All-optical tunable delay line based on soliton self-frequency shift and filtering broadened spectrum due to self-phase modulation

doi:10.1364/OE.14.007895

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All-optical tunable delay line based on soliton self-frequency shift and filtering broadened spectrum due to self-phase modulation

Shoichiro Oda and Akihiro Maruta »View Author Affiliations

Optics Express, Vol. 14, Issue 17, pp. 7895-7902 (2006)
http://dx.doi.org/10.1364/OE.14.007895

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▸ Article Outline
– Abstract
1. Introduction
2. Principle of all-optical TDL
3. Experiment
4. Conclusion
– References and links

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Abstract

We propose a novel all-optical tunable delay line based on soliton self-frequency shift and filtering broadened spectrum due to self-phase modulation to compensate for the frequency shift. We experimentally demonstrate the proposed all-optical tunable delay line and achieve a continuous temporal shift up to 19.2 ps for 0.5 ps pulse, corresponding to a delay-to-pulse-width ratio of 38.4.

1. Introduction

To meet strong demands for the Internet, spread all over the world with an explosive pace, an all-optical network is expected to be constructed, in which data is carried from a source to a destination in optical format without any optical-to-electrical (O/E) and electrical-to-optical (E/O) conversions. In such a network, the realization of an all-optical router is required to transfer data optically. For the purpose, the research on many subsystems has been actively conducted, e.g. header recognition [1

K. Kitayama, N. Wada, and H. Sotobayashi, “Architectural considerations for photonic IP router based upon optical code correlation,” IEEE/OSA J. Lightwave Technol. 18, 1834–1844 (2000). [CrossRef]

], wavelength conversion [2

K. Inoue and H. Toba, “Wavelength conversion experiment using fiber four-wave mixing,” IEEE Photon. Tech-nol. Lett. 4, 69–71 (1992). [CrossRef]

], data regeneration [3

P. V. Mamyshev, “All-optical data regeneration based on self-phase modulation effect,” in Proceedings of the European Conference on Optical Communication (ECOC) (IEEE, 1998) pp. 475–476.

], and so on. Optical buffers have been recognized as an indispensable subsystem for an all-optical router, whereas the development has been hindered due to difficulty in realizing an optical random access memory (RAM). Recently, much attention has been given to the use of optical tunable delay lines (TDLs) instead of optical RAM for implementing optical buffers. To date, well-known techniques are to utilize gain-dependent variations of the group refractive index based on stimulated Brillouin scattering (SBS) [4

K. Y. Song, M. G. Herraez, and L. Thevenaz, “Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering,” Opt. Express 13, 82–88 (2005). [CrossRef] [PubMed]

], stimulated Raman scattering (SRS) [5

J. E. Sharping, Y. Okawachi, and A. L. Gaeta, “Wide bandwidth slow light using a Raman fiber amplifier,” Opt. Express 13, 6092–6098 (2005). [CrossRef] [PubMed]

], and parametric process (OPA) [6

D. Dahan and G. Eisenstein, “Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering,” Opt. Express 13, 6234–6249 (2005). [CrossRef] [PubMed]

]. While they are interesting works, they may have inherent limitations on waveform distortion, narrow bandwidth, or relatively small temporal shift. On the other hand, a hybrid system consisting of a wavelength conversion and a fiber dispersion has been also proposed [7

J. E. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. E. Willner, and A. L. Gaeta, “All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion,” Opt. Express 13, 7872–7877 (2005). [CrossRef] [PubMed]

, 8

J. van Howe and C. Xu, “Ultrafast optical delay line using soliton propagation between a time-prism pair,” Opt. Express 13, 1138–1143 (2005). [CrossRef] [PubMed]

], which may be able to overcome the limitations of the above mentioned techniques. Such a hybrid system, however, may be complicated compared with the techniques based on SBS, SRS, or OPA.

In this paper, we propose a novel all-optical TDL consisting of intensity-dependent delay induced by soliton self-frequency shift (SSFS) and compensation for the frequency shift by filtering broadened spectrum due to self-phase modulation (SPM). Remarkable features of the proposed TDL are i) without waveform distortion, ii) simple configuration, and iii) compatibility with less than 1ps pulse. We experimentally demonstrate a continuous temporal shift up to 19.2 ps for an optical pulse with a pulse width of 0.5 ps, which corresponds to a delay-to-pulse-width ratio of 38.4. The delay-to-pulse-width ratio of 38.4 is more than 10 times larger than the ratio obtained in [4

, 5

J. E. Sharping, Y. Okawachi, and A. L. Gaeta, “Wide bandwidth slow light using a Raman fiber amplifier,” Opt. Express 13, 6092–6098 (2005). [CrossRef] [PubMed]

, 6

] and comparable with the ones reported in [7

, 8

J. van Howe and C. Xu, “Ultrafast optical delay line using soliton propagation between a time-prism pair,” Opt. Express 13, 1138–1143 (2005). [CrossRef] [PubMed]

2. Principle of all-optical TDL

2.1. Soliton self-frequency shift

We consider that an optical pulse is launched into an anomalous dispersion fiber (ADF). Assuming that the pulse width of the launched pulse is less than 1 ps, the behavior of the optical pulse propagating in the ADF is described by

i \frac{\partial E}{\partial z} - \frac{β_{2}}{2} \frac{\partial^{2} E}{\partial t^{2}} + γ {∣ E ∣}^{2} E = - i g E + i \frac{β_{3}}{6} \frac{\partial^{3} E}{\partial t^{3}} + γ T_{R} E \frac{\partial {∣ E ∣}^{2}}{\partial t} .

(1)

We here summarize the units of the quantities appeared in Eq. (1).

z [m]	:	propagation distance,
t [s]	:	time moving at the group velocity,
E(z,t) (\|E\|²[W])	:	complex envelope of electric field,
$β_{2} [\frac{s^{2}}{m}] = - \frac{λ^{2} D}{2 π c}$	:	group velocity dispersion,
$β_{3} [\frac{s^{3}}{m}] = - \frac{(S λ + 2 D) λ^{3}}{{(2 π c)}^{2}}$	:	third order dispersion,
$γ [\frac{1}{(m ∙ W)}] = \frac{2 π n_{2}}{λ A_{eff}}$	:	nonlinearity,
T_R [s]	:	Raman time constant,
$g [\frac{1}{m}] = - \frac{\log_{e} 10}{20} α$	:	fiber loss,

where D[s/m²], S[s/m³], λ[m], n ₂[m²/W], A _eff[m²], and α[dB/km] are the dispersion parameter, the dispersion slope, the wavelength, the Kerr coefficient, the effective core area, and the power loss, respectively. c[m/s] represents the velocity of light in vacuum. Normalizing the variables in Eq. (1) in terms of the characteristic values of power P ₀[mW], time t_s [ps], and distance z_d [km] and introducing the non-dimensional quantities as,

q = \frac{E [\sqrt{mW}]}{\sqrt{P_{0} [mW]}}, T = 1.763 \frac{t [ps]}{t_{s} [ps]}, Z = \frac{z [km]}{z_{d} [km]},

(2)

we obtain the perturbed nonlinear Schrödinger equation (P-NLSE),

i \frac{\partial q}{\partial Z} + \frac{1}{2} \frac{\partial^{2} q}{\partial T^{2}} + {∣ q ∣}^{2} q = τ_{R} q \frac{\partial {∣ q ∣}^{2}}{\partial T},

(3)

where

{\begin{matrix} P_{0} = 262.5 \frac{{(λ [μ m])}^{3} D [\frac{\frac{ps}{nm}}{km}]}{\frac{n_{2}}{A_{eff}} [\times \frac{10^{- 9}}{W}] {(t_{s} [ps])}^{2}}, \\ z_{d} = 0.6062 \frac{{(t_{s} [ps])}^{2}}{{(λ [μ m])}^{2} D [\frac{\frac{ps}{nm}}{km}]}, \\ τ_{R} = 1.763 \frac{T_{R} [ps]}{t_{s} [ps]} . \end{matrix}

Note that the third order dispersion and fiber loss are ignored to simplify the discussion hereafter. It is well-known that we can analyze Eq. (3) with several perturbation methods [9

A. Hasegawa and Y. Kodama, “Solitons in optical communications,” in Chapter 5 (Oxford University Press, Oxford, 1995).

]. The perturbed soliton solution of Eq. (3) is assumed as

q (Z, T) = η (Z) sech [η (Z) {T - T_{0} (Z)}] \exp {- i κ (Z) T + i θ (Z)},

(4)

where η(Z), κ(Z), T ₀(Z), and θ(Z) are the amplitude, the frequency, the time position, and the phase of the soliton. By applying the perturbation method to Eq. (3) with the anzats of Eq. (4), the Z-dependence of η, κ, and T ₀ can be obtained by following dynamical equations,

\frac{d η}{d Z} = 0, \frac{d κ}{d Z} = - \frac{8}{15} τ_{R} η^{4}, \frac{d T_{0}}{d Z} = - κ .

(5)

Integrating Eqs. (5), we obtain

η (Z) = η_{0}, κ (Z) = - \frac{8}{15} τ_{R} η_{0}^{4} Z, T_{0} (Z) = \frac{4}{15} τ_{R} η_{0}^{4} Z^{2},

(6)

where we set η(Z = 0) = η ₀ and κ(Z = 0) = T ₀(Z = 0) = 0. As can be seen in Eqs. (6), the amplitude of the soliton is constant and the frequency is down-shifted proportional to Z and

η_{0}^{4}

, that is, a square of the peak power of the soliton. Such a phenomenon is called soliton self-frequency shift (SSFS) [10

F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986). [CrossRef] [PubMed]

, 11

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986). [CrossRef] [PubMed]

]. Moreover, it is notable that the time position also changes depending on a square of the peak power of the soliton. We apply this peculiar feature to an all-optical TDL.

Let us consider the evolution of the following initial pulse,

q (Z = 0, T) = A sech (T) .

(7)

Fig. 1. Schematic diagram of the proposed all-optical TDL.

Here A represents the initial amplitude. Setting the right-hand side of Eq. (3) to be zero, the soliton amplitude η can be analytically obtained by using the inverse scattering transformation [12

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

] as

η = 2 A - 1, (0.5 \leq A < 1.5) .

(8)

Since the soliton amplitude η is linearly proportional to the initial amplitude A, the temporal shift based on SSFS occurs proportional to A ⁴. Note that the condition of initial amplitude, 0.5 ≤ A < 1.5 should be satisfied, because for A < 0.5 no soliton is formed and for A ≥ 1.5 a higher order soliton formed, resulting in a fission of solitons [13

K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. 13, 392–394 (1988). [CrossRef] [PubMed]

2.2. All-optical TDL based on SSFS and filtering broadened spectrum due to SPM

Figure 1 shows the schematic diagram of the proposed all-optical TDL, which consists of two stages. In the first stage, an optical pulse with a central frequency of ω ₀ is launched into an ADF. Assuming that the pulse width of the launched pulse is less than 1ps, the central frequency of the pulse is down-shifted to ω_s due to SSFS and simultaneously we obtain a time delay of ∆t. The frequency shift ∆ω is proportional to a square of the input peak power as mentioned above, which means that an all-optical TDL can be achieved by varying the input peak power to the ADF. In the second stage, the delayed optical pulse with a time delay of ∆t and a central frequency of ω_s is launched into a normal dispersion flattened fiber (NDFF) for SPM-induced spectral broadening. At the output of the NDFF, the broadened spectrum is filtered out by an optical bandpass filter (OBPF) with a central frequency of ω ₀. Consequently, we obtain the delayed optical pulse with a time delay of ∆t and an original central frequency of ω ₀ by controlling the input peak power to the ADF. Note that the time delay may be slightly changed by filtering the spectrum due to frequency chirp.

Intensity-controllers for the proposed all-optical TDL can be implemented by either all-optical devices such as a nonlinear optical loop mirror [14

K. J. Blow, N. J. Doran, B. K. Nayar, and B. P. Nelson, “Two-wavelength operation of the nonlinear optical loop mirror,” Opt. Lett. 15, 248–250 (1990). [CrossRef] [PubMed]

] or electrical devices as a lithium niobate Mach-Zehnder intensity modulator, and so on. Since an optical pulse that is less than 1 ps is required to generate SSFS efficiently, the proposed all-optical TDL would be well-suited for the application of more than 160 Gbps systems in the coming future.

Fig. 2. Experimental setup for all-optical TDL.

Table 1. Parameters of HNLFs @1550 nm.

	HNLF1	HNLF2
Dispersion [ps/nm/km]	2.2	-0.6
Dispersion Slope [ps/nm²/km]	0.032	0.03
Loss [dB/km]	0.59	0.6
Nonlinearity [W^-1 ∙ km^-1]	12	20
Length [km]	2.4	0.5

3. Experiment

Figure 2 shows the experimental setup for the proposed all-optical TDL. A Gaussian-shaped pulse sequence with a repetition rate of 25 MHz and a central wavelength of 1545 nm from a fiber laser (FL) is launched into a highly nonlinear fiber (HNLF)1 for a temporal shift based on SSFS. The pulse width of the input pulse is 0.5 ps and the input peak power to the HNLF1 is controlled by a variable optical attenuator (VOA). After transmitting in the HNLF1, the optical pulses are sent to a HNLF2 for SPM-induced spectral broadening. The broadened spectrum is filtered out by an OBPF with a bandwidth of 5 nm at 1545 nm. The parameters of HNLFs are summarized in Table 1. The filtered pulse is detected by a photo detector (PD) with a bandwidth of 50 GHz and its waveform is observed by a sampling oscilloscope (OSC).

Figures 3(a) show the experimentally observed spectra at the input and output of the HNLF1 for the input peak power to the HNLF1 of 2.5, 4, and 5 W. The output and input spectra are illustrated by solid and dotted lines, respectively. As one can see, the central wavelength is shifted to longer side with increasing the input peak power. The experimentally measured central wavelengths as a function of the input peak power are plotted with open circles in Fig. 4. To compare with the experimental results, we numerically calculated Eq. (1) using the same parameters of the HNLF1 and the input pulse to the HNLF1 as the experimental ones. We set the Raman time constant T_R to 3 femtosecond (fs). The central wavelengths versus the input peak power obtained by the numerical simulation are plotted with open squares, which agree with the experimental result. Figures 3(b) show the experimentally observed spectra at the output of the HNLF2. When the input peak power is 2.5 W, the spectrum shown in the upper of Figs. 3(b) is slightly broadened and split to two spectral components due to SPM. The middle ofFigs. 3(b) shows the flatly broadened spectrum, i.e., supercontinuum spectrum compared with the spectrum at the output of the HNLF1 for the input peak power of 4 W. For the input peak power of 5 W, the further spectral broadening is observed in the lower of Figs. 3(b). Their broadened spectral width is enough to filter out at the original central wavelength for the purpose of compensating for the frequency shift. Figures 5(a) show the experimentally observed waveforms at the output of the OBPF for the input peak power to the HNLF1 of 2.5, 4, and 5 W. The insets shows autocorrelation traces obtained by an autocorrelator (AC). One can see that the pulses are delayed with increasing the input peak power from 2.5 W to 5 W. Moreover, the delayed pulses have almost same shapes, intensities, and autocorrelation traces. Although remnants of the pulse that do not contribute to SSFS is observed in the case of the input peak power of 5 W, it could be removed by adding an optical 2R regenerator (e.g. [3

P. V. Mamyshev, “All-optical data regeneration based on self-phase modulation effect,” in Proceedings of the European Conference on Optical Communication (ECOC) (IEEE, 1998) pp. 475–476.

]) after the proposed TDL. Note that the waveforms observed by the OSC are broadened due to the bandwidth limit of the PD used in the experiment. The spectra at the output of the OBPF and the input of the HNLF1 are illustrated by solid and dotted lines in Figs. 5(b), respectively. It is observed that the original central wavelength is successfully recovered. Note that in the case of the input peak power of 2.5 W, the spectrum is split into two components due to SPM. The spectral split can be mitigated by using a HNLF with slightly larger normal dispersion for the generation of a flatly broadened spectrum. Figure 6 summarizes the temporal shift from the time position of the pulse with an input peak power of 2.5 W as a function of the input peak power. Open circles and squares show the results by the experiment and the numerical simulation, respectively. In the simulation, the same parameters of the HNLFs and the input pulse to the HNLF1 are used. The increment of the temporal shift is proportional to a square of the input peak power, which qualitatively agrees with the theoretical prediction [11

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986). [CrossRef] [PubMed]

]. Furthermore, the experimental results are in good agreement with ones by numerical simulation. In the experiment, the maximum temporal shift of 19.2 ps is successfully achieved, which corresponds to a delay-to-pulse-width ratio of 38.4. The pulse width and the average power at the output of OBPF are plotted by open circles and squares in Fig. 7, respectively. While the pulse widths measured from 3 W to 5 W range between 0.69 ps and 0.76 ps, we obtain the broadened pulse with a pulse width of over 0.8 ps below the input peak power of 3 W. To recovery the original pulse width, it may be required that the bandwidth of an OBPF is carefully chosen and the frequency chirp accumulated in HNLFs1 and 2 is appropriately compensated for. The variation of the average power for each input peak power is below 1.5 dB. Taking into account of the variation of the input power of 3 dB (from 2.5 W to 5 W), the variation of 1.5 dB at the output is fairly small.

Fig. 3. Experimentally observed spectra at the output of HNLF1 and 2.

Fig. 4. Central wavelength versus input peak power.

Fig. 5. Experimentally observed waveforms and spectra at the output of OBPF.

Fig. 6. Experimentally measured temporal shift versus input peak power.

Fig. 7. Experimentally measured pulse width and average power at the output of OBPF.

4. Conclusion

In this paper, we have described the principle of the all-optical TDL based on SSFS and filtering broadened spectrum due to SPM. We have experimentally demonstrated a continuous temporal shift up to 19.2 ps for an optical pulse with a pulse width of 0.5 ps without any significant waveform distortion, which corresponds to a delay-to-pulse-width ratio of 38.4. Experimentally measured temporal shift is good agreement with numerically predicted shift. These results allow us to confirm that the proposed TDL has notable advantages of delaying an optical pulse that is less than 1 ps for future more than 160 Gbps system. We believe that the ratio of more than 100 is easily achieved by using a longer HNLF with a larger dispersion. Since the proposed TDL consists of only two kind of fibers, an intensity- controller, and an OBPF, the TDL would be easily implemented in an actual system.

References and links

1.	K. Kitayama, N. Wada, and H. Sotobayashi, “Architectural considerations for photonic IP router based upon optical code correlation,” IEEE/OSA J. Lightwave Technol. 18, 1834–1844 (2000). [CrossRef]
2.	K. Inoue and H. Toba, “Wavelength conversion experiment using fiber four-wave mixing,” IEEE Photon. Tech-nol. Lett. 4, 69–71 (1992). [CrossRef]
3.	P. V. Mamyshev, “All-optical data regeneration based on self-phase modulation effect,” in Proceedings of the European Conference on Optical Communication (ECOC) (IEEE, 1998) pp. 475–476.
4.	K. Y. Song, M. G. Herraez, and L. Thevenaz, “Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering,” Opt. Express 13, 82–88 (2005). [CrossRef] [PubMed]
5.	J. E. Sharping, Y. Okawachi, and A. L. Gaeta, “Wide bandwidth slow light using a Raman fiber amplifier,” Opt. Express 13, 6092–6098 (2005). [CrossRef] [PubMed]
6.	D. Dahan and G. Eisenstein, “Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering,” Opt. Express 13, 6234–6249 (2005). [CrossRef] [PubMed]
7.	J. E. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. E. Willner, and A. L. Gaeta, “All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion,” Opt. Express 13, 7872–7877 (2005). [CrossRef] [PubMed]
8.	J. van Howe and C. Xu, “Ultrafast optical delay line using soliton propagation between a time-prism pair,” Opt. Express 13, 1138–1143 (2005). [CrossRef] [PubMed]
9.	A. Hasegawa and Y. Kodama, “Solitons in optical communications,” in Chapter 5 (Oxford University Press, Oxford, 1995).
10.	F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986). [CrossRef] [PubMed]
11.	J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986). [CrossRef] [PubMed]
12.	V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
13.	K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. 13, 392–394 (1988). [CrossRef] [PubMed]
14.	K. J. Blow, N. J. Doran, B. K. Nayar, and B. P. Nelson, “Two-wavelength operation of the nonlinear optical loop mirror,” Opt. Lett. 15, 248–250 (1990). [CrossRef] [PubMed]

OCIS Codes
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.5650) Nonlinear optics : Raman effect

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 27, 2006
Revised Manuscript: August 9, 2006
Manuscript Accepted: August 9, 2006
Published: August 21, 2006

Citation
Shoichiro Oda and Akihiro Maruta, "All-optical tunable delay line based on soliton self-frequency shift and filtering broadened spectrum due to self-phase modulation," Opt. Express 14, 7895-7902 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7895

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References

K. Kitayama, N. Wada, and H. Sotobayashi, "Architectural considerations for photonic IP router based upon optical code correlation," J. Lightwave Technol. 18,1834-1844 (2000). [CrossRef]
K. Inoue and H. Toba, "Wavelength conversion experiment using fiber four-wave mixing," IEEE Photon. Technol. Lett. 4, 69-71 (1992). [CrossRef]
P. V. Mamyshev, "All-optical data regeneration based on self-phase modulation effect," in Proceedings of the European Conference on Optical Communication (ECOC) (IEEE, 1998) pp. 475-476.
K. Y. Song, M. G. Herraez, and L. Thevenaz, "Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering," Opt. Express 13, 82-88 (2005). [CrossRef] [PubMed]
J. E. Sharping, Y. Okawachi, and A. L. Gaeta, "Wide bandwidth slow light using a Raman fiber amplifier," Opt. Express 13, 6092-6098 (2005). [CrossRef] [PubMed]
D. Dahan and G. Eisenstein, "Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering," Opt. Express 13, 6234-6249 (2005). [CrossRef] [PubMed]
J. E. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. E. Willner, and A. L. Gaeta, "All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion," Opt. Express 13, 7872-7877 (2005). [CrossRef] [PubMed]
J. van Howe and C. Xu, "Ultrafast optical delay line using soliton propagation between a time-prism pair," Opt. Express 13, 1138-1143 (2005). [CrossRef] [PubMed]
A. Hasegawa and Y. Kodama, "Solitons in optical communications," (Oxford University Press, Oxford, 1995), Chap. 5.
F. M. Mitschke and L. F. Mollenauer, "Discovery of the soliton self-frequency shift," Opt. Lett. 11, 659-661 (1986). [CrossRef] [PubMed]
J. P. Gordon, "Theory of the soliton self-frequency shift," Opt. Lett. 11, 662-664 (1986). [CrossRef] [PubMed]
V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).
K. Tai, A. Hasegawa, and N. Bekki, "Fission of optical solitons induced by stimulated Raman effect," Opt. Lett. 13, 392-394 (1988). [CrossRef] [PubMed]
K. J. Blow, N. J. Doran, B. K. Nayar, and B. P. Nelson, "Two-wavelength operation of the nonlinear optical loop mirror," Opt. Lett. 15, 248-250 (1990). [CrossRef] [PubMed]

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