as overlap and save buffers  1005 ) that may buffer 256 symbols, for example. Each of overlap and save buffers  1005  may receive some of the first symbols or another number of such symbols at a time from a corresponding one of bits to symbol circuits  1004 . Thus, overlap and save buffers  1005  may combine new symbols from bits to symbol circuits  1005 , with the previous symbols received from bits to symbol circuits  1005 . 
     Each overlap and save buffer  1005  supplies an output, which is in the time domain, to a corresponding one of fast Fourier Transform (FFT) circuits  1006 - 0  to  1006 - 31  (collectively referred to as “FFTs  1006 ”). Each of FFTs  1006  converts the received symbols to the frequency domain using or based on, for example, a fast Fourier transform. Each of FFTs  1006  may provide the frequency domain data to bins and switches blocks  1021 - 0  to  1021 - 31 . As discussed in greater detail below, bins and switches blocks  1021  include, for example, memories or registers, also referred to as frequency bins (FB) or points, that store frequency components associated with each subcarrier SC. 
     Selected frequency bins FB are shown in  FIG.  9   . Groups of such frequency bins FB are associated with corresponding subcarriers. Accordingly, for example, a first group of frequency bins, FB 0 - 0  to FB 0 - n , is associated with subcarrier  1  and a second group of frequency bins FB 19 - 0  to FB 19 - n  with subcarrier  31  (where n is a positive integer). As further shown in  FIG.  9   , each of frequency bins FB is further coupled to a respective one of switches SW. For example, each of frequency bins FB 0 - 0  to FB 0 - n  is coupled to a respective one of switches SW 0 - 0  to SW 0 - n , and each of FB 31 - 0  to FB 31 - n  is coupled to a respective one of switches or switch circuits SW 31 - 0  to SW 31 - n.    
     Each switch SW selectively supplies either frequency domain data output from one of FFT circuits  1006 - 0  to  1006 - 31  or a predetermined value, such as 0. In order to block or eliminate transmission of a particular subcarrier, the switches SW associated with the group of frequency bins FB associated with that subcarrier are configured to supply the zero value to corresponding frequency bins. Accordingly, for example, in order to block subcarrier  1 , switches SW 0 - 0 ′ to SW 0 - n ′ supply zero (0) values to a respective one of frequency bins FB 0 - 0  to FB 0 - n . Further processing, as described below, of the zero (0) values by replicator components  1007  as well as other components and circuits in DSP  704  result in drive signals supplied to modulators  910 , such that subcarrier  1  is omitted from the optical output from the modulators. 
     On the other hand, switches SW may be configured to supply the outputs of FFTs  1006 , i.e., frequency domain data FD, to corresponding frequency bins FB. Further processing of the contents of frequency bins FB by replicator components  1007  and other circuits in DSP  704  result in drive signals supplied to modulators  910 , whereby, based on such drive signals, optical subcarriers are generated that correspond to the frequency bin groupings associated with that subcarrier. 
     In the example discussed above, switches SW 0 - 0 ′ to SW 0 - n ′ supply frequency domain data FD 0 - 0  to FD-n from FFT  1006 - 0  to a respective one of switches SW 0 - 0  to SW 0 - n . These switches, in turn, supply the frequency domain data to a respective one of frequency bins FB 0 - 0  to FB 0 - n  for further processing, as described in greater detail below. 
     Each of replicator components or circuits  1007 - 0  to  1007 - 31  may replicate the contents of the frequency bins FB and store such contents (e.g., for T/2 based filtering of the subcarrier) in a respective one of the plurality of replicator components. Such replication may increase the sample rate. In addition, replicator components or circuits  1007 - 0  to  1007 - 31  may arrange or align the contents of the frequency bins to fall within the bandwidths associated with pulse shaped filter circuits  1008 - 0  to  1008 - 31  described below. 
