1, CH2, CH3) are shown at the bottom of the graph as solid lines. As shown, the net group-delay spectrum of one cell (e.g., cell 314) has a staircase-like shape, leading to different propagation velocities among the different channels, while keeping the velocities of all spectral components within each channel very similar.

Thus, in GDM nonlinear medium 308, different WDM channels propagate with different group velocities, whereas various frequency components of the same channel travel at the same velocity. High phase- and group-velocity mismatch between the channels (inter-channel dispersion) dramatically suppresses FWM and XPM. At the same time, dispersion within each channel's band is kept small to preserve the pulse integrity for efficient spectral broadening by SPM. Although GDM medium 308 can be implemented using off-the-shelf PGDDs, for practical use, PGDDs 312 should be compact and have low insertion loss. Highly-nonlinear silicon-nitride or chalcogenide waveguides and microring networks have a potential for integrating such a GDM medium entirely on a chip. However, at present, MIT is unmatched in terms of high nonlinearity/loss ratio (γ/α). Therefore, in some implementations, HNLF can be used for nonlinear elements and PGDDs can be implemented on a low-loss planar lightwave circuit chip; although, it should be appreciated that optical processor 300 is not limited to just, these components.

Over the last decade, however, the telecom industry's interest has shifted from intensity-modulation direct detection (IMDD) data formats to more advanced modulation formats, where the information is encoded not only into amplitude but also in the phase of the wave (e.g., quadrature phase-shift keying or QPSK, M-PSK, quadrature amplitude modulation such as 16-QAM, starM-QAM formats, etc.), which allows for increased link capacity without requiring larger bandwidth. It will be appreciated that Mamyshev 2R regenerators (e.g., regenerator 100) tend to distort the phase of an incoming signal; hence, these types of regenerators are not desired for advanced modulation formats, owing to strong nonlinear-optical transfer of noise from amplitude to phase (e.g., known as that Gordon-Mollenauer effect). A few other types of all-optical 2R regenerators, such as those based on nonlinear-optical loop mirrors (NOLMs), or nonlinear amplifying loop mirrors (NALMs), can regenerate amplitude without disturbing the phase (known as phase-preserving amplitude regenerators, or PPARs). As a result, PPARs can work with the aforementioned advanced formats and, in principle, are compatible with a GDM nonlinear medium (e.g., GDM nonlinear medium 308) for multi-channel operation.

Referring now FIG. 5, a diagram of an example NALM 500 is shown, according to some implementations. In some implementations, NALM 500 is imbalanced, e.g., by an input/output fiber coupler, an asymmetrically placed bi-directional amplifier (Bi-EDFA), an asymmetrically placed attenuator, a directional attenuator, a directional amplifier, or an asymmetrically placed dispersive element. NALM 500 further includes a nonlinear medium 502, such as highly nonlinear fiber. In some implementations, NOLM- and NALM-based regenerators, such as NALM 500, can be configured as PPARs and can potentially accommodate multiple regenerating levels. Replacing nonlinear medium 502 with the GDM medium described above in FIG. 3 (e.g., GDM medium 308) can allow NALM 500 to process multiple channels simultaneously. However, the fiber-loop nature of these regenerators makes them very susceptible to phase noise from Rayleigh scattering and reflections from any discontinuities in the loop. Hence, NOLMs and NALMs have not been widely adapted for multi-channel regeneration of advanced modulation formats.

As mentioned above, a problem with practical use of Mamyshev 2R regenerators is nonlinear amplitude-to-phase noise transfer, degrading any modulation format that uses phase encoding. For example, a simulation for 8-PSK-modulated data is shown in FIG. 6, according to some implementations. Specifically, FIG. 6 shows simulated constellation diagrams (i.e., color-coded histograms of complex-valued electric fields) of 8-PSK modulation formats at the input (left) and output of an ideal PPAR (middle) and dispersion-less Mamyshev 2R regenerator (right). Corresponding power histograms are shown in the bottom row. The average nonlinear phase shift at the peak of RZ pulse in Mamyshev 2R case is 2.41. As shown, while the amplitude is well regenerated by the dispersion-less Mamyshev 2R regenerator, the phase levels become overlapped and virtually undistinguishable.

All-Optical Multi-Channel Signal Amplitude Processor

Referring now to FIG. 7, a block diagram of an all-optical, multi-channel signal amplitude processor 700 is shown, according to some implementations. As described herein, processor 700 may be configured for processing of multiple input channels simultaneously. More specifically, processor 700 performs 2R regeneration of one or more optical signal channels (e.g., inputs). As described above, 2R regeneration generally includes re-amplification and nonlinear optical reshaping of a probability density function of input signals received from the one or more optical signal channels. Notably, processor 700 addresses certain drawbacks described above with other electro-optical and optical regenerators and processors (e.g., regenerator 100, optical processor 300, NALM 500).

Processor 700 is first shown to include input port(s) 702 which receive input signals from one or more optical channels. In some implementations, at least one of input port(s) 702 is configured to receive multiple WDM channels. In some implementations, the multiple WDM channels may provide at least a first optical signal and a second optical signal, and the power level of the first optical signal may be greater than a power level of the second signal.

As shown, instead of a single nonlinear-optical processor, processor 700 includes two nonlinear-optical processing stages with a device performing optical phase conjugation (OPC), also known as spectral inversion, positioned between them. In particular, processor 700 includes a first nonlinear optical processor stage 704. In some implementations, first nonlinear optical processor stage 704 is a Mamyshev 2R, such as regenerator 100, or a modified Mamyshev 2R regenerator, such as optical processor 300. In some implementations, first nonlinear optical processor stage 704 is a NOLM or NALM, such as NALM 500. In some such implementations, the NOLM or NALM is imbalanced by one or more of an input/output fiber coupler, an asymmetrically placed amplifier, an asymmetrically placed attenuator, a directional attenuator, a directional amplifier, or an asymmetrically placed dispersive element, as described above with respect to FIG. 5. In some implementations, first nonlinear optical processor stage 704 may be configured to perform partial amplitude regeneration of a signal—in particular, a signal received from input port(s) 702. In some implementations, first nonlinear optical processor stage 704 also causes nonlinear transfer of noise from amplitude to phase, where positive amplitude fluctuations translate into positive phase shifts. In other words, the input to first nonlinear optical processor stage 704 (e.g., an optical signal) generally has a nonlinear relationship to an output of first nonlinear optical processor stage 704. In some implementations, this nonlinear relationship is due to the nonlinear medium (e.g., GDM medium 308) within first nonlinear optical processor stage 704. For example, as described above with respect to FIG. 3, first nonlinear optical processor stage 704 may contain one or more sections of nonlinear medium (e.g., cells 314). In some implementations, first nonlinear optical processor stage 704 includes an optical bandpass filter (e.g., output comb filter 306) at its output, which locks in the partial amplitude regeneration. In some implementations, the output optical bandpass filter is a periodic optical bandpass filter.

Processor 700 is shown to further include a second nonlinear optical processor stage 708. In some implementations, second nonlinear optical processor stage 708 is a Mamyshev 2R, such as regenerator 100, or a modified Mamyshev 2R regenerator, such as optical processor 300. In some implementations, second nonlinear optical processor stage 708 is a NOLM or NALM, such as NALM 500. In some such implementations, the NOLM or NALM is imbalanced by one or more of an input/output fiber coupler, an asymmetrically placed amplifier, an asymmetrically placed attenuator, a directional attenuator, a directional amplifier, or an asymmetrically placed dispersive element, as described above with respect to FIG. 5. Like, first nonlinear optical processor stage 704, second nonlinear optical processor stage 708 may cause nonlinear transfer of noise from amplitude to phase, in some implementations (e.g., the input to second nonlinear optical processor stage 708 generally has a nonlinear relationship to an output of second nonlinear optical processor stage 708). In some implementations, this nonlinear relationship is due to the nonlinear medium (e.g., GDM medium 308) within second nonlinear optical processor stage 708. Accordingly, in some implementations, second nonlinear optical processor stage 708 may contain one or more sections of nonlinear medium (e.g., cells 314). In some implementations, second nonlinear optical processor stage 708 is configured to compensate for phase noise arising from nonlinear amplitude-to-phase noise transfer in first nonlinear optical processor stage 704. In some implementations, second nonlinear optical processor stage 708 includes an optical bandpass filter (e.g., output comb filter 306) at its output. In some implementations, the output optical bandpass filter is a periodic optical bandpass filter.

As described herein, both first nonlinear optical processor stage 704 and second nonlinear optical processor stage 708 may be multi-channel nonlinear optical processors. Accordingly, in some implementations, both first nonlinear optical processor stage 704 and second nonlinear optical processor stage 708 are, or include, one of processor 300. As mentioned above, each of first nonlinear optical processor stage 704 and second nonlinear optical processor stage 708 generally include one or more sections of nonlinear medium, which may be the same nonlinear medium (e.g., GDM medium 308) or different mediums. As described above, each section of nonlinear medium generally includes a highly nonlinear material (e.g., GDM or HLNF) and at least one PGDD. In some implementations, where input signals are at different power levels (e.g., a first input signal is at a higher power level than a second input signal), at least one of first nonlinear optical processor stage 704 or second nonlinear optical processor stage 708 is configured to invert the power levels of the signals.

The nonlinear medium of one or both processor stages can impart, a number of different characteristics and/or effects. For example, the nonlinear medium can retain intra-channel nonlinear-optical effects. In another example, the nonlinear medium of one or both processor stages suppresses nonlinear interactions among a plurality of WDM channels. In yet another example, the nonlinear medium of one or both processor stages spectrally broadens the one or more optical signal channels. In yet another example, the nonlinear medium of one or both processor stages suppresses interactions among the one or more optical signal channels. In some implementations, a group delay is accumulated between center frequencies of adjacent WDM channels in one or more of the nonlinear medium sections of first nonlinear optical processor stage 704 and second nonlinear optical processor stage 708. In some such implementations, an absolute value of the group delay accumulated between the center frequencies of the adjacent WDM channels may be substantially larger than an absolute value of the group delay accumulated between any two frequency components of either one of the adjacent channels. Processor 700 is also shown to include output port(s) 710 which receive signals from second nonlinear optical processor stage 708 after having been fully processed.

Between first nonlinear optical processor stage 704 and second nonlinear optical processor stage 708 is an OPC 706. Accordingly, OPC 706 generally receives signals from first nonlinear optical processor stage 704 after initial processing and passes signals to second nonlinear optical processor stage 708. In some implementations, OPC 706 is configured to perform complex conjugation of an electric field of the output signal of first nonlinear optical processor stage 704. In some implementations, OPC 706 is configured to invert an optical spectrum of the output signal of first nonlinear optical processor stage 704. In some implementations, OPC 706 causes positive phase fluctuations (e.g., due to positive power fluctuations) to become negative, such that second nonlinear optical processor stage 708 can bring these phase fluctuations back to zero due to subsequent accumulation of nonlinear phase shift. Conditions under which this occurs can include:

γΔ
⁢

P
1

⁢

L
1

=

γ
⁢
Δ
⁢

P
2

⁢

L
2

and
/
or

Δ
⁢

P
1

Δ
⁢

P
2

=

γ
2

⁢

L
2

γ
1

⁢

L
1

where γ is the nonlinear constant, ΔP is the power fluctuation, and L is the effective length of nonlinear medium marked by a subscript (e.g., 1 or 2).

It will be appreciated that the aforementioned functionality of OPC 706 ensures that nonlinear phase shifts due to power fluctuations in first nonlinear optical processor stage 704 and second nonlinear optical processor stage 708 cancel each other out. In some implementations, a negative slope of the power transfer function of one of the stages (e.g., one of first nonlinear optical processor stage 704 and second nonlinear optical processor stage 708) can be leveraged to effectively achieve negative nonlinear constant γ (i.e., obtain negative phase nonlinear shifts for positive power fluctuations). Thus, OPC 706 is generally configured to cancel nonlinear phase shifts in a strongly nonlinear device (e.g., which relies on high nonlinearity for its optical processing functionality). In some implementations, for M-PSK and starM-QAM modulation formats, there is no need to regenerate zero level; hence the optical bandpass filter at the output of first nonlinear optical processor stage 704 and/or second nonlinear optical processor stage 708 may not be off-centered.

In some implementations, if the average power of the signal and/or parameters of the nonlinear medium of first nonlinear optical processor stage 704 and second nonlinear optical processor stage 708 is selected to satisfy the equations above, or another condition for cancellation of the nonlinear phase noise is met, the output power curve of second nonlinear optical processor stage 708 can closely match the “ideal” curve shown in FIG. 2A. With reference to FIG. 8, a plurality of example constellation diagrams and corresponding power histograms for an input signal that is processed using processor 700 are shown which illustrate this point, according to some implementations. Specifically, FIG. 8 shows simulated constellation diagrams of 8-PSK modulation formats corresponding to an input signal (e.g., the input signal shown in FIG. 5) after first nonlinear optical processor stage 704 (top left), after OPC 706 (top middle), and after second nonlinear optical processor stage 708 (top right). In this example, the average nonlinear phase shift at the peak of RZ pulse is 1.66 for the first stage and 2.9 for the second stage. Corresponding power histograms are shown in the bottom row, which illustrate the outputs of first nonlinear optical processor stage 704 (bottom left) and second nonlinear optical processor stage 708 (bottom right). It is understood that the outputs of first nonlinear optical processor stage 704 and OPC 706 generally have similar power distributions.

In some implementations, processor 700 is fully compatible with WDM operation if the nonlinear media in both first nonlinear optical processor stage 704 and second nonlinear optical processor stage 708 is GDM media (e.g., as described above with respect to FIG. 3). In some implementations, OPC 706 can also be implemented in a multi-channel-compatible way by using optical parametric amplification in an HNLF with low dispersion slope, pumped slightly on the positive side of its zero-dispersion wavelength. In some implementations, OPC 706 can also be implemented in periodically poled lithium niobate waveguides and other nonlinear materials. It should be appreciated, as well, that the operation of processor 700 is not limited to Mamyshev 2R regeneration; rather, the scheme described herein can apply to NOLM or NALM-based regenerators, as well as to XPM-based optical switches, wavelength converters, or logic gates, all-optical data format converters or inverters, etc.

Several examples of how to separate the conjugated idler channels from the input signal channels and the pump are shown in FIGS. 9-10B. For example, FIG. 9 illustrates optical phase conjugation (i.e., spectral inversion) of multiple WDM channels positioned, e.g., on Δf=100 GHz-spaced ITU grid. In this example, the darkly shaded channels represent an input to OPC 706, and the lightly shaded channels represent an output of OPC 706, which can be separated from the input channels and the pump by a low-pass filter. FIG. 10A and FIG. 10B illustrate two examples of alternative channel arrangements, in which input channels are located on both sides of the pump and can be separated from the conjugated channels at the output by an optical interleaver (e.g., a 100-to-200-GHz interleaver, in this example). The transmission function of the interleaver is illustrated with a dashed line.

Additional implementations Using a Single Nonlinear Optical Processor

As has been described in detail, processor 700 generally includes two nonlinear optical processors (e.g., first nonlinear optical processor stage 704 and second nonlinear optical processor stage 708) and an optical phase conjugator (e.g., OPC 706) positioned between them. Each nonlinear optical processor could be an all-optical 2R regenerator, which may require a high-power optical amplifier, nonlinear medium (e.g., possibly a GDM medium for multi-channel operation), and/or optical filters or couplers. As mentioned above, however, removing one of the two nonlinear optical processors would practically halve the cost, space, and power consumption of the all-optical, multi-channel signal amplitude processor. In this regard, variations of processor 700 are described below in which the two nonlinear optical processors are replaced by a single nonlinear optical processor in which both the original signal and the phase-conjugated signal (also known as idler) share propagation.

Referring now to FIG. 11, a first variation of the all-op multi-channel signal amplitude processor described herein (e.g., processor 700) is shown. This first variation—shown as all-optical, multi-channel signal amplitude processor 1100—generally utilizes a nonlinear optical processor, such as Mamyshev 2R regenerator, that is compatible with bidirectional propagation of independent signals. Thus, processor 1100 is shown to include only one nonlinear optical processor 1104, which may be the same as, or functionally equivalent to, first nonlinear optical processor stage 704 and/or second nonlinear optical processor stage 708. In other words, nonlinear optical processor 1104 can be a Mamyshev 2R, such as regenerator 100, or a modified Mamyshev 2R regenerator, such as optical processor 300. Processor 1100 is also shown to include a pair of WDM couplers 1102, 1106 for multiplexing and demultiplexing signal and idler beams; however, the number of WDM couplers is not intended to be limiting. WDM couplers 1102, 1106 are described in greater detail below. Likewise, processor 1100 can include an OPC 1108, which may be the same as, or functionally equivalent to, OPC 706 as described above.

During operations, a signal (e.g., a beam) is first received via WDM coupler 1102. After passing through nonlinear optical processor 1104, as shown, the signal then passes through WDM coupler 1106 and OPC 1108. At the output of OPC 1108, the conjugated signal (referred to herein as an idler) and non-conjugated signal are directed to WDM coupler 1106. As mentioned above, WDM coupler 1106 can multiplex and demultiplex the signal and idler beams. It should be noted that, throughout FIGS. 11-15D, an original/received signal is indicated by a solid line, a conjugated signal (referred to herein as an idler) is indicated by a dashed or “broken” line, and pump beams are indicated by dot-dashed lines. Additionally, optional components are identified with dotted borders.

In some implementations, WDM coupler 1106 blocks the non-conjugated signal while allowing the idler to enter nonlinear optical processor 1104. The idler can then travel through nonlinear optical processor 1104 in a direction opposite that of the original signal. Thus, the signal and idler share the same nonlinear optical processor (e.g., nonlinear optical processor 1104) by counter-propagating through it. Counter-propagation significantly reduces any chance of linear or nonlinear interaction (e.g., crosstalk) between the signal and idler beams. Since signal and idler wavelengths are different, WDM coupler 1106 can easily separate or combine them.

To this point, in some implementations where the signal and idler wavelength bands do not overlap (e.g., as shown in FIG. 9), WDM coupler 1106 is a WDM band combiner/splitter. In some implementations where the signal and idler wavelengths are interleaved (e.g., as shown in FIG. 10), WDM coupler 1106 is an optical interleaver/de-interleaver. In either case, a WDM band combiner/splitter and/or an optical interleaver/de-interleaver are inexpensive passive (e.g., not energy consuming) devices with very low insertion losses. In other implementations, WDM coupler WDM coupler 1106 is a programmable filter based on liquid crystal on silicon (LCoS) technology. In some implementations, the counter-propagating signal and idler can alternatively be combined or separated using an optical circulator rather than a WDM coupler. It should be understood that WDM coupler 1102 is generally the same as, or functionally equivalent to, WDM coupler 1106; thus, the preceding description of WDM coupler 1106 is equally applicable to WDM coupler 1102 but is not repeated herein for the sake of brevity.

Referring now to FIG. 12, a second variation of the all-optical, multi-channel signal amplitude processor described herein (e.g., processor 700) is shown. In this second variation—shown as all-optical, multi-channel signal amplitude processor 1200 the nonlinear optical processor (e.g., nonlinear optical processor 1104) is a NOLM or other nonlinear optical processor that is not necessarily compatible with bidirectional propagation. In some such implementations, as shown, the signal and idler beams are configured to co-propagate. A co-propagating geometry is more prone to nonlinear signal-idler interaction, but such interaction can be minimized, for example, by employing GDM nonlinear medium inside nonlinear optical processor 1104.

Referring now to FIG. 13, a third variation of the all-optical, multi-channel signal amplitude processor described herein (e.g., processor 700) is shown. In this third variation—shown as all-optical, multi-channel signal amplitude processor 1300—the nonlinear optical processor is, specifically, a Mamyshev 2R regenerator (e.g., optical processor 300). In this configuration, the signal and idler beams counter-propagate in a conventional or GDM nonlinear medium 1302 and are separated or combined by WDM couplers 1102, 1106. In addition, processor 1300 is shown to include optical bandpass filters (OBPFs) 1304 before and after nonlinear medium 1302. However, if the signal and idler wavelengths are interleaved, e.g., as was shown in FIG. 10, then WDM couplers 1102, 1106 can be implemented in the form of optical interleavers/de-interleavers, which also automatically perform periodic optical bandpass filtering of the signals and idlers. In that case, separate OBPFs (e.g., OBPFs 1304) are not included in processor 1300, further simplifying the scheme. In practice, it might still be beneficial to use OBPFs 1304 even in the cause of utilizing optical interleavers/de-interleavers for WDM couplers 1102, 1106, because an optical interleave/de-interleaver may have some linear crosstalk (e.g., leakage) between the signal and idler, which can be suppressed by the additional OBPF stage. In addition, processor 1300 is shown to include a set of high-power optical amplifiers 1306 for amplifying an input signal and/or the idler. In some implementations, at OPC 1108, pump(s) further provide additional pump beams, which can subsequently be filtered out by an additional WDM coupler, shown as WDM coupler 1308. For the purposes of this description, WDM coupler 1308 may generally be implemented using technology similar to one of those described for implementation of WDM couplers 1102 and/or 1106 above.

Referring now to FIG. 14, a fourth variation of the all-optical, multi-channel signal amplitude processor described herein (e.g., processor 700) is shown. In this fourth variation—shown as all-optical, multi-channel signal amplitude processor 1400—the nonlinear optical processor is, again, a Mamyshev 2R regenerator (e.g., optical processor 300). Thus, the signal and idler beams counter-propagate in a conventional or GDM nonlinear medium 1302 and are separated or combined by WDM couplers 1102, 1106. It should be understood, however, that some of the most expensive and power-consuming components in processor 1300 (e.g., described above) are high-power optical amplifiers 1306. It is possible to further simplify the device, in turn reducing cost and power consumption, by eliminating one of high-power optical amplifiers 1306 from the configuration shot-vii in FIG. 13. To this point, processor 1400 is shown to include a single one of high-power optical amplifiers 1306 for amplifying both signal and idler beams.

It is also possible to use high-power optical amplifier 1306 (e.g., a single optical amplifier) to boost the pump power. In some such implementations, processor 1400 can further include two or more pump add/drop multiplexers 1406. As shown, one of pump add/drop multiplexers 1406 is posited after a WDM coupler 1402 and a second is positioned before a WDM coupler 1404. FIGS. 15A-15D illustrate examples of narrowband pump add/drop multiplexers, which can be used for pump add/drop multiplexers 1406. Specifically, FIGS. 15A and 15B illustrate a narrowband pump add multiplexer and FIGS. 15C and 15D illustrate a narrowband pump drop multiplexer. In each configuration, the narrowband pump add/drop multiplexers are defined by a combination of a three-port circulator 1502 (circle with an arrow) and a fiber Bragg grating (FBG) 1504. In three-port circulator 1502, light can only propagate from Port 1 to Port 2 and from Port 2 to Port 3. FBG 1504 reflects narrowband light around the pump wavelength.

Experimental Proof-of-Principle Demonstration

Referring now to FIG. 16, an experimental setup for demonstrating use of a Mamyshev regenerator as a phase-preserving amplitude regenerator (PPAR) is shown. While described briefly herein, additional details of this experimental setup are described in C. Guo and M. Vasilyev, “Phase-preserving amplitude regeneration in a Mamyshev regenerator with mid-stage optical phase conjugation,” 2023 Optical Fiber Communications Conference and Exhibition (OFC), San Diego, CA, USA, 2023, pp. 1-3, doi: 10.1364/OFC.2023.W2A.31, which is incorporated herein by reference in its entirety.