Modal theory of slow light enhanced third-order nonlinear effects in photonic crystal waveguides

doi:10.1364/OE.20.020043

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Modal theory of slow light enhanced third-order nonlinear effects in photonic crystal waveguides

Tao Chen, Junqiang Sun, and Linsen Li »View Author Affiliations

Optics Express, Vol. 20, Issue 18, pp. 20043-20058 (2012)
http://dx.doi.org/10.1364/OE.20.020043

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▸ Article Outline
– Abstract
1. Introduction
2. Modal theory for third-order nonlinear effects
3. Parameters of coupled-mode equations
4. Theoretical analysis and numerical simulation
5. Conclusion
– References and links

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Abstract

In this paper, we derive the couple-mode equations for third-order nonlinear effects in photonic crystal waveguides by employing the modal theory. These nonlinear interactions include self-phase modulation, cross-phase modulation and degenerate four-wave mixing. The equations similar to that in nonlinear fiber optics could be expanded and applied for third-order nonlinear processes in other periodic waveguides. Based on the equations, we systematically analyze the group-velocity dispersion, optical propagation loss, effective interaction area, slow light enhanced factor and phase mismatch for a slow light engineered silicon photonic crystal waveguide. Considering the two-photon and free-carrier absorptions, the wavelength conversion efficiencies in two low-dispersion regions are numerically simulated by utilizing finite difference method. Finally, we investigate the influence of slow light enhanced multiple four-wave-mixing process on the conversion efficiency.

1. Introduction

Periodic dielectric structures have long been known to exhibit peculiar propagation characteristics such as slow light and band gap. Recently, the photonic crystal waveguides (PCWs) have attracted huge interest for the development of compact optical devices owing to slow light and strong field confinement [1

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008). [CrossRef]

–3

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 344–356 (2010). [CrossRef]

]. The slow light reduces the propagation velocity of the electric-field energy, thus it may enhance the interactions between the lights and nonlinear material [4

A. Baron, A. Ryasnyanskiy, N. Dubreuil, P. Delaye, Q. Vy Tran, S. Combrié, A. De Rossi, R. Frey, and G. Roosen, “Light localization induced enhancement of third order nonlinearities in a GaAs photonic crystal waveguide,” Opt. Express 17, 552–557 (2009). [CrossRef] [PubMed]

]. Meanwhile, the strong field confinement leads to small effective interaction area, resulting in an efficient nonlinear interaction. The enhanced nonlinear effects in PCWs, such as self-phase modulation (SPM) [5

S. Combrié, Q. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95, 221108 (2009). [CrossRef]

], cross-phase modulation (XPM) [6

K. Inoue, H. Oda, N. Ikeda, and K. Asakawa, “Enhanced third-order nonlinear effects in slow-light photonic-crystal slab waveguides of line-defect,” Opt. Express 17, 7206–7216 (2009). [CrossRef] [PubMed]

], two-photon absorption (TPA) [2

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17, 2944–2953 (2009). [CrossRef] [PubMed]

], stimulated Raman scattering [7

J. McMillan, M. Yu, D. Kwong, and C. Wong, “Observation of spontaneous Raman scattering in silicon slow-light photonic crystal waveguides,” Appl. Phys. Lett. 93, 251105 (2008). [CrossRef]

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B 82, 041308 (2010). [CrossRef]

], third-harmonic generation (THG) [9

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009). [CrossRef]

,10

C. Monat, C. Grillet, B. Corcoran, D. J. Moss, B. J. Eggleton, T. P. White, and T. F. Krauss, “Investigation of phase matching for third-harmonic generation in silicon slow light photonic crystal waveguides using Fourier optics,” Opt. Express 18, 6831–6840 (2010). [CrossRef] [PubMed]

], four-wave mixing (FWM) [11

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express 18, 15484–15497 (2010). [CrossRef] [PubMed]

], spontaneous FWM for quantum entangled resource [12

C. Xiong, C. Monat, A. Clark, C. Grillet, G. Marshall, M. Steel, J. Li, L. O’Faolain, T. Krauss, J. Rarity, and B. J. Eggleton, “Slow-light enhanced correlated photon pair generation in a silicon photonic crystal waveguide,” Opt. Lett. 36, 3413–3415 (2011). [CrossRef] [PubMed]

], generation and propagation of temporal solitons [13

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics 4, 862–868 (2010). [CrossRef]

], and so on have already been demonstrated.

