Supercontinuum generation in sign alternating dispersion waveguide segments with reduced noise

Abstract

A waveguide structure includes a plurality of alternating segments of normal dispersion (ND) waveguide segments and anomalous dispersion (AD) waveguide segments along a length of the waveguide structure. The alternating segments are configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the ND waveguide segments and the AD waveguide segments. The AD waveguide segments are each further configured to shape the optical pulses into higher-order soliton pulses that are compressed to a soliton inversion point to thereby cause a temporal width and a spectral width of the higher-order soliton pulses to oscillate between maximum and minimum values as the higher-order optical pulses traverse each of the AD waveguide segments.

Description

Docket No. 3262-002US1SUPERCONTINUUM GENERATION IN SIGN ALTERNATING DISPERSION WAVEGUIDE SEGMENTS WITH REDUCED NOISECROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of US Provisional Application Serial No.63 / 740038, filed December 30, 2024, the contents of which are incorporated herein by reference.BACKGROUND

[0002] Wide-bandwidth continuous spectral generation from the Kerr nonlinearity of a material, named supercontinuum generation is pivotal in a vast plethora of fields in laser science. Supercontinuum generation is inherent in wide-bandwidth (spectrally) coherent laser sources and frequency combs that can be used in LIDAR, metrology, spectroscopy and optical coherent tomography. Spectral bandwidths of these sources can span several thousand nanometers, all generated from relatively narrowband sources, e.g., within the typical range 8-60 nm. While, other light sources exhibit vast bandwidths, such as halogen lamps, the emitted optical radiation is incoherent, i.e., there is no constant phase and amplitude relationship between frequency components (over a minimal interval of time). In contrast, supercontinuum light sources have a spectral range that is generated through a deterministic process, directly scalable to the intensity of optical radiation, and not through stochastic processes such as black body radiation and spontaneous emission. Therefore, SCG sources can circumvent the incoherence problem in wide-bandwidth optical sources.SUMMARY

[0003] In accordance with one aspect of present disclosure, a waveguide structure includes a plurality of alternating segments of normal dispersion (ND) waveguide segments and anomalous dispersion (AD) waveguide segments along a length of the waveguide structure. The alternating segments are configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the ND waveguide segments and the AD waveguide segments. The AD waveguide segments are each further configured to shape the optical pulses into higher-order soliton pulses that are compressed to a solitonDocket No. 3262-002US1inversion point to thereby cause a temporal width and a spectral width of the higher-order soliton pulses to oscillate between maximum and minimum values as the higher-order optical pulses traverse each of the AD waveguide segments.

[0004] In another aspect of the present disclosure, the AD waveguide segments are further configured so that two extrema in bandwidth fluctuations of an incoming optical pulse train are mapped to both sides of the soliton inversion point.

[0005] In yet another aspect of the present disclosure, the ND waveguide segments have a length that minimizes spectral bandwidth differences between two extrema and minimizes temporal durations of the two extrema such that a sign of a temporal chirp between the two extrema is the same.

[0006] In yet another aspect of the present disclosure, the ND waveguide segments are further configured to reduce modulation instability arising in a previous AD waveguide segment traversed by the optical pulses. In general, the ND segments accomplish four things in the alternating structure: (1) It maintains the bandwidth convergence done by the previous AD segment in terms of the two extrema; (2) At the end of the ND segment the bandwidth duration are the same between the two extrema. (3) The temporal chirp is the same; (4) The modulations in the spectrum are greatly reduced due to the novel optical wavebreaking in the ND Segment.

[0007] In yet another aspect of the present disclosure, the modulation instability is reduced by a wave breaking mechanism that spectrally broadens and flattens the optical pulses. This consists of an optical shock front that passes through each sub-pulse in the temporal domain merging them into the main pulse and providing a global monotonically increasing phase profile. This reduces both the temporal modulation (i.e., by merging everything into one pulse instead of a group of pulses) and produces a flat spectrum as spectral interferences from subIn pulses are now substantially reduced.

[0008] In yet another aspect of the present disclosure, the plurality of alternating segments includes a number of alternating segments to optimize a lower extremum of spectral bandwidth and duration fluctuations in the optical pulses such that an optical pulse at another extremum is initially disfavored for bandwidth generation, making its larger initial bandwidth increase more slowly compared to lower extremum optical pulses. The difference in bandwidth generation rates as a function of segment number results in both bandwidthsDocket No. 3262-002US1converging to the same value at a segment in the SADW structure. This causes the other extremum to converge to the lower extremum in the SADW structure, therefore reducing the fluctuation noise.

[0009] In yet another aspect of the present disclosure, soliton coupling is avoided in initial ones of the AD waveguide segments while bandwidth generation continues. At a given segment, soliton coupling to the fundamental soliton occurs. Moreover, this coupling would occur at the same segment in the SADW structure, over the range of input fluctuations of the pulses, by design. Therefore, since both extrema would be mapped to the same fundamental soliton, pulse convergence over the fluctuating input pulses has occurred, minimizing noise. Past this fundamental soliton coupling, the pulses all have the same dynamics for further bandwidth generation segments in the SADW. The advantage here is that this solution makes the soliton coupling length constraint for the AD segment (where soliton coupling occurs) less specific, allowing for more variation in design, at the trade-off of longer overall SADW structures with more segments and the possibility of a lower bandwidth generation from the SADW structure.

[0010] In yet another aspect of the present disclosure, a waveguide structure includes at least one anomalous dispersion (AD) waveguide segment that is configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the AD waveguide segment. The AD waveguide segment is further configured to shape the optical pulses into higher-order soliton pulses that are compressed to a soliton inversion point to thereby cause a temporal width and a spectral width of the higher-order soliton pulses to oscillate between maximum and minimum values as the higher-order optical pulses traverse the AD waveguide segment. The AD waveguide segment is further configured so that two extrema in bandwidth fluctuations of an incoming optical pulse train are mapped to both sides of the soliton inversion point.

[0011] In yet another aspect of the present disclosure, a waveguide structure includes at least one normal dispersion (ND) waveguide segment that is configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the ND waveguide segment. The ND waveguide segment has a length that minimizes spectral bandwidth differences between two extrema. The ND waveguide segment is further configured to reduce modulation instability of incoming optical pulses by a wave breaking mechanism that spectrally broadens and flattens the optical pulses.Docket No. 3262-002US1

[0012] In yet another aspect of the present disclosure, a waveguide structure includes a normal dispersion (ND) waveguide segment and an anomalous dispersion (AD) waveguide segment extending along a length of the waveguide structure. The AD and ND waveguide segments are configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the ND waveguide segment and the AD waveguide segment. The AD waveguide segment is further configured to shape the optical pulses into higher-order soliton pulses that are compressed to a soliton inversion point to thereby cause a temporal width and a spectral width of the higher-order soliton pulses to oscillate between maximum and minimum values as the higher-order optical pulses traverse the AD waveguide segment.

[0013] In yet another aspect of the present disclosure, a waveguide structure includes a normal dispersion (ND) waveguide segment and an anomalous dispersion (AD) waveguide segment extending along a length of the waveguide structure. The AD and ND segments are configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the ND waveguide segment and the AD waveguide segment. The AD waveguide segment is further configured to shape the optical pulses into higher-order soliton pulses that are compressed to a soliton inversion point. The AD waveguide segment is further configured so that two extrema in bandwidth fluctuations of an incoming optical pulse train are mapped to both sides of the soliton inversion point. At the end of the AD segment the bandwidth converges between the two extrema. The ND segment then is configured such that the remaining temporal chirp inversion between the two extrema is corrected such that they converge to the same chirp profile at the end of the ND segment, also maintaining that the bandwidth generation between the two extrema in the ND segment also converge. The ND segment is also configured such that the modulations in the temporal and spectral profiles of the pulses are reduced through the optical wavebreaking mechanism employed in the ND segment.

[0014] In yet another aspect of the present disclosure, a method of supercontinum generation (SCG) and nose reduction includes: receiving an optical pulse train at an input to a waveguide structure that includes alternating segments of normal dispersion (ND) waveguide segments and anomalous dispersion (AD) waveguide segments that extend along a propagation direction in the waveguide structure, the alternating segments being configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguideDocket No. 3262-002US1structure is affected in the ND waveguide segments and the AD waveguide segments; spectrally broadening optical pulses in the optical pulse train by SCG as the optical pulses traverse the ND waveguide segments and the AD waveguide segments; shaping the optical pulses into higher-order soliton pulses that are compressed to a soliton inversion point to thereby cause a temporal width and a spectral width of the higher-order soliton pulses to oscillate between maximum and minimum values as the higher-order optical pulses traverse each of the AD waveguide segments; and emitting the optical pulses from the output of the waveguide structure.

[0015] In yet another aspect of the present disclosure, extrema in bandwidth fluctuations of the optical pulse train is mapped to both sides of the soliton inversion point in the AD waveguide segments.

[0016] In yet another aspect of the present disclosure, spectral bandwidth differences between two extrema is minimized, and the temporal durations of the two extrema are minimized such that a sign of temporal chirp between the two extrema is the same.

[0017] In yet another aspect of the present disclosure, modulation instability arising in a previous AD waveguide segment traversed by the optical pulses is reduced.

[0018] In yet another aspect of the present disclosure, an optical arrangement includes a laser source for generating an ultrafast pulse train. The optical arrangement also includes a waveguide structure that includes alternating segments of normal dispersion (ND) waveguide segments and anomalous dispersion (AD) waveguide segments that extend along a propagation direction in the waveguide structure. The waveguide structure has an input configured to receive the ultrafast pulse train from the laser source. The waveguide structure is further configured to use nonlinear dynamics to reduce bandwidth and durational fluctuations in the ultrafast pulse train.

[0019] This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. Furthermore, the claimed subject matter is not limited to implementations that solve any or all disadvantages noted in any part of this disclosure.Docket No. 3262-002US1BRIEF DESCRIPTION OF THE DRAWINGS

[0020] FIG. la shows an example of the temporal evolution of the three salient cases of nonlinear pulse dynamics in anomalous dispersion waveguides; FIG. lb illustrates the spectral bandwidth development of the solitons in FIG. la and the dispersive case; and FIG.1c is an illustrative example of modulation instability gain from intensity noise modulations on a continuous wave carrier.

[0021] FIG. 2 depicts an illustrative example showing how a higher-order soliton nonlinearly propagates in an AD waveguide.

[0022] FIG. 3 is an illustrative example depicting how two pulses are shaped to solitons and converge to the same soliton.

[0023] FIG. 4 illustrates the process of optical wave breaking.

[0024] FIG. 5 shows a plot of power versus pulse time at different propagation coordinates, to illustrate the coherent combining of main and satellite pulses through optical wave breaking (OWB).

[0025] FIG. 6a illustrates the temporal dynamics (pulse duration versus propagation coordinate in an AD waveguide) of a chirped pulse entering an AD segment from a previous ND segment; and FIG. 6b is a corresponding illustration of the bandwidth versus propagation coordinate in an AD waveguide.

[0026] FIG. 7a illustrates the convergence over differing pulses (with different bandwidths) after the initial segments of the SADW structure; FIG. 7b shows the temporal duration versus propagation distance within the AD segment of FIG. 7a for two pulses, pulse 1 (higher bandwidth at entrance of SADW) and pulse 2 (lower bandwidth); and FIG. 7c shows the spectral development corresponding to the AD segment over the two pulses of FIG. 7b.

