Optical communication with some compensation of nonlinear optical effects

We disclose an optical transport system configured to reduce nonlinear signal distortions using an electronic phase rotation, the phase value of which is determined using pre-filtering, e.g., via a low-pass filter, of the digital samples representing an optical communication signal prior to applying a squaring operation to the digital samples. In some embodiments, the phase value used in the electronic phase rotation can be determined using double filtering of the digital samples that, in addition to the pre-filtering, employs post-filtering, e.g., via another low-pass filter, of the digital samples generated by the squaring operation. The electronic phase rotation can be implemented as part of a backward-propagation algorithm that, in addition to reducing the nonlinear signal distortions, provides at least partial dispersion compensation. In various embodiments, the corresponding backward-propagation module can be incorporated into the transmitter's digital signal processor (DSP) or the receiver's DSP.

BACKGROUND

Field

The present disclosure relates to optical communication equipment and, more specifically but not exclusively, to at least partial compensation of nonlinear optical effects.

Description of the Related Art

After propagating through a length of optical fiber, the received optical signal may be distorted due to linear impairments, such as chromatic dispersion (CD) and polarization mode dispersion (PMD), and nonlinear impairments, such as the Kerr effect, self-phase modulation (SPM), cross-phase modulation (XPM), and four-wave mixing (FWM). These distortions typically cause a detrimental increase in the bit-error rate (BER). Optical and electrical signal-processing techniques that can reduce this BER penalty are therefore desirable.

SUMMARY OF SOME SPECIFIC EMBODIMENTS

Disclosed herein are various embodiments of an optical transport system configured to reduce nonlinear signal distortions using an electronic phase rotation, the phase value of which is determined using pre-filtering, e.g., via a low-pass filter, of the digital samples representing an optical communication signal prior to applying a squaring operation to the digital samples. In some embodiments, the phase value used in the electronic phase rotation can be determined using double filtering of the digital samples that, in addition to the pre-filtering, employs post-filtering, e.g., via another low-pass filter, of the digital samples generated by the squaring operation. The electronic phase rotation can be implemented as part of a backward-propagation algorithm that, in addition to reducing the nonlinear signal distortions, provides at least partial dispersion compensation. In various embodiments, the corresponding backward-propagation module can be incorporated into the transmitter's digital signal processor (DSP) or the receiver's DSP.

According to one embodiment, provided is an apparatus comprising: an optical front-end circuit configured to transmit or receive an optical communication signal; and a signal processor operatively connected to the optical front-end circuit and configured to: apply an electronic phase rotation to digital samples representing the optical communication signal; and determine a phase value for the electronic phase rotation using pre-filtering of the digital samples performed by a low-pass filter prior to applying a squaring operation to the digital samples.

According to another embodiment, provided is an apparatus comprising: an optical front-end circuit configured to transmit or receive an optical communication signal; and a first electronic nonlinear-compensation module operatively connected to the optical front-end circuit to process digital samples corresponding to the optical communication signal; and wherein the first electronic nonlinear-compensation module comprises: a first digital filter configured to digitally filter a first sequence of the digital samples to generate a second sequence of the digital samples; and a first squaring circuit configured to generate a squared absolute value of each digital sample of the second sequence to generate a third sequence of the digital samples; and wherein the first electronic nonlinear-compensation module is configured to: apply a phase rotation to the first sequence of the digital samples; and determine a phase value used in the phase rotation using the third sequence of the digital samples.

DETAILED DESCRIPTION

FIG. 1shows a block diagram of an optical transport system100according to an embodiment. System100has an optical transmitter110and an optical receiver190coupled to one another via an optical transport link140. In an example embodiment, optical transport link140can be implemented using one or more spans of optical fiber or fiber-optic cable. For illustration purposes and without any implied limitations, optical transport link140is shown inFIG. 1as being an amplified optical link having a plurality of optical amplifiers144configured to amplify the optical signals that are being transported through the link, e.g., to counteract signal attenuation. In an alternative embodiment, optical transport link140that has only one or even no optical amplifiers144can similarly be used.