The FWM process for wavelength conversion has been used extensively in all-optical signal processing, for example in optical regeneration [14

M. Ebnali-Heidari, C. Monat, C. Grillet, and M. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express 17, 18340–18353 (2009). [CrossRef] [PubMed]

]. In conventional PCWs, the strong dispersion of slow light can cause the phase mismatch that result in a narrow bandwidth of less than 1 or 2 nm for the FWM process. In order to overcome this problem, the dispersion engineered PCWs with flat-band and low-dispersion slow light have been proposed [15

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008). [CrossRef] [PubMed]

,16

S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010). [CrossRef]

]. There are two low-dispersion regions in the slow light engineered PCWs. One is the high frequency region with smaller group index and very low propagation loss. In this situation, the enhanced nonlinear interaction is mainly due to the small effective interaction area. Based on this region, the FWM has been observed in a 1.3 mm-long GaInP PCW [17

V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett. 35, 1440–1442 (2010). [CrossRef] [PubMed]

], and the time demultiplex and parametric gain using FWM have been investigated in 1.5-mm long dispersion engineered GaInP PCWs [18

I. Cestier, A. Willinger, V. Eckhouse, G. Eisenstein, S. CombriÚ, P. Colman, G. Lehoucq, and A. De Rossi, “Time domain switching/demultiplexing using four wave mixing in GaInP photonic crystal waveguides,” Opt. Express 19, 6093–6099 (2011). [CrossRef] [PubMed]

–20

P. Colman, I. Cestier, A. Willinger, S. Combrié, G. Lehoucq, G. Eisenstein, and A. De Rossi, “Observation of parametric gain due to four-wave mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett. 36, 2629–2631 (2011). [CrossRef] [PubMed]

]. The other is the low-frequency slow light region with relatively low group velocity and large propagation loss, the enhancement of the FWM process may scale with the fourth power of the slowdown factor. Utilizing the slow light in this region, the enhanced FWM and time demultiplex have been demonstrated in the short (80 μm or 96 μm) dispersion-engineered silicon PCWs [21

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express 18, 22915–22927 (2010). [CrossRef] [PubMed]

, 22

B. Corcoran, M. D. Pelusi, C. Monat, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Ultracompact 160Gbaud all-optical demultiplexing exploiting slow light in an engineered silicon photonic crystal waveguide,” Opt. Lett. 36, 1728–1730 (2011). [CrossRef] [PubMed]

], and the optimized device length for the conversion efficiency in the presence of propagation losses has been studied [23

J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express 19, 4458–4463 (2011). [CrossRef] [PubMed]

For the FWM process in PCWs, the group index changes so rapidly as a function of frequency that the slow light enhanced factors, propagation losses and mode overlap integrals can not be treat as constants like in slab waveguides or fibers [24

N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 257–266 (2010). [CrossRef]

]. In order to research and compare the FWM in the two low-dispersion regions with different group indices, the accurate calculation of above nonlinear effective coefficients is necessary. Moreover, the guided-mode fields in PCWs are periodic along the propagation direction, the period of PCW has to be taken into account. Santagiustina et.al have derived the nonlinear propagation equations for the FWM in PCW firstly [25

M. Santagiustina, C. Someda, G. Vadala, S. Combrie, and A. De Rossi, “Theory of slow light enhanced four-wave mixing in photonic crystal waveguides,” Opt. Express 18, 21024–21029 (2010). [CrossRef] [PubMed]

], where the slow light enhanced factors and mode field superpositions are contained in the relative effective volumes. Recently, they have investigated the relative effective volumes in detail [26

S. Roy, M. Santagiustina, P. Colman, S. Combrie, and A. De Rossi, “Modeling the Dispersion of the Nonlinearity in Slow Mode Photonic Crystal Waveguides,” IEEE Photon. J. 4, 224–233 (2012). [CrossRef]

]. In the conventional nonlinear fiber optics, however, the effective interaction areas which describe the mode field overlap are the widely used parameters. Based on these parameters, one can directly compare the results with those in different other systems such as nanowire, ridge waveguide and fiber.