     Each of pulse shape filter circuits  1008 - 0  to  1008 - 31  may apply a pulse shaping filter to the data stored in the  512  frequency bins of a respective one of the plurality of replicator components or circuits  1007 - 0  to  1007 - 31  to thereby provide a respective one of a plurality of filtered outputs, which are multiplexed and subject to an inverse FFT, as described below. Pulse shape filter circuits  1008 - 1  to  1008 - 31  calculate the transitions between the symbols and the desired subcarrier spectrum so that the subcarriers can be packed together spectrally for transmission, e.g., with a close frequency separation. Pulse shape filter circuits  1008 - 0  to  1008 - 31  also may be used to introduce timing skew between the subcarriers to correct for timing skew induced by links between nodes shown in  FIG.  1   , for example. Multiplexer component  1009 , which may include a multiplexer circuit or memory, may receive the filtered outputs from pulse shape filter circuits  1008 - 0  to  1008 - 31 , and multiplex or combine such outputs together to form an element vector. 
     Next, IFFT circuit or component  1010 - 1  may receive the element vector and provide a corresponding time domain signal or data based on an inverse fast Fourier transform (IFFT). In one example, the time domain signal may have a rate of 64 GSample/s. Take last buffer or memory circuit  1011 - 1 , for example, may select the last 1024 samples, or another number of samples, from an output of IFFT component or circuit  1010 - 1  and supply the samples to DACs  904 - 1  and  904 - 2  (see  FIG.  9   ) at 64 GSample/s, for example. As noted above, DAC  904 - 1  is associated with the in-phase (I) component of the X pol signal, and DAC  904 - 2  is associated with the quadrature (Q) component of the Y pol signal. Accordingly, consistent with the complex representation XI+jXQ, DAC  904 - 1  receives values associated with XI and DAC  904 - 2  receives values associated with jXQ. As indicated by  FIG.  9   , based on these inputs, DACs  904 - 1  and  904 - 2  provide analog outputs to MZMD  906 - 1  and MZMD  906 - 2 , respectively, as discussed above. 
     As further shown in  FIG.  9   , each of bits-to-symbol circuits  1004 - 0  to  1004 - 31  outputs a corresponding one of symbols indicative of data carried by the Y polarization component of the polarization multiplexed modulated optical signal output on fiber  916 . As further noted above, these symbols may have the complex representation YI+j*YQ. Each such symbol may be processed by a respective one of overlap and save buffers  1015 - 0  to  1015 - 31 , a respective one of FFT circuits  1016 - 0  to  1016 - 31 , a respective one of replicator components or circuits  1017 - 0  to  517 - 31 , pulse shape filter circuits  1018 - 0  to  1018 - 31 , multiplexer or memory  1019 , IFFT  1010 - 2 , and take last buffer or memory circuit  1011 - 2 , to provide processed symbols having the representation YI+j*YQ in a manner similar to or the same as that discussed above in generating processed symbols XI+j*XQ output from take last circuit  1011 - 1 . In addition, symbol components YI and YQ are provided to DACs  904 - 3  and  904 - 4  ( FIG.  9   ), respectively. Based on these inputs, DACs  904 - 3  and  904 - 4  provide analog outputs to MZMD  906 - 3  and MZMD  906 - 4 , respectively, as discussed above. 
     While  FIG.  9    shows DSP  704  as including a particular number and arrangement of functional components, in some implementations, DSP  704  may include additional functional components, fewer functional components, different functional components, or differently arranged functional components. In addition, typically the number of overlap and save buffers, FFTs, replicator circuits, and pulse shape filters associated with the X component may be equal to the number of switch outputs, and the number of such circuits associated with the Y component may also be equal to the number of switch outputs. However, in other examples, the number of switch outputs may be different from the number of these circuits. 
     As noted above, based on the outputs of MZMDs  906 - 1  to  906 - 4 , a plurality of optical subcarriers SC 0  to SC 19  may be output onto optical fiber  916  ( FIG.  9   ), which is coupled to the primary node  110 . 
     Consistent with an aspect of the present disclosure, the number of subcarriers transmitted from primary node  110  to secondary nodes  112  may vary over time based, for example, on capacity requirements at the primary node and the secondary nodes. For example, if less downstream capacity is required initially at one or more of the secondary nodes, transmitter  202  in primary node  110  may be configured to output fewer optical subcarriers. On the other hand, if further capacity is required later, transmitter  202  may provide more optical subcarriers. 
     In addition, if based on changing capacity requirements, a particular secondary node  112  needs to be adjusted, for example, the output capacity of such secondary node may be increased or decreased by, in a corresponding manner, increasing or decreasing the number of optical subcarriers output from the secondary node. 
     As noted above, by storing and subsequently processing zeros (0s) or other predetermined values in frequency bin FB groupings associated with a given subcarrier SC, that subcarrier may be removed or eliminated. To add or reinstate such subcarrier, frequency domain data output from the FFTs  1006  may be stored in frequency bins FB and subsequently processed to provide the corresponding subcarrier. Thus, subcarriers may be selectively added or removed from the optical outputs of primary node transmitter  202  and secondary node transmitter  304 , such that the number of subcarriers output from such transmitters may be varied, as desired. 
     In the above example, zeros (0s) or other predetermined values are stored in selected frequency bins FBs to prevent transmission of a particular subcarrier SC. Such zeroes or values may, instead, be provided, for example, in a manner similar to that described above, at the outputs of corresponding replicator components  1007  or stored in corresponding locations in memory or multiplexer  1009 . Alternatively, the zeroes or values noted above may be provided, for example, in a manner similar to that described above, at corresponding outputs of pulse shape filters  1008 . 
     In a further example, a corresponding one of pulse shape filters  1008 - 1  to  1008 - 31  may selectively generate zeroes or predetermined values that, when further processed, also cause one or more subcarriers to be omitted from the output of either primary node transmitter  202 . In particular, as shown in  FIG.  10   , pulse shape filters  1008 - 0  to  1008 - 31  are shown as including groups of multiplier circuits M 0 - 0  to M 0 - n  . . . M 19 - 0  to M 19 - n  (also individually or collectively referred to as M). Each multiplier circuit M constitutes part of a corresponding butterfly filter. In addition, each multiplier circuit grouping is associated with a corresponding one of subcarriers SC. 
     Each multiplier circuit M receives a corresponding one of output groupings RDO- 0  to RDO-n . . . RD 31 - 0  to RD 31 - n  from replicator components  1007 . In order to remove or eliminate one of subcarriers SC, multiplier circuits M receiving the outputs within a particular grouping associated with that subcarrier multiply such outputs by zero (0), such that each multiplier M within that group generates a product equal to zero (0). The zero products then are subject to further processing similar to that described above to provide drive signals to modulators  910  that result in a corresponding subcarrier SC being omitted from the output of the transmitter (either transmitter  202  or  304 ). 
     On the other hand, in order to provide a subcarrier SC, each of the multiplier circuits M within a particular groping may multiply a corresponding one of replicator outputs RD by a respective one of coefficients C 0 - 0  to C 0 - n  . . . C 31 - 0  to C 31 - n , which results in at least some non-zero products being output. Based on the products output from the corresponding multiplier grouping, drive signals are provided to modulators  910  to output the desired subcarrier or subcarriers from the transmitter  202 . 
     Accordingly, for example, in order to block or eliminate subcarrier  1 , each of multiplier circuits M 0 - 0  to M 0 - n  (associated with subcarrier  1 ) multiplies a respective one of replicator outputs RDO- 0  to RDO-n by zero (0). Each such multiplier circuit, therefore, provides a product equal to zero, which is further processed, as noted above, such that resulting drive signals cause modulators  910  to provide an optical output without subcarrier  1 . In order to reinstate  1 , multiplier circuits M 0 - 0  to M 0 - n  multiply a corresponding one of appropriate coefficients C 0 - 0  to C 0 - n  by a respective one of replicator outputs RDO- 0  to RDO-n to provide products, at least some of which are non-zero. Based on these products, as noted above, modulator drive signals are generated that result in subcarrier  1  being output. 
     The above examples are described in connection with generating or removing the X component of a subcarrier. The processes and circuitry described above is employed or included in DSP  704  and optical circuitry used to generate the Y component of the subcarrier to be blocked. For example, switches and bins circuit blocks  1022 - 0  to  1022 - 31 , have a similar structure and operate in a similar manner as switches and bins circuit blocks  1021  described above to provide zeroes or frequency domain data as the case may be to selectively block the Y component of one or more subcarriers SC. Alternatively, multiplier circuits, like those described above in connection with  FIG.  10    may be provided to supply zero products output from selected pulse shape filters  1018  in order to block the Y component of a particular subcarrier or, if non-zero coefficients are provided to the multiplier circuits instead, generate the subcarrier. 
     Thus, the above examples illustrate mechanisms by which subcarriers SC may be selectively blocked from or added to the output of transmitter  202 . Since, as discussed below, DSPs and optical circuitry provided in secondary node transmitters are similar to that of primary node transmitter  202 , the processes and circuitry described above is provided, for example, in the secondary node transmitters to selectively add and remove subcarriers from the outputs of the secondary node transmitters. Moreover, consistent with the present disclosure, the circuitry described above in connection with  FIGS.  9  and/or  10    may be configured so that a first number of optical subcarriers are output from the transmitter (in either the primary node  110  or the secondary node  112 ) during a first period of time based on initial capacity requirements. Later, during a second period of time, a second number of optical subcarriers can be output from the hub and/or leaf transmitters based on capacity requirements different than the first capacity requirements. 
     Thus, as noted above, the number of subcarriers output from the hub node as well the probabilistic shaping of each subcarrier can be adjusted based on capacity requirements a leaf node and the distance of that leaf node from the hub. 
       FIG.  11    shows another example of an optical communication network  1100  consistent with the present disclosure. Optical communication system  1100  is similar to network  100  described above. In network  1100 , however, splitter  114  is replaced with a wavelength selective switch (WSS)  1102 . 
     Moreover, in the example shown in  FIG.  11   , three groups of optical subcarriers, G 1 , G 2 , and G 3  (see  FIG.  12   ) are provided to WSS  1102  from primary or hub node  110 . As shown in  FIG.  12   , which is a power spectral density plot showing subgroups of optical subcarriers, subgroup G 1  includes optical subcarriers  1 - 8 , subgroup G 2  includes optical subcarriers  9 - 17 , and subgroup G 3  includes optical subcarriers  18 - 24 . As further shown in  FIG.  12   , a spectral or frequency gap GP 1  separates subcarrier  8  (the subcarrier having the highest frequency in subgroup G 1 ) from subcarrier  9  (the subcarrier having the lowest frequency in subgroup G 2 ). In addition, a second spectral or frequency gap GP 2  separates subcarrier  16  (the subcarrier having the high frequency in subgroup G 2 ) from subcarrier  17  (the subcarrier having the lowest frequency in subgroup G 3 ). Accordingly, subgroup G 1  is spectrally separated from subgroup G 2  and subgroup G 2  is further spectrally separated from subgroup G 3 . Frequency gaps, such as gaps GP 1  and GP 2  may be beneficial to spectrally confine subcarriers to within passbands of filters included in the WSS. If subcarriers were provided in the gaps, such subcarriers would be outside the passband and may therefore be substantially attenuated. Thus, preferably, the gaps cause the frequencies of the transmitted subcarriers to fall within the passbands of the WSS filters. 
     Consistent with a further aspect of the present disclosure, each of secondary nodes  112 - j  to  112 - 1  may have different capacity requirements and may be located a different distance from hub node  110 . With system described herein, however, the capacity and SE may be tailored for optimal performance. For example, assuming the capacity of node  112 - j  300 Gbit/s, node  112 - k  is 100 Gbit/s, and node  112 - l  is 20 Gbit/s. Moreover, assuming that the hub capacity is 800 Gbit/s, and each subcarrier has a maximum capacity of 25 Gbit/s,  12  subcarriers may be allocated to node  112 - j . Moreover, if node  112 - j  is located relatively far from primary node  110 , a probabilistic shaping distribution may be selected with a lower SE. In that case, additional subcarriers may be transmitted to node  112 - j  such that the aggregate capacity of the subcarriers transmitted to this node equals 300 Gbits. Thus, the present disclosure provides flexibility of bandwidth and flexibility of reach for optical subcarriers transmitted from the hub node to the leaf node. 
     In a further example, leaf node  112 - 1  may have a capacity requirement of 15 Gbit/s. If this node is relatively far away from the hub, two subcarriers may be transmitted to the leaf node, each with a low SE and associated probabilistic shaped distribution whereby the outer symbols are transmitted with a lower probability. On the other hand, if the node is close to the hub, a higher SE may be provided with a different distribution whereby the outer symbols are transmitted with a higher probability, i.e., the symbols are transmitted more frequently. In that case, one subcarrier may be transmitted to the leaf node. As noted above, the present disclosure also provides for selectively activating and deactivating subcarriers to accommodate different leaf node capacity requirements, for example. 
       FIG.  13    shows an example optical communications network  100  consistent with a further aspect of the present disclosure. The optical communications network  100  includes multiple network nodes that are communicatively coupled to one another by an access ring  102 . 
     In this example, the network nodes include a hub node  104  (“Hub”) and N leaf nodes  106   a - 106   n  (“Leaf- 1 ,” “Leaf- 2 ,” . . . “Leaf-n”). Each of the network nodes can include one or more respective computer devices (e.g., server computers, client computers, etc.). In some implementations, the network nodes can be configured such that the hub node  104  transmits and/or receives data from each of the leaf nodes  106   a - 106   n . For example, the hub node  104  can receive data (e.g., from another network node) that is intended for one of the leaf nodes  106   a - 106   n , and route the data to that leaf node  106   a - 106   n . As another example, a leaf node  106   a - 106   n  can generate data that is intended for another network device, and route the data to the hub node  104  for delivery to the intended network device. Although a single hub node  104 , this is merely an illustrative example. In practice, an optical communications network can include any number of hub nodes. Similarly, an optical communications network can include any number of leaf nodes. 
     As shown in  FIG.  13   , the network nodes are communicatively coupled to one another using an access ring  102 . In this example, the access ring  102  includes two optical paths  108   a  and  108   b  (which may also be referred to as optical communication paths). The first optical path  108   a  communicatively couples the hub node  104  and the leaf nodes  106   a - 106   n  in a sequence in a first direction (e.g., a clockwise direction). The second optical path  108   b  communicatively couples the hub node  104  and the leaf nodes  106   a - 106   n  in a sequence in a second direction (e.g., a counter-clockwise direction). Each of the optical paths  108   a  and  108   b  can be implemented using one or more optical links (e.g., optical fiber) and/or equipment interconnecting the optical links (e.g., line system components). 
     In the example shown in  FIG.  13   , hub node  104  is configured to supply a first group of optical subcarriers to optical path  108   a  via at least one WSS provided in the hub. In addition, hub node  104  is configured to supply a second group of optical subcarriers to optical path  108   b  via another WSS or the same WSS, as discussed in greater detail below with reference to  FIG.  14   . 
     During an example data transmission operation of the hub node  104 , a Tx processor  450  of the hub node  104  receives optical data D 1  to D 8  (intended for the leaf nodes  106   a - 106   h , respectively) using an optical signal processor (DSP)  402 . The data D 1  to D 8  is transmitted from the DSP  402  to an optical to analog converter (D/A)  404  (which also may be referred to as a digital-to-analog conversion circuitry). The D/A  404  converts the optical data into corresponding analog signal. The analog signals are provided to a laser driver  406  (which may also be referred to as driver circuitry). The driver  406  generates optical signals based on the analog signals. The generated optical signals are provided to a modulator  408 , which modulates the optical signal with a carrier optical signal output by a laser  410  and an optical splitter  412 . As an example, the modulated optical signal can include data modulated according to each of the optical subcarriers SC 1  to SC 16 . 
     The modulated optical signal is provided to an optical splitter  414 , which splits the modulated optical signal between two wavelength selective switches (WSSes)  416   a  and  416   b  (e.g., splits the modulated optical signal, such that the power of the optical signal is split among the WSSes  416   a  and  416   b ). The WSS  416   a  selects wavelengths of the modulated optical signal corresponding to a subset of the optical subcarriers (e.g., the optical subcarriers SC 1 -SC 8 ), and injects the selected wavelengths of the modulated optical signal into the first optical signal path  108   a  (e.g., the “hub working Tx” path). The other WSS  416   b  selects wavelengths of the modulated optical signal corresponding to the other subset of the optical subcarriers (e.g., the optical subcarriers SC 9 -SC 16 ), and injects the selected wavelengths of the modulated optical signal into the second optical signal path  108   b  (e.g., the “hub protect Tx” path). 
     During an example data receipt operation of the hub node  104 , the hub node  104  receives a first optical signal from the first optical path  108   a  (e.g., the “hub protect Rx” path) using a WSS  416   c , and receives a second optical signal from the second optical path  108   b  (e.g., the “hub working Tx” path) using a WSS  416   d . The first optical signal can include, for example, a first instance of data D 1 ′-D 8 ′ transmitted by the leaf nodes  106   a - 106   h , respectively. Further, the second optical signal can include a second instance of the data D 1 ′-D 8 ′ transmitted by the leaf nodes  106   a - 106   h , respectively. The WSS  416   c  selects wavelengths of the first optical signal corresponding to a subset of the optical subcarriers (e.g., the optical subcarriers SC 9 ′-SC 16 ′), and provides the selected wavelengths to an optical combiner  420 . Similarly, the WSS  416   d  selects wavelengths of the second optical signal corresponding to another subset of the optical subcarriers (e.g., the optical subcarriers SC 1 ′-SC 8 ′), and provides the selected wavelengths to the optical combiner  418 . 
     The optical combiner  418  combines the selected wavelengths of light, and provides the combined wavelengths of light to receiver Rx, which, in turn, proves the received optical subcarriers and provides data D 1 ′ to D 16 ′. 
     Although  FIG.  14    shows an example hub node  104  having four WSSes  416   a - 416   d , in some implementations, a hub node  104  can include a fewer number of WSSes. For example, referring to  FIG.  4 B , a hub node  104  can include two WSSes  416   e  and  416   f . The WSS  416   e  can be configured to select wavelengths of light for transmission using the optical path  108   a , and to select wavelengths of light received from the optical path  108   b . The WSS  416   f  can be configured to select wavelengths of light for transmission using the optical path  108   b , and to select wavelengths of light received from the optical path  108   a.    
     As described above, during normal operation (e.g., when both the first optical path  108   a  and the second optical path  108   b  are intact and do not have any malfunctioning optical links or equipment), each of the leaf nodes  106   a - 106   h  can recover the respective data D 1 -D 8  from the optical signal received from the first optical path  108   a  (e.g., by demodulating the optical signal received over that optical path, in particular the optical subcarrier that was assigned to or allotted to that leaf node). For example, the leaf node  106   a  can recover the data D 1 , the leaf node  106   b  can recover the data D 2 , and so forth. Similarly, the hub node  104  can recover the data D 1 ′-D 8 ′ from the optical signal received from the second optical path  108   b  (e.g., by demodulating the optical signal received over that optical path, in particular the optical subcarriers that are assigned to or allotted to each of the leaf nodes). For example, the hub node  104  can recover the data D 1 ′-D 8 ′. 
     As described above, the optical communications network  100  can be configured to mitigate the effects of severed and/or malfunctioned optical links in the access ring  102 . 
     For example, the hub node  104  can be configured to transmit multiple instances of a particular portion of data to one of the more of the leaf nodes  106   a - 106   n  concurrently using the optical paths  108   a  and  108   b . For instance, the hub node  104  can receive data intended for each of the leaf nodes  106   a - 106   n  (e.g., eight portions of data D 1 -D 8  intended for eight leaf nodes  106   a - 106   h , respectively). The hub node  104  can generate a first optical signal, modulate the first optical signal based on the data D 1 -D 8  (e.g., using respective optical subcarriers assigned to or allotted to the leaf nodes  106   a - 106   h ), and transmit the first optical signal over the first optical path  108   a . With reference to this data transmission, the first optical path  108   a  may be referred to as the “hub working Tx” path or the “leaf working Rx” path. 
     Further, the hub node  104  can generate a second optical signal, modulate the second optical signal based on the data D 1 -D 8  (e.g., using respective optical subcarriers assigned to or allotted to the leaf nodes  106   a - 106   h ), and transmit the second optical signal over the second optical path  108   b , concurrently with the transmission of the first optical signal over the optical path  108   a . With reference to this data transmission, the second optical path  108   b  may be referred to as the “hub protect Tx” path or the “leaf protect Rx” path. 
     In some implementations, the information transmitted by the hub node  104  along the first optical path  108   a  can be identical to the information transmitted by the hub node  104  along the second optical path  108   b.    
     In some implementations, the information transmitted by the hub node  104  along the first optical path  108   a  can be different from the information transmitted by the hub node  104  along the second optical path  108   b . For example, the first information and the second information can include the same data modulated according to different digital subcarriers (e.g., as described above). As another example, the first information and the second information can include the same data transmitted according to different forward error correction (FEC) schemes (e.g., by including different FEC codes or bits). For instance, as shown in  FIG.  2 A , the length of the first optical path  108   a  from the hub node  104  to the leaf node  106   b  (e.g., 60 km) can be different from the length of the second optical path  108   b  from the hub node  104  and to leaf node  106   b  (40 km). Due to this difference, the hub node  104  can transmit data intended to the second leaf  106   b  according to different FEC schemes (e.g., by including different FEC codes or bits), depending on the optical path this is used. 
     During normal operation (e.g., when both the first optical path  108   a  and the second optical path  108   b  are intact and do not have any malfunctioning optical links or equipment), each of the leaf nodes  106   a - 106   h  can recover the respective data D 1 -D 8  from the optical signal received from the first optical link path  108   a  (e.g., by demodulating the optical signal received over that optical path, in particular the optical subcarrier that was assigned to or allotted to that leaf node). For example, referring to  FIG.  2 A , the leaf node  106   a  can recover the data D 1 , the leaf node  106   b  can recover the data D 2 , and so forth. 
     However, if the first optical path  108   a  includes severed or malfunctioning optical links or equipment, each of the leaf nodes  106   a - 106   h  can recover the respective data D 1 -D 8  from the optical signal received from the second optical path link  108   b  (e.g., by demodulating the optical signal received over that optical path, in particular the optical subcarrier that was assigned to or allotted to that leaf node. Accordingly, the connectivity between the hub node  104  and each of the leaf network  106   a - 106   n  can be maintained, despite malfunctioning optical links or equipment. 
     Further, each of the leaf nodes  106   a - 106   n  also can be configured to transmit multiple instances of a particular portion of data to the hub node  104  concurrently using the optical paths  108   a  and  108   b . For instance, each of the leaf nodes  106   a - 106   n  can receive respective data D 1 ′-D 8 ′ intended for the hub node  104 . Each of the leaf nodes  106   a - 106   n  can generate a first optical signal, modulate the first optical signal based on a respective one of the data D 1 ′-D 8 ′ (e.g., using respective optical subcarriers assigned to or allotted to the leaf nodes  106   a - 106   n ), and transmit the first optical signal over the second optical path  108   b . With reference to this data transmission, the second optical path  108   b  may be referred to as the “leaf working Tx” path or the “hub working Rx” path. 
     Further, each of the leaf nodes  106   a - 106   n  can generate a second optical signal, modulate the second optical signal based on a respective one of the data D 1 ′-D 8 ′ (e.g., using respective optical subcarriers assigned to or allotted to the leaf nodes  106   a - 106   n ), and transmit the second optical signal over the first optical path  108   a , concurrently with the transmission of the first optical signal over the second optical path  108   b . With reference to this data transmission, the first optical path  108   a  may be referred to as the “leaf protect Tx” path or the “hub protect Rx” path. 
     Similarly, in some implementations, the information transmitted by a leaf node  106   a - 106   h  along the first optical path  108   a  can be identical to the information transmitted by the leaf node  106   a - 106   n  along the second optical path  108   b.    
     In some implementations, the information transmitted by a leaf node  106   a - 106   h  along the first optical path  108  can be different from the information transmitted by the leaf node  106   a - 106   h  along the second optical path  108   b . For example, the first information and the second information can include the same data modulated according to different digital subcarriers (e.g., as described above). As another example, the first information and the second information can include the same data transmitted according to different forward error correction (FEC) schemes (e.g., by including different FEC codes or bits). For instance, the length of the first optical path  108   a  from the leaf node  106   b  to the hub node  104  (e.g., 40 km) can be different from the length of the second optical path  108   b  from the leaf node  106   b  to the hub node  104  (60 km). Due to this difference, the leaf node  106   b  can transmit data intended to the hub node  104  according to different FEC schemes (e.g., by including different FEC codes or bits), depending on the optical path this is used. 
     Similarly, during normal operation (e.g., when both the first optical path  108   a  and the second optical path  108   b  are intact and do not have any malfunctioning optical links or equipment), the hub node  104  can recover the data D 1 ′-D 8 ′ from the optical signal received from the second optical path link  108   b  (e.g., by demodulating the optical signal received over that optical path, in particular the optical subcarriers that are assigned to or allotted to each of the leaf nodes). For example, referring to  FIG.  2 A , the hub node  104  can recover the data D 1 ′-D 8 ′. 
     However, if the second optical path  108   b  includes severed or malfunctioning optical links or equipment, the hub node  104  can recover the data D 1 ′-D 8 ′ from the optical signal received from the first optical path link  108   a  (e.g., by demodulating the optical signal received over that optical path, in particular the optical subcarriers that are assigned to or allotted to each of the leaf nodes). Accordingly, the connectivity between the hub node  104  and each of the leaf network  106   a - 106   n  can be maintained, despite malfunctioning optical links or equipment. 
     Consistent with a further aspect of the present disclosure, optical subcarriers transmitted from hub node  104  to more distant leaf nodes  106  may have an associated distribution, as described above, and corresponding probabilistic shaping such that a corresponding SE associated with such optical subcarrier is relatively low (see, for example,  FIG.  5   b   ). Optical subcarriers transmitted to leaf nodes  106  that are located closer to the hub node may have a relatively higher SE based on probabilistic shaping and distribution wherein outer symbols are transmitted more frequently. See, for example,  FIG.  5     b.    
       FIG.  15    shows examples of WSS filter passbands PB 1  and PB 2  consistent with a further aspect of the present disclosure. As shown in  FIG.  15   , passband PB 1  encompasses frequencies associated with subcarriers SC 1  to SC 12 , and passband PB 2  encompasses frequencies associated with subcarriers SC 13  to SC 24 . No subcarriers are transmitted within the gap that is four subcarrier frequencies wide, in this example. As noted above, such subcarriers would be provided outside the passbands and, therefore, would be substantially attenuated. 
     As noted above, one or subcarriers within each group of optical subcarriers shown in  FIG.  15    may have a different probabilistic shaping that results in an optimal SE based on a distance from the hub to an associated leaf node. 
     Other embodiments will be apparent to those skilled in the art from consideration of the specification. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit of the invention being indicated by the following claims.