In operation, transmitter110receives a digital electrical input stream102of payload data and applies it to a digital signal processor (DSP)112. DSP112processes input data stream102to generate digital signals1141-1144. In an example embodiment, DSP112may perform, inter alia, one or more of the following: (i) de-multiplex input stream102into two sub-streams, each intended for optical transmission using a respective one of the orthogonal (e.g., X and Y) polarizations of an optical output signal130; (ii) encode each of the sub-streams using a suitable code, e.g., to prevent error propagation and enable error correction at receiver190; (iii) convert each of the two resulting sub-streams into a corresponding sequence of constellation symbols; and (iv) perform digital signal pre-distortion, e.g., to mitigate the adverse effects imposed by an electrical-to-optical (E/O) converter (also sometimes referred to as a front-end circuit)116of transmitter110, optical transport link140, and/or a front-end circuit172of receiver190. In each signaling interval (also referred to as a symbol period or time slot), signals1141and1142carry digital values that represent the in-phase (I) component and quadrature (Q) component, respectively, of a corresponding (possibly pre-distorted) constellation symbol intended for transmission using a first (e.g., X) polarization of light. Signals1143and1144similarly carry digital values that represent the I and Q components, respectively, of the corresponding (possibly pre-distorted) constellation symbol intended for transmission using a second (e.g., Y) polarization of light.

E/O converter116operates to transform digital signals1141-1144into a corresponding modulated optical output signal130. More specifically, drive circuits1181and1182transform digital signals1141and1142, as known in the art, into electrical analog drive signals IXand QX, respectively. Drive signals IXand QXare then used, in a conventional manner, to drive an I-Q modulator124X. In response to drive signals IXand QX, I-Q modulator124Xoperates to modulate an X-polarized beam122Xof light supplied by a laser source120as indicated inFIG. 1, thereby generating a modulated optical signal126X.

Drive circuits1183and1184similarly transform digital signals1143and1144into electrical analog drive signals IYand QY, respectively. In response to drive signals IYand QY, an I-Q modulator124Yoperates to modulate a Y-polarized beam122Yof light supplied by laser source120as indicated inFIG. 1, thereby generating a modulated optical signal126Y. A polarization beam combiner128operates to combine modulated optical signals126Xand126Y, thereby generating optical output signal130. Optical output signal130is then applied to optical transport link140.

After propagating through optical transport link140, optical signal130becomes optical signal130′, which is applied to receiver190. Optical signal130′ may differ from optical signal130because optical transport link140typically adds noise and imposes various linear and nonlinear signal distortions, such as the above-mentioned Kerr effect, CD, PMD, SPM, XPM, and FWM.

Front-end circuit172of receiver190comprises an optical-to-electrical (O/E) converter160, analog-to-digital converters (ADCs)1661-1664, and an optical local-oscillator (OLO) source156. O/E converter160has (i) two input ports labeled S and R and (ii) four output ports labeled1through4. Input port S receives optical signal130′ from optical transport link140. Input port R receives an OLO signal158generated by OLO source156. OLO signal158has an optical-carrier frequency (wavelength) that is sufficiently close to that of signal130′ to enable coherent (e.g., intradyne) detection of the latter signal. OLO signal158can be generated, e.g., using a relatively stable tunable laser whose output wavelength (frequency) is approximately the same as the carrier wavelength (frequency) of optical signal130′.

In an example embodiment, O/E converter160operates to mix input signal130′ and OLO signal158to generate eight different mixed (e.g., by interference) optical signals (not explicitly shown inFIG. 1). O/E converter160then converts the eight mixed optical signals into four electrical signals1621-1624that are indicative of complex values corresponding to the two orthogonal-polarization components of signal130′. For example, electrical signals1621and1622may be an analog I signal and an analog Q signal, respectively, corresponding to a first (e.g., horizontal, h) polarization component of signal130′. Electrical signals1623and1624may similarly be an analog I signal and an analog Q signal, respectively, corresponding to a second (e.g., vertical, v) polarization component of signal130′. Note that the orientation of the h and v polarization axes at receiver190may not coincide with the orientation of the X and Y polarization axes at transmitter110.

Each of electrical signals1621-1624generated by O/E converter160is converted into digital form in a corresponding one of ADCs1661-1664. Optionally, each of electrical signals1621-1624may be amplified in a corresponding electrical amplifier (not explicitly shown) prior to the resulting signal being converted into digital form. Digital signals1681-1684produced by ADCs1661-1664are then processed by a DSP170to recover the data of the original input data stream102applied to transmitter110. In an example embodiment, DSP170may perform, inter alia, one or more of the following: (i) perform signal processing directed at dispersion compensation; (ii) perform signal processing directed at compensation of nonlinear distortions; (iii) perform electronic polarization de-multiplexing; and (iv) perform error correction based on the data encoding applied at DSP112. Example embodiments of DSP170are described in more detail below in reference toFIGS. 2-5.