In this paper, we derive the modal Eqs. for the slowly varying amplitudes of underlying fast oscillating Bloch modes. These coupled-mode Eqs., which are presented in the similar style like that in nonlinear fiber optics [27

G. Agrawal, Nonlinear Fiber Optics and Applications of Nonlinear Fiber Optics , 4th ed. (Elsevier Science, New York, 2007).

], are the most appropriate description. In addition, the Eqs. could also be adopted for the third-order nonlinear processes in other periodic waveguides, e.g. photonic crystal coupled-resonator optical waveguides (CROW) [28

N. Matsuda, T. Kato, K. Harada, H. Takesue, E. Kuramochi, H. Taniyama, and M. Notomi, “Slow light enhanced optical nonlinearity in a silicon photonic crystal coupled-resonator optical waveguide,” Opt. Express 19, 19861–19874 (2011). [CrossRef] [PubMed]

], and one-dimensional grating waveguides with flat-band slow light [29

C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett. 24, 7–9 (2012). [CrossRef]

]. Based on the formulaes of our model, the nonlinear effective coefficients are calculated over the whole guide-mode region for a slow light engineered Si PCW. According to the parameters, we theoretically investigate the FWM processes of the two low-dispersion regions in detail.

2. Modal theory for third-order nonlinear effects

Since the nonlinearity is small, the third-order nonlinear interaction produce a perturbative polarization for the Maxwell’s equations of linear PCW. We start from the the linear Bloch modes of the PCW for the fixed frequencies and initial conditions, the unperturbed electric and magnetic fields are as followed

\begin{array}{l} E_{1} = e_{p, s, i} (x, y, z) exp [i k_{p, s, i} (ω) x], \\ H_{1} = h_{p, s, i} (x, y, z) exp [i k_{p, s, i} (ω) x] . \end{array}

(1)

These fields propagate along the positive x direction in the PCW. The p,s,i refer to the pump, signal and idler waves for degenerate FWM (2ω_p = ω_s + ω_i). e_p,s,i(x, y, z) and h_p,s,i(x, y, z) denote the lattice-periodic electric and magnetic field component. The z dependence of e_p,s,i(x, y, z) and h_p,s,i(x, y, z) are similar to the fundamental mode of slab waveguide mode, the x and y dependence are consistent with the modulation behavior of PCW in the x and y directions [30

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B 81, 155115 (2010). [CrossRef]

]. k_p,s,i(ω) is the respective Bloch vectors.

In order to exploit the conjugated form of the Lorentz reciprocity theorem [31

A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).

], under non-linear polarization P_NL we define the perturbed electromagnetic fields in the time domain

\begin{array}{l} E_{2} = & \frac{1}{2} a_{p} (x, t) e_{p} (x, y, a) exp (i k_{p} x) exp (- i ω_{p} t) + \frac{1}{2} a_{s} (x, t) e_{s} (x, y, z) exp (i k_{s} x) exp (- i ω_{s} t) \\ + \frac{1}{2} a_{i} (x, t) e_{i} (x, y, z) exp (i k_{i} x) exp (- i ω_{i} t) + c . c ., \\ H_{2} = & \frac{1}{2} a_{p} (x, t) h_{p} (x, y, z) exp (i k_{p} x) exp (- i ω_{p} t) + \frac{1}{2} a_{s} (x, t) h_{s} (x, y, z) exp (i k_{s} x) exp (- i ω_{s} t) \\ + \frac{1}{2} a_{i} (x, t) h_{i} (x, y, z) exp (i k_{i} x) exp (- i ω_{i} t) + c . c . \end{array}

(2)

where the ω_p, ω_s, ω_i are the central frequencies of the three waves, and k_p, k_s, k_i are the corresponding central Bloch vectors. a_p(x, t), a_s(x, t), a_i(x, t) denote the slowly varying envelopes. The P_NL, E₂, H₂ satisfy the Maxwell’s equations, and we can obtain their relationship for non-absorptive materials [32

D. Michaelis, U. Peschel, C. Wächter, and A. Bräuer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E 68, 065601–065601 (2003). [CrossRef]

]

\int_{S} (E_{2} \times H_{1}^{*} + E_{1}^{*} \times H_{2}) d A = i ω \int_{V} E_{1}^{*} \cdot P_{N L} d V .

