Optimized wavelength conversion in silicon waveguides based on “off-Raman-resonance” operation: extending the phase mismatch formalism

doi:10.1364/OE.19.018810

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Optimized wavelength conversion in silicon waveguides based on “off-Raman-resonance” operation: extending the phase mismatch formalism

Yannick Lefevre, Nathalie Vermeulen, Christof Debaes, and Hugo Thienpont »View Author Affiliations

Optics Express, Vol. 19, Issue 20, pp. 18810-18826 (2011)
http://dx.doi.org/10.1364/OE.19.018810

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Abstract

We present a generic approach to determine the phase mismatch for any optical nonlinear process. When applying this approach, which is based on the evaluation of local phase changes, to Raman- and Kerr-based four-wave mixing in silicon waveguides, we obtain an expression for the phase mismatch which is more accurate as compared to the conventional definition; and which contains additional contributions due to the dispersion of the four-wave-mixing processes. Furthermore, starting from the general propagation equations for the involved pump, Stokes and anti-Stokes waves, we investigate the impact of this four-wave-mixing dispersion in silicon waveguides and examine how it is influenced by changing the frequency difference between the pump and Stokes input waves. We show by means of numerical simulations that, by detuning this frequency difference slightly away from Raman resonance, the four-wave-mixing conversion efficiency can be more than doubled, but can also lead to a decrease in efficiency of more than 10 dB. We also discuss how the pump-Stokes frequency difference that is optimal for wavelength conversion varies with the length of the silicon waveguides and with their dispersion characteristics. Finally, starting from the newly introduced phase mismatch formula we simplify the set of propagation equations such that they are less computationally intensive to solve while still giving accurate estimates of the optimal pump-Stokes frequency difference and the corresponding wavelength conversion efficiency.

1. Introduction

Over the past decade, the strong third-order nonlinear optical effects displayed by silicon waveguides have been successfully investigated for, amongst others, wavelength conversion in the near-infrared [1

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express 11, 2862–2872 (2003). [CrossRef] [PubMed]

–15

N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman-and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron. 17, 1078–1091 (2011). [CrossRef]

]. Generally this functionality can be accomplished by means of two four-wave-mixing (FWM) processes, namely Kerr four-wave-mixing (KFWM) and coherent anti-Stokes Raman scattering (CARS). The latter is sometimes referred to as Raman-resonant FWM. It originates from the interaction of light with the vibrations of the silicon atoms and converts long-wavelength Stokes-photons into short-wavelength anti-Stokes photons, provided that the Stokes and anti-Stokes frequencies each are shifted with 15.6 THz (the so-called Raman shift) with respect to the pump frequency. The non-resonant KFWM in its turn is due to interactions between the incident light and the electrons of the material, and will in a degenerate three-wave scheme occur even when the pump-Stokes and pump-anti-Stokes frequency differences are detuned from Raman resonance [15

]. We point out that, apart from the two commonly used conversion processes described above, there exist also other wavelength conversion schemes that are based on e.g. non-degenerate KFWM employing two pump waves [14

S. Gao, E. Tien, Q. Song, Y. Huang, and O. Boyraz, “Ultra-broadband one-to-two wavelength conversion using low-phase-mismatching four-wave mixing in silicon waveguides,” Opt. Express 18, 11898–11903 (2010). [CrossRef] [PubMed]

To obtain efficient wavelength conversion with either the degenerate KFWM or CARS, the involved waves require to be phase-matched, i.e. the linear dispersion and the dispersion due to the nonlinear effects need to be matched, such that the phases of the waves relatively to one another remain the same. Nowadays the phase mismatch between the involved waves is in silicon waveguides most commonly expressed by a formula that was originally derived from analytical solutions of propagation equations describing only the Kerr effect [16

G. Agrawal, Nonlinear Fiber Optics , 3rd ed. (Academic Press, 2001).

] or both the Kerr and the Raman effect in optical fibers [17

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990). [CrossRef]

]. These analytical solutions have been obtained by making the undepleted pump approximation, which is often valid when working with optical fibers.

When considering silicon waveguides, however, the case is completely different: any pump wave up to a wavelength of 2.2 μm will experience severe losses due to two-photon absorption and free-carrier absorption in silicon [18

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006). [CrossRef]

] and hence will experience depletion. As a consequence the analytical solutions as obtained in the undepleted-pump approximation are in general not valid for silicon waveguides and do not always provide an accurate calculation of the phase mismatch in these structures.

In this paper, we introduce a generic approach to deduce the phase mismatch out of any given set of propagation equations, which will bring forward an extra dispersion term due to the Kerr and Raman FWM processes, which we will call the “FWM dispersion”. Furthermore, we discuss the impact of this term on the wavelength conversion mechanism and show how it can be adjusted by changing the pump-Stokes frequency detuning. We investigate by means of numerical simulations the impact of this frequency difference on the total wavelength conversion efficiency, and determine for different dispersion characteristics of silicon waveguides which frequency difference value optimizes the efficiency. In this way, we show that for certain detunings away from Raman resonance the four-wave-mixing conversion efficiency can be doubled as compared to on-Raman-resonance operation, while other detunings can lead to a more than 10 dB decrease in efficiency. Finally, the use of the full phase mismatch formalism will lead to a set of propagation equations which are computationally less demanding while yielding accurate solutions.

2. Theoretical model for wavelength conversion in the strong pump approximation

In the wavelength conversion configuration under study we assume that the electromagnetic radiation consists out of three spectral components–a pump, a Stokes and an anti-Stokes component–at the frequencies ω_p , ω_s and ω_a (see Fig. 1), which because of energy conservation [15

] satisfy the relation:

ω_{a} - ω_{p} = ω_{p} - ω_{s} \equiv Δ Ω .

(1)

Here ΔΩ is the pump-Stokes and anti-Stokes-pump frequency difference.

Fig. 1 Spectrum of a typical wavelength conversion set-up. A strong pump and a weak Stokes spectral component, of which the angular frequencies ω_p and ω_s are detuned by ΔΩ = ω_p – ω_s , are used as input (full lines). Due to FWM interactions an anti-Stokes component will be generated at ω_a = ω_p + ΔΩ (broken line).

Assuming that the three components are propagating in the same direction–which we define as the z-direction–along the waveguide and that all electromagnetic components have the same polarization–along which we orientate the x-axis–we can write the total electric field in the frequency domain as [19

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

\tilde{E} (r, ω) = \tilde{F} (x, y, ω) \tilde{A} (z, ω) 1_{x} .

(2)

F̃ governs here the mode profiles in the plane transverse to the propagation direction z and Ã represents the complex mode amplitude. In the time domain the corresponding amplitude A can be written as:

A (z, t) = \frac{1}{2} \sum_{j = p, s, a} (A_{j} (z, t) e^{i ω_{j} t} + A_{j}^{*} (z, t) e^{- i ω_{j} t}) .

(3)

A_j represent the component amplitudes normalized in such a way that |A_j |² correspond to the intensities I_j of the different components. Note that the phase factors e ^{−iβ
_0,j
z}, with β _0,j the propagation constants describing the linear dispersion, are still included in the complex terms A_j .

Due to the strong losses caused mainly by free-carrier absorption (FCA), the undepleted pump approximation (|A_p (z)|² ≈ I_p (0)) often used in silica fibers [16

G. Agrawal, Nonlinear Fiber Optics , 3rd ed. (Academic Press, 2001).

, 17

], is in general not valid when describing nonlinearities in silicon waveguides (see Section 1). However, also the formalisms for fibers that do take pump depletion into account [20

Y. Chen, “Combined processes of stimulated Raman scattering and four-wave mixing in optical fibers,” J. Opt. Soc. Am. B 7, 43–52 (1990). [CrossRef]

–23

T. Sylvestre, H. Maillotte, E. Lantz, and P. Tchofo Dinda, “Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber,” Opt. Lett. 24, 1561–1563 (1999). [CrossRef]

] cannot be “simply” extended to the case of silicon waveguides as the presence of two-photon absorption (TPA) in silicon makes the considered problem more intricate. Indeed, this additional nonlinearity does not only introduce severe nonlinear losses (TPA and TPA-induced FCA) and additional nonlinear dispersion through the free-carrier index change (FCI), but also gives rise to an imaginary part of the KFWM term. The case of silicon waveguides thus requires a separate investigation.

To ensure that our approach for silicon waveguides does not become too complex, we will make use here of the strong pump approximation (|A_p |² ≫ |A_s |², |A_a |²) [6

Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006). [CrossRef] [PubMed]

, 19

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

] which is for silicon waveguides met in most cases. This approximation is far less stringent than the undepleted-pump approximation, as it allows the pump to actually deplete while only assuming it remains at all times stronger than the other waves present.

Taking this strong pump assumption into account, the propagation of continuous wave (CW) pump, Stokes, and anti-Stokes waves can then in the near-infrared be described by the following set of nonlinear Schrödinger equations [19

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

\frac{\partial A_{p}}{\partial z} = - i β_{0, p} A_{p} - \frac{1}{2} (α_{p} + α_{f, p}) A_{p} - i \frac{ω_{p}}{c} n_{f, p} A_{p} - i (γ_{K} + γ_{R, p}) {| A_{p} |}^{2} A_{p},

(4)

\begin{array}{l} \frac{\partial A_{s}}{\partial z} & = & - i β_{0, s} A_{s} - \frac{1}{2} (α_{s} + α_{f, s}) A_{s} - i \frac{ω_{s}}{c} n_{f, s} A_{s} \\ - i (2 γ_{K} + γ_{R, s} + γ_{R, s} H_{R} (- Δ Ω)) {| A_{p} |}^{2} A_{s} - i (γ_{K} + γ_{R, s} H_{R} (- Δ Ω)) A_{p}^{2} A_{a}^{*}, \end{array}

(5)

\begin{array}{l} \frac{\partial A_{a}}{\partial z} & = & - i β_{0, a} A_{a} - \frac{1}{2} (α_{a} + α_{f, a}) A_{a} - i \frac{ω_{a}}{c} n_{f, a} A_{a} \\ - i (2 γ_{K} + γ_{R, a} + γ_{R, a} H_{R} (Δ Ω)) {| A_{p} |}^{2} A_{a} - i (γ_{K} + γ_{R, a} H_{R} (Δ Ω)) A_{p}^{2} A_{s}^{*} . \end{array}

(6)

Here α_j represents the linear loss and γ_K the Kerr nonlinear parameter that describes the TPA, self- and cross-phase modulation (SPM and XPM), and KFWM. γ_K is given by [19

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

γ_{K} = ξ_{K} (\frac{n_{2} ω_{p}}{c} - i \frac{β_{T}}{2}) .

(7)

with ξ_K a polarization coefficient, n ₂ the Kerr coefficient and β_T the TPA coefficient. The coefficients α_f,j and n_f,j stand for the FCA and FCI coefficients. Around λ_r = 1550 nm it is common to approximate these two free-carrier coefficients as [19

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

α_{f, j} = σ_{a, j} N,

(8)

n_{f, j} = σ_{n e, j} N + σ_{n h, j} N^{0.8},

(9)

with σ_a,j = 14.5 × 10⁻¹⁸ (ω_r /ω_j )² cm², σ_ne,j = −8.8 × 10⁻²² (ω_r /ω_j )² cm³, σ_nh,j = −8.5 × 10⁻¹⁸ (ω_r /ω_j )² cm³ and where N is the free-carrier density generated by the TPA. In the CW regime and for the strong pump approximation this last quantity is given by [19

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

N = \frac{β_{T} τ_{c}}{2 \bar{h} ω_{p}} {| A_{p} |}^{4},

(10)

where τ _c stands for the effective free-carrier lifetime. The coefficient γ_R,j indicates the Raman nonlinear parameter at a frequency ω_j and H_R the Raman gain function which together describe in Eqs. (5) and (6) respectively the stimulated Stokes Raman scattering (SSRS) and stimulated anti-Stokes Raman scattering (SARS) processes (see terms function of |A_p |² A_s and |A_p |² A_a , respectively), and also the CARS process (see terms function of

A_{p}^{2} A_{a}^{*}

and

A_{p}^{2} A_{s}^{*}

, respectively). The Raman gain function H_R is for silicon approximated as [19

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

H_{R} (Δ Ω) = \frac{Ω_{R}^{2}}{Ω_{R}^{2} - Δ Ω^{2} + 2 i Γ_{R} Δ Ω},

(11)

with Γ _R the Raman linewidth and Ω _R the Raman angular frequency shift. Finally we define the Raman nonlinear parameter γ_R,j at the frequency ω_j as:

γ_{R, j} = \frac{ξ_{R} g_{R, r e f} Γ_{R}}{Ω_{R}} \frac{ω_{j}}{ω_{r e f}} .

(12)

Here, ξ_R is a polarization factor and g_R,ref the Raman gain at the reference frequency ω_ref , as for instance in [24

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003). [CrossRef] [PubMed]

]. Note that we explicitly included in this definition the linear dependency of the Raman gain g_R = g_R,refω_j /ω_ref with the frequency [19

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

], as opposed to other definitions where γ_R and g_R are approximated as being constant over the considered frequency range [6

Even though the two considered FWM processes, KFWM and CARS, are two distinct physical processes, the terms they introduce in Eqs. (5)–(6) are quite similar. Moreover, one is actually interested in the total effect of the two FWM interactions, rather than in the separate effects of KFWM and CARS on the amplitude evolutions. Therefore, we here investigate the net effect of all the FWM processes. As such, we define the total complex FWM gain G_FWM,j :

G_{F W M, j} \equiv - i (γ_{K} + γ_{R, j} H_{R} (ω_{j} - ω_{p})) .

(13)

G_FWM,s and G_FWM,a then correspond to the complex gain factors found in the last terms of respectively Eq. (5) and Eq. (6).

3. Extended phase mismatch formalism and four-wave-mixing dispersion

Since the FWM processes–KFWM and CARS–involve nonlinear interactions between three different coherent waves, the waves need to be phase-matched in order to obtain efficient wavelength conversion [3

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Lightwave Technol. 23, 2094–2102 (2005). [CrossRef]

, 6

This requirement becomes clear by looking at the intensity gains for the Stokes and the anti-Stokes fields due to FWM which are easily deduced by combining Eq. (6) with

\partial I_{j} / \partial_{z} = 2 Re [A_{j}^{*} \frac{\partial A_{j}}{\partial z}]

\frac{\partial I_{s}}{\partial z} |_{F W M} = 2 Re [G_{F W M, s} A_{p}^{2} A_{a}^{*} A_{s}^{*}],

(14)

\frac{\partial I_{a}}{\partial z} |_{F W M} = 2 Re [G_{F W M, a} A_{p}^{2} A_{a}^{*} A_{s}^{*}] .

(15)

These expressions show that a limited phase evolution of

A_{p}^{2} A_{s}^{*} A_{a}^{*}

over a certain length is required in order to enable a considerable gain due to FWM over that length. To quantify this condition, it is common to define the phase mismatch κ [3

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Lightwave Technol. 23, 2094–2102 (2005). [CrossRef]

, 6

, 19

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

], which is implicitly present in the set of Eqs. (4)–(6), as:

κ = Δ β_{0} + \frac{1}{c} (n_{f, s} ω_{s} + n_{f, a} ω_{a} - 2 n_{f, p} ω_{p}) + I_{p} Re [2 γ_{K} + (γ_{R, s} + γ_{R, a}) H_{R} (Δ Ω)] .

(16)

Here Δβ ₀ = β _0, _s + β _0, _a – 2β _0, _p represents the linear phase mismatch, while the other terms indicate the contributions from the FCI and the phase modulation due to the Kerr and Raman effects. This definition of the phase mismatch is based on analytic derivations of the propagation equations which took an undepleted pump into account [16

G. Agrawal, Nonlinear Fiber Optics , 3rd ed. (Academic Press, 2001).

, 17

]. From such analytic solutions one can indeed derive the phase mismatch definition of Eq. (16) provided that the linear phase mismatch is small compared to the phase mismatch due to nonlinear dispersion [1

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express 11, 2862–2872 (2003). [CrossRef] [PubMed]

, 2

D. Dimitropoulos, V. Raghunathan, R. Claps, and B. Jalali, “Phase-matching and nonlinear optical processes in silicon waveguides,” Opt. Express 12, 149–160 (2004). [CrossRef] [PubMed]

, 17

Furthermore, because the Stokes and anti-Stokes frequencies are located symmetrically around the pump frequency, the linear phase mismatch depends only on the even order dispersion parameters [6

Δ β_{0} = β_{2, p} Δ Ω^{2} + \frac{β_{4, p}}{12} Δ Ω^{4} + \dots,

(17)

with β_m,j = ∂β ₀/∂ω|_{ω=ω
_j} the mth order dispersion parameter at ω_j .

Neglecting the higher order terms, the phase mismatch as defined in Eq. (16) thus becomes:

\begin{array}{l} κ \approx β_{2, p} Δ Ω^{2} + \frac{β_{4, p}}{12} Δ Ω^{4} + \frac{1}{c} (n_{f, s} ω_{s} + n_{f, a} ω_{a} - 2 n_{f, p} ω_{p}) \\ + I_{p} Re [2 γ_{K} + (γ_{R, s} + γ_{R, a}) H_{R} (Δ Ω)] . \end{array}

(18)

This definition of the phase mismatch, however, does not comprise all of the dispersion effects occurring in a silicon waveguide due to the nonlinear effects present in the set of Eqs. (4)–(6). For instance, the Stokes FWM term in Eq. (5),

G_{F W M, s} A_{p}^{2} A_{a}^{*}

, does not necessarily have the same phase as the amplitude A_s to which it contributes. If their phases differ, or in other words if

G_{F W M, s} A_{p}^{2} A_{a}^{*} A_{s}^{*}

is not strictly real, the Stokes wave will experience dispersion due to the FWM term. Similarly, the anti-Stokes will experience a phase change due to FWM if

G_{F W M, a} A_{p}^{2} A_{s}^{*} A_{a}^{*}

has an imaginary part. Furthermore, since the total FWM gains G_FWM,a and G_FWM,s have a different phase (see Eq. (11) and Eq. (13)), both FWM terms,

G_{F W M, a} A_{p}^{2} A_{s}^{*} A_{a}^{*}

and

G_{F W M, s} A_{p}^{2} A_{a}^{*} A_{s}^{*}

, cannot be real at the same time, such that there will always be, for non-zero fields, dispersion due to nonlinear FWM.

To estimate the phase mismatch including this so-called FWM dispersion, we write the fields as:

A_{j} (z) = | A_{j} (z) | e^{i ϕ_{j} (z)},

(19)

where we defined ϕ_j (z) as the phase of the wave j at the position z. By taking the spatial derivative of this equation, we obtain:

\frac{\partial A_{j}}{\partial z} = i \frac{\partial ϕ_{j}}{\partial z} A_{j} + \frac{\partial | A_{j} |}{\partial z} \frac{A_{j}}{| A_{j} |} .

(20)

We now define the local dispersion parameter β_loc,j as:

β_{l o c, j} = - \frac{\partial ϕ_{j}}{\partial z},

(21)

such that β_loc,j represents the local phase evolution which is the total of the linear dispersion β _0,j and the dispersion due to the nonlinear effects.

As long as A_j ≠ 0 one can easily derive from Eqs. (20)–(21) the following expression for β_loc,j :

β_{l o c, j} = - Im [\frac{1}{A_{j}} \frac{\partial A_{j}}{\partial z}] .

(22)

The complete phase mismatch κ′ can then be deduced by combining this definition with Eqs. (4)–(6):

κ' = β_{l o c, s} + β_{l o c, a} - 2 β_{l o c, p}

(23)

\begin{array}{l} = β_{2, p} Δ Ω^{2} + \frac{β_{4, p}}{12} Δ Ω^{4} + \frac{1}{c} (n_{f, s} ω_{s} + n_{f, a} ω_{a} - 2 n_{f, p} ω_{p}) \\ + I_{p} Re [2 γ_{K} + (γ_{R, s} + γ_{R, a}) H_{R} (Δ Ω)] - Im [G_{F W M, s} A_{p}^{2} \frac{A_{a}^{*}}{A_{s}}] - Im [G_{F W M, a} A_{p}^{2} \frac{A_{s}^{*}}{A_{a}}] . \end{array}

(24)

The two last terms in this equation represent the FWM dispersion and should be left out when A_a ,A_s = 0. Furthermore we again approximated the linear dispersion Δβ ₀ by the first two terms of Eq. (17). We note that the conventional definition of κ (see Eq. (16)) remains correct in case the aforementioned analytic solutions of the propagation equations are adequate. In all other cases, the more elaborate phase mismatch definition of Eq. (24) provides a more complete overview of all dispersion effects present in the set of Eqs. (4)–(6), hence offering a better physical insight in how the fields can be effectively phase-matched and how the conversion efficiency can be maximized. Moreover, unlike the conventional definition of the phase mismatch κ, our definition of κ′ using Eqs. (22)–(23) is fully generic, and can as such be generalized to wavelength ranges other than the near-infrared, to materials other than silicon, and this regardless of the nature of the nonlinearities involved. Even in case of a weak-pump situation, the formalism holds.

To investigate the dispersion due to the FWM terms we define the relative phase difference Δϕ as:

Δ ϕ = 2 ϕ_{p} - ϕ_{s} - ϕ_{a} .

(25)

From Eqs. (22) and (23) it is clear that the evolution of Δϕ is given by:

\frac{\partial Δ ϕ}{\partial z} = κ'

(26)

\begin{array}{l} = β_{2, p} Δ Ω^{2} + \frac{β_{4, p}}{12} Δ Ω^{4} + \frac{1}{c} (n_{f, s} ω_{s} + n_{f, a} ω_{a} - 2 n_{f, p} ω_{p}) \\ + I_{p} Re [2 γ_{K} + (γ_{R, s} + γ_{R, a}) H_{R} (Δ Ω)] \\ - Im [G_{F W M, s} e^{i Δ ϕ}] I_{p} \frac{| A_{a} |}{| A_{s} |} - Im [G_{F W M, a} e^{i Δ ϕ}] I_{p} \frac{| A_{s} |}{| A_{a} |}, \end{array}

(27)

in which we have substituted

p h a s e [A_{p}^{2} A_{a}^{*} / A_{s}] = p h a s e [A_{p}^{2} A_{s}^{*} / A_{a}] = Δ ϕ

. It should be noted that terms similar to the FWM terms in this equation have already been reported before in relation to FWM in optical fibers [20

Y. Chen, “Combined processes of stimulated Raman scattering and four-wave mixing in optical fibers,” J. Opt. Soc. Am. B 7, 43–52 (1990). [CrossRef]

, 21

G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991). [CrossRef]

, 23

]. However, the effects of these terms have, according to our knowledge, not yet been investigated in detail. To do so, we first rewrite the expressions for the intensity gain for the Stokes and the anti-Stokes fields due to FWM of Eqs. (14) and (15) in terms of Δϕ as:

\frac{\partial I_{s}}{\partial z} |_{F W M} = 2 Re [G_{F W M, s} e^{i Δ ϕ}] I_{p} | A_{a} | | A_{s} |,

(28)

\frac{\partial I_{a}}{\partial z} |_{F W M} = 2 Re [G_{F W M, a} e^{i Δ ϕ}] I_{p} | A_{s} | | A_{a} | .

(29)

Now, the evolutions of the real and imaginary parts of G _FWM,a e ^iΔϕ as a function of Δϕ clearly show the effects of the FWM dispersion (see Fig. 2 for an example with λ_p =1550 nm, at perfect Raman resonance and with the silicon nonlinear parameters given in Table 1): the anti-Stokes FWM dispersion term (i.e. –Im [G _FWM,a e ^iΔϕ], the last term in Eq. (27)) drives the phase difference Δϕ to higher values when positive, and to lower values when negative (see Eq. (27)). As such it drives Δϕ towards the value –phase[G_FWM,a ], at which the waves will be phase-matched. The anti-Stokes wave then experiences a maximal FWM intensity gain (i.e. Re [G _FWM,a e ^iΔϕ] in Eq. (29) is maximal) while the FWM dispersion is 0 (i.e. –Im [G _FWM,a e ^iΔϕ] = 0). Similarly, the Stokes FWM dispersion term (i.e. –Im [G _FWM,s e ^iΔϕ] in Eq. (27)) drives Δϕ towards the value –phase[G_FWM,s ], at which the FWM intensity gain will be maximal for the Stokes wave. When the FWM intensity gain is maximal for either the Stokes or the anti-Stokes wave, the other wave in its turn will not only experience FWM dispersion, but also intensity loss due to FWM.

Fig. 2 The variation for both the Stokes and the anti-Stokes waves of (a) the FWM dispersion factors –Im [G _FWM,j e ^iΔϕ] and (b) the FWM intensity gain factor Re [G _FWM,j e ^iΔϕ] versus the phase difference Δϕ, as computed for TE-polarized light at perfect Raman resonance and λ_p =1550 nm using the silicon nonlinear parameters given in Table 1. As can be seen in (a), the FWM dispersion due to each wave drives the Δϕ towards the value –phase [G_FWM,j ] that corresponds to that wave.

Table 1 List of Values of Nonlinear Silicon Parameters near λ = 1550 nm for TE-Polarized Light

Parameter	Value	Reference
ξ_K	1.135	[19 Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed] ]
n ₂	6×10⁻⁵ cm²/GW	[6 Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006). [CrossRef] [PubMed] ]
β_T	0.5 cm/GW	[6 Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006). [CrossRef] [PubMed] ]
ξ_R	1	[19 Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed] ]
g_R,ref	20 cm/GW at λ_ref = 1542.3 nm	[24 R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003). [CrossRef] [PubMed] ]
Γ _R	2π × 52.5 GHz	[6 Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006). [CrossRef] [PubMed] ]
Ω _R	2π × 15.6 THz	[6 Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006). [CrossRef] [PubMed] ]

Also the factor |A_s |/|A_a | plays an important role for the nature of the FWM dispersion. For instance, when the anti-Stokes (Stokes) intensity is much weaker than the Stokes (anti-Stokes) intensity, the anti-Stokes (Stokes) FWM dispersion term will dominate in Eq. (27) and, as discussed above, drive Δϕ to the value –phase[G_FWM,a ] (–phase[G_FWM,s ]) and thus to optimal anti-Stokes (Stokes) FWM intensity gain. This is for instance the case at the beginning of waveguides (z ≈ 0) in typical wavelength conversion set-ups, for which a Stokes wave and a strong pump are used as inputs. Since there is no anti-Stokes seed (A_a ≈ 0), one can approximate Eq. (6) there as:

\frac{\partial A_{a}}{\partial z} (z \approx 0) = G_{F W M, a} A_{p}^{2} A_{s}^{*} .

(30)

As such, the phase of the initially generated anti-Stokes wave is equal to the phase of the complete FWM term:

p h a s e [A_{a} (z \approx 0)] = p h a s e [G_{F W M, a} A_{p}^{2} A_{s}^{*}] .

(31)

This corresponds to Δϕ ≈ –phase[G_FWM,a ], the same value as predicted by the FWM dispersion description.

An important characteristic of the FWM dispersion is the phase difference between the two complex FWM gains which we will call the FWM phase difference Δϕ_FWM :

Δ ϕ_{F W M} = - p h a s e [G_{F W M, a}] - (- p h a s e [G_{F W M, s}]) = p h a s e [G_{F W M, s} G_{F W M, a}^{*}] .

(32)

This quantity completely describes the signs of the FWM dispersion and the FWM intensity gain for both waves in function of the value of Δϕ relative to –phase[G_FWM,a ] or –phase[G_FWM,s ], as will be shown further on.

By devising a generic formalism to extract the phase mismatch out of any given set of equations and applying it on the set described in Section 2, we have ascertained and quantified an additional dispersion term. The impact that this dispersion term has on the nonlinear propagation and thus on the wavelength conversion efficiency is described by its main characteristic, the FWM phase difference Δϕ_FWM . A means to affect Δϕ_FWM –and thus the FWM processes–is by changing the frequency difference ΔΩ (see Eq. (11)). Doing so of course also influences the other nonlinear processes. We therefore now investigate the impact of ΔΩ on all the relevant nonlinear processes.

4. Impact of the pump-Stokes frequency difference ΔΩ

SSRS and SARS Raman processes are most effective when working at perfect Raman resonance. The imaginary part of H (ΔΩ) has, in absolute value, indeed its maxima at ΔΩ = |Ω _R | (see Eq. (11)) while the real part is at the same time 0 for these values, such that at the perfect Raman resonance the conversion of pump to Stokes and of anti-Stokes to pump due to respectively SSRS and SARS will be maximal while there will be no SPM or XPM dispersion caused by the Raman scattering. Note that, as discussed above, there will still be FWM dispersion due to the Raman effect present.

This situation changes however when allowing a frequency difference ΔΩ ≠ Ω _R . The Raman spectrum H_R (ΔΩ) is then no longer purely imaginary, such that there is dispersion due to Raman XPM introduced. By choosing ΔΩ carefully one can obtain Raman dispersion that effectively counteracts the total of the experienced dispersion, thus augmenting the efficiency of the wavelength conversion.

A value of ΔΩ that is off Raman resonance will also affect the magnitude of the FWM intensity gain as well as the FWM dispersion. Its influence on the former can be seen for instance in the beginning of a waveguide where no anti-Stokes input is present (the situation described at the end of Section 3). In this case the anti-Stokes intensity evolution at z ≈ 0 follows directly from Eq. (30):

\frac{\partial I_{a}}{\partial z} (z \approx 0) = 2 Real [G_{F W M, a} A_{p}^{2} A_{s}^{*} A_{a}^{*}] = 2 | G_{F W M, a} | I_{p} | A_{s} | | A_{a} | .

(33)

Here we substituted the fields’ phases using Eq. (31) to obtain the last expression. This equation implies that the efficiency of the initial anti-Stokes generation is determined by the absolute value of G_FWM,a . Similarly, when using an anti-Stokes and a strong pump wave as input, the initial Stokes generation is found to be proportional to |G_FWM,s |. When working at λ_p = 1550 nm with TE-polarized waves, |G_FWM,a | is not maximal for Ω _R , but for a frequency difference ΔΩ that is 80 GHz smaller than the Raman resonance Ω _R (see Fig. 3a), which suggests that by working off Raman resonance we might improve the wavelength conversion efficiency. Though at first sight the resonance offset of 80 GHz might seem small, it is still several times larger than 12.5 GHz, the smallest frequency spacing up to which the standard spectral grid for dense wavelength division multiplexing applications is specified [25

ITU-T Rec. G.694.1, “Spectral grids for WDM applications: DWDM frequency grid” (2002).

Fig. 3 (a) The initial FWM anti-Stokes intensity gain |G_FWM,a | and (b) the FWM phase difference Δϕ_FWM between the complex FWM gains G_FWM,j which characterizes the FWM dispersion, both in function of the frequency resonance off-set ΔΩ – Ω _R and the Stokes wavelength resonance off-set λ_s – λ _s0, where λ _s0 is the Stokes wavelength at perfect Raman resonance, as computed for TE-polarized light at a fixed λ_p =1550 nm using the silicon nonlinear parameters given in Table 1.

The effect that a working point detuned from Raman resonance has on the FWM dispersion can in its turn be investigated by means of Δϕ_FWM , the difference in phase of the Stokes and anti-Stokes FWM gains as given in Eq. (32): Δϕ_FWM increases monotonically with ΔΩ around the Raman resonance (see Fig. 3b). Around ΔΩ ≈ Ω _R – 200 GHz this phase difference goes across the value π/2, such that for lower ΔΩ the Stokes FWM intensity gain (see Eq. (28)) will be positive–and thus the Stokes wave will actually experience gain due to FWM–when the anti-Stokes optical intensity gain is maximal (i.e. the anti-Stokes is phase matched: Δϕ = –phase[G_FWM,a ]). In contrast, for higher ΔΩ the Stokes wave will in that case experience loss due to FWM. Furthermore this loss increases with ΔΩ until it becomes maximal at ΔΩ ≈ Ω _R + 100 GHz where Δϕ_FWM = π, after which it again becomes smaller.

The total impact of working off Raman resonance on the conversion efficiency is clearly an intricate interplay of various effects and is thus best investigated by means of numerical simulations.

5. Numerical simulation results

5.1. The potential of working off Raman resonance

To ensure accurate numerical simulation results, we used a set of nonlinear Schrödinger equations related to Eqs. (4)–(6), but in which all the relevant terms were taken into account without making a strong pump assumption. We used this approach to ensure that at all times the losses were not underestimated. By using the split-step Fourier method [16

G. Agrawal, Nonlinear Fiber Optics , 3rd ed. (Academic Press, 2001).

], we then solved the resulting set of equations. In the remainder of this paper we refer to this set of equations as the weak-pump equations.

The CW simulation results obtained in this way clearly show the benefit of allowing a detuning away from Raman resonance: for instance over a 2cm-long silicon-on-insulator (SOI) nano-waveguide characterized by β _2,0 = −1.14 × 10⁻⁴ ps²/cm and β _4,p = −8.93 × 10⁻⁸ ps⁴/cm and with an input of I _p,0 = 0.2 GW/cm² and I _0,s = 0.2 MW/cm², a doubling of the conversion efficiency I_a /I_s (0)–this quantity is also referred to here as the normalized anti-Stokes intensity–can be obtained when working at ΔΩ = Ω _R – 250 GHz instead of at perfect Raman resonance (see Fig. 4a). Moreover, the danger of using an inaccurate λ_s also becomes apparent as for the same waveguide the wavelength conversion efficiency decreases with about 40% when ΔΩ = Ω _R + 50 GHz (equivalent with λ_s – λ_s ₀ < 0.1 nm) and with more than 10 dB when ΔΩ = Ω _R + 250 GHz (λ_s – λ_s ₀ < 0.4 nm), both as compared to ΔΩ = Ω _R . It should be noted that the considered frequency detunings are all within the Raman linewidth Γ _R (see Table 1).

Fig. 4 Normalized intensity evolutions of (a) anti-Stokes I_a /I_s (0), (b) Stokes I_s /I_s (0) and (c) the pump I_p /I_p (0), and evolutions of the corresponding (d) phase difference Δϕ and (e) the phase mismatch, and this for several frequency differences ΔΩ as simulated for CW TE-polarized light, with pump and Stokes input intensities of I_p (0) = 0.2 GW/cm² and I_s (0) = 0.2 MW/cm², along a SOI nano-waveguide with β _2,p = −1.14 × 10⁻⁴ ps²/cm and β _4,p = −8.93 × 10⁻⁸ ps⁴/cm, at the fixed pump wavelength λ_p =1550 nm. The nonlinear parameters of Table 1 were used together with a linear loss α =0.2 dB/cm [6

]. In (e) the phase mismatch is both computed as κ′, including the FWM dispersion (full lines), and according to the conventional definition as κ, excluding the FWM dispersion (dotted lines).

The signal amplification or normalized Stokes intensity I_s /I_s (0) does also change with the frequency detuning (see Fig. 4b), although the variations are here less pronounced: the largest and smallest obtained amplifications differ only by a factor 5. It should furthermore be noted that efficient wavelength conversion occurs for high Stokes amplitudes, while the weakest anti-Stokes generation corresponds also with the weakest Stokes generation. This corresponds with the fact that a higher Stokes amplitude results in a higher FWM intensity gain for the anti-Stokes wave (see Eq. (29)). The differences in I_a and I_s evolutions are also reflected in the evolution of the pump intensity I_p /I_p (0) (see Fig. 4c). At the frequency detuning for which the Stokes and anti-Stokes generation is the weakest, the pump is at the end of the waveguide the least depleted as less energy was converted from the pump into the Stokes and anti-Stokes waves. Similarly, the pump depletion is the strongest when the largest Stokes and anti-Stokes amplitudes are obtained.

All these intensity evolutions can be explained by studying the phase difference Δϕ along the waveguide (see Fig. 4d). As predicted in Section 3, the anti-Stokes generation becomes increasingly less efficient as Δϕ diverges from the value –phase[G_FWM,a ], and the anti-Stokes wave will even experience loss when Δϕ tends towards –phase[G_FWM,a ] – π/2. Furthermore as explained in Sections 3 and 4, the Stokes wave will experience gain due to FWM as long as Δϕ differs no more than π/2 from the value –phase[G_FWM,s ] = –phase[G_FWM,a ] – Δϕ_FWM . By comparison of Figs. 3b and 4d, it is clear that this is mostly the case for the more negative frequency detunings, while it is not for ΔΩ ≥ Ω _R . For ΔΩ < Ω _R , the Stokes wave experiences thus gain due to FWM, while, for ΔΩ ≥ Ω _R , FWM acts as a loss mechanism for the Stokes wave. This explains the observed differences in signal amplification.

To evaluate how a change in ΔΩ affects the evolution of Δϕ as shown in Fig. 4d, we have to investigate the phase mismatch κ′, as defined in Eq. (24) (see full lines in Fig. 4e). By tuning ΔΩ – Ω _R to more negative values, one introduces positive Raman XPM dispersion, which dominates in the beginning of the waveguide where the pump intensity is large. This positive dispersion effectively counteracts the negative linear dispersion which dominates further down the waveguide were I_p has decayed to lower values, and as such the cumulative dispersion is, after that distance, closer to zero as compared to the case of perfect Raman resonance for which the dispersion is at all times negative. It is because of this dispersion compensation that the wavelength conversion is most efficient at ΔΩ = Ω _R – 250 GHz, even though the anti-Stokes FWM intensity gain (|G_FWM,a |) is actually smaller than for higher ΔΩ (see Fig. 3a).

Furthermore, the importance and significance of FWM dispersion is clear by comparing the evolutions of the phase mismatch κ′ as defined in Eq. (24) including FWM and the phase mismatch κ as conventionally defined in Eq. (18) excluding FWM (see respectively the full and the dotted lines in Fig. 4e). κ is only affected by the pump depletion [21

G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991). [CrossRef]

], and thus follows a purely monotone trend. The evolution of κ′ is more complicated: at the beginning of the waveguide the FWM dispersion mitigates the phase mismatch by approximately a factor two, while after a certain distance it even causes κ′ to slope strongly upwards instead of following the monotone trend of κ. As a consequence, the phase mismatch values of κ′ lay for the different ΔΩ not only closer to each other, but also closer to zero for the largest part of the waveguide than was originally predicted by the conventional phase mismatch κ. Hence, when using the conventional phase mismatch κ, one underestimates the efficiency of the wavelength conversion along the waveguide. For example, the conversion efficiency graphs in Fig. 4a exhibit an increasing positive slope towards the end of the waveguide (z ≈ 2 cm), and this increase matches with the tendency of κ′ of evolving to zero again around z ≈ 2 cm, whereas κ does not hint at such an increase as it, for most frequency differences, moves further away from zero around z ≈ 2 cm.

It is clear that to obtain the most efficient wavelength conversion, the frequency difference ΔΩ needs to be finely tuned to a certain optimal value. If not, or if tuned wrongly, the efficiency can decrease significantly as a result. Moreover, this optimal value in turn depends on the length of the waveguide, and additionally on the waveguide’s dispersion characteristics. In the following section we will therefore further investigate both of these dependencies.

5.2. Optimal frequency difference ΔΩ_Opt

As discussed in Section 3, phase matching of the pump, Stokes and anti-Stokes waves is essential for efficient wavelength conversion. For instance, a phase mismatch of, in absolute values, 1.6 × 2π/cm will result in a decrease of 10 to 20 dB in conversion efficiency as compared to a phase mismatch close to zero [3

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Lightwave Technol. 23, 2094–2102 (2005). [CrossRef]

]. Now, SOI waveguides for which the dispersion characteristics are of the order |β _2,p| ≈ 10⁻³ ps²/cm and |β _4,p| ≈ 10⁻⁸ ps⁴/cm, will, according to Eq. (17), have at perfect Raman resonance a linear phase mismatch of about |Δβ ₀| ≈ 1.6 × 2π/cm. Since this is much larger than the nonlinear phase mismatch at the considered pump intensity levels (for instance I_p Re [2γ_K ] ≈ 0.09 × 2π/cm for I_p = 0.2 GW/cm²), such waveguides will be poor wavelength convertors and the dominating process will instead be amplification of the Stokes wave by SSRS. Therefore, we here only consider waveguides that are characterized by |β _2,p| < 10⁻³ ps²/cm (see Table 2 for all relevant dispersion characteristics), such that the FWM interaction will certainly play a significant role.

Table 2 Dispersion Characteristics of the Numerically Simulated Waveguides

Waveguide	β _2,p [ps²/cm]	β _4,p [ps⁴/cm]	Δβ ₀ at ΔΩ= Ω _R [×2π/cm]
1	5.56×10⁻⁴	−3.86×10⁻⁸	0.80
2	6.45×10⁻⁵	−4.71×10⁻⁸	0.041
3	−7.28×10⁻⁶	−4.72×10⁻⁸	−0.069
4	−1.14×10⁻⁴	−8.93×10⁻⁸	−0.28
5	−6.94×10⁻⁴	−5.62×10⁻⁸	−1.13

Now, to describe the dependence of the optimal frequency difference with the distance we define the function ΔΩ _Opt (z) for a given waveguide as the value of ΔΩ which gives the best conversion efficiency after traveling a distance z along that waveguide. This function is easily determined out of the numerical simulations by comparing at each position z the anti-Stokes intensities obtained for the different ΔΩ values.

By subsequently comparing the evolution of ΔΩ _Opt along the considered set of SOI nano-waveguides, several observations can be made (see Fig. 5): firstly, the optimal frequency difference is in the beginning of the waveguides independent of the waveguide characteristics and its value corresponds to the Ω _R – 80 GHz, the same value as for which the anti-Stokes intensity gain |G_FWM,a | is maximum, as discussed in Section 4. Secondly, further along the waveguide ΔΩ _Opt increases or decreases depending on the waveguide’s second-order dispersion parameter β _2,p. The more negative β _2,p, the faster the optimal ΔΩ decreases as stronger Raman XPM dispersion is needed to compensate the linear dispersion (as discussed in Section 5.1). Likewise, for waveguides with a large positive β _2,p, ΔΩ _Opt increases with the distance: the larger β _2,p the faster the increase. For very small positive β _2,p, this general tendency of an increasing ΔΩ _Opt is only present towards the end of the waveguide, while initially there is a small decrease in ΔΩ _Opt . The origins of this last behavior lay in the complicated interplay of the various effects discussed in Section 5.1, but its exact details fall outside the scope of this paper, and are as such not discussed here. Thirdly, for the waveguides with the larger |β _2,p|, the optimal ΔΩ starts oscillating after a certain distance. This is because the poor phase-matching due to the large dispersion makes that the anti-Stokes intensity effectively starts to experience loss. As a consequence the whole process becomes difficult to predict–a strong decay of the anti-Stokes wave will for instance generate a spike in the FWM dispersion since the amplitude A_a occurs in the numerator of the last term in Eq. (23). From the point of view of efficient wavelength conversion this is a far from favorable situation and is thus not further investigated here.

Fig. 5 Evolution of ΔΩ _Opt – Ω _R , the frequency resonance off-set that gives the optimal CW wavelength conversion after a certain distance, in function of that distance, for several SOI nano-waveguides of which the dispersion characteristics are given in Table 2. The pump wavelength is fixed at λ_p =1550 nm, with the same input and material parameters as before. ΔΩ _Opt is determined by comparing at each position z the anti-Stokes intensities as numerically simulated for a set of ΔΩ with a resolution of 0.5 GHz.

By deducing the general tendencies of ΔΩ _Opt (z) for different classes of dispersion characteristics, we have determined a first guideline to estimate the optimal frequency difference for a given waveguide of a given length. Since the wavelength conversion efficiency is very sensitive to minor changes of ΔΩ (see Section 5.2), a far more accurate estimate is needed in practice. One option is to apply the same method that is used in this section to deduce the ΔΩ _Opt (z), but this is far from desirable given the extensive computing power required by this method. In the following section we will therefore derive a less computationally demanding approach to estimate ΔΩ _Opt .

5.3. Estimation of the optimal frequency difference ΔΩ

Since wavelength conversion is an intricate interplay of various nonlinear effects, devising a rule of thumb to estimate the optimal frequency difference ΔΩ _Opt for a given length of an arbitrary waveguide is hardly possible. It seems that the only way to obtain this information is by numerically simulating the field propagation for various ΔΩ values. Within these boundaries, however, we can benefit from a simplification of the propagation equations so that we can numerically solve them with much less computing power than when solving the weak-pump equations as we did to obtain the numerical results in Sections 5.1 and 5.2. For instance, the set of Eqs. (4)–(6) for which the strong pump assumption was made, can easily be rewritten as the following set of three complex equations:

\frac{\partial {\bar{A}}_{p}}{\partial z} = - \frac{1}{2} (α_{p} + α_{f, p}) {\bar{A}}_{p} - i \frac{ω_{p}}{c} n_{f, p} {\bar{A}}_{p} - i (γ_{K} + γ_{R, p}) {| {\bar{A}}_{p} |}^{2} {\bar{A}}_{p},

(34)

\begin{array}{l} \frac{\partial {\bar{A}}_{s}}{\partial z} & = & - \frac{1}{2} (α_{s} + α_{f, s}) {\bar{A}}_{s} - i \frac{ω_{s}}{c} n_{f, s} {\bar{A}}_{s} - i (2 γ_{K} + γ_{R, s} + γ_{R, s} H_{R} (- Δ Ω)) {| {\bar{A}}_{p} |}^{2} {\bar{A}}_{s} \\ - i (γ_{K} + γ_{R, s} H_{R} (- Δ Ω)) e^{i Δ β_{0} z} {\bar{A}}_{p}^{2} {\bar{A}}_{a}^{*}, \end{array}

(35)

\begin{array}{l} \frac{\partial {\bar{A}}_{a}}{\partial z} & = & - \frac{1}{2} (α_{a} + α_{f, a}) {\bar{A}}_{a} - i \frac{ω_{a}}{c} n_{f, a} {\bar{A}}_{a} - i (2 γ_{K} + γ_{R, a} + γ_{R, a} H_{R} (Δ Ω)) {| {\bar{A}}_{p} |}^{2} {\bar{A}}_{a} \\ - i (γ_{K} + γ_{R, a} H_{R} (Δ Ω)) e^{i Δ β_{0} z} {\bar{A}}_{p}^{2} {\bar{A}}_{s}^{*}, \end{array}

(36)

in which we introduced Ā_j = A _j e ^{iβ
_j
z}. This set of equations can be numerically solved with a by far smaller spatial resolution because the phases of these new field amplitudes A_j will not vary as rapidly as those of the original Ā_j . Moreover, this set of equations is commonly used to describe the nonlinear propagation along waveguides and fibers in which FWM plays an important role [17

, 20

Y. Chen, “Combined processes of stimulated Raman scattering and four-wave mixing in optical fibers,” J. Opt. Soc. Am. B 7, 43–52 (1990). [CrossRef]

–22

Y. Chen and A.W. Snyder, “Four-photon parametric mixing in optical fibers: effect of pump depletion,” Opt. Lett. 14, 87–89 (1989). [CrossRef] [PubMed]

Another set of equations can be obtained by writing Eqs. (4)–(6) in terms of the absolute value of the amplitudes using the expression

2 \partial | A_{j} | / \partial z = Re [\sqrt{(A_{j}^{*} / A_{j})} \frac{\partial A_{j}}{\partial z}] = Re [e^{- i ϕ_{j}} \frac{\partial A_{j}}{\partial z}]

and subsequently introducing in this set of equations the newly developed phase mismatch formalism, or more specifically, the phase difference Δϕ as defined in Eq. (25):

\frac{\partial I_{p}}{\partial z} = - (α_{p} + α_{f, p}) I_{p} - ξ_{K} β_{T} I_{p}^{2},

(37)

\begin{array}{l} \frac{\partial | A_{s} |}{\partial z} & = & - \frac{1}{2} (α + α_{f, s}) | A_{s} | + (- ξ_{K} β_{T} + γ_{R, s} Im [H_{R} (- Δ Ω)]) I_{p} | A_{s} | \\ + Re [G_{F W M, s} e^{i Δ ϕ}] I_{p} | A_{a} |, \end{array}

(38)

\begin{array}{l} \frac{\partial | A_{a} |}{\partial z} & = & - \frac{1}{2} (α + α_{f, a}) | A_{a} | + (- ξ_{K} β_{T} + γ_{R, a} Im [H_{R} (Δ Ω)]) I_{p} | A_{a} | \\ + Re [G_{F W M, a} e^{i Δ ϕ}] I_{p} | A_{s} | . \end{array}

(39)

which form together with Eq. (27) a set of four real equations. Similar sets of equations are often used to describe FWM problems in optical fibers [20

Y. Chen, “Combined processes of stimulated Raman scattering and four-wave mixing in optical fibers,” J. Opt. Soc. Am. B 7, 43–52 (1990). [CrossRef]

, 21

G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991). [CrossRef]

, 23

The set of real equations considered here leads to more accurate numerical simulation results while at the same time requiring less spatial points–and thus allowing a faster calculation–than the set of three complex Eqs. (34)–(36): for instance, the anti-Stokes intensity after propagating 1 cm along the waveguide and for the ΔΩ described in Sec. 5.1, features a five times smaller relative error when computed by numerically solving the set of four real equation using 200 equally spaced spatial points as compared to numerically solving the set of three complex equations using 2000 equally spaced spatial points (see Fig. 6). After 2 cm, this relative error generated by solving the four real equations is still about four times smaller than the one produced by solving the three complex equations.

Fig. 6 The approximation error relative to the solutions found with the weak-pump description which is depicted in Fig. 4 of the anti-Stokes intensity evolutions, under the same conditions as those in Fig. 4, as made by numerically solving on the one hand the set of three complex Eqs. (34)–(36) using 2000 equally spaced spatial points (dash-dotted lines), and on the other hand the set of the four real Eqs. (25), (37)–(39) using 200 equally spaced spatial points (full lines).

Important to note is that the simulation results for both methods were acquired with a simulation time of more than three orders of magnitude smaller than the time needed to solve the weak-pump equations. This clearly illustrates the significant decrease in required computing power offered by both sets of equations.

The derived set of Eqs. (37)–(39) is thus a practical means to not only quickly estimate the optimal frequency difference ΔΩ _Opt (z) of waveguides, but also to estimate and compare the corresponding wavelength conversion efficiencies.

6. Conclusion

We have derived a generic formalism to extract the phase mismatch out of any given set of propagation equations. In contrast with prior definitions, this formalism does not require any assumptions to be made about the waves involved. As such, it is applicable to a wider range of situations, in particular to FWM processes for which the pump depletion is not negligible.

Furthermore, by applying this formalism to the set of propagation equations which describe wavelength conversion in silicon waveguides in the near-infrared, we have determined a novel dispersion term that is due to the FWM nonlinear effects. We showed that this so-called FWM dispersion is deeply intertwined with the intensity gain (or loss) experienced by the Stokes and anti-Stokes waves, and will dominate all the other dispersion effects if one of these two waves is much stronger than the other. Our numerical simulations also show that for a Stokes and anti-Stokes wave which are of the same order of magnitude, the FWM dispersion has a considerable mitigating effect on the phase mismatch and as such causes better efficiency to be attained than would be predicted by the conventional definition of the phase mismatch.

One way to manipulate the FWM dispersion is by tuning the pump-Stokes frequency difference, although this also affects the various other nonlinear Raman effects involved. We showed by means of numerical simulations that the wavelength conversion efficiency is very sensitive to this frequency difference. As such, working slightly off Raman resonance can result under certain conditions both in an improvement of the wavelength conversion efficiency with a factor of two, or in a decrease of more than 10 dB, depending on the exact value of the pump-Stokes frequency difference. The frequency difference that is optimal for the wavelength conversion depends both on the length of the considered waveguide, and on its dispersion characteristics. Using our numerical simulations, we determined the general tendencies of the spatial evolution of the optimal frequency difference for several classes of dispersion characteristics. Moreover we derived a set of real propagation equations that are less computationally intensive to solve as compared to conventional sets of equations. Still they are able to accurately estimate the optimal frequency difference along the waveguide, as well as the corresponding wavelength conversion efficiency.

The novel formalism proposed here for determining the phase mismatch clearly offers important additional insights regarding the nonlinear FWM interactions at near-infrared wavelengths in silicon nano-waveguides and allows to improve the operational characteristics of a variety of nonlinear optical devices.

Acknowledgments

This work was financially supported in part by FWO-Vlaanderen which provides an Aspirant grant for Y. Lefevre and Postdoctoraal Onderzoeker grants for N. Vermeulen and C. Debaes, and in part by Methusalem, IAP, VUB-GOA, and VUB-OZR.

References and links

1.	R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express 11, 2862–2872 (2003). [CrossRef] [PubMed]
2.	D. Dimitropoulos, V. Raghunathan, R. Claps, and B. Jalali, “Phase-matching and nonlinear optical processes in silicon waveguides,” Opt. Express 12, 149–160 (2004). [CrossRef] [PubMed]
3.	V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Lightwave Technol. 23, 2094–2102 (2005). [CrossRef]
4.	H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express 13, 4629–4637 (2005). [CrossRef] [PubMed]
5.	R. Espinola, J. Dadap, J. Osgood, S. McNab, and Y. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express 13, 4341–4349 (2005). [CrossRef] [PubMed]
6.	Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006). [CrossRef] [PubMed]
7.	Y. Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, and O. Cohen, “Demonstration of wavelength conversion at 40 Gb/s data rate in silicon waveguides,” Opt. Express 14, 11721–11726 (2006). [CrossRef] [PubMed]
8.	M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006). [CrossRef] [PubMed]
9.	M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express 15, 12949–12958 (2007). [CrossRef] [PubMed]
10.	W. Mathlouthi, H. Rong, and M. Paniccia, “Characterization of efficient wavelength conversion by four-wave mixing in sub-micron silicon waveguides,” Opt. Express 16, 16735–16745 (2008). [CrossRef] [PubMed]
11.	P. Koonath, D. R. Solli, and B. Jalali, “High efficiency CARS conversion in silicon,” in Conference on Lasers and Electro-Optics and on Quantum Electronics and Laser Science , (2008), pp. 1–2.
12.	N. Vermeulen, C. Debaes, and H. Thienpont, “The behavior of CARS in anti-Stokes Raman converters operating at exact Raman resonance,” IEEE J. Quantum Electron. 44, 1248–1255 (2008). [CrossRef]
13.	A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express 18, 1904–1908 (2010). [CrossRef] [PubMed]
14.	S. Gao, E. Tien, Q. Song, Y. Huang, and O. Boyraz, “Ultra-broadband one-to-two wavelength conversion using low-phase-mismatching four-wave mixing in silicon waveguides,” Opt. Express 18, 11898–11903 (2010). [CrossRef] [PubMed]
15.	N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman-and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron. 17, 1078–1091 (2011). [CrossRef]
16.	G. Agrawal, Nonlinear Fiber Optics , 3rd ed. (Academic Press, 2001).
17.	E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990). [CrossRef]
18.	B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006). [CrossRef]
19.	Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]
20.	Y. Chen, “Combined processes of stimulated Raman scattering and four-wave mixing in optical fibers,” J. Opt. Soc. Am. B 7, 43–52 (1990). [CrossRef]
21.	G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991). [CrossRef]
22.	Y. Chen and A.W. Snyder, “Four-photon parametric mixing in optical fibers: effect of pump depletion,” Opt. Lett. 14, 87–89 (1989). [CrossRef] [PubMed]
23.	T. Sylvestre, H. Maillotte, E. Lantz, and P. Tchofo Dinda, “Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber,” Opt. Lett. 24, 1561–1563 (1999). [CrossRef]
24.	R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003). [CrossRef] [PubMed]
25.	ITU-T Rec. G.694.1, “Spectral grids for WDM applications: DWDM frequency grid” (2002).

OCIS Codes
(130.4310) Integrated optics : Nonlinear
(160.4330) Materials : Nonlinear optical materials
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(190.5650) Nonlinear optics : Raman effect

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 6, 2011
Revised Manuscript: August 7, 2011
Manuscript Accepted: August 11, 2011
Published: September 13, 2011

Citation
Yannick Lefevre, Nathalie Vermeulen, Christof Debaes, and Hugo Thienpont, "Optimized wavelength conversion in silicon waveguides based on “off-Raman-resonance” operation: extending the phase mismatch formalism," Opt. Express 19, 18810-18826 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-18810

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References

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express11, 2862–2872 (2003). [CrossRef] [PubMed]
D. Dimitropoulos, V. Raghunathan, R. Claps, and B. Jalali, “Phase-matching and nonlinear optical processes in silicon waveguides,” Opt. Express12, 149–160 (2004). [CrossRef] [PubMed]
V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Lightwave Technol.23, 2094–2102 (2005). [CrossRef]
H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express13, 4629–4637 (2005). [CrossRef] [PubMed]
R. Espinola, J. Dadap, J. Osgood, S. McNab, and Y. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express13, 4341–4349 (2005). [CrossRef] [PubMed]
Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express14, 4786–4799 (2006). [CrossRef] [PubMed]
Y. Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, and O. Cohen, “Demonstration of wavelength conversion at 40 Gb/s data rate in silicon waveguides,” Opt. Express14, 11721–11726 (2006). [CrossRef] [PubMed]
M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature441, 960–963 (2006). [CrossRef] [PubMed]
M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express15, 12949–12958 (2007). [CrossRef] [PubMed]
W. Mathlouthi, H. Rong, and M. Paniccia, “Characterization of efficient wavelength conversion by four-wave mixing in sub-micron silicon waveguides,” Opt. Express16, 16735–16745 (2008). [CrossRef] [PubMed]
P. Koonath, D. R. Solli, and B. Jalali, “High efficiency CARS conversion in silicon,” in Conference on Lasers and Electro-Optics and on Quantum Electronics and Laser Science, (2008), pp. 1–2.
N. Vermeulen, C. Debaes, and H. Thienpont, “The behavior of CARS in anti-Stokes Raman converters operating at exact Raman resonance,” IEEE J. Quantum Electron.44, 1248–1255 (2008). [CrossRef]
A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express18, 1904–1908 (2010). [CrossRef] [PubMed]
S. Gao, E. Tien, Q. Song, Y. Huang, and O. Boyraz, “Ultra-broadband one-to-two wavelength conversion using low-phase-mismatching four-wave mixing in silicon waveguides,” Opt. Express18, 11898–11903 (2010). [CrossRef] [PubMed]
N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman-and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron.17, 1078–1091 (2011). [CrossRef]
G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).
E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron.26, 1815–1820 (1990). [CrossRef]
B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron.12, 412–421 (2006). [CrossRef]
Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express15, 16604–16644 (2007). [CrossRef] [PubMed]
Y. Chen, “Combined processes of stimulated Raman scattering and four-wave mixing in optical fibers,” J. Opt. Soc. Am. B7, 43–52 (1990). [CrossRef]
G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B8, 824–838 (1991). [CrossRef]
Y. Chen and A.W. Snyder, “Four-photon parametric mixing in optical fibers: effect of pump depletion,” Opt. Lett.14, 87–89 (1989). [CrossRef] [PubMed]
T. Sylvestre, H. Maillotte, E. Lantz, and P. Tchofo Dinda, “Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber,” Opt. Lett.24, 1561–1563 (1999). [CrossRef]
R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express11, 1731–1739 (2003). [CrossRef] [PubMed]
ITU-T Rec. G.694.1, “Spectral grids for WDM applications: DWDM frequency grid” (2002).

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Optimized wavelength conversion in silicon waveguides based on “off-Raman-resonance” operation: extending the phase mismatch formalism

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Abstract

1. Introduction

2. Theoretical model for wavelength conversion in the strong pump approximation

3. Extended phase mismatch formalism and four-wave-mixing dispersion

4. Impact of the pump-Stokes frequency difference ΔΩ

5. Numerical simulation results

5.1. The potential of working off Raman resonance

5.2. Optimal frequency difference ΔΩ_Opt

5.3. Estimation of the optimal frequency difference ΔΩ

6. Conclusion

Acknowledgments

References and links

References

IEEE J. Quantum Electron.

IEEE J. Sel. Top. Quantum Electron.

J. Lightwave Technol.

J. Opt. Soc. Am. B

Nature

Opt. Express

Opt. Lett.

Other

Cited By

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OpticsInfoBase

Optics Express

Optimized wavelength conversion in silicon waveguides based on “off-Raman-resonance” operation: extending the phase mismatch formalism

Article Tools

Article Navigation

Abstract

1. Introduction

2. Theoretical model for wavelength conversion in the strong pump approximation

3. Extended phase mismatch formalism and four-wave-mixing dispersion

4. Impact of the pump-Stokes frequency difference ΔΩ

5. Numerical simulation results

5.1. The potential of working off Raman resonance

5.2. Optimal frequency difference ΔΩOpt

5.3. Estimation of the optimal frequency difference ΔΩ

6. Conclusion

Acknowledgments

References and links

References

IEEE J. Quantum Electron.

IEEE J. Sel. Top. Quantum Electron.

J. Lightwave Technol.

J. Opt. Soc. Am. B

Nature

Opt. Express

Opt. Lett.

Other

Cited By

Figures

5.2. Optimal frequency difference ΔΩ_Opt