Soliton propagation with cross-phase modulation in silicon photonic crystal waveguides

Matthew Marko; Xiujian Li; Jiangjun Zheng

doi:10.1364/JOSAB.30.002100

1. INTRODUCTION

Optical solitons are undistorted standing waves that result as part of the interplay between the nonlinear Kerr effect and anomalous dispersion [1,2]. One method of studying solitons numerically is to use the split-step Fourier nonlinear Schrödinger equation (NLSE) [1–4]. The NLSE is one of the most effective methods for numerically analyzing soliton pulse propagation in a nonlinear dispersive waveguide.

Photonic crystal waveguides (PhCWGs) have extraordinary nonlinear effects [5]; measurements of the dispersion have shown an increase of 5 orders of magnitude in the group-velocity dispersion (GVD) compared to the material dispersion [6,7]. PhCWGs, with their vastly increased nonlinear effects, are an ideal medium for reducing the nonlinear length scales from the meters and kilometers typical of fibers to submillimeter PhCWGs. A soliton is formed by the interplay of anomalous GVD, as well as the third-order Kerr nonlinear optical effect. When the Kerr nonlinearities are the result of the pulse’s own intensity, it is known as self-phase modulation (SPM); whereas, when Kerr nonlinearities occur by means of a second coupled pulse, it is referred to as cross-phase modulation (XPM) [1,2]. In addition to having large GVDs, PhCWGs can be fabricated out of highly third-order nonlinear dielectrics, dramatically reducing nonlinear length scales.

The first such experimental measurement of soliton pulse compression in a PhCWG was conducted using a gallium indium phosphate PhCWG [7], which has the advantage of not having intensity-dependent two-photon absorption (TPA) and TPA-induced free-carrier absorption (FCA) [8]. The applications of this technology are numerous, as two-dimensional PhCWG may be fabricated into optical waveguides on a device the size of a computer chip, to be utilized for all-optical computing [9]. Optical computing has the advantage of significantly greater bandwidth compared to current solid-state devices, with reduced power and cooling, which can allow vastly improved computational performance [10].

Another highly nonlinear dielectric material is silicon [11,12], which may also potentially be used to fabricate chip-scale optical waveguides. Silicon is advantageous, as it is cheap, durable, readily available, and there already exists an infrastructure to mass-produce silicon devices. Pulse compression was previously observed experimentally in novel silicon nanowires [13,14], but previous analytical and numerical studies have demonstrated that with SPM alone, TPA and FCA will prevent a pulse from self-focusing itself sufficiently to generate a soliton-like wave [8]. Even if the soliton pulse were to be compressed, it would simply dissipate after propagating relatively short nonlinear lengths. One of the advantages of XPM is that a pulse can be focused without needing to be highly energetic and intense [1]; therefore there is interest in seeing whether a second pump pulse could be used to focus the primary signal pulse, to induce the soliton. If XPM can be used to overcome the limitations of TPA and FCA and enable chip-scale soliton propagation in silicon, it may very well develop into practical all-optical computing.

2. NONLINEAR SCHRÖDINGER EQUATION

In order to accurately model the optical soliton, two crucial nonlinear characteristics are necessary, the GVD and the nonlinear Kerr effect [1]. The GVD is determined from the derivatives of the refractive index over the angular frequency. The other component of soliton pulse propagation is the nonlinear Kerr effect, which results in a change of refractive index proportional to the square of the electric field, or proportional to the optical intensity. The SPM or XPM will cause the leading edge frequencies to be lowered, and the trailing edge frequencies to be raised. At the same time, the anomalous GVD will independently cause the lowered leading edge frequencies to slow down, and the raised trailing edge frequencies to speed up. When there is strong SPM and/or XPM interplaying with strong anomalous GVD, these two phenomena result in the pulse evolving temporally into a soliton pulse [1,2].

The GVD is determined by knowing the derivatives of the refractive index over the angular frequency. These terms are found by using the following equations [1,2,4]:

β (ω) = n (ω) \frac{ω}{c_{0}},

β_{M} (ω) = \frac{d^{M} β (ω)}{d ω^{M}},

β_{1} (ω) = \frac{1}{v_{G}} = \frac{n_{G}}{c_{0}} = \frac{1}{c_{0}} (n (ω) + ω \frac{d n (ω)}{d ω}),

β_{3} (ω) = \frac{d β_{2} (ω)}{d ω} = \frac{d^{2} β_{1} (ω)}{d ω^{2}} = \frac{d^{3} β (ω)}{d ω^{3}},

where

n (ω)

is the refractive index as a function of angular velocity,

ω (rad / s)

is the angular frequency,

c_{0} (m / s)

is the speed of light,

v_{G} (m / s)

is the group velocity, and

n_{G}

is the group index. These values can be either determined experimentally [11,15], approximated with the Sellmeier equation [1,2], or engineered to specific values with geometric dispersion, as is the case of photonic crystals [5,6].