(3)

where S is the surface of the volume V. The three waves are TE guided modes. Although the TE mode of PCW have a polarization component along the x direction, it contains only a small fraction of incident power, and the nonlinear effects are dominated by the transverse y component. For this reason, we simplify the problem considerably by adopting a scalar approach and writing the total electric field the third-order nonlinear polarization P_NL for degenerate FWM in the form

P_{N L} = \frac{1}{2} [P_{p} exp (i k_{p} x - i ω_{p} t) + P_{s} exp (i k_{s} x - i ω_{s} t) + P_{i} exp (i k_{i} x - i ω_{i} t)] + c . c .

(4)

The P_p, P_s, P_i are the envelopes of the polarization at ω_p, ω_s, ω_i and given by

\begin{array}{l} P_{p} = \frac{3 ε_{0}}{4} χ [{| a_{p} e_{p} |}^{2} a_{p} e_{p} + 2 ({| a_{s} e_{s} |}^{2} + {| a_{i} e_{i} |}^{2}) a_{p} e_{p} + 2 a_{p}^{*} a_{s} a_{i} e_{p}^{*} e_{s} e_{i} exp [i (k_{s} + k_{i} - 2 k_{p}) x], \\ P_{s} = \frac{3 ε_{0}}{4} χ [{| a_{s} e_{s} |}^{2} a_{s} e_{s} + 2 ({| a_{p} e_{p} |}^{2} + {| a_{i} e_{i} |}^{2}) a_{s} e_{s} + a_{i}^{*} a_{p} a_{p} e_{i}^{*} e_{p} e_{p} exp [- i (k_{s} + k_{i} - 2 k_{p}) x], \\ P_{i} = \frac{3 ε_{0}}{4} χ [{| a_{i} e_{i} |}^{2} a_{i} e_{i} + 2 ({| a_{p} e_{p} |}^{2} + {| a_{s} e_{s} |}^{2}) a_{i} e_{i} + a_{s}^{*} a_{p} a_{p} e_{s}^{*} e_{p} e_{p} exp [- i (k_{s} + k_{i} - 2 k_{p}) x] . \end{array}

(5)

The available third-order tensors of Si are

χ_{2222}^{(3)}

and

χ_{1122}^{(3)}

, but the x-component electric field is relatively small. For simplicity, we neglect the

χ_{1122}^{(3)}

tensor, and employ

χ = χ_{2222}^{(3)} (x, y, z)

and e_p,s,i = e_p,s,i(x, y, z). χ depicts the varied third-order nonlinearity as a function of three-dimensional coordinate (x, y, z). The first two terms on the right-hand side describe the nonlinear phase shift due to SPM and XPM. The last term describes the energy transfer between the interacting fields. The THG and other FWM are neglected by assuming that they will not be phase matched.

For a W1 PCW which is formed by removing a row of nearest neighbor holes to create a defect waveguide along the ΓK direction, the period in the x direction is lattice constant a. Figure 1 depicts the electric-field energy density ε|E|² profile for even TE guided mode by using 3D plane wave expansion (PWE) method [33

S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). [CrossRef] [PubMed]

]. It is clearly seen that the field is varied with the confinement behavior of PCW along x direction in one period, thus the spatial integrals should be over the unit cell Ω in Fig. 1. The slowly varying envelopes change slowly over many periods, and can be moved out of the integrals [34

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008). [CrossRef]

, 35

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81, 023820 (2010). [CrossRef]

]. We transform the electromagnetic fields of Eq. (2) into frequency domain by using Fourier transform, and perform the integration of Eq. (3) by combining the unperturbed fields in Eq. (1). Considering the sufficiently small translational period over a in x direction, we can obtain the differential equations under the rotating-wave approximation (RWA) [32

D. Michaelis, U. Peschel, C. Wächter, and A. Bräuer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E 68, 065601–065601 (2003). [CrossRef]

, 34

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008). [CrossRef]