[0027] FIG. 8 is a schematic diagram illustrating supercontinuum generation in a generic waveguide structure.Docket No. 3262-002US1DETAILED DESCRIPTIONIntroduction

[0028] As discussed in U. S. Pat. No. 11,520,214, sign-alternating dispersion waveguides (SADW) have been shown to be effective in generating spectral bandwidth through supercontinuum generation (SCG). The present disclosure further shows that by properly selecting the lengths of the anomalous dispersion (AD) and normal dispersion (ND) segments in the SADW structure, SCG noise, such as durational and bandwidth noise fluctuations in the incoming pulse train, as well as modulation instability noise, can be reduced relative to the SADW structures shown in the aforementioned patent application. A normal dispersion segment refers to a waveguide segment having an overall group velocity dispersion (GVD) profile that is positive in value, usually expressed in units of s2 / m versus angular frequency co. An anomalous dispersion segment 104 refers to a waveguide segment having an overall group velocity dispersion (GVD) profile that is negative in value, also usually expressed in units of s2 / m versus angular frequency co. FIG. 8 is a schematic diagram illustrating supercontinuum generation in a generic waveguide structure 100 (e.g., integrated or fiber waveguide) having a chain of segments of alternating normal dispersion (ND) 102 and anomalous dispersion (AD) 104 as one progresses along the overall length of the waveguide structure 100. Note that while the figure shows an ND segment 102 initially preceding an AD segment 104, the disclosure is not so limited, and such will be disclosed and discussed in more detail subsequently.

[0029] SCG occurs in both the AD and ND segments of the SADW structure. SCG occurs as a result of the interaction between self-phase modulation (SPM) and the anomalous dispersion of the waveguide segment. This process temporally compresses the pulses while also generating spectral bandwidth. However, after propagating a certain distance in the AD segment, the effects of SPM balance out the effects of the anomalous dispersion, the pulses are shaped into solitons and no further bandwidth is generated. The pulse can couple into either a fundamental soliton or a higher-order soliton.

[0030] When coupling to a higher-order soliton, the duration of the pulse initially compresses rapidly and the spectral bandwidth increases. At the point of maximum compression, referred to as the soliton inversion point, the temporal phase of the pulse is reversed and as a consequence the duration of the pulse increases and the spectral bandwidth decreases until the pulse returns to its initial profile. Past the soliton inversion point the process repeats as theDocket No. 3262-002US1duration of the pulse again compresses rapidly and the spectral bandwidth again increases. In other words, the spectral and temporal widths of the pulse periodically oscillate between minimum and maximum values as the pulse continues to propagate through the AD waveguide segment.

[0031] In accordance with one aspect of the present disclosure, the oscillations in the spectral and temporal widths of the pulses that arise in the AD segment of the SADW structure are used to reduce the noise in an incoming pulse train and increase spectral bandwidth generation. In this way spectral and temporal fluctuations among the pulses in an incoming pulse train are reduced so that the output pulses from the AD segment are more similar in their spectral profde and duration, while also increasing spectral generation. This is accomplished by designing the AD segment so that the two extrema in the bandwidth fluctuations of the pulse train are mapped to both sides of the soliton inversion point.

[0032] Although the AD segment can reduce certain types of noise in the incoming pulse trains, one type of noise that is generated in these segments arises from the amplification of the intensity variations in the input optical pulse train. This noise, referred to as modulation instability, is manifested as modulations that appear in the temporal intensity profile of the pulses, resulting in the formation of sub-pulses in the time domain. Thus, the overall impact of the AD segment is to minimize amplitude fluctuations in the pulses while causing modulation instability noise.

[0033] Turning to the ND segments of the SADW structure, when an input pulse enters this segment its spectral bandwidth and duration initially increases, but saturates as the pulse continues to expand and the effects of SPM decrease. Accordingly, the output from the ND segment is a temporally broad, spectrally saturated pulse. The noise dynamics in the ND segments arise from different mechanisms than in the AD segment and are therefore complimentary. In the ND segment, an effect known as wave breaking occurs when selfphase modulation generates a frequency chirp in which the wings (past the inflection point of the pulse’s intensity profile, where the intensity asymptotically approaches zero) of the pulse generates lower frequencies while the region just before the inflection point of the pulse generates higher frequencies. The normal dispersion causes the higher frequency shifted region before the wings to move faster than the lower frequency wings. The leading regions of the pulse eventually "overtakes" the trailing wings, causing an optical shock wave to occur. The optical shock wave induces wave breaking, which means that the pulse overlaps with itsDocket No. 3262-002US1wings. This results in a pulse that has increased spectral broadening due to the high gradients of the shock wave and after the effects of further dispersion is still flattened into a gaussian-like profile. If the pulse train enters the ND segment from an AD segment, which as discussed above causes the pulses to have sub-pulses in the time domain as a result of modulation instability, the spectral broadening and flattening of the pulse eliminates the modulation instability noise as the shock wave moves through the sub-pulses, reorganizing the frequency chirp to monotonically increase and combine the sub-pulses into one pulse. This reduces the temporal modulations (the sub-pulse structure) and the spectral modulations (from spectral interferences from the sub-pulses).

[0034] In summary, the AD segment of SADW structure generates spectral bandwidth while reducing fluctuations in spectral bandwidth and duration, but with an increase in modulation instability noise. When the pulse then enters and ND segment, spectral broadening continue to occur (until saturation is reached) and the modulation instability noise is eliminated without causing spectral bandwidth and duration fluctuations. The overall effect of the two segments is to generate spectral bandwidth as a result of SCG while reducing multiple types of noise. This process may continue in subsequent sign-alternating dispersion waveguide segments, further increasing SCG and further reducing noise.

[0035] It will be shown below that sign-alternating dispersion waveguides (SADW) used in supercontinuum generation can exhibit the lowest pulse-to pulse fluctuations in duration and spectral bandwidth of any SCG system. Also, it will be shown that SADW SCG waveguides can reduce the inherent durational and bandwidth fluctuations of an input pulse train, making them useful as noise decreasing devices utilizing nonlinear effects to reduce pulse to pulse fluctuations. A number of illustrative embodiments of SADW structures will be described that minimize pulse to pulse fluctuations in pulse trains.1. ANOMALOUS DISPERSION SUPERCONTINUUM AND NOISE ANALYSIS 1.1 Dominant Noise Source

[0036] Supercontinuum generation in anomalous dispersion waveguides is governed by the interaction of self-phase modulation (SPM) with the anomalous dispersion of the waveguide.Docket No. 3262-002US1Here, since the temporal chirp that exists across a pulse due to SPM is aligned with the normal dispersion chirp case (red frequencies in the front, blue frequencies in the back of the pulse), the anomalous dispersion would act to compress the chirp formed from SPM. This compresses the pulse temporally, raising its peak power and generating spectral bandwidth. This process starts off rapidly generating large bandwidth increase versus length, due to the positive reinforcement feedback of the bandwidth generation with peak power, which further compresses the pulse. However, after a certain length, the pulse shapes itself to an invariant solution, called soliton, of the corresponding nonlinear Schrodinger equation (GNLSE), that describes the nonlinear propagation dynamics, and bandwidth generation ceases with the generation of solitons. The pulse can either couple into fundamental solitons, a higher order soliton, or neither if the energy of the pulse is too low. In the first case, no bandwidth generation occurs past this coupling, the fundamental solitons do not increase their duration (but walk-off in time delay relative to each other). In the second case, the bandwidth fluctuates between maximum and minimum values periodically along the propagation direction, and the duration fluctuations at n phase in relation to bandwidth fluctuations (e.g. at min bandwidth, the duration is maximized). In the third case the bandwidth decreases approaching a lower constant bandwidth than the input bandwidth, in the asymptotic limit of arbitrarily long waveguide lengths, and displays a monotonically increasing pulse duration as a function of propagation. So, for all three cases, the pulse couples into a solution where the bandwidth is clamped to a maximum possible level, in other words, is clamped in bandwidth generation versus propagation length. The spectral evolution of the three cases described, along with the temporal evolution of the pulse duration is shown in FIGs. la and lb.

[0037] FIG. la shows an illustrative temporal evolution of the three salient cases of nonlinear pulse dynamics in anomalous dispersion waveguides. The first case describes the qualitative temporal dynamics of the higher order soliton (IV > 1), the temporal dynamics of the fundamental soliton case (N = 1) and the case where the pulse energy is below the threshold for soliton coupling, i.e., linear dispersive broadening dominated (N < 1). Here a transform limited pulse is first shaped into the solitons through the characteristic Lshapinglength (which can be different between the solitons) then once coupled into the soliton, Lsis the characteristic length of the higher order soliton between its maximum duration and minimum duration. The higher order soliton exhibits a temporal chirp across its profile as it propagates in the waveguide except when it reaches its maximum and minimum duration where the pulse is transform limited. In contrast, the fundamental soliton is transform limited. In addition,Docket No. 3262-002US1whilst the pulse is shaping into the solitons, it is dispersing radiation in the form of dispersive waves and side pulses to match the profde (of the main pulse) to the soliton. FIG. lb illustrates the spectral bandwidth development of the above solitons and the dispersive case. Here, slight spectral narrowing transpires in the dispersive case (N < 1) due to the reversal of dispersive temporal phase (chirp) in relation to the intensity slope of the pulse. This result in a reduction in instantaneous frequencies (and thus bandwidth) from SPM. FIG. 1c is an illustrative example of modulation instability gain from intensity noise modulations on a continuous wave carrier. The modulations are treated like a series of pulses that nonlinearly compress to increase the peak amplitude of the modulation.

[0038] The positive feedback loop of peak power, pulse duration and bandwidth generation in anomalous dispersion with SPM, can also amplify intensity variations in the input optical waveform as it makes these variations “sharper” in time. The amplification of intensity noise in the form of modulations that occur in the input optical waveform is schematically shown in FIG. 1c. These modulations can also occur through side-band generation by spontaneous and then amplified four- wave mixing and other quantum effects.

[0039] The modulations introduce a time-variant amplitude profde, that generates instantaneous frequencies through SPM (i.e., SPM is proportional to n2, n2is the nonlinear index coefficient of the material, I is the intensity.) This positive feedback quickly compresses these local variations in the optical waveform to a group of pulses that are separated by the modulation period. This results in the amplification of intensity components around the modulation frequency (FIG. la). This modulation instability (MI) gain coefficient is approximately given as,Eq. 1 .PAD

[0040] With modulation instability length given as4Eq. 2.In the above equations / 3ADis the group velocity dispersion around the carrier frequency of the optical waveform, a>mis the modulation (angular) frequency in the intensity profile, y is the waveguide’s nonlinear coefficient, Pois the peak power of the input optical waveform.Docket No. 3262-002US1

[0041] Modulation instability amplification, is the most dominant noise term in AD SCG, causing modulations that appear in the temporal intensity profile and in the spectrum due to the grouping of sub-pulses that are formed in the time domain. As the original modulation to be amplified is bom out of random noise, the modulations are then also random and can be fluctuating in time.

[0042] In the case of higher order solitons, the SPM induced phase of the wings interacting with the anomalous dispersion during the nonlinear compression, creates side lobes which can even undergo fission off of the rest of the higher-order soliton due to third-order dispersion. The modulation being an effect in the wings of the pulse is because, there, the frequency chirp goes back to central carrier frequency and does not continue the increasing (or decreasing) monotonic trend of the corresponding side of the pulse where the wing is located. Thus, the wings move away from the main pulse creating side-lobs. This creates a series of sub-pulses (in the case of soliton fission (fundamental) soliton pulses) each at different central wavelengths, with a chirp profile aligned with the normal chirp before the point of soliton inversion (where the minimum duration of the soliton is reached) and then aligned as anomalous chirp after the soliton inversion point. In the spectral domain, this creates modulations due to spectral interference that occurs from pulses at different time delays. Moreover, the placement of these pulses varies due to differing input conditions in energy, duration, chirp, etc (see next sections on the discussion on AD segment lengths and higher order soliton coupling).

[0043] FIG. 2 depicts an illustrative example showing how a higher-order soliton nonlinearly propagates in an AD waveguide. The arrows show the direction of movement of the instantaneous frequencies in relation to the center of the pulse. Relative magnitudes of arrows indicate the relative speed, i.e., GV associated to the instantaneous frequency. At the center the pulse is compressed to the lowest duration, it is also at its transform limited duration. This point is called the point of soliton inversion. Past this point, the temporal phase is reversed, and instantaneous frequencies are opposite as on the left-hand side of the diagram prior to compression. The pulse then ends up in its original transform limited start duration and the cycle is repeated. There is a mirror symmetry in spectral amplitude shape and temporal amplitude shape between pulses past the minimum duration location and before it (i.e., mirror symmetry about the soliton inversion point).Docket No. 3262-002US1

[0044] For ultrashort pulses, it is important to understand what happens to peak to peak or duration fluctuations when they couple to solitons. In these cases, as the pulse shapes to the fundamental or higher order soliton, the pulse sheds radiation in the form of narrowband dispersive waves and sub pulses that peel off the main pulse (see FIGs. 1 and 2 for reference). Therefore, the dynamics of the higher order solitons (described below) can also contribute to MI gain.1.2 AD Segment Noise reduction

[0045] The AD segments contribute to modulation instability noise; however, they minimize amplitude fluctuations of the main pulse that is shaped and outputted from this segment type. It will be shown in this section that a wide range of amplitude fluctuations and input spectral bandwidth can converge to a singular output pulse, if certain conditions are met. The convergence to the same output pulse profile occurs because the end pulse of the AD waveguide converges to a single soliton type regardless of the input conditions (within a certain range).

[0046] In essence, regardless of input pulse conditions, if the pulse is shaped to the same order of soliton in the waveguide, then past that point the pulse would follow the soliton dynamics regardless of the input conditions. This result is summarized in FIG. 3, which is an example illustrating how two pulses are shaped to solitons and conveige to the same soliton, provided Eq. 4 is respected. The Figure shows a higher order soliton (order 2) and a fundamental soliton. The characteristic lengths are shown, which is the shaping length (defined as Lshor equivalently Lshapi) and then the soliton nonlinear compression lengths).

[0047] The general energy equation for solitons is given as:Where E is the pulse energy (fixed quantity), N is the soliton order,is the second order dispersion, y is the waveguide nonlinear coefficient and t0is the maximum soliton duration.

[0048] A pulse entering with a duration txwill decrease in duration due to the combination of SPM and anomalous dispersion. The final duration would correspond to the greatest N such that t0is belowt0determines which soliton order is shaped into (i.e., N) and is theDocket No. 3262-002US1duration of the soliton at its largest duration point. In the case of and N = 1 soliton, t0is the duration of the soliton for all propagation distance after the soliton is produced, i.e., the soliton duration does not change.

[0049] For higher-order solitons the duration periodically fluctuates along the propagation length between t0and tmin. This is also true for the spectral bandwidth, reaching its maximum when tminoccurs. Though the start duration is larger, as the order increases, so does the bandwidth at minimum duration and as well, tmindecreases. The larger bandwidth versus soliton order occurs because, t0, the start duration, is larger for higher order solitons, therefore, there is more length for nonlinear generation to occur until the characteristic dispersion length (Lo) balances with the characteristic nonlinear length (Lnl) and generation slows down, accumulating more chirp, such that generation can continue past the balancing of the nonlinear and dispersion length, to even smaller pulse durations (and larger spectrum) than the fundamental soliton (where these lengths are equivalent at all propagation). The caveat is, that since Lni> LDat minimum pulse duration, the higher order soliton starts to increase its duration and inverts its chirp (FIG. 2).

[0050] Returning to Eq. 3, the final duration (and spectrum) of the pulse is only related to the pulse energy and not input duration. It can then be derived that the input transform limited pulse duration can fluctuate by a factor of:t0< G < t.0- ( / V« —+-1)2, v Eq. 4 „.Under constant pulse energy, if the input pulse duration increases past this inequality, then the soliton duration at the end would jump in a step wise fashion to the duration corresponding to the next highest order soliton (which would be smaller) or the lower order soliton (longer duration). Otherwise, all input durations that satisfy this inequality would get mapped to the same soliton in the waveguide and thus same pulse bandwidth and duration. Thus, we show that it is possible for the input pulse to converge to always a given output pulse, within a range of input pulse durations, given by Eq. 4.

[0051] In the above analysis, it is shown that coupling into one soliton is possible within a range of input durations and peak powers. However, the length needed to couple into the soliton is indeed dependent on the input pulse conditions. Therefore, the soliton can exit atDocket No. 3262-002US1different occurrences along its periodic cycle at the output of the waveguide, dependent on these input conditions and the length of the waveguide.

[0052] For shaping to a fundamental soliton (N = 1) this characteristic length is less of an issue, as any longer length would maintain the same soliton. In contrast, for higher order solitons N > 1 this length must be considered as the soliton periodically varies in duration and bandwidth and is not stagnant. Another length of importance is the length at which the minimum duration of a higher order soliton is arrived at [agrawal], which is given by:Where Porefers to the peak power of the soliton (at duration t0). In the case of a higher order soliton, for any L = Ls+ AL > Lsthe soliton’s pulse profile would be the same as at L = Ls— AL, i.e., with the exception that the phase chirp would be inverted, i.e., the pulse[t2profiles evolves periodically, with period length Lp= 2 I0, and therefore, symmetricallyabout Ls. See FIG. 3 for reference.

[0053] While Lsis the same for different input durations, the shaping length before the higher order soliton is coupled varies depending on the input pulse conditions. Therefore, the overall AD segment length must be chosen such that the overall length difference effect between differing input pulses is minimized.

[0054] A procedure to account for this in the segment design will be shown in section 3 detailing how the noise reduction effects of both ND and AD can be combined in the signalternating dispersion waveguide used for SCG. The reduction of modulation instability induced oscillations in the spectrum will be accomplished in the ND segments, while the duration and bandwidth fluctuations in the pulse train will be reduced in the AD segments.2. NORMAL DISPERSION SUPERCONTINUUM NOISE SOURCES

[0055] The noise dynamics in ND SCG arise from different mechanisms than AD SCG and are therefore complimentary. Thus, a brief discussion of noise in ND SCG is presented to clearly show how Patterned Sign-Alternating Dispersion Waveguides (SADW) SCG can reduce noise in many dimensions (spectral bandwidth fluctuations, duration fluctuations, and even (main) pulse energy fluctuations). The Optical Wave Breaking effect (OWB) in NDDocket No. 3262-002US1SCG reduces spectral modulations and renders a flatter SCG spectrum than AD SCG. It thus eliminates modulation instability noise by absolutely decreasing any modulation.

[0056] To explain this effect in more detail, we turn back to the case where SPM is present with no waveguide dispersion across the frequency range. As the name implies, self-phase modulation imposes spectral modulations in the spectral domain. This concept is integral in the description of how ND SCG reduces noise and spectral modulations. SPM introduces a modulated spectrum from the instantaneous frequency distribution of the wings of the pulse (i.e., up to the infliction point of e.g., a Gaussian pulse - see FIG. 4). These wings have decreasing phase (instantaneous frequencies) back to the carrier frequency, due to the decreased intensity derivatives across this region. This introduces a repetition of instantaneous frequency, therefore introducing a beat modulation between the Fourier transform of these envelope sections in the frequency domain - causing the modulations associated with SPM.

[0057] Now going back to the case where the waveguide has normal dispersion, under this influence, the region of the pulse envelope around the maximum instantaneous frequency (occurring at the inflection point of a Gaussian pulse ~l / e) travels faster than the region just past this inflection point where the instantaneous frequency is less. This creates an optical shock wave to appear, where local compression of the envelope occurs around these regions. This increases the derivative, increasing the instantaneous frequency in this region and the optical shock increases, both in central frequency and in velocityOn the other side of the shock wave, the pulse is stretching.

[0058] FIG. 4 illustrates the process of optical wave breaking. The wings of the pulse when just considering SPM can be seen at the left most pulse at the start of the propagation in the ND segment. Here the left most pulse and associated instantaneous phase vs pulse time plot is shown (where positive pulse time means earlier arrival times). The instantaneous phase goes back to the carrier frequency while the center portion of the pulse has a linearly decreasing instantaneous phase due to SPM. This matches the sign of the phase induced by dispersive broadening in ND, so the pulse accelerates its dispersive broadening. However, the side of the main pulse (at ~ amplitude 1 / e) compresses with the decreasing frequencies of the wing forming an optical shock as can be seen in the next snapshot of the pulse. This can be seen by the edge of the pulses as they propagate. The optical shock then changes the wing decreasing inst. phase to an increasing instantaneous phase more rapidly than the main pulse due to theDocket No. 3262-002US1high intensity derivative. Eventually, the optical shock flattens out due to the accelerated dispersive broadening accompanying the high instantaneous frequencies there- along with dispersive wave generation away from the shock due to higher order dispersion in the segment. The flattening of the pulse into a gaussian like profile again, can be seen by the right most pulse. The inst. frequencies are monotonically decreasing across pulse times now, so this means the modulation instability originating from the wings has been reduced.

[0059] This optical shock imposes an increasing frequency profile in the region of the wings, changing the profile from the case of SPM with no dispersion. Also, since the optical shock travels faster than the region of the pulse near the peak, it creates a trapezoidal pulse profile, causing the intensity derivatives near the pulse center to become flatter (stopping any further frequency broadening there). The optical shock creates side lobs in the frequency domain as it flattens the derivative before it, reducing frequency generation elsewhere but in the shock region. Further,, as the decreasing frequency is reduced in the wing region of the pulse, the spectral modulations are reduced such that the spectrum is flat.

[0060] After the characteristic optical wave breaking length, the optical shock induces wave breaking, meaning that it overlaps with the wings of the pulse, inducing an average increasing frequency in this region. The pulse then undergoes normal dispersion temporal broadening with an increasing chirp profile. Therefore, the overall spectrum is flatter than AD SCG, with no modulation instability noise- or spectral modulations imposed by SPM. Essentially OWB prevents the pulse to break up into sub-pulses, or even into different sub-regions within one pulse profile (like in the case of just SPM), creating a smooth broad temporal profile as opposed to that of the AD SCG, and thus a smooth spectrum.

[0061] In terms of modulation instability- in the general sense, there is no MI gain in normal dispersion, because ND SCG does not nonlinearly compress the SCG pulse, but is always followed by temporal stretching of the pulse. Therefore, since solitons cannot form here, no amplification of modulations in an input waveform can exist.

[0062] However, while modulation instability is eliminated, peak to peak fluctuations of the input pulse do have an impact on the SCG pulse. As there is no coupling into dispersive waves or convergence to a fundamental soliton through nonlinear compression and pulse shaping these peak fluctuations directly translate to temporal and spectral fluctuations.Therefore, in ND SCG peak fluctuations of ultrashort pulses are the pertinent noise term,Docket No. 3262-002US1while modulations in input spectra are highly reduced, thus modulation noise is decreased relative to the input.3. SIGN-ALTERNATING DISPERSION WAVEGUIDE STRUCTURES FOR SCG

[0063] Generally, in SADW structures that give rise to SCG, the pulse undergoes spectral broadening in both the ND and AD segments. Segments are alternated repeatedly as a function of propagation coordinate in the waveguide. They are connected by adiabatic transitions between segments. In ND segments, due the accelerated dispersive broadening of pulses that accompanies the SCG, the bandwidth saturates. Similarly, in AD segments, the bandwidth development ceases after coupling into a soliton, in SADW structures, when the pulses peak power is lost and duration is increased due to dispersive broadening, in ND segments, the pulse then enters an AD segment where the pulse is recompressed and recovers a higher peak power and smaller duration (due to bandwidth generation). When the pulse undergoes soliton self-compression (or anywhere before per design), the pulse exits the AD segment and goes to another ND segment where the soliton formation is disrupted. Therefore, spectral generation can remain on going in the ND and AD segments and the saturation mechanisms are overcome versus just one AD or ND waveguide. These dynamics make highly efficient SCG in SADW structures compared to conventional SCG in a singular dispersion type.4. SIGN-ALTERNATING DISPERSION WAVEGUIDES FOR SCG NOISE REDUCTION

[0064] The above-described noise sources that occur in SCG are unique and complementary for each dispersion type. The modulation instability and temporal phase noise that occurs in an AD waveguide can be reduced, under certain circumstances, in a preceding ND waveguide. Likewise, the peak-to peak fluctuation that remains in ND SCG can be compensated for in a preceding AD segment. Therefore, the iterative process of alternating the sign of dispersion for SCG can have a net effect of lowering SCG durational and bandwidth noise, and even lowering intensity noise present in the original pulse train, across the main SCG pulses. The conditions for this to happen are outlined below.Docket No. 3262-002US15. REDUCING THE EFFECTS OF MODULATIONS IN AN ND SEGMENT WITH INPUT FROM AD SEGMENT

[0065] The output from an AD segment, i.e., the input conditions into the ND segment, is critical in reducing modulation instability noise by the SCG process in the ND segment. The AD segment output usually consists of a main pulse and a series of satellite pulses that develop from the modulation instability gain or higher order soliton coupling that occurs in the AD segment. These pulses have central frequencies that are chirped in time aligned the same way as typical normal dispersion chirp, typically because the AD segment length is terminated before the higher order soliton inverts its phase, and they break off from the higher order soliton as it is compressing. This is illustrated in FIG. 2.

[0066] In general, the phase profde across these pulses are:1) Transform limited, negligible envelope phase over all pulses in output. Each satellite pulse has its own central wavelength (carrier frequency).2) Chirped, with satellite pulses having a chirp that runs contrary to main pulse and with central wavelengths away from main pulses’ (close to the endpoints of the 1 / e bandwidth frequency range).3) Include one main pulse with (just) a SPM like phase profde - i.e., this can occur for short AD segments or low dispersion magnitudes. This is not a pure case of MI, but the spectrum is self-modulated. In general, modulation in the spectrum occurs when instantaneous frequencies have several locations in time. This causes the spectral interference that creates the spectral modulation. Optical wave breaking creates an edge that compresses the waveform in the wings, resulting in an amplitude function with an increased instantaneous frequency (in relation to the main part of the pulse). As the optical shock travels, the wing instantaneous frequencies change from decreasing to the central frequency of the pulse to increasing away from it. This then removes spectral modulation.

[0067] The common pulse profdes that can be found at the entrance of the ND segments described above is summarized in FIG. 2, where pulses on the left-hand side of the compressed central pulse have frequency distribution along the ND dispersion chirp. These pulses would temporally broaden in the ND segment. For pulses on the right-hand side, they exhibit the opposite chirp. Therefore, they would temporally compress, but spectrally narrow as the phase induced by SPM is contrary to the chirp. Therefore, the sub-pulses would comeDocket No. 3262-002US1together in one main pulse, albeit at a smaller spectrum than the central compressed pulse in FIG. 2, in the beginning length portion of the ND segment.

[0068] This satellite pulse structure, before traversing the ND segment, directly have modulations in the spectrum, through spectral interference that is generated from the distribution of pulses. Moreover, the instantaneous phase distribution at the wings of each of these sub-pulses multiply the SPM induced modulations in the spectrum (described, for example in FIG. 4 for one pulse).

[0069] By tuning the parameters such as the optical wave breaking length of a normal dispersion fiber and the dispersion length, given an input pulse peak power and duration, the effects of modulation instability built up in the AD segment within the neighboring ND segment can be eliminated. Reducing the effects of modulation instability means reducing both the number of sub-pulses, coherently combining them into one pulse with a monotonic chirp and / or reducing the spectral modulations that transpire from the modulated temporal profile (e.g., the sub pulses in time). We do this using a new nonlinear Kerr based mechanism in normal dispersion, utilizing optical wave breaking (OWB) to combine two pulses in a phase matched manner or utilizing OWB within a calculated ND segment length to reduce the dominant wing contributions to MI.

[0070] The basic concept of coherently combining the main and satellite pulses through OWB are illustrated in FIG. 5 This ensures that the edge of the main pulse has, ideally, the higher frequency than the edge of the satellite pulse when the two pulses meet. Then, at these edges, the two pulses can add coherently, raising a new edge. This edge then undergoes optical wave breaking in the satellite pulse, increasing the instantaneous frequency distribution in this region, such that all instantaneous frequencies across the time region of main pulse + satellite pulse is unique. Ultimately, this minimizes spectral interference and results in a smooth spectrum, regardless of the original modulations. Therefore, compensating for MI noise in the ND segment.

[0071] FIG. 5 shows a plot of power versus pulse time at different propagation coordinates, to illustrate the coherent combining of main and satellite pulses through OWB. The pulses’ shock fronts eventually meet as seen at different propagation coordinates. If the instantaneous frequency, of the main pulse shock is greater in absolute value than the neighbouring pulses shock front, the main pulse’s shock front will go through the satellite pulses. This joins theDocket No. 3262-002US1pulses in a fashion that the instantaneous, frequency distribution is now a monotonic function as in FIG. 4. This reduces the spectral modulations from spectral interferences from the satellite pulses and has the net effect of reducing modulation instability both spectrally and temporally (one pulse is found at the end of the propagation).5.1 Scenario 1: Pulses Have High Degree Of Frequency Space Overlap, Central Frequencies Are Marginally Close

[0072] In this scenario all pulses and sub-pulses have a high degree of overlapped in frequency space, whereby the central frequencies of neighbouring pulses are very close together. Pulses are at a spacing T greater than T = √2to, i.e., at high modulation spectral frequency. Under this scenario, wave breaking occurs per pulse and the OWB effect of smoothing out the decreasing instantaneous frequency distribution at the wings of the pulse is utilized per pulse. If pulses are sufficiently spaced apart temporally, then, if pulses partially overlap due to dispersive broadening, an undesired beat modulation (in the temporal domain) can occur, and thus this should be avoided. These temporal modulations can seed MI in subsequent AD segments. However, due to the non-existent MI gain in the ND SCG it is expected that they do not add substantial additional modulations in the spectral development in this segment type.

[0073] In this case, we allow OWB to occur up until the OWB shock front does not overlap with the neighbouring pulse. The OWB characteristic length, which describes when the shock front has passed through the wings of the pulse is given as:LDLWB - ~ Eq. 6(4N2e(-3 / 2)- 1)0.5Where LD=N2= — which is also the soliton order (in AD segments), Ln, = —.LD= to² / βND, Lnl= 1 / γPo

[0074] It is apparent from Eq. 6 that the LWBis a function of the nonlinear length Lnl, which in turn is a function of the highest peak power sub-pulse (which is the main pulse). Therefore, we choose LWBto be the minimum length of the pulse set, i.e., the main pulse’s LWB.Therefore, the length of the ND segment is chosen (under this scenario) to be:Docket No. 3262-002US1LWB Eq.7Tois the (smallest) spacing between main pulse and neighbor pulse (s). Eq. 7 limits the length to a length at which the main pulse and sub pulse are about to overlap temporally. Derived under the approximation that the neighboring pulse has the same duration and undergoes the same OWB, which makes the bounding length a conservative estimation. In general, dependent on the peak power ratio of the main to neighboring pulse the upper bound length (T0- zj tcould be up to ωsatis the bandwidth of the main pulse after it has traversed alength of LWBand when the shock wave is at t0from the pulse center (time zero), t0is theintensity duration. ωsatis more or less the start of bandwidth saturation in the ND segment. As the bandwidth is actually increasing, the bounding length on the right is decreased from the calculation involving this static bandwidth, but restricting it with the factor of 2 in the denominator is a good overall compromise of choosing the ND segment length such that sub-pulses under scenario 1 (high degree of overlap in the frequency domain and separated by a large separation) do not interfere temporally in a manner that then creates temporal modulations.5.2 Scenario 2: Two Pulses At Increased Frequency Separation, With Or Without Remaining Spectral Overlap (Modulations Present)

[0075] For the following analysis it is important to note that the two pulses, main and satellite, are at slightly differing central frequencies, such that their spectral content overlap (i.e., modulations are present) but not completely. In this case, the higher instantaneous frequency shock edge of one pulse can overlap coherently with the instantaneous frequency edge of the satellite pulse (can be a shock front due to OWB as well). If this is not the case, i.e., if they are at the same central frequency, the edges would have instantaneous frequencies with opposite sign, resulting in modulations induced now in the time domain. This structure would then yield the opposite dynamic: would enhance (instead of suppress) spectral modulations.Docket No. 3262-002US1

[0076] In general, if the instantaneous frequency of the main shock edge is higher (in magnitude away from the central frequency of the main pulse, i.e., abs(ωinst,1− ωo,1) > abs(ωinst,2− ωo),( 1: main pulse 2: satellite pulse) at edge overlap, the optical shock front can propagate through the satellite pulse without breaking up into an intensity modulated signal, imposing the frequency distribution set by the instantaneous frequency of the propagating shock front. The following general conditions need to be satisfied.|ωinst,1(Δt1,z0)| > |ωinst,2(Δt2,z0)| ∧ ∀ωinst,2(|t| > Δt2) Eq. 8 a, b, c |Ati| + |At2| = At0sgn(ωinst,1(Δt1,z0) − ωo,1) = sgn(ωinst,2(Δt1,z0) − ωo,1) ωinst,1,2is the instantaneous envelope frequency of the shock front of either the main pulse (1) or the satellite pulse (2), At12is the location of these instantaneous frequencies, At0is the peak-to-peak time interval between main and satellite pulse. To solve for these conditions, one would have to simulate the nonlinear pulse propagation using the generalized Schoedinger Equation (GNLSE). However, a closed form expression can be obtained for the case Lni« Ld(nonlinear length and dispersion length) and the two pulses main and satellite are at a central (angular) frequency ofand a>2< a)1— 2 jn t|qjs case. the edge instantaneous frequency can be described as, for pulse 1 and 2:AoijC^inst. edge,l > <^1 Y^1Z’ Eq.9A(Z)2C^inst. edge, 2 < ^2 T + Y^2ZWhere is the power temporal derivative at the edge. And for the time location of the edge (At1) as a function of z:ΔT1= ∫zz₀≈0β2(ω1− Δω1 / √2 − γṖ1z)dz,β2(ω1− Δω1 / √2 − γṖ1z)dz, Eq. 10z0≈0And analogously for the edge of the second pulse.Solving |ΔT1| + |ΔT2| = Δt0with the assumption of Eq. 8, yields, a quadratic equationDocket No. 3262-002US1β2(2ω1ez + ½γ(3Ṗ1− Ṗ2)z²) = ΔT0, Eq. 11.We use the approximation that P2obeys the same scaling to I as the gaussian like original pulses obey, namely,Ṗ2= (P2 / P1)(T1 / T2)Ṗ1,Ṗ2= (P2 / P1)(T1 / T2)Ṗ1, Eq. 12Yielding,β2(2ω1ez + ½γṖ1(3 − (Ṗ2 / P1)(T1 / T2))z²) = ΔT0, Aβ2(2ω1ez + ½γṖ1(3 − (Ṗ2 / P1)(T1 / T2))z²) = ΔT0, Eq. 13\z\ ' i 'A' / /

[0077] The above can be solved by the quadratic equation to obtain the real roots of z, where only one would be real and physically relevant. We call this solution Zv, where the two edges are in union.

[0078] Knowing this Zv, we go back and substitute it into Eq. 9 to verify Eq. 8. If the solution does not satisfy Eq. 8. There are several parameters than can be changed to specification. The dispersion coefficient of the waveguide, (by geometrical variation), y again by reducing the effective area through geometry.5.3 Scenario 3: Pulses Do Not Overlap In Frequency Space But They Do After Spectral Generation In Main Pulse

[0079] In this scenario, the technique described herein of shock front intersection and further propagation of the dominant shock front can be applied to reduce the number of satellite pulses and to coherently combine them into one pulse in the ND segment as described in scenario 2. In terms of spectral modulation, since the sub-pulses do not overlap in frequency space, this would not reduce spectral modulations, but this coherent combining would reduce temporal modulations. The reduction of temporal modulations ultimately leads to reduction of spectral modulations, as subsequent SCG in further segments would eventually overlap the spectra of these pulses - by reducing the number of sub-pulses then would reduce this eventual possibility of occurring.

[0080] Finally, there are scenarios where the conditions of Eq. 8 are not met, or the pulses can never temporally overlap due to the frequency separation, temporal separation of the pulses and amount of bandwidth generation (being inadequate to compensate). Under suchDocket No. 3262-002US1circumstances, the modulations are inherently reduced to begin with, but the temporal structure would remain.6. GLOBAL NOISE REDUCTION

[0081] In this section we will discuss how to design SADW structures that reduce amplitude, bandwidth, and modulation fluctuations in the original pulse train, thereby lowering these noise sources in the output pulse train. The noise sources that can exist in the original pulse train or during SCG are:1. Peak power fluctuations due to bandwidth fluctuations (and duration fluctuations) 2. Peak power fluctuations due to chirped pulses3. Peak power fluctuations due to energy fluctuations4. Spectral modulations and associated pulse to pulse spectral fluctuations6.1 Peak Power Fluctuations Due To Bandwidth Fluctuations (And Duration Fluctuations)

[0082] Bandwidth fluctuations at the entrance of an AD segment would couple to the same final soliton and therefore bandwidth at the output of the AD segment. However, in general, bandwidth fluctuations at the entrance of an AD segment would result in a different temporal compression length to the start of pulse shaping into the soliton that the pulse would couple into. This change in length must be mitigated such that the same convergent bandwidth is reached even with these input fluctuations.

[0083] In this section, the pulse train is described as being composed of pulses fluctuating between a range of bandwidths that are between a maximum bandwidth, Δf1and a minimum bandwidth Δf2. The length to the start of soliton pulse shaping is defined as the length at which the original chirped phase of the stretched pulse entering the AD segment reaches a minimum, and couples into the soliton. The length is defined as LSHand is composed of the sum of the original dispersion dominated compression length of the chirped pulse entering the segment, L dispersive and then the shaping length to the soliton, when the pulse is close to compressiontransform limited at the end of L dispersive, labelled as Lsha. FIG. 6a shows the compressiontemporal dynamics of the chirped pulse entering the AD segment coupling into a soliton, where these lengths are indicated. FIG. 6b shows the general spectral dynamics that occur in this process.Docket No. 3262-002US1

[0084] In particular, FIG. 6a illustrates the temporal dynamics (pulse duration versus propagation coordinate in an AD waveguide) of a chirped pulse entering an AD segment (from a previous ND segment). The regions are shown are the first dispersive compression region, followed by the nonlinear dominated shaping to soliton region and the nonlinear compression of the soliton. Examples are shown for a higher order soliton (IV = 2) and the fundamental soliton. FIG. 6b is a corresponding illustration of the bandwidth versus propagation coordinate in an AD waveguide. Spectral generation occurs in all regions but is accelerated in the nonlinear shaping and soliton compression.

[0085] In general, the length LSHdepends on the chirp exiting the ND segment, and>^SHAfz ■ 'qcreasons for the length increase for higher bandwidth at the SADW entrance first starts with the assumption that Ldispersivecompression dominates (> Lshaping) in the overall LSH. Ldispersivecompression decreases for lower bandwidth because the chirp for pulses with a smaller bandwidth coming from the ND segment would be less than pulses with a larger bandwidth. In contrast, the nonlinear shaping length for lower bandwidth would be longer, due to the smaller generated bandwidth before reaching this regime, as compared to the more chirped larger bandwidth input pulse to the SADW structure. However, in general, Ldispersive compression dominates and LSHis greater for the input pulses (to the beginning of the SADW structure) with higher bandwidth, at the AD segment that follows the first ND segment.

[0086] In more detail, for smaller bandwidth pulses entering the SADW, the initial ND dispersion length would be longer and the ratio — would be higher, so less dispersive phase nlis added versus phase due to SPM. Therefore, pulses with a smaller bandwidth would reach the saturation bandwidth (which is albeit less) in the ND segment with less dispersive chirping effects (less durational increase in relation to transform limit), where SPM induced chirping would be more predominant. In contrast the ND segment would be longer than what is needed to reach saturation bandwidth for a larger bandwidth input, if designing the SADW structure for the lower bandwidth input pulse.

[0087] Now the AD segment will be designed such that the soliton fission length presented in the last section is,Ls= N° ~ LSH,Δf₂, Eq. 14Docket No. 3262-002US1tWhere,2LD= t0² / βADand N is the soliton number. The AD segment length will be chosen as:’0PADLLAD= LSH,Δf₁+ LSH,Δf₂ / 2 + Ls, Eq. 15

[0088] Due to the symmetry of the periodic soliton, with this AD length definition pulses at Δf1would be mapped to the same spectral bandwidth and pulse profile as pulses at A2, as shown in FIG. 7, described below. Also, sincerequire longerthey would be mapped to the left side of the soliton inversion point, and pulses closer to A2with hs / / , A / 2would be mapped to the right side of the soliton inversion point.

[0089] This can be extended across the fluctuation bandwidth range such that pulses with original bandwidths that result in placement on the left side of the soliton minimum, defined as the soliton inversion point, would have the same amplitude profile as pulses on the right side of the soliton minimum (soliton inversion point). This effectively reduces the bandwidth by a factor of 2 (in the general case).

[0090] Therefore, the distribution of bandwidth variation is now between2

[0091] At the next ND segment, the pulses coming from the side of the distribution that crossed the soliton minimum in the previous AD segment would have a reversed temporal chirp than pulses (on the Δf2side, e.g. with bandwidths within the range(Δf2, Δf1− ΔA / 2) that did not. Spectral narrowing, and duration decrease would happen at the start of the ND segment for the pulses with the reversed chirp, e.g. pulses within Δf1−However once the temporal phase is minimized generation would start again from the compressed pulse, in an approximately symmetric fashion, accounting for the initial decrease. Pulses from the Δf2side would be chirped and start to broaden both temporally and spectrally. If the ND segment is much larger than this initial region of compression forpulses, at the end of the ND segment, the net effect would be a slightly more temporally broadened and chirped Δf2pulses than Δf1side pulses, as illustrated in FIG. 7. However, the spectral bandwidth ratio (Δf1 / Δf2) would be the same or greatly reduced than the input distribution to the entrance of the alternating waveguide. Since the relative effective length difference is small (prop. to LD / Lnl) this chirp difference is negligible and is omitted from the analysis.Docket No. 3262-002US1

[0092] FIG. 7a illustrates the convergence over differing pulses (with different bandwidths) after the initial segments of the SADW structure, i.e., ND-AD-ND iterations. At the first ND segment, the pulse duration versus propagation is plotted for the two pulses with differing bandwidths, along with the spectral bandwidth development. Underneath the AD segment is described how the AD segment length maps the higher bandwidth to left side of the soliton inversion point and the lower bandwidth pulse to the right side in a symmetric fashion. The pulses here only differ by an inversion of temporal phase. In the last ND segment, the pulse duration and spectral bandwidth development versus propagation coordinate in the segment is shown. Here both the durations and spectral profile converge. Essentially, the two pulses are closer to being identical and are approximately the same in this illustration. Past this point the pulses develop the same way in the SADW structure, effectively removing the original deviation in bandwidth. FIG. 7b shows the temporal duration versus propagation distance within the AD segment of FIG. 7a for the two pulses, pulse 1 (higher bandwidth at entrance of SADW) and pulse 2 (lower bandwidth). The arrows indicate the relative length of the dispersive compression for both pulses as well as the relative lengths for the nonlinear shaping. FIG. 7c shows the spectral development corresponding to the AD segment over the two pulses, corresponding to FIG. 7b. The point of soliton inversion is indicated by a dashed line and the mirror symmetry mapping of the two pulses are indicated by stars on the figures.

[0093] When entering a second AD segment, the bandwidth distribution would be approximately halved, and the same procedure used in selecting the length is chosen, i.e., Eq.15. In this fashion, at every AD segment, the bandwidth variation is approximately halved. Obtaining,Δf1,i / Δf2,i− 1 = (1 / 2m_AD)(Δf1,o / Δf2,oEq. 16− 1)Where the indices o indicates the output bandwidth of the pulse train and i is the input bandwidth of the pulse train to the SADW waveguide. mADis the number of AD segments in the sign-alternating dispersion SCG waveguide. Also, the temporal durations would converge after both the AD and ND segments (shown in FIG. 7a in the ND plot of duration vs propagation coordinate). Therefore, both the bandwidth and durations would converge, obtaining a convergent pulse profile from an initial distribution of profiles entering the SADW waveguide. It is important to note that after each ND waveguide (after the first) thereDocket No. 3262-002US1is a reversal in regard to which pulse type has lower bandwidth (i.e., initially within(Δf1, Δf1− ΔA / 2) at input to the SADW waveguide, or within (Δf2, Δf2+ ΔA / 2).

[0094] It is also important to note that to illustrate the method of the present disclosure, we present the general case where the lower bandwidth pulses require a larger AD segment length, but this is not always the case. However, the analysis is general enough to cover differing cases. It is only necessary to define Δf1as those pulses that would require longer AD segment lengths to reach their minimum duration.

[0095] In conclusion, by using this AD design process, the SADW structure can be used to minimize spectral bandwidth differences in a pulse train, and temporal differences, not only in the SCG increase of the bandwidth but lower than the original bandwidth fluctuations across the pulse train.6.1.1 Special Case Where Pulses Have Δf1= 20.5Δf2and Ln« LDin the ND Segments

[0096] The above analysis can be applied to a special case where the nonlinear length of pulses (Ln) is much lower than the dispersion length (LD) in the ND segment.

[0097] In such a case it can be shown that the spectral bandwidth per pulse exiting the ND segment will be approximately the same, but would be temporally broadened differently. Pulses with bandwidth close to Δf2would be linearly broadened (i.e., with negligible nonlinear spectral generation) over a distance Lnless than pulses close to Δf1. If Lnl« LD,ADthen this is negligible, and the bandwidth variation is eliminated in this pulse train by the spectral generation in the first ND segment. If not, the AD segment length, is now given asPNDLs> LSH,Δf₂· Lnl,NDEq. 17where the / 3s represent the group velocity dispersion coefficients of the segments. Under these conditions,(βND / βAD)LAD= LSH,Δf₂+ 0.5Lnl,ND+ Eq. 18PAD'and so forth.Docket No. 3262-002US16.1.2 Subcase Where A Fundamental Soliton Is The Converged Profile In The AD Segment

[0098] If N = 1, then a fundamental soliton would be the convergent profile at the end of the AD segment. Here, the soliton does not fluctuate as a function of propagation distance, maintaining a constant frequency and temporal profile, save for a global propagation coordinate dependent phase factor. Therefore, in this case, the special considerations that needed to be considered in Eq. 14-18 do not need to be applied. It is sufficient that the AD segment length LADbe greater than LSH^2the shaping length into the soliton for the higher bandwidth pulse - or in the general case the length associated with pulses with the longest shaping length from the input pulse train. I.e., LAD> LSH,Δf₂.6.2 Peak Power Fluctuations Due To Chirped Pulses

[0099] In this section we generalize the case where the pulse duration can fluctuate due to spectral phase changes and not only bandwidth changes. The spectral phase directly gives rise to a different spectral bandwidth, stretched pulse duration and frequency chirp at the output of the first ND segment. The analysis of the previous case equally applies here though. After this first segment, the pulses entering the next ND segment after the following AD segment, would not exhibit the spectral phase differences that existed at the entrance of the SADW structure and would follow the governing dynamics of the higher order soliton as described in the previous section. Therefore, this source of noise is reduced in the SADW structure in the supercontinuum generation.

[0100] There is a subcase of these chirped pulses that is important to consider as it yields analytic insight, when applying the methodology of the previous section. The subcase is when LD< LN. As well, the duration fluctuations of the chirped input pulses have a variation of Tmax / Tmin=When this condition of the characteristic dispersion and nonlinear length is met, the chirped pulses (assuming spectral bandwidth differences are negligible) have an effective dispersive phase path length difference of LDbetween pulses near Tmaxand Tmin. The design of the segment lengths in the next AD segment then becomes:Eq. 19.Docket No. 3262-002US1Again, as in the previous section, LSH Twas used above as for the usual general case LSH,τis the longer length, but in general this term in Eq.19 should be the longer length (with a sign change in front of the second term if the longer length is not LSH T).6.3 Peak Power Fluctuations Due To Energy Fluctuations

[0101] To understand the energy scaling and ultimately how the SADW structures used in SCG can minimize pulse energy fluctuations, we start by going back to Eq. 3 describing soliton coupling in the AD segment. According to Eq. 3, the pulse duration of the soliton, at a specific order, is directly inversely proportional to energy and therefore decreases for increasing pulse energy, τo∝ 1 / E. It can also be derived that the maximal spectral bandwidth is linearly proportional to pulse energy, Δω ∝ E. Therefore, there is a higher generation of bandwidth before coupling into the soliton that occurs with higher pulse energies to match the soliton duration. Also, there is more generation of bandwidth in the higher energy case overall, even after coupling to the soliton due to the linear dependence of bandwidth with energy.

[0102] Another effect that should be considered is that as the pulse is shaping to a soliton, and even as the soliton is compressing, the pulse is shedding energy through the emission of dispersive waves. These dispersive waves are composed of generated frequencies that surpass the anomalous dispersion bandwidth of the AD waveguide segment. As energy is lost in the emission of these dispersive waves, the main pulse energy decreases, changing its soliton duration. Therefore, in considering the case where an input signal exhibits energy fluctuations, after the AD segment due to dispersive wave shedding, the distribution of energies within the main pulses would be closer together. Overall, the total energy remains the same, but in the main pulse the energy converges.

[0103] Simulations with the GNLSE equation can provide better AD segment lengths that result in an energy and bandwidth profile that is matched in the main pulse across the energy range of the input pulse train. This is a new result that has not been used for the purpose of reducing energy variations across a pulse train. The noise reduction described in the AD segment carries through the whole SADW structure, iteratively reducing the energy fluctuations with the addition of more AD segments. Therefore, energy shedding into dispersive waves shows that SADW structures used for SCG also can be used to reduce energy fluctuations in input pulses through nonlinear noise reduction, as described here.Docket No. 3262-002US16.4 Spectral Modulations And Associated Pulse To Pulse Spectral Fluctuations (Global Soliton Reduction)

[0104] In this section we address how spectral modulations and modulation fluctuations across pulses can be reduced by the SCG process in the SADW structures. There are various effects that reduce modulations in SADW SCG, each of which will be discussed in this section.

[0105] To counter the emergence of modulated spectra that fluctuate pulse to pulse, we start by looking at the ND segments, which in this case become important in the SADW SCG structure. As outlined in section 5, in the ND segments such a pulse train will undergo optical wave breaking, where a global linear chirp across each pulse would form, as well as an optical shock front that moves through the wings of the pulse (forming frequency side lobs) and transforms the phase into a linear phase. These optical fronts from each pulse at a different central frequency would eventually meet (see FIG. 5). Therefore, the side pulses merge with the main pulse to form one continuous chirped pulse, where instantaneous frequencies do not repeat at delayed times. This naturally flattens the spectrum, and modulations are not present anymore. This is found by the semi -analytic expressions in section 5, or by using the GNLSE simulation.

[0106] The ND segments are chosen such that the length is greater than the characteristic length needed for these optical fronts to meet across all pairs of pulses produced by modulation instability gain / soliton fission. The meeting of optical fronts to converge to one pulse has not been discussed in the context of lowering modulation instability noise in SCG.

[0107] A specific case should also be noted for OWB in the ND segments. This case is defined when a chirped pulse enters the ND segment with reversed chirp compared to the chirp induced by SPM (e.g., the pulses that are past the soliton inversion point in section 6.1).Also, the pulse exhibits side pulses due to the higher order soliton dynamics. Here the pulse would temporally compress, due to the reversed chirp, and its side pulses, centerd at different frequencies along with it would come together. At the same time the spectrum of the pulse and sub-pulses would narrow due to the opposing phase addition of SPM. The side pulses would come closer to the main pulse, and spectrally narrow. Therefore, modulations in theDocket No. 3262-002US1spectral domain would be reduced, both in frequency and in that the sub-pulses and main pulse would have less spectral overlap (due to the narrowing for each pulse). After the main pulse approaches the transform limit, spectral broadening due to ND SCG would occur and then the procedure of section 5 can be implemented for coherent overlap of the subsequent shock fronts.

[0108] Therefore, a system can be designed where, in the AD segment, the length is chosen such that output pulses are past the soliton inversion point. These pulses then go into the next ND segment, where the ND segment reduces the modulation instability from the previous AD segment, as the sub-pulse structure would move closer together in (spectrally narrowed) pulses that then undergo subsequent spectral broadening akin to ND SCG with OWB. The ND length is chosen as outlined in the scenarios of section 5.6.4.1 Global Soliton Reduction

[0109] SADW waveguides can also be configured such that the shaping to the minimum duration of the higher-order soliton is avoided in the AD segments or even the full nonlinear shaping to a higher order soliton is avoided in the first place. It is even allowed the SADW structure can iteratively increases the bandwidth by generation primarily only in the first chirp compression region in the AD waveguide and the ND segments. This would translate to a linearly increasing bandwidth across the segments of the waveguide. This is in stark contrast to other SCG schemes where a negligible spectrum would be generated under this scenario.

[0110] In this scenario, per AD segment, the amount of bandwidth generation is not maximized to the upper limit of what it can be. Therefore, invariably these SADW structures would contain more segments and would be possibly longer. However, the main advantage here is that the per segment length of nonlinear generation and shaping has decreased, so the modulation instability gain has decreased as well. Furthermore, because of prior bandwidth generation at the entrance of subsequent AD segments, the transform limited duration is smaller of the chirped pulse entering the segment. This means that the characteristic soliton duration is decreasing as well according to Eq. 3, i.e., at subsequent segments, the order of soliton that can be coupled into the AD segment is decreasing, until only a fundamental soliton remains as the choice. As the order of the soliton decreases, so does the number of side-lobs and modulation instabilities that can form due to stochastic processes. Therefore, at subsequent segments, the modulation instability gain drops to a negligible value. This meansDocket No. 3262-002US1that nonlinear generation in the AD segment can be extended more and more in the nonlinear shaping regime (LSH) and the soliton self-compression regime (into Lsregion) without succumbing to modulation instability.

[0111] At even further segments, the transform limited duration dips even below the condition for a fundamental soliton. In this domain, there is very little modulation instability gain as the pulse transform limited durations are too low for a soliton to exist and all generation comes from the chirped pulse compression region of the AD segment with no nonlinear shaping or soliton self-compression length.

[0112] Therefore, modulation instability is reduced by two mechanisms in the alternating waveguide segments. The first mechanism sets the ND segment such that the shock front from optical wave breaking meets the front of the neighbouring pulses emitted from the previous AD segment. The second mechanism uses only the generation length prior to soliton coupling into the AD segments and to decrease the soliton order as a function of segment number.7. PATTERNED ALTERNATING DISPERSION ROBUSTNESSGlobal Convergence

[0113] In general, for SADW structures with many iterations there is a global convergence of the input train of pulses to a set output pulse, i.e., spectrum and duration. For this convergence to occur, the SADW structure is to be optimized for the worst case pulse profile across the pulse train, for example for longest duration and lowest bandwidth.

[0114] Under these conditions, at the second segment, pulses with lower durations (a greater initial bandwidth) would be negatively impacted, generating less spectral bandwidth, while the ideal pulses in the pulse train would undergo more spectral generation (a greater relative increase in bandwidth). Therefore, there is an inversion where pulses starting with a higher bandwidth have relatively lower bandwidth generation than pulses starting at the lower bandwidth, for which the SADW structure is originally optimized. Because of this inversion, at the end of a given segment in the SADW structure, the two pulses would have similar durations and spectral bandwidth. From this point on the two pulses undergo all the same dynamics, to finally converge to a set spectral bandwidth and duration at the output of the SADW structure, therefore minimizing the original fluctuations in duration and spectral bandwidth. Therefore, simply by optimizing the SADW structure for the higher duration,Docket No. 3262-002US1lower bandwidth arrangement, other pulse arrangements (duration, bandwidth) would be forced to converge because for these other pulse arrangements the SADW structure exhibits less efficient generation, converging the extrema of the range of input profiles together in one output profile from the SADW structure.

[0115] Therefore, there is a range of input pulse durations whereby the SADW structure maps the final output to the same values. This robustness is unique to the process and found primarily by first optimizing the structure for the highest duration, narrowest spectrum allowed in the original pulse train. Once the optimized structure is obtained through GNLSE simulations, the duration and spectral bandwidth are varied to obtain the range of convergence and the robustness of the structure.

[0116] The above analysis (sections 6.1,6.2, 6.3) are specific cases of this general convergence. For example, in the case that Ln« Ldand the AD segment is arranged according to Eq. 18, after just the second segment of the SADW structure, there would be convergence and robustness in design over pulses, with bandwidths increased by up to a factor of A / 2, i.e, the SADW structure will give the same output spectrum with pulse bandwidths up to a factor of 1.4 of the original bandwidth.

[0117] Another approach is to use the convergence to a fundamental soliton as described in section 6.1.2. Here the dynamics in the AD segments are controlled such that soliton compression is not carried out (or negligibly carried out). Therefore, the spectral broadening is less per segment, but over many segments, the profile of the pulse would couple into a fundamental soliton. The structure is extended such that convergence to a fundamental soliton is obtained for both extremes of the input pulse train’s duration / bandwidth range and the SADW structure is designed such that soliton convergence occurs at the same AD segment for both extrema of the range. Past that point, when the fundamental soliton is converged the pulse dynamics would be the same for the entire range of input pulses being considered. In subsequent ND and AD segments, bandwidth generation would solely be in the dispersive compression length of the AD segment, and the ND SCG in the ND segment.8. COMPARISON TO UNIFORM STRUCTURESDocket No. 3262-002US1

[0118] In contrast to what was presented above for SADW structures, uniform structures do not demonstrate a robustness or ability to account for input pulse fluctuations such as bandwidth and duration fluctuations.

[0119] For ND SCG there is some robustness in bandwidth fluctuations, which have been exploited in the SADW ND segments. However, the presence of AD segments after the ND segments can shape pulses into the same profile, while they can be differing coming out of the ND segment. Therefore, not having this extra ability, i.e., only using uniform ND SCG, limits the robustness of the structure to spectral pulse to pulse bandwidth and duration fluctuations. As well, ND SCG spectral bandwidth generation is low, thus the function of SCG is restricted in this type of nonlinear device.

[0120] In contrast, for uniform SCG in AD segments, pulses can indeed converge to the same soliton profile. However, modulation instability is not countered by OWB in the ND segments. Also, pulses can be mapped to a reversed temporal phase coming out of the segment. Therefore, input pulse fluctuations translate fluctuating temporal phase in the pulse train. Further, instead of higher order soliton coupling, fundamental soliton coupling as described above can work to converge the pulses. But the spectral bandwidth generation, i.e., the other function of AD SCG would be low.

[0121] Therefore, the iterative approach to the SADW structure extends the robustness of ND SCG, along with fixing the temporal phase fluctuations and modulation instability in AD segments. It also extends the generation to larger bandwidths, more than both segments could provide on their own. Finally, it extends the amount of dispersive wave generation, which can be used as a mechanism to balance pulse energy in the main SCG pulse, and therefore also offering a means of lower energy fluctuations.9. ILLUSTRATIVE ADVANTAGES OF SADW STRUCTURES WITH REDUCED NOISE

[0122] The SADW structures described herein provided a number of important advantages, some of which are described below.

[0123] For instance, when designed according to the present disclosure, SADW SCG can reduce SCG noise, such as durational and bandwidth noise fluctuations. In addition, Modulation Instability noise, relative to conventional SCG in uniform AD and ND structuresDocket No. 3262-002US1can also be reduced. SADW structures can also reduce input pulse train durational, bandwidth fluctuations. SADW structures for SCG can use nonlinear optical generation to reduce noise and stabilize a pulse train.

[0124] An SADW structure as described herein that includes two or more segments can reduce MI in an AD SCG device. Additionally, an ND SCG segment based on embodiments described herein can reduce AD SCG MI in another AD SCG device.

[0125] SADW structures as described herein can be optimized to demonstrate new physics, such as global soliton reduction to reduce durational and bandwidth noise. These SADW structures also can be optimized to reduce energy fluctuations in the main pulses of a pulse train by coupling higher energy pulses to dispersive wave generation, which is a constant CW-like low temporal intensity waves away from main pulse.

[0126] The new SADW structural design can be iterated over many ND and AD segments10. SUMMARY OF ILLUSTRATIVE EMBODIMENTS10.1 Using SADW Structures To Reduce Bandwidth And Duration Fluctuations In Original Pulse Train And In A SCG Device, By Nonlinear SCG Using Sign-Alternating Dispersion SCG

[0127] In one embodiment, an SADW structure has AD segments that are designed such that the two extrema in bandwidth fluctuations of the pulse train are mapped to both sides of the soliton inversion point as indicated in FIG. 8. Therefore, they exhibit the same spectral bandwidth and temporal duration coming out of the AD segment. The corresponding ND segment is long enough such that they converge to the same spectrum and duration at the end of the ND segment (found from GNLSE simulations).• The implementation of this procedure would reduce the relative deviations by a factor of two at each AD segment.• Length of the ND segment is a given, or is found to minimize spectral bandwidth difference between two extrema. Through GNLSE simulation.• Length of the AD segment could be found from GNLSE simulations to accommodate this condition or, where Lsis found from Eq.4• LAD= + Ls, Eq. 15Docket No. 3262-002US1• Lsis found using main text Eq. 4, Ls= t02and AD waveguide attributes such asnonlinear coefficient and second order group velocity dispersion coefficient, LSH's are found from GNLSE simulation or analytic 2nd order dispersion expression for pulse duration [] based on input pulse duration and bandwidth coming from ND segment.• Special case where pulses have= 2°'5A / 2and Ln« LDin the ND segments.• Working principle: Ln< 0.5 * LD, then pulse input can fluctuate in bandwidth such that max^bandwidth') « mm(bandwidth).• True for higher pulse energies as Lnoc peak power.• Spectral and durational convergence happens in first ND segment. Segment designed for this i t by geometrical variation (or setting input pulse energy and duration) to optimize Ln= —, LD= -22-• If not AD segment length can be found through GNLSE or using an expression as:• LAD= (LSHAf2+ 0.5Ln iVD* gs) + Ls, Eq. 18• Similar to ND segment convergence there is also AD segment convergence: Subcase where a fundamental soliton is the converged profile in the AD segment• AD segment is engineered through geometric variation with its y, f> (or setting input pulse energy, duration) to have a convergent solution N = 1, i.e., a fundamental soliton. Then both bandwidth extrema converge to a non periodically varying solution where both spectrum and duration do not change.• ^AD ^SH, f2In another embodiment, the S ADW structure has AD segments that are designed such that the two extrema in durational fluctuations of the pulse train are mapped to both sides of the soliton inversion point as indicated in FIG. 8. Therefore, they exhibit the same spectral bandwidth and temporal duration coming out of the AD segment. The corresponding ND segment is long enough such that they converge to the same spectrum and duration at the end of the ND segment (found from GNLSE simulations)• Found through GNLSE simulations to seek the AD length necessary for durational and bandwidth convergence after the AD segment for the two extrema in duration of the pulse train • particular subcase when LD< LNand Tmax= -Tminin terms of the relation between maximum and minimum pulse duration.• Design of ND segment, given minimum peak power of pulse train going in. Or accommodate pulse train to have this condition in ND segment

[0001] AD segment length design is then:PND^AD — ( LsH, Tmin+ 0.5 LDEq.119.PADDocket No. 3262-002US1• AD segment length can be chosen by varying geometricallyIn another embodiment, a general SADW structure has AD segments that are designed such that the two extrema in durational fluctuations and bandwidth fluctuations of the pulse train are mapped to both sides of the soliton inversion point as indicated in FIG. 8. Therefore, they exhibit the same spectral bandwidth and temporal duration coming out of the AD segment. The corresponding ND segment is long enough such that they converge to the same spectrum and duration at the end of the ND segment (found from GNLSE simulations).• The following procedure can be implemented in the AD segments such that the relative deviation in pulse duration and bandwidth can be reduced by a factor of 2 at each AD segment.• Pulses can be described by a coordinate pair (duration, bandwidth) per each pulse.= max^bandwidth), f2= min(bandwidth), AT) = max duration). T2= min(duration). The following algorithm is applied:1. ND segment length is chosen such that spectral saturation occurs with all pulse coordinate pair combinations. As well, Convenient arrangement of the segment characteristic length values can be such that max(Ln) < 0.5min (fD). where min and max refer to the global min, max of the multiplicity of max / min extrema combinations: (A / , ATX). (A / , AT2). (A^. AT, (A / 2, AT2).2. GNLSE simulations are performed to find the AD segment length, corresponding to the minimum duration at output (soliton point of inversion) for each combination listed in step 1.3. The minimum length is found and labelled as LSH minand the maximum length is found and labelled as LSH max.4. The AD segment length then becomes:3- ^AD ~ + ^SH,min) + Ls,5. The length of the ND segment is chosen such that the spectral saturation is reached for all pulse pair combinations.6. Procedure is repeated for every AD segment.10.2. Using SADW Structures To Reduce Energy Fluctuations In Original Pulse Train And In A SCG Device By Nonlinear SCG Using Sign-Alternating Dispersion SCGIn one embodiment, the SADW structure may be used to generate dispersive waves such that the amount of optical energy in the main pulse (pulse with highest peak power) converges to a constant, despite the original pulse train having energy deviations.• Using AD dispersive wave generation to remove energy in main pulse through dispersive wave generation:Docket No. 3262-002US11. Dispersive wave phase matching condition oc yP where P is the peak power of the main pulse. The dispersive wave is reached with the higher pulse energies, labelled as E1but there is no phase matching for E2. Therefore, there is more energy shedding for higher energies.2. Matching yADand the dispersive profile with higher order dispersion (HOD) to accommodate the above dispersive wave matching condition3. This is iterated through all AD segments in the waveguides such as that the main pulse energy distribution converges to E2.4. OWB also generates dispersive waves in ND with the presence of HOD. Use ND segments as well to energy shed from the main pulse.5. The segment number would increase to reduce the energy deviation of the original pulse train.10.3. Using SADW SCG To Reduce Modulation Instability In SCG DevicesOWB is used in ND to reduce MI in AD segments, while pulse shape fluctuations is reduced in AD segments in the SADW structure.Choosing the ND segment length to exploit OWB to smooth spectrum results in:Scenario 1:• Pulses have high degree of frequency space overlap, central frequencies are marginally close • Length of ND segment is chosen such that the main pulse (highest peak power) completes OWB smoothing and no intra pulse overlap occurs. The ND length follows:Qr0- • LWB< LND< —— — -, Eq. 7Scenario 2: Two pulses at increased frequency separation, with remaining spectral overlap (modulations present) or without• The optical shock front of the main pulse passes through the secondary pulse 1. Induces an increasing instantaneous frequency distribution2. No repeating frequencies at different delays3. Reduced modulation in the spectral domain4. One uniform pulse is obtained from the group of pulses with a monotonic derivative of the spectral phase function5. This is modelled by the GNLSE to exploit this effect, Eq. 8,9 describes the salient bounding condition6. Eq. 10 is an exemplary analytic methodologySADW structure for general technique for reducing modulation instability in AD SCGDocket No. 3262-002US1• A SADW structure can be used in general as a technique to reduce modulation instability as described in the scenarios above. The AD SCG is then fed to an ND segment, where OWB according to the specific scenario is carried out and ND segment length is set. The ND segment following the AD segment can reduce modulations that exist with pulses entering who are past the soliton inversion point at output of the AD segment:• A specific case should also be noted for OWB in the ND segments. The case is defined when there is a chirped pulse entering the ND segment with reversed chirp compared to the chirp induced by SPM (e.g., the pulses that are past the soliton inversion point in section 6.1). As well, the pulse exhibits side pulses due to the higher order soliton dynamics. Here the pulse would temporally compress, due to the reversed chirp, and its side pulses, centered at different frequencies along with it would come together. At the same time the spectrum of the pulse and sub-pulses would narrow due to the opposing phase addition of SPM. The side pulses would come closer to the main pulse, and spectrally narrow. Therefore, modulations in the spectral domain would be reduced, both in frequency and that the subpulses and main pulse would have less spectral overlap (due to the narrowing for each pulse). After the main pulse reaches near transform limit, spectral broadening due to ND SCG would occur and then the procedure of section 5 can be implemented for coherent overlap of the shock fronts subsequently.1. This case can be applied for the SADW structures that reduce duration / bandwidth described above• a system can be designed where, in the AD segment, the length is chosen such that pulses outputted are past the soliton inversion point. These pulses then go into the next ND segment, where the ND segment reduces the modulation instability from the previous AD segment, as the sub-pulse structure would move closer together in (spectrally narrowed) pulses that then undergoes subsequent spectral broadening akin to ND SCG with OWB. The ND length is chosen as outlined in the scenarios of section 5 (main text).• Can be over just one AD segment followed by ND segment, using SADW here to specifically reduce MI in AD SCG in a device arrangement.-A new phenomenon over the SADW structure AD segments can reduce modulation instability. The phenomenon is called global soliton reduction.• AD segment lengths are chosen such that soliton shaping or soliton nonlinear compression is limited.o Therefore, invariably these SADW structures would contain more segments and would be possibly longer. However, the main advantage here is that the per segment length of nonlinear generation and shaping has decreased, so the modulation instability gain has decreased as well.Docket No. 3262-002US1• Furthermore, because of prior bandwidth generation at the entrance of subsequent AD segments, the transform limited duration is smaller of the chirped pulse entering the segment. This means that the characteristic soliton duration is decreasing as well according to Eq. 3. 1.e., at subsequent segments, the order of soliton that can be coupled into the AD segment is decreasing, until only a fundamental soliton remains as the choice. As the order of soliton decreases, so does the amount of side-lobs and modulation instabilities that can form due to stochastic processes. Therefore, at later segments, the modulation instability gain drops to a negligible amount. This means that, nonlinear generation in the AD segment can be extended more and more in the nonlinear shaping regime (LSH) and the soliton self-compression regime (into Lsregion) without succumbing to modulation instability.E = N2β2 / γtoo E = N2β2 / γto, Eq. 3, where E is constant.• At even later segments, the transform limited duration dips even below the condition for a fundamental soliton. In this domain, there is very little modulation instability gain as the pulse transform limited durations are too low for a soliton to exist and all generation comes from the chirped pulse compression region of the AD segment with no nonlinear shaping or soliton self-compression length.o The spectral bandwidth increases linearly in the ND and AD segments, but there is no modulation instability effect from soliton generation• Therefore, modulation instability is reduced by two mechanisms in the alternating waveguides. The first mechanism is by setting the ND segment such that the shock front from optical wavebreaking meets the front of the neighboring pulses emitted from the previous AD segment. The next mechanism is by using only the generation length prior to soliton coupling into the AD segments and to decrease the soliton order as a function of segment number. 10.4 Patterned Alternating Dispersion Robustness- Global Convergence• simply by optimizing the SADW structure for the higher duration lower bandwidth arrangement, would force convergence for other pulse arrangements (duration, bandwidth) due to the SADW structure exhibiting less efficient generation for these other pulse arrangements, converging the extrema of the range of input profiles together in one output profde from the SADW structure.1. SADW structure is designed for higher duration, lower bandwidth2. Structure segment iterations are increased such that convergence is found between the extrema of input duration, bandwidth3. This robustness is unique to the process and found primarily by optimizing the structure firstly for the highest duration, narrowest spectrum allowed in the original pulseDocket No. 3262-002US1train. Once the optimized structure is obtained through GNLSE simulations [] the duration and spectral bandwidth are varied to obtain the range of convergence and the robustness of the structure.• The above analysis (sections 6.1,6.2, 6.3) are specific cases of this general convergence. For example, in the case that Ln« Ldand the AD segment is set according to Eq. 18, after just the second segment of the SADW structure, there would be convergence and robustness in design over pulses with bandwidths increased by up to a factor of V2, i.e, the SADW structure will give the same output spectrum with pulse bandwidths up to a factor of 1.4 the original bandwidth.1. Here the SADW is designed to tolerate a robustness of 1.4X the minimum bandwidth it was designed for.• Another approach is to use the convergence to a fundamental soliton as described in section 6.1.2.1. AD segment is designed such that both extrema in input pulse train map to fundamental soliton in AD segment. Then solutions have converged and are identical (save for global phase constant)2. -Or- SADW structure avoids nonlinear shaping and generation for entire pulse train.• Over many AD segments in the SADW structure, fundamental soliton can be coupled into.o There the pulse train would be identical after the specific AD segment • Further spectral dynamics in the SADW structure would be identical and will exhibit a linearly increasing bandwidth (neglecting waveguide losses) -General limits for Robustness, a summary of example possible structures:• SADW structures can be found, from above described structures and methods, but are not limited to being robust in bandwidth:• Can work with convergent output spectrum for input bandwidths up to √2Δf , the design bandwidth• Can work with transform limited pulse durations(Ao+ l)2to < G < t0- A 7722- ’EC1'4'Set from the spectrum entering the first AD segment. Nois found as the maximum soliton order supported given the transform limited duration and is found from Eq. 3 main text (substituting increasing values of N until the maximum pulse duration t0is found such that it is still less than.• SADW structures can be designed to tolerate a duration increase of √2 tiDocket No. 3262-002US1where is the incident intensity duration. This can be durations calculated from chirped input pulses or transform limited ones.

[0128] It should be noted that in the examples presented above the SADW structures generally begin with an ND segment. However, the SADW structure alternatively may begin with an AD segment to arrive at an equally valid result. In that case the general definition of LSH,max / min is used, since the bandwidth relation with LSHlength would change. Note that the generalization of the AD length analysis is presented above). Furthermore, Some embodiments of the SADW structure may have just one AD segment. Likewise, some embodiments may have just one ND segment for the convergence after one segment under the special case of section 6.1.1 and for the ND OWB analysis.

[0129] The foregoing description, for the purpose of explanation, has been described with reference to specific embodiments. However, the illustrative discussions above are not intended to be exhaustive or to limit the invention to the precise forms disclosed. Many modifications and variations are possible in view of the above teachings. The embodiments were chosen and described in order to best explain the principles of the embodiments and its practical applications, to thereby enable others skilled in the art to best utilize the embodiments and various modifications as may be suited to the particular use contemplated. Accordingly, the present embodiments are to be considered as illustrative and not restrictive, and the invention is not to be limited to the details given herein, but may be modified within the scope and equivalent of the appended claims.

Claims

1. Docket No. 3262-002US1Claims:

1. A waveguide structure, comprisinga plurality of alternating segments of normal dispersion (ND) waveguide segments and anomalous dispersion (AD) waveguide segments along a length of the waveguide structure, the alternating segments being configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the ND waveguide segments and the AD waveguide segments; andwherein the AD waveguide segments are each further configured to shape the optical pulses into higher-order soliton pulses that are compressed to a soliton inversion point to thereby cause a temporal width and a spectral width of the higher-order soliton pulses to oscillate between maximum and minimum values as the higher-order optical pulses traverse each of the AD waveguide segments.

2. The waveguide structure of claim 1, wherein the AD waveguide segments are further configured so that two extrema in bandwidth fluctuations of an incoming optical pulse train are mapped to both sides of the soliton inversion point.

3. The waveguide structure of claim 1, wherein the ND waveguide segments have a length that minimizes spectral bandwidth differences between two extrema and minimizes temporal durations of the two extrema such that a sign of a temporal chirp between the two extrema is the same.

4. The waveguide structure of claim 3, wherein the ND waveguide segments are further configured to reduce modulation instability arising in a previous AD waveguide segment traversed by the optical pulses.

5. The waveguide structure of claim 4, wherein the modulation instability is reduced by a wave breaking mechanism that spectrally broadens and flattens the optical pulses.

6. The waveguide structure of claim 1, wherein the plurality of alternating segments includes a number of alternating segments to optimize a lower extremum of spectral bandwidth and duration fluctuations in the optical pulses such that an optical pulse at another extremum is initially disfavored for bandwidth generation.Docket No. 3262-002US17. The waveguide structure of claim 1, wherein soliton coupling is avoided in initial ones of the AD waveguide segments while bandwidth generation continues.

8. A waveguide structure, comprisingat least one anomalous dispersion (AD) waveguide segment being configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the AD waveguide segment;wherein the AD waveguide segment is further configured to shape the optical pulses into higher-order soliton pulses that are compressed to a soliton inversion point to thereby cause a temporal width and a spectral width of the higher-order soliton pulses to oscillate between maximum and minimum values as the higher-order optical pulses traverse the AD waveguide segment, the AD waveguide segment being further configured so that two extrema in bandwidth fluctuations of an incoming optical pulse train are mapped to both sides of the soliton inversion point.

9. A waveguide structure, comprisingat least one normal dispersion (ND) waveguide segment being configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the ND waveguide segment, the ND waveguide segment having a length that minimizes spectral bandwidth differences between two extrema; andwherein the ND waveguide segment is further configured to reduce modulation instability of incoming optical pulses by a wave breaking mechanism that spectrally broadens and flattens the optical pulses.

10. A waveguide structure, comprisinga normal dispersion (ND) waveguide segment and an anomalous dispersion (AD) waveguide segment extending along a length of the waveguide structure, the AD and ND waveguide segments being configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the ND waveguide segment and the AD waveguide segment; andwherein the AD waveguide segment is further configured to shape the optical pulses into higher-order soliton pulses that are compressed to a soliton inversion point to thereby cause a temporal width and a spectral width of the higher-order soliton pulses to oscillateDocket No. 3262-002US1between maximum and minimum values as the higher-order optical pulses traverse the AD waveguide segment.

11. A waveguide structure, comprisinga normal dispersion (ND) waveguide segment and an anomalous dispersion (AD) waveguide segment extending along a length of the waveguide structure, the AD and ND segments being configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the ND waveguide segment and the AD waveguide segment; andwherein the AD waveguide segment is further configured to shape the optical pulses into higher-order soliton pulses that are compressed to a soliton inversion point, wherein the AD waveguide segment is further configured so that two extrema in bandwidth fluctuations of an incoming optical pulse train are mapped to both sides of the soliton inversion point. At the end of the AD segment the bandwidth has converged between the two extrema.

12. A method of supercontinum generation (SCG) and noise reduction, comprising: receiving an optical pulse train at an input to a waveguide structure that includes alternating segments of normal dispersion (ND) waveguide segments and anomalous dispersion (AD) waveguide segments that extend along a propagation direction in the waveguide structure, the alternating segments being configured such that supercontinuum generation (SCG) of optical pulses traversing the waveguide structure is affected in the ND waveguide segments and the AD waveguide segments;spectrally broadening optical pulses in the optical pulse train by SCG as the optical pulses traverse the ND waveguide segments and the AD waveguide segments;shaping the optical pulses into higher-order soliton pulses that are compressed to a soliton inversion point to thereby cause a temporal width and a spectral width of the higher-order soliton pulses to oscillate between maximum and minimum values as the higher-order optical pulses traverse each of the AD waveguide segments; andemitting the optical pulses from the output of the waveguide structure.

13. The method of claim 1, further comprising mapping extrema in bandwidth fluctuations of the optical pulse train to both sides of the soliton inversion point in the AD waveguide segments.Docket No. 3262-002US114. The method of claim 1, further comprising minimizing spectral bandwidth differences between two extrema, and minimizing the temporal durations of the two extrema such that a sign of temporal chirp between the two extrema is the same.

15. The method of claim 3, further comprising reducing modulation instability arising in a previous AD waveguide segment traversed by the optical pulses16. An optical arrangement, comprising:a laser source for generating an ultrafast pulse train; anda waveguide structure that includes alternating segments of normal dispersion (ND) waveguide segments and anomalous dispersion (AD) waveguide segments that extend along a propagation direction in the waveguide structure, the waveguide structure having an input configured to receive the ultrafast pulse train from the laser source, the waveguide structure being further configured to use nonlinear dynamics to reduce bandwidth and durational fluctuations in the ultrafast pulse train.

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