The other component of generating soliton pulse propagation and compression is the nonlinear Kerr effect. The change in refractive index is determined by

Δ n = n_{2} \cdot (I_{1} + 2 \cdot I_{2}),

I = \frac{{| A |}^{2}}{2 (η_{0} / n)},

where

n_{2} (m^{2} / W)

is the nonlinear Kerr coefficient,

η_{0}

is the vacuum impedance (

\approx 120 π Ω

),

n

is the refractive index,

A (V / m)

is the electric field, and

I_{1}

and

I_{2} (W / m^{2})

are, respectively, the intensities of the pulse and another coupled pulse. It is clear that the effect on the effective refractive index from the Kerr nonlinearity is twice that from another coupled pulse (XPM) as compared to a pulse’s own intensity (SPM). For this reason, there is interest in determining if XPM could effectively maintain a soliton pulse when SPM alone is infeasible.

The nonlinear Kerr coefficient $n_{2}$ can be determined from the third-order nonlinear susceptibility, which is also responsible for four-wave mixing [2,16]:

n_{2} = 3 η_{0} \cdot \frac{χ^{(3)}}{ε_{0} \cdot n},

where

ε_{0}

is the electric permittivity in a vacuum (

8.854 \times 10^{- 12} F / m

), and

χ^{(3)} (m^{2} / V^{2})

is the third-order nonlinear susceptibility coefficient [2]. Throughout this study, a Kerr nonlinearity of

4 \times 10^{- 18} m^{2} / W

[6,11,17] is used. The nonlinear phase shift is thus [1]

Δ ϕ_{NL} = n_{2} \cdot k_{0} \cdot L \cdot (I_{1} + 2 \cdot I_{2}) = n_{2} \cdot \frac{2 π}{λ} \cdot L \cdot (I_{1} + 2 \cdot I_{2}),

where

k_{0} (m^{- 1})

is the optical wave-number,

λ

(meters) is the wavelength, and

L

(meters) is the optical propagation length. These two nonlinear optical effects can be combined to form the NLSE, which in its complete form is represented as

\frac{\partial A}{\partial z} + \frac{1}{v_{G}} \frac{\partial A}{\partial t} + \frac{j}{2} β_{2} \frac{\partial^{2} A}{\partial t^{2}} + \frac{j}{6} β_{3} \frac{\partial^{3} A}{\partial t^{3}} = j γ {| A |}^{2} A \cdot \exp (- α z),

γ = \frac{2 π n_{2}}{A_{e} \cdot λ} \cdot {(\frac{n_{g}}{n})}^{2},

where

β_{2} (s^{2} / m)

is the second-order GVD coefficient,

β_{3} (s^{3} / m)

is the third-order dispersion coefficient,

A_{e} (m^{2})

is the effective cross-sectional area of the waveguide,

γ {(m \cdot W)}^{- 1}

is the nonlinear parameter,

α (m^{- 1})

is the linear loss coefficient,

j = \sqrt{- 1}

, and

A (V / m)

is the pulse electric field envelope function.

The NLSE is also known as the split-step Fourier method because each incremental distance step involves two fundamental steps. The first step is to account for the GVD. A code will convert the pulse information into the spectral domain by using the fast Fourier transform method [1,4,18,19], apply the GVD, and then convert the pulse back from the spectral domain to the temporal domain to calculate the Kerr nonlinearity. The equation for the GVD is as follows [1,4]:

\bar{A} (z, ω) = \bar{A} (0, ω) \cdot \exp [(\frac{i}{2} \cdot β_{2} \cdot ω^{2} \cdot z) + (\frac{i}{6} \cdot β_{3} \cdot ω^{3} \cdot z) + \dots],