, 35

\begin{array}{l} \frac{\partial {\tilde{a}}_{p} (x, ω)}{\partial x} - i [k_{p} (ω) - k_{p}] {\tilde{a}}_{p} (x, ω) = \frac{i ω}{2 a S_{p}} exp [i (k_{p} (ω) - k_{p})] \int_{Ω} e_{p}^{*} \cdot P_{N L} exp [- i k_{p} (ω) x] d V, \\ \frac{\partial {\tilde{a}}_{s} (x, ω)}{\partial x} - i [k_{s} (ω) - k_{s}] {\tilde{a}}_{s} (x, ω) = \frac{i ω}{2 a S_{s}} exp [i (k_{s} (ω) - k_{s})] \int_{Ω} e_{s}^{*} \cdot P_{N L} exp [- i k_{s} (ω) x] d V, \\ \frac{\partial {\tilde{a}}_{i} (x, ω)}{\partial x} - i [k_{i} (ω) - k_{i}] {\tilde{a}}_{i} (x, ω) = \frac{i ω}{2 a S_{i}} exp [i (k_{i} (ω) - k_{i})] \int_{Ω} e_{i}^{*} \cdot P_{N L} exp [- i k_{i} (ω) x] d V . \end{array}

(6)

where ã_p(x, ω), ã_s(x, ω), ã_i(x, ω) are the Fourier transforms of the three-waves slowly varying envelopes.

S_{p, s, i} = \frac{1}{2} Re \int e_{p, s, i} \times h_{p, s, i}^{*} d A

is the time-average modal Poynting vector flux the y–z plane boundary of per unit cell along x direction. Due to the orthogonality relation of PCW modes, S_p,s,i is for the unperturbed system and unrelated to x. We expand the k_p,s,i(ω) by a Taylor series about the respective central Bloch vector at frequency domain as

k_{p, s, i} (ω) = k_{p, s, i} + (ω - ω_{p, s, i}) β_{p, s, i}^{1} + \frac{1}{2} {(ω - ω_{p, s, i})}^{2} β_{p, s, i}^{2} + \dots

(7)

where

β_{p, s, i}^{n} = {[\frac{\partial^{n} k_{p, s, i} (ω)}{\partial ω^{n}}]}_{ω_{p, s, i}}

, (n = 1, 2,...). The

β_{p, s, i}^{1}

and

β_{p, s, i}^{2}

stand for the reciprocal value of group velocity (GV) and group velocity dispersion (GVD), respectively. We transform Eq. (6) back to the time domain, insert Eq. (5) to it and replace (ω − ω_p,s,i) with

i \frac{\partial}{\partial t}

, we obtain

\begin{matrix} \frac{\partial a_{p} (x, t)}{\partial x} + β_{p}^{1} \frac{\partial a_{p} (x, t)}{\partial t} + \frac{i}{2} β_{p}^{2} \frac{\partial a_{p}^{2} (x, t)}{\partial t^{2}} = [{| a_{p} |}^{2} a_{p} \int_{Ω} χ e_{p}^{*} {| e_{p} |}^{2} e_{p} d V + 2 {| a_{s} |}^{2} a_{p} \int_{Ω} χ e_{p}^{*} {| e_{s} |}^{2} e_{p} d V \\ + 2 {| a_{i} |}^{2} a_{p} \int_{Ω} χ e_{p}^{*} {| e_{i} |}^{2} e_{p} d V + 2 a_{p}^{*} a_{s} a_{i} exp (i Δ k x) \int_{Ω} χ e_{p}^{*} e_{p}^{*} e_{s} e_{i} d V] \frac{i ω_{p}}{2 a S_{p}} \frac{3 ε_{0}}{8}, \\ \frac{\partial a_{s} (x, t)}{\partial x} + β_{s}^{1} \frac{\partial a_{s} (x, t)}{\partial t} + \frac{i}{2} β_{s}^{2} \frac{\partial a_{s}^{2} (x, t)}{\partial t^{2}} = [{| a_{s} |}^{2} a_{s} \int_{Ω} χ e_{s}^{*} {| e_{s} |}^{2} e_{s} d V + 2 {| a_{p} |}^{2} a_{s} \int_{Ω} χ e_{s}^{*} {| e_{p} |}^{2} e_{s} d V \\ + 2 {| a_{i} |}^{2} a_{s} \int_{Ω} χ e_{s}^{*} {| e_{i} |}^{2} e_{s} d V + a_{i}^{*} a_{p} a_{p} exp (- i Δ k x) \int_{Ω} χ e_{s}^{*} e_{i}^{*} e_{p} e_{p} d V] \frac{i ω_{s}}{2 a S_{s}} \frac{3 ε_{0}}{8}, \\ \frac{\partial a_{i} (x, t)}{\partial x} + β_{i}^{1} \frac{\partial a_{i} (x, t)}{\partial t} + \frac{i}{2} β_{i}^{2} \frac{\partial a_{i}^{2} (x, t)}{\partial t^{2}} = [{| a_{i} |}^{2} a_{i} \int_{Ω} χ e_{i}^{*} {| e_{i} |}^{2} e_{i} d V + 2 {| a_{p} |}^{2} a_{i} \int_{Ω} χ e_{i}^{*} {| e_{p} |}^{2} e_{i} d V \\ + 2 {| a_{s} |}^{2} a_{i} \int_{Ω} χ e_{i}^{*} {| e_{s} |}^{2} e_{i} d V + a_{s}^{*} a_{p} a_{p} exp (- i Δ k x) \int_{Ω} χ e_{i}^{*} e_{s}^{*} e_{p} e_{p} d V] \frac{i ω_{i}}{2 a S_{i}} \frac{3 ε_{0}}{8} . \end{matrix}