In some embodiments, the signal processing directed at dispersion compensation and/or compensation of nonlinear distortions can be performed at DSP112instead of being performed at DSP170. In this case, this signal processing can be used to pre-distort optical output signal130in a manner that causes optical signal130′ to be less distorted than in the absence of this signal processing.

FIG. 2shows a block diagram of DSP170(FIG. 1) according to an embodiment. Digital signals1681-1684and output data stream102are also shown inFIG. 2to better illustrate the relationship between the circuits shown inFIGS. 1 and 2.

Ideally, digital signals1681and1682represent the I and Q components, respectively, of the horizontal polarization component of optical signal130′, and digital signals1683and1684represent the I and Q components, respectively, of the vertical polarization component of that optical signal. However, various transmission impairments, front-end implementation imperfections, and configuration inaccuracies generally cause each of digital signals1681-1684to be a convoluted signal that has various signal distortions and/or contributions from different signal components originally generated at transmitter110(FIG.1). The train of signal processing implemented in DSP170is generally directed at reducing the adverse effects of signal distortions and de-convolving digital signals1681-1684so that the transmitted data can be properly recovered to generate output data stream102.

DSP170comprises a signal-pre-processing module210configured to receive digital signals1681-1684. One of the functions of module210may be to adapt the signal samples received via digital signals1681-1684to a form that is more-suitable for the signal-processing algorithms implemented in the downstream modules of DSP170. For example, module210may be configured to (i) resample digital signals1681-1684such that each of these signals carries two samples per symbol period and (ii) convert the resulting signal samples into the corresponding complex-valued signal samples. The resulting complex-valued digital signals generated by signal-pre-processing module210are labeled2121-2122.

DSP170further comprises a backward-propagation (BP) module210that converts digital signals2121and2122into digital signals2221and2222, respectively. In an example embodiment, BP module210implements signal processing directed at causing digital signals2221and2222to represent a good approximation of the undistorted optical signal130applied by transmitter110to optical transport link140(seeFIG. 1). This signal processing can be based, e.g., on a numerical model that applies the negative sign to certain signal-propagation parameters, such as the dispersion coefficient D and the nonlinear coefficient γ, to mathematically invert the non-linear Schrödinger equation that describes the forward signal propagation through optical transport link140. The resulting inverse non-linear Schrödinger equation can be solved, e.g., using the split-step Fourier method (SSFM), to generate digital signals2221and2222.

Example embodiments of BP module220are described in more detail below in reference toFIGS. 3-5. The general principles of digital BP and SSFM, as applied to optical communications, are reviewed in a paper by Rameez Asif, Chien-Yu Lin, and Bernhard Schmauss, entitled “Digital Backward Propagation: A Technique to Compensate Fiber Dispersion and Non-Linear Impairments,” published as Chapter 2 in the e-book “Applications of Digital Signal Processing,” Dr. Christian Cuadrado-Laborde (Ed.), InTech, DOI: 10.5772/25410, which paper is incorporated herein by reference in its entirety. Certain embodiments of BP module220may benefit from the use of some aspects of the digital BP techniques disclosed, e.g., in U.S. Pat. Nos. 8,036,541, 8,805,209, and 9,225,455, all of which are incorporated herein by reference in their entirety.

Digital signals2221and2222are applied to a 2×2 MIMO (multiple-input/multiple-output) equalizer230for MIMO-equalization processing therein, and the resulting equalized signals are complex-valued digital signals232Xand232Y. In an example embodiment, equalizer230can be a butterfly equalizer configured to perform electronic polarization demultiplexing and reduce residual inter-symbol interference (ISI). Example 2×2 MIMO equalizers that can be used to implement equalizer230are disclosed, e.g., in U.S. Pat. No. 9,020,364 and U.S. Patent Application Publication No. 2015/0372764, both of which are incorporated herein by reference in their entirety.

Digital signals232Xand232Ygenerated by equalizer230are applied to a carrier-recovery module240that is configured to perform signal processing generally directed at (i) compensating the frequency mismatch between the carrier frequencies of OLO signal158and input optical signal130′ and/or (ii) reducing the effects of phase noise. Various signal-processing techniques that can be used to implement the frequency-mismatch-compensation processing in carrier-recovery module240are disclosed, e.g., in U.S. Pat. Nos. 7,747,177 and 8,073,345, both of which are incorporated herein by reference in their entirety. Example signal-processing techniques that can be used to implement phase-error-correction processing in carrier-recovery module240are disclosed, e.g., in U.S. Pat. No. 9,112,614, which is incorporated herein by reference in its entirety.

Digital signals242Xand242Ygenerated by carrier-recovery module250are applied to a symbol-detection module250. In an example embodiment, symbol-detection module250is configured to use the complex values conveyed by digital signals242Xand242Yto appropriately map each complex value onto an operative constellation to determine the corresponding received symbol and, based on said mapping, determine the corresponding bit-word encoded by the symbol. Symbol-detection module250then concatenates the determined bit-words to generate data streams252Xand252Y.

In some embodiments, data streams252Xand252Ycan be applied to an optional forward-error-correction (FEC) decoder260that performs digital signal processing that implements error correction based on data redundancies (if any) in optical signal130. FEC decoder260appropriately multiplexes the resulting error-corrected data streams to generate output data stream102. Many FEC methods suitable for this purpose are known in the art. Several suitable FEC methods that can be used to implement FEC decoder260are disclosed, e.g., in U.S. Pat. Nos. 7,734,191, 7,574,146, 7,424,651, 7,212,741, and 6,683,855, all of which are incorporated herein by reference in their entirety.

FIG. 3shows a block diagram of BP module220(FIG. 2) according to an embodiment. Digital signals2121-2122and2221-2222are also shown inFIG. 3to better illustrate the relationship between the circuits shown inFIGS. 2 and 3.

In the shown embodiment, BP module220comprises dispersion-compensation modules3101and3102and a nonlinear-compensation (NLC) module330. Dispersion-compensation module3101precedes NLC module330, and dispersion-compensation module3102follows the NLC module in the chain of signal processing as indicated inFIG. 3. Each of modules310and330has two signal-processing paths, each configured to process digital samples corresponding to a different respective polarization component of optical signal130′.

In operation, dispersion-compensation modules3101and3102perform signal processing that tends to reduce the deleterious effects of chromatic dispersion accrued by optical signal130′ in optical transport link140. For example, the total amount of chromatic dispersion, CDt, compensated by dispersion-compensation modules3101and3102can be expressed as follows:
CDt=De×L0(1)
where Deis the effective dispersion coefficient; and L0is the length of optical fiber used in optical transport link140. Dispersion-compensation module3101is configured to compensate a first fraction, e.g., ρ<1, of the total chromatic dispersion CDt; and dispersion-compensation module3102is configured to similarly compensate a second fraction, e.g., (1−ρ)<1, of the total chromatic dispersion CDt. In some embodiments, the values of both the dispersion coefficient Deand the fraction ρ can be adjustable parameters of the dispersion-compensation algorithm and can be selected in a manner that optimizes the overall performance of BP module220, e.g., by minimizing the receiver's BER. In some embodiments, the value of the fraction ρ can be 0.5. In some other embodiments, the value of the fraction ρ can be 0 or 1, in which case one of dispersion-compensation modules3101and3102can be omitted.

In various embodiments, dispersion-compensation modules3101and3102can be implemented using digital time-domain (e.g., finite impulse response, FIR) filters or digital frequency-domain filters. In an example embodiment, dispersion-compensation modules3101and3102can be configured to have the transfer functions HPREand HPOST, respectively, approximated by Eqs. (2a)-(2b):
HPRE(f)=exp(−jρCDtπ(λ0f)2/c)  (2a)
HPOST(f)=exp(−j(1−ρ)CDtπ(λ0f)2/c)  (2b)
where λ0is the carrier wavelength; f is the frequency; and c is the speed of light.

Digital signals3201and3202generated by dispersion-compensation module3101are applied to NLC module330for being digitally processed therein, e.g., as further described below. The resulting digital signals3701and3702generated by NLC module330are then applied to dispersion-compensation module3102.

NLC module330comprises multipliers3601and3602configured to generate digital signals3701and3702, respectively, by applying a phase rotation to the complex-valued digital samples supplied by digital signals3201and3202. In an example embodiment, this phase rotation can be implemented in accordance with Eqs. (3a) and (3b):
{tilde over (x)}k=xkexp(jΦk)  (3a)
{tilde over (y)}k=ykexp(jΦk)  (3b)
where {tilde over (x)}kand {tilde over (y)}kare the digital samples carried in the k-th time slot of digital signals3701and3702, respectively; xkand ykare the digital samples carried in the k-th time slot of digital signals3201and3202, respectively; and Φkis the phase value used for the phase rotation in the k-th time slot.

NLC module330further comprises a complex-value generator350that operates to compute the exponential factor exp(jΦk) in each time slot and then apply the computed exponential factor to multipliers3601and3602as indicated inFIG. 3. For example, complex-value generator350can first compute the value of the phase Φkand then use the computed value of the phase Φkto compute the exponential factor exp(jΦk), which is then applied to multipliers3601and3602. The value of the phase Φkcan be computed, e.g., in accordance with Eq. (4):
Φk=γeIk(4)
where γeis the effective nonlinear coefficient representing the nonlinear impairments imposed onto optical signal130′ by optical transport link140; and Ikis the effective signal intensity (power) in the k-th time slot. In some embodiments, the effective nonlinear coefficient γecan be an adjustable parameter of the back-propagation algorithm whose value can be selected in a manner that optimizes the overall performance of BP module220. In some other embodiments, the effective nonlinear coefficient γecan be a fixed parameter whose value can be obtained using the pertinent technical characteristics of the optical fiber and other optical elements used in optical transport link140.

NLC module330further comprises squaring circuits3361and3362, digital filters3321,3322,3401, and3402, and an adder346that are operatively connected to each other and to other elements of the NLC module as indicated inFIG. 3to compute the values of Ikapplied to complex-value generator350. Digital filters3321and3401are located upstream and downstream, respectively, of squaring circuit3361. Digital filters3322and3402are similarly located upstream and downstream, respectively, of squaring circuit3362.

In various embodiments, digital filters3321,3322,3401, and3402can be implemented using digital time-domain filters or digital frequency-domain filters. A person of ordinary skill in the art will understand that both time-domain and frequency-domain implementations can be designed to have substantially equivalent transfer characteristics. For illustration purposes and without any implied limitations, the subsequent description of digital filters3321,3322,3401, and3402is given in reference to a time-domain implementation in which each of these digital filters is or comprises an FIR filter.

In an example embodiment, squaring circuits3361and3362, digital filters3321,3322,3401, and3402, and adder346implement the signal processing that causes the effective signal intensity Ikin the k-th time slot to be computed in accordance with Eq. (5):

Ik=∑n=0N⁢gX-POST,n·Px,k-n2+∑n=0N⁢gY-POST,n·Py,k-n2(5)
where (N+1) is the total number of taps in each of the digital filters3401and3402; n is an index that is used to consecutively number the taps of the digital filter, where 0≤n≤N; gX-POSTis the transfer function of digital filter3401; gY-POSTis the transfer function of digital filter3402; {Px,k} is the sequence of digital samples applied by digital filter3321to squaring circuit3361by way of a digital signal3341; and {Py,k} is the sequence of digital samples applied by digital filter3322to squaring circuit3362by way of a digital signal3342.

Digital filters3321and3322operate to generate the individual complex-valued digital samples of the sequences {Px,k} and {Py,k}, respectively, in accordance with Eqs. (6a)-(6b):

Px,k=∑m=0M⁢gX-PRE,m·xk-m(6⁢a)Py,k=∑m=0M⁢gY-PRE,m·yk-m(6⁢b)
where (M+1) is the total number of taps in each of the digital filters3321and3322; m is an index that is used to consecutively number the taps of the digital filter, where 0≤m≤M; gX-PREis the transfer function of digital filter3321; gY-PREis the transfer function of digital filter3322; and xkand ykare the digital samples carried in the k-th time slot of digital signals3201and3202, respectively (also see Eqs. (3a)-(3b)).

Squaring circuit3361operates to (i) generate a square of the absolute value of each complex-valued digital sample of the sequence {Px,k} received via digital signal3341from digital filter3321and (ii) apply the resulting sequence {|Px,k|2} to digital filter3401by way of a digital signal3381. Squaring circuit3362similarly operates to (i) generate a square of the absolute value of each complex-valued digital sample of the sequence {Py,k} received via digital signal3341from digital filter3321and (ii) apply the resulting sequence {|Px,k|2} to digital filter3402by way of a digital signal3382.

Digital filters3401and3402operate to generate digital signals3421and3422, respectively, and apply these digital signals to adder346. Digital signal3421carries the sequence {Ix,k}, the individual digital samples of which are generated by digital filter3401in accordance with Eq. (7a):

Ix,k=∑n=0N⁢gX-POST,n·Px,k-n2(7⁢a)
Digital signal3422carries the sequence {Iy,k}, the individual digital samples of which are similarly generated by digital filter3402in accordance with Eq. (7b):

Adder346operates to sum the digital values conveyed by digital signals3421and3422in each time slot, thereby generating the sequence {Ik} in accordance with Eq. (8):
Ik=Ix,k+Iy,k(8)
The generated sequence {Ik} is then applied to complex-value generator350as indicated inFIG. 3.

In an example embodiment, digital filters3321,3322,3401, and3402can be designed and configured such that each of the transfer functions gX-PRE, gY-PRE, gX-POST, and gY-POSTapproximates or is functionally equivalent to a frequency response of a low-pass filter. As known in the pertinent art, a low-pass filter is a filter that passes the signals with a frequency lower than a cutoff frequency and attenuates or blocks the signals with frequencies higher than the cutoff frequency.

In an alternative embodiment, digital filters3321,3322,3401, and3402can be designed and configured such that each of the transfer functions gX-PRE, gY-PRE, gX-POST, and gY-POSTapproximates or is functionally equivalent to a frequency response of a band-pass filter. As known in the pertinent art, a band-pass filter is a filter that passes frequencies within a certain range and rejects or attenuates frequencies outside that range.

In various embodiments, the frequency envelopes corresponding to the transfer functions gX-PRE, gY-PRE, gX-POST, and gY-POSTcan be selected from a variety of suitable spectral shapes, such as a rectangular shape, a triangular shape, a trapezoid shape, etc. In general, the spectral shapes and the cutoff frequencies of digital filters3321,3322,3401, and3402can be selected at the design stage in a manner that optimizes the performance characteristics of BP module220for the intended application.

Note that BP module220employs a single NLC module330. In this configuration, the resulting BP method in effect uses a relatively coarse size for the backward propagation step. At this size, the effects of CD and nonlinear distortions become convoluted and cannot be cleanly separated in the corresponding mathematical model. This problem is addressed, at least in part, by embodiments of NLC module330, wherein digital filters3321,3322,3401, and3402configured to operate, e.g., as described above, help to cancel at least some of the spectral artifacts of the mathematical model, thereby significantly improving the accuracy of the linear and nonlinear compensation. For example, the use of digital filters3321,3322,3401, and3402in BP module220enables the BP module to achieve a better level of performance compared to that achieved when some or all of digital filters3321,3322,3401, and3402are not present in the corresponding NLC module. Advantageously, a typical level of performance provided by BP module220can be obtained at a much lower cost than a comparable level of performance provided by a conventional BP circuit.

Although example embodiments of BP module220are described above as being incorporated into receiver DSP170(FIG. 1), embodiments of the invention are not so limited. In some embodiments BP module220can alternatively be incorporated into transmitter DSP112to implement digital signal pre-distortion thereat as known in the pertinent art. For example, general principles of such pre-distortion are reviewed, e.g., in the above-cited paper by Rameez Asif, Chien-Yu Lin, and Bernhard Schmauss, entitled “Digital Backward Propagation: A Technique to Compensate Fiber Dispersion and Non-Linear Impairments.”

FIG. 4shows a block diagram of BP module220(FIG. 2) according to an alternative embodiment. In the shown embodiment, BP module220comprises: (i) (L−1) serially connected compensation stages410, where L is an integer greater than one; and (ii) an NLC module330Lconnected at the downstream end of the series. Each of the compensation stages410iincludes an NLC module330iand a dispersion-compensation module310i, where 1≤i≤(L−1). Therein, NLC module330iprecedes dispersion-compensation module310i. For illustration purposes and without any implied limitations, only one of the (L−1) compensation stages410is explicitly shown inFIG. 4. The minimum number of compensation stages410is one, which corresponds to L=2. The possible maximum number of compensation stages410is limited by practical considerations, such as the resulting circuit complexity and cost, and can be on the order of ten or greater than ten.

In an example embodiment, compensation stage410iis configured to carry out BP signal processing corresponding to the i-th section of optical transport link140, with different compensation stages410being configured to perform the BP signal processing corresponding to different respective sections of the optical transport link. NLC module330Lis configured to reduce residual nonlinear distortions that are left uncompensated by the preceding compensation stage(s)410. Based on the description provided above in reference toFIG. 3and Eqs. (1)-(8), a person of ordinary skill in the art will understand how to configure NLC modules330iand dispersion-compensation modules310iof each compensation stage410iand further understand how to configure NLC module330Lto cause the corresponding embodiment of BP module220to have desired performance characteristics.

FIG. 5shows a block diagram of BP module220(FIG. 2) according to another alternative embodiment. In the shown embodiment, BP module220comprises: (i) (L−1) serially connected compensation stages510, where L is an integer greater than one; and (ii) a dispersion-compensation module310Lconnected at the downstream end of the series. Each of the compensation stages510iincludes a dispersion-compensation module310iand an NLC module330i, where 1≤i≤(L−1). Therein, NLC module330ifollows dispersion-compensation module310i. For illustration purposes and without any implied limitations, only one of the (L−1) compensation stages510is explicitly shown inFIG. 5. The minimum number of compensation stages510is one, which corresponds to L=2 (also seeFIG. 3). The possible maximum number of compensation stages510is limited by practical considerations and can be on the order of ten or greater than ten.

In an example embodiment, compensation stage510iis configured to carry out BP signal processing corresponding to the i-th section of optical transport link140, with different compensation stages510being configured to perform the BP signal processing corresponding to different respective sections of the optical transport link. Dispersion-compensation module310Lis configured to reduce the residual effects of chromatic dispersion that are left uncompensated by the preceding compensation stage(s)510. Based on the description provided above in reference toFIG. 3and Eqs. (1)-(8), a person of ordinary skill in the art will understand how to configure dispersion-compensation modules310iand NLC modules330iof each compensation stage510iand further understand how to configure dispersion-compensation module310Lto cause the corresponding embodiment of BP module220to have desired performance characteristics.

Although example embodiments of BP module220are described above as being designed and configured for processing polarization-division-multiplexed (PDM) signals, a person of ordinary skill in the art will understand how to modify optical transport system100and a disclosed embodiment of BP module220to make them suitable for processing communication signals in which both polarizations carry the same sequences of constellation symbols.

Some embodiments can be adapted for use in an optical wavelength-division-multiplexed (WDM) transport system. For example, each WDM channel of such system can be provided with a separate instance (nominal copy) of appropriately configured BP module220.

In some embodiments, a single pre-filter functionally analogous to digital filter332can be used for two or more WDM channels.

Although example embodiments of BP module220are described above as employing low-pass and/or band-pass filters332and340, other suitable types of filters can be used in some alternative embodiments.

In some embodiments, BP module220can be designed and configured for processing space-division-multiplexed (SDM) signals.

According to an example embodiment disclosed above in reference toFIGS. 1-5, provided is an apparatus (e.g.,100,FIG. 1) comprising: an optical front-end circuit (e.g.,116or172,FIG. 1) configured to transmit or receive an optical communication signal (e.g.,130or130′,FIG. 1); and a signal processor (e.g.,112or170,FIG. 1) operatively connected to the optical front-end circuit (e.g., as indicated inFIGS. 1-3) and configured to: apply an electronic phase rotation (e.g., using330,FIG. 3) to digital samples representing the optical communication signal; and determine a phase value (e.g., Φk, Eqs. (3a)-(3b)) for the electronic phase rotation using pre-filtering of the digital samples performed by a low-pass filter (e.g.,332,FIG. 3) prior to applying a squaring operation to the digital samples.

In some embodiments of the above apparatus, the signal processor is further configured to determine the phase value using post-filtering performed by another low-pass filter (e.g.,340,FIG. 3) after the squaring operation.

According to another example embodiment disclosed above in reference toFIGS. 1-5, provided is an apparatus (e.g.,100,FIG. 1) comprising: an optical front-end circuit (e.g.,116or172,FIG. 1) configured to transmit or receive an optical communication signal (e.g.,130or130′,FIG. 1); and a first electronic nonlinear-compensation module (e.g.,330,FIG. 3) operatively connected to the optical front-end circuit (e.g., as indicated inFIGS. 1-3) to process digital samples corresponding to the optical communication signal; and wherein the first electronic nonlinear-compensation module comprises: a first digital filter (e.g.,3321,FIG. 3) configured to digitally filter a first sequence (e.g., {xk}, Eq. (3a);3201,FIG. 3) of the digital samples to generate a second sequence (e.g., {Px,k}, Eq. (6a);3341,FIG. 3) of the digital samples; and a first squaring circuit (e.g.,3361,FIG. 3) configured to generate a squared absolute value of each digital sample of the second sequence to generate a third sequence (e.g., {|Px,k|2};3381,FIG. 3) of the digital samples; and wherein the first electronic nonlinear-compensation module is configured to: apply a phase rotation (e.g., in accordance with Eq. (3a)) to the first sequence of the digital samples; and determine a phase value (e.g., Φk, Eqs. (3a)-(3b)) used in the phase rotation using the third sequence of the digital samples.

In some embodiments of the above apparatus, the first digital filter is configured to operate as a low-pass filter or as a band-pass filter.

In some embodiments of any of the above apparatus, the apparatus comprises an optical receiver (e.g.,190,FIG. 1) configured to receive the optical communication signal; and wherein the optical front-end circuit and the first electronic nonlinear-compensation module are parts of the optical receiver.

In some embodiments of the above apparatus, the first electronic nonlinear-compensation module further comprises: a second digital filter (e.g.,3322,FIG. 3) configured to digitally filter a fourth sequence (e.g., {yk}, Eq. (3b);3202,FIG. 3) of the digital samples to generate a fifth sequence (e.g., {Py,k}, Eq. (6b);3342,FIG. 3) of the digital samples; and a second squaring circuit (e.g.,3362,FIG. 3) configured to generate a squared absolute value of each digital sample of the fifth sequence to generate a sixth sequence (e.g., {|Py,k|2};3382,FIG. 3) of the digital samples; and wherein the first electronic nonlinear-compensation module is further configured to determine the phase value used in the phase rotation using the sixth sequence of the digital samples.

In some embodiments of the above apparatus, the first sequence of the digital samples corresponds to a first polarization (e.g., X or h) of the optical communication signal; and wherein the fourth sequence of the digital samples corresponds to a different second polarization (e.g., Y or v) of the optical communication signal.

In some embodiments of the above apparatus, the first electronic nonlinear-compensation module is configured to apply a phase rotation (e.g., in accordance with Eq. (3b)) to the fourth sequence of the digital samples using the phase value.

In some embodiments of the above apparatus, the second digital filter is configured to operate as a low-pass filter or as a band-pass filter.

In some embodiments of the above apparatus, the first electronic nonlinear-compensation module further comprises a second digital filter (e.g.,3401,FIG. 3) configured to digitally filter the third sequence of the digital samples to generate a fourth sequence (e.g., {IX,k}, Eq. (7a);3421,FIG. 3) of the digital samples; and wherein the first electronic nonlinear-compensation module is further configured to determine the phase value used in the phase rotation using the fourth sequence of the digital samples.

In some embodiments of the above apparatus, the first digital filter comprises a finite-impulse-response filter.

In some embodiments of the above apparatus, the apparatus further comprises a first electronic dispersion-compensation module (e.g.,3101,FIG. 3) configured to generate the first sequence of the digital samples.

In some embodiments of the above apparatus, the apparatus further comprises a second electronic dispersion-compensation module (e.g.,3102,FIG. 3) configured to receive a sequence (e.g., {{tilde over (x)}k}, Eq. (3a);3701,FIG. 3) of the digital samples generated by the first electronic nonlinear-compensation module using the phase rotation.

In some embodiments of the above apparatus, the apparatus further comprises a dispersion-compensation module (e.g.,3101,FIG. 4) configured to receive a sequence (e.g., {{tilde over (x)}k}, Eq. (3a);3701,FIG. 3) of the digital samples generated by the first electronic nonlinear-compensation module using the phase rotation.

In some embodiments of the above apparatus, the apparatus further comprises a second electronic nonlinear-compensation module (e.g.,330L,FIG. 4) serially connected with the first electronic nonlinear-compensation module and the dispersion-compensation module (e.g., as indicated inFIG. 4).

In some embodiments of the above apparatus, the second electronic nonlinear-compensation module is a nominal copy of the first electronic nonlinear-compensation module.

In some embodiments of the above apparatus, the first electronic nonlinear-compensation module is a part of a digital signal processor (e.g.,112or170,FIG. 1) configured to implement (e.g., using220,FIG. 2) a backward-propagation algorithm corresponding to the optical communication signal; and wherein the backward-propagation algorithm uses the phase rotation.

In some embodiments of the above apparatus, the optical front-end circuit is configured to receive the optical communication signal from an optical transport link (e.g.,140,FIG. 1); and wherein the digital signal processor is configured to recover data encoded in the optical communication signal using the backward-propagation algorithm.

In some embodiments of the above apparatus, the optical front-end circuit is configured to apply the optical communication signal to an optical transport link (e.g.,140,FIG. 1) for propagation therethrough; and wherein the backward-propagation algorithm is configured to cause a pre-distortion of the optical communication signal that tends to be removed by the propagation.

While this disclosure includes references to illustrative embodiments, this specification is not intended to be construed in a limiting sense. Various modifications of the described embodiments, as well as other embodiments within the scope of the disclosure, which are apparent to persons skilled in the art to which the disclosure pertains are deemed to lie within the principle and scope of the disclosure, e.g., as expressed in the following claims.

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 disclosure may be made by those skilled in the art without departing from the scope of the disclosure, e.g., as expressed in the following claims.