\bar{A} (z, ω) = \int_{- \infty}^{\infty} A (z, T) \cdot \exp [- 2 π i ω T] d T,

T = t - \frac{z}{v_{G}},

where

A (z, T)

and

\bar{A} (z, ω)

are the pulse envelope functions in the temporal and spectral domains,

T

is the normalized time,

z

is the propagation distance, and

ω

is the normalized angular frequency. The second step is to take the pulse, now in the temporal domain, and apply the effects of SPM or XPM. The equation for the nonlinear phase shift is [1,4]

A_{1} (z, T) = A_{1} (0, T) \cdot \exp [i \cdot n_{2} \cdot \frac{2 π}{λ} \cdot L \cdot \frac{({| A_{1} (0, T) |}^{2} + 2 \cdot {| A_{2} (0, T) |}^{2})}{2 (η_{0} / n)}] .

After completing these two steps, one can apply the effects of linear and nonlinear loss for the specified distance increment. By following this algorithm, soliton pulse propagation can be simulated and studied.

3. TWO-PHOTON ABSORPTION

One big challenge to soliton pulse propagation in silicon is the nonlinear effect of multiphoton absorption [8]. Dielectric materials have an electronic bandgap, where photons with energy greater than the electronic bandgap become absorbed by the material [20]. The electronic bandgap for silicon is 1.1 eV [7,8]. As it is desired to use light pulses at a wavelength of 1550 nm, this corresponds to a photon energy of 0.8 eV, and thus silicon is transparent at this wavelength. Multiphoton absorption, however, is the result of multiple photons interacting, and their combined energy exceeding the electronic bandgap. For example, at 1550 nm, two photons will result in a total energy of $0.8 \times 2 = 1.6 eV$ , which exceeds the 1.1 eV electronic bandgap of silicon. Therefore, silicon is subject to TPA at 1550 nm.

To calculate the effect of TPA on silicon, one must find the coefficient of TPA at the given wavelength by means of the following equations [21]:

β_{TPA} (ω) = K \cdot \sqrt{\frac{21}{n_{0}^{2} \cdot E_{g}^{3}}} \cdot F_{2} (\frac{2 π ℏ \cdot ν}{E_{g}}),

F_{2} (x) = \frac{{(2 x - 1)}^{\frac{3}{2}}}{{(2 x)}^{5}},

2 π ℏ \cdot ν = \frac{1240}{λ (nm)} (eV) .

where

ℏ

is Plank’s constant over

2 π

,

ν

is the frequency,

n_{0}

is the refractive index, and

E_{g}

is equal to the 1.1 eV electronic bandgap of silicon [7,8,12]. The constant

K

was calculated based on the experimentally measured value of

4.4 \times 10^{- 12} m / W

at 1550 nm [22]. With the value of

β_{TPA}

, the TPA can be simulated with [12,23]

I (z, t) = I (0, t) \cdot \exp [- β_{TPA} \cdot I (0, t) \cdot z] .

Once the TPA coefficient is determined, one can determine the effect of the free carriers generated by the TPA. For ultrashort pulses, where the pulse is substantially shorter than the free-carrier lifetime, the FCA tends to be substantially less than the effects of TPA [12]. The model calculates the FCA at each distance step separately, assuming that there were no free carriers there at the beginning. This is an approximation, as carriers will in fact drift over to the different distance steps, but this is difficult to accurately predict. For this reason, the assumption that carriers stay localized will be used. After the SPM or XPM and dispersion were applied to the pulse, the model predicts the rate of free-carrier generation from the TPA [12,22–24],

\frac{d N (z, t)}{d t} = G - \frac{N (z, t)}{τ_{FCA}},

G = \frac{β_{TPA} \cdot I^{2}}{4 π ℏ ν},

where

β_{TPA} (m / W)

is the coefficient of TPA,

I (W / m^{2})

is the optical intensity (either from the signal or coupled pulse),

ν

is the frequency,

ℏ

is Planks constant over

2 π

,

G

is the rate of free-carrier generation,

τ_{TPA}

(seconds) is the free-carrier lifetime, and

N (z, t)

is the free-carrier density. Once the rate of free-carrier generation over time has been determined at the specific distance step, the next step is to calculate the density of free carriers. While the simulation goes through the process of generating carriers over time, it also reduces carriers based on the specified free-carrier lifetime. After the carrier density is determined as a function of time (at the given distance increment), the appropriate FCA is applied to the pulse in the temporal domain. The equation is as follows [12,23]:

I (z) = I (0) \cdot \exp [- \frac{1}{2} \cdot (1.45 \cdot 10^{- 21}) \cdot N (t) \cdot z] .

By implementing these equations within the NLSE simulation, an accurate analysis of soliton pulse propagation in a silicon waveguide subjected to TPA and FCA can be achieved.

4. SIMULATIONS OF CROSS-PHASE MODULATION

The NLSE model that will be used throughout this study was first set to a single pulse with SPM. First, the model propagated a 10 pJ hyperbolic secant squared pulse down a 10 cm silicon nanowire; the results of this simulation are shown in Fig. 1(a). A compression ratio factor of 7 was clearly observed. The model then scaled the pulse energy and waveguide length to match the soliton number [7] and nonlinear length [1] as the nanowire study, using the following relations:

L_{NL} = \frac{τ^{2}}{| β_{2} |},

L^{'} = L \cdot | \frac{β_{2}}{β_{2}^{'}} |,

E^{'} = E \cdot | \frac{β_{2}^{'}}{β_{2}} | \cdot [\frac{A_{e}^{'}}{A_{e}}] \cdot [\frac{n_{G} \cdot n^{'}}{n_{G}^{'} \cdot n}],

where

L_{NL}

is the nonlinear length,

L

(meters) is the waveguide length,

E

(joules) is the input pulse energy,

β_{2} (s^{2} / m)

is the GVD coefficient,

A_{e} (m^{2})

is the waveguide effective cross-sectional area, and

n_{G}

and

n

are the group and refractive indexes, respectively. In practical applications, most chip-scale PhCWGs, being less than a millimeter in length, are typically 0.1 nonlinear lengths for a picosecond pulse, the equivalent nonlinearity of a meter of optical fiber. Based on these relations, it is clear that by using a PhCWG, the soliton length scales can become reduced dramatically. The caveat is that it is necessary to use higher pulse energies, which in turn have greater TPA and FCA, which attenuate the soliton and disturb the pulse, as observed in Fig. 1(b). While compression has been observed at the early onset of soliton propagation within a PhCWG, these compressed pulses simply cannot sustain themselves in the presence of intense TPA and FCA.

Fig. 1. SPM-induced soliton propagation, in (a) a silicon nanowire and (b) a silicon PhCWG. Both simulations are conducted for equivalent nonlinear lengths, as defined in Eq. (22).

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The NLSE model was modified to study both XPM, as well as the effects of cross-TPA, and the resulting FCA [12]. In this XPM simulation the process is completed twice, first for the primary pulse, and second for the secondary pulse. It uses a modified split step, since there are two coupled pulses, to take into account differences in the group velocity that result from the different wavelengths between the pump and the signal.

Part of the modified aspect is a loop for the nonlinear effects that require considering both pulses; the two pulses include XPM, cross-TPA, and FCA induced from cross-TPA. The split-step calculations are used in a normalized time domain [1]; however, with different wavelengths and thus different group velocities, this normalized time is not always applicable. In fact, one concern with XPM is the case of walk-off, where a faster traveling pulse fully clears away from the slower pulse within the waveguide [25]. A time shift factor is determined, based on the difference in group velocities, defined as $δ T$ ,

δ T = (n_{G}^{'} - n_{G}) \cdot (\frac{δ z}{δ t \cdot c_{0}}),

where

δ z

is the distance increment,

δ t

is the time step used in the simulation, and

c_{0}

is the speed of light in vacuum. The value of

δ T

is rounded down, as it is an integer number of time steps to shift, and added to calculate the effects of pulse 2 on pulse 1 (and subtracted for pulse 1 on pulse 2).

5. SOLITON PROPAGATION WITH CROSS-PHASE MODULATION

This model was run using the parameters experimentally observed in silicon PhCWG [6], with a negligibly low pulse energy (to avoid SPM) of 10 fJ pulse for the signal pulse near the highly dispersive photonic bandgap wavelength at 1544 nm, coupled temporally with an intense 105 pJ pump pulse at 1536 nm. In this particular example, the GVD at the signal wavelength was a highly dispersive anomalous $- 4 {ps}^{2} / mm$ , whereas the pump wavelength had a normal GVD coefficient of $100 {ps}^{2} / m$ . The signal was also set to a third-order dispersion coefficient of $0.1 {ps}^{3} / mm$ , a value typical of these engineered PhCWGs [7]. While higher-order dispersion is known to affect soliton propagation, especially at wavelengths very close to the highly dispersive photonic crystal bandgap [26], there was negligible difference in outcomes for the practical lengths modeled in this study. Finally, a linear loss of $2.4 dB / mm$ was added to the simulation.

Temporal compression by the phenomena of XPM was observed; the signal pulse underwent compression by a factor of 5 after propagating through 120 μm of the PhCWG. This result is displayed both as a function of normalized time and propagation distance in Fig. 2(a), as well as a comparison of the normalized temporal intensity function in Fig. 2(b). Finally, the pump pulse did not experience any compression, as it was subjected to much less GVD; this is demonstrated in Fig. 2(c). As a result of this, the simulation clearly demonstrates that the compression is the result of XPM, as no compression is observed when the same 10 fJ signal pulse propagates through the waveguide without the pump pulse.

Fig. 2. XPM-induced compression: (a) signal pulse compression as function of waveguide propagation distance and time, (b) waveguide output of weak signal pulse under XPM, and (c) waveguide output of weak signal pulse without XPM, thus proving that the compression is the result of XPM.

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When running the simulation, a numerical scale was developed to represent XPM-induced solitons. Much like a SPM soliton [7,8], the XPM soliton number $N$ is a ratio of the fundamental soliton energy:

N = \sqrt{\frac{E_{P}}{E_{f}}},

E_{f} = \frac{2 | β_{2} |}{| γ | \cdot τ},

where

β_{2} (s^{2} / m)

is the GVD coefficient,

γ {(W m)}^{- 1}

is the nonlinear Kerr coefficient described in Eq. (10),

τ

(seconds) is the temporal duration of the signal pulse,

E_{f}

(joules) is the fundamental soliton energy, and

E_{p}

(joules) is the pump pulse energy. The pulse compression ratio requires higher XPM soliton numbers than with SPM; this is expected because of the effect of pulse walk-off, which becomes more profound at longer waveguide lengths. This is demonstrated in Fig. 3, where the soliton pulse compression is clearly more profound with SPM (maximum compression of 5.5) in Fig. 3(a) than with XPM (maximum compression of 1.3) in Fig. 3(b), given the same waveguide parameters, a soliton number of 10, and a propagation of 0.3 nonlinear lengths. At the practical length scale of 0.1 nonlinear lengths (similar to the length studied with SPM), which equated to 0.4 mm (a typical PhCWG [6–8,11]), a pulse compression of nearly 3 was observed from an XPM-soliton number of 15; the results of these studies can be seen in Fig. 4(a). The time–bandwidth product (TBP) was tabulated and is represented by Fig. 4(b). Under the practical conditions mentioned above, the TBP remained relatively unchanged, which is evidence of XPM-induced soliton pulse propagation. At longer length scales, the pulse eventually underwent broadening, and the TBP increased; this can be attributed to TBP attenuating the pump pulse, as well as pulse walk-off. The nonlinear loss (in decibels) was also observed in Fig. 4(c). In all cases, the longer the nonlinear propagation length, the more loss was observed; this is to be expected. It is interesting to note that, in shorter propagation lengths, higher XPM soliton numbers had an ability to actually increase the peak energy as the signal pulse was compressed by XPM. In all cases studied, the optical pulses were all by-definition dissipative solitons; it was demonstrated that XPM can sustain the pulse over much longer propagation length scales than SPM alone.

Fig. 3. Comparative study between SPM and XPM, with a soliton number of 10 and a waveguide length of 0.3 nonlinear lengths, for both (a) SPM and (b) XPM.

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Fig. 4. Results of numerical XPM studies: (a) signal pulse compression factor, (b) output signal pulse TBP, and (c) nonlinear peak intensity loss (decibels).

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6. PRACTICAL APPLICATIONS OF CROSS-PHASE MODULATION

The previous study was based on the assumption that both the pump and the signal wavelengths were subjected to TPA and FCA; this may be practically realized by utilizing a spectrally broad pulse and narrow-bandpass filters [25]. While this may be simple to implement, it has the limitation of significant power attenuation from the filters. Another potential option to generate optical pulses of different wavelengths coupled temporally and spatially is to use an optical parametric oscillator (OPO), where a broad range of wavelengths may be generated by adjusting the length of the resonating cavity [27,28]. In an OPO, signal and idler pulses are generated with a combined wavelength energy equal to the pump wavelength energy:

\frac{1}{λ_{pump}} = \frac{1}{λ_{signal}} + \frac{1}{λ_{idler}} .

By utilizing this technology, one may realistically generate pulses coupled temporally and spectrally, where the pump pulse (not to be confused with the pump pulse for the OPO) has a wavelength not subjected to TPA within silicon. In this next study, the authors investigate pulse compression resulting from a low-powered signal pulse at 1550 nm; and a pump pulse at a wavelength of 2300 nm, which is not subjected to TPA due to silicon’s electronic bandgap. This can be achieved with a properly aligned OPO pumped at 926 nm, a realistic wavelength that can be achieved in a titanium sapphire laser [29].

The study was conducted with the same dispersion and energy parameters as before, but without the pump pulse subjected to TPA, and a compression ratio over 5 was observed at 90 μm of propagation. This is shorter than the 120 μm of propagation needed for the study in Fig. 2; and the results of this study can be observed in both Figs. 5(a) and 5(b). It was interesting to observe that the generation of free carriers, as shown in Fig. 5(c), actually increased as the signal pulse, which is subjected to TPA, was compressed and increased in temporal intensity because of the XPM from the pump pulse. In addition, it was observed that the compression remained consistent for even the longer propagation studies. This study was repeated for a host of different XPM soliton numbers and nonlinear length ratios, and the compression [Fig. 6(a)], TBP [Fig. 6(b)], and nonlinear losses [Fig. 6(c)] were characterized. These nonlinear losses were very close to the previous non-OPO study [Fig. 4(c)]. The data clearly show that by utilizing a substantially longer wavelength pulse, practical soliton pulse propagation in silicon can be both achieved and sustained for low-energy pulses at wavelengths subjected to intensity-dependent TPA.

Fig. 5. Results of numerical XPM studies, with a low-energy C-band pulse coupled with a high-energy 2300 nm pump pulse: (a) signal pulse compression as function of waveguide propagation distance and time, (b) numerical predicted free-carrier density as a function of waveguide propagation distance and time, and (c) waveguide output of weak signal pulse under XPM.

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Fig. 6. Results of numerical XPM studies, with a low-energy C-band pulse coupled with a high-energy 2300 nm pump pulse: (a) signal pulse compression factor, (b) output signal pulse TBP, and (c) nonlinear peak intensity loss (decibels).

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7. CONCLUSION

The simulations conducted have demonstrated the physical ability of soliton pulse compression and propagation via XPM in a silicon PhCWG with TPA and FCA present at both the pump and signal wavelengths. Use of a pump pulse and XPM to focus the pulse enables the soliton phenomena to occur on pulses with much lower intensities and thus much lower TPA. One challenge that was successfully overcome was to develop a NLSE split-step Fourier algorithm that would take into account the differences in group velocity and pulse walk-off that will occur between two pulses of different wavelengths. The simulation demonstrated that, through the use of XPM, temporal soliton pulse propagation can be realized over practical length scales with low-intensity ultrashort pulses in a silicon PhCWG with TPA. This phenomenon might be practically used to sustain a data pulse within a silicon waveguide, a capability with numerous potential technological applications.

ACKNOWLEDGMENTS

Sources of funding for this effort include the Navy Air Systems Command (NAVAIR)-4.0T Chief Technology Officer Organization as an Independent Laboratory In-House Research (ILIR) Basic Research Project (Nonlinear Analysis of Ultrafast Pulses with Modeling and Simulation and Experimentation); a National Science Foundation (NSF) award (1102257), a National Science Foundation of China (NSFC) award (61070040), and a Science Mathematics And Research for Transformation (SMART) fellowship. The authors thank Chee Wei Wong, Jiali Liao, James McMillan, Tingyi Gu, Kishore Padmaraju, Noam Ofir, and the laboratory of Karen Bergman for fruitful discussions.

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Soliton propagation with cross-phase modulation in silicon photonic crystal waveguides

Abstract

1. INTRODUCTION

2. NONLINEAR SCHRÖDINGER EQUATION

3. TWO-PHOTON ABSORPTION

4. SIMULATIONS OF CROSS-PHASE MODULATION

5. SOLITON PROPAGATION WITH CROSS-PHASE MODULATION

6. PRACTICAL APPLICATIONS OF CROSS-PHASE MODULATION

7. CONCLUSION

ACKNOWLEDGMENTS

REFERENCES

Cited By

Figures (6)

Equations (28)

Journal of the Optical Society of America B