(8)

where Δk = k_s + k_i − 2k_p expresses the phase mismatch. Owing to the equivalence of energy and group velocity, the time-average modal Poynting vector S_p,s,i depend on the group velocity

(v_{p, s, i}^{g} = 1 / β_{p, s, i}^{1} = c / n_{p, s, i}^{g})

, where

n_{p, s, i}^{g}

is the group index. Normalizing the fields via

A_{p, s, i} (x, t) = a_{p, s, i} (x, t) \sqrt{S_{p, s, i}}

, we can obtain

\begin{matrix} \frac{\partial A_{p} (x, t)}{\partial x} + β_{p}^{1} \frac{\partial A_{p} (x, t)}{\partial t} + \frac{i}{2} β_{p}^{2} \frac{\partial A_{p}^{2} (x, t)}{\partial t^{2}} = \frac{i n_{2} ω_{p}}{c} [{(m_{p})}^{2} {| A_{P} |}^{2} A_{P} f_{p p p p} + 2 m_{p} m_{s} {| A_{s} |}^{2} A_{P} f_{p s p s} \\ + 2 m_{p} m_{i} {| A_{i} |}^{2} A_{P} f_{p i p i} + 2 m_{p} \sqrt{m_{s}} \sqrt{m_{i}} A_{p}^{*} A_{s} A_{i} exp (i Δ k x) f_{p p s i}], \\ \frac{\partial A_{s} (x, t)}{\partial x} + β_{s}^{1} \frac{\partial A_{s} (x, t)}{\partial t} + \frac{i}{2} β_{s}^{2} \frac{\partial A_{s}^{2} (x, t)}{\partial t^{2}} = \frac{i n_{2} ω_{s}}{c} [{(m_{s})}^{2} {| A_{s} |}^{2} A_{s} f_{s s s s} + 2 m_{s} m_{p} {| A_{p} |}^{2} A_{s} f_{s p s p} \\ + 2 m_{s} m_{i} {| A_{i} |}^{2} A_{s} f_{s i s i} + m_{p} \sqrt{m_{s}} \sqrt{m_{i}} A_{i}^{*} A_{p} A_{p} exp (- i Δ k x) f_{s i p p}], \\ \frac{\partial A_{i} (x, t)}{\partial x} + β_{i}^{1} \frac{\partial A_{i} (x, t)}{\partial t} + \frac{i}{2} β_{i}^{2} \frac{\partial A_{i}^{2} (x, t)}{\partial t^{2}} = \frac{i n_{2} ω_{i}}{c} [{(m_{i})}^{2} {| A_{i} |}^{2} A_{i} f_{i i i i} + 2 m_{p} m_{i} {| A_{p} |}^{2} A_{i} f_{i p i p} \\ + 2 m_{s} m_{i} {| A_{s} |}^{2} A_{i} f_{i s i s} + m_{p} \sqrt{m_{s}} \sqrt{m_{i}} A_{s}^{*} A_{p} A_{p} exp (- i Δ k x) f_{i s p p}] . \end{matrix}

(9)

where

n_{2} = \frac{3 χ_{2222}^{(3)}}{4 c n^{2} ε_{0}}

is the nonlinear Kerr coefficient with the effective index n ≈ n_p ≈ n_s ≈ n_i [36

A. Yariv, Photonics: Optical Electronics in Modern Communications , 6th ed. (Oxford University Press, USA, 2007).

], the n_p,s,i is refractive index. Due to the low group velocity in PCW,

m_{p, s, i} = \frac{n_{p, s, i}^{g}}{n_{p, s, i}}

is the well-known slow light factor [37

T. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D: Appl. Phys. 40, 2666–2670 (2007). [CrossRef]

, 38

M. Soljacic and J. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–220 (2004). [CrossRef] [PubMed]

]. |A_p,s,i(x, t)|² is the powers of the three waves which pass through the x-dependence unit cell. The overlap of fields is calculated by

f_{1234} = \frac{a {(ε_{n 1} ε_{n 2} ε_{n 3} ε_{n 4})}^{1 / 2} {(χ_{1111}^{(3)})}^{- 1} \int_{Ω} χ e_{1}^{*} e_{2}^{*} e_{3} e_{4} d V}{{[\int_{Ω} ε_{1} (r) {| e_{1} |}^{2} d V \int_{Ω} ε_{2} (r) {| e_{2} |}^{2} d V \int_{Ω} ε_{3} (r) {| e_{3} |}^{2} d V \int_{Ω} ε_{4} (r) {| e_{4} |}^{2} d V]}^{1 / 2}} .

(10)

where ε_np, ε_ns, ε_ni depict the dielectric constants of medium for three waves, and ε_p(r), ε_s(r), ε_i(r) illustrate the varied dielectric constants as a function of three-dimensional coordinate r. The effective interaction area is defined by

A_{eff}^{1234} = \frac{1}{f_{1234}}

. Similarly, we define the slow light enhanced factor as

τ_{1234} = \sqrt{m_{1} m_{2} m_{3} m_{4}} .

(11)

Fig. 1 Schematic of the modified W1 PCW (the modified geometry will be shown in next figure). Green region indicates the electric-field energy density of TE mode with group index 98. The rectangle outline depicts the real unit cell Ω. The y–z plane boundary of the rectangle is the area for the calculation of time-average modal Poynting vector.

Thus the Eqs. (9) can be expressed as

\begin{array}{l} \frac{\partial A_{p}}{\partial x} + β_{p}^{1} \frac{\partial A_{p}}{\partial t} + \frac{i}{2} β_{p}^{2} \frac{\partial A_{p}^{2}}{\partial t^{2}} = & i [γ_{p p p p} {| A_{P} |}^{2} A_{p} + 2 γ_{p p s s} {| A_{s} |}^{2} A_{p} + 2 γ_{p p i i} {| A_{i} |}^{2} A_{p} \\ + 2 γ_{p p s i} A_{p}^{*} A_{s} A_{i} exp (- i Δ k x)], \\ \frac{\partial A_{s}}{\partial x} + β_{s}^{1} \frac{\partial A_{s}}{\partial t} + \frac{i}{2} β_{s}^{2} \frac{\partial A_{s}^{2}}{\partial t^{2}} = & i [γ_{s s s s} {| A_{s} |}^{2} A_{s} + 2 γ_{p p s s} {| A_{p} |}^{2} A_{s} + 2 γ_{s s i i} {| A_{i} |}^{2} A_{s} \\ + γ_{p p s i} A_{i}^{*} A_{p} A_{p} exp (i Δ k x)], \\ \frac{\partial A_{i}}{\partial x} + β_{i}^{1} \frac{\partial A_{i}}{\partial t} + \frac{i}{2} β_{i}^{2} \frac{\partial A_{i}^{2}}{\partial t^{2}} = & i [γ_{i i i i} {| A_{i} |}^{2} A_{i} + 2 γ_{p p i i} {| A_{p} |}^{2} A_{i} + 2 γ_{s s i i} {| A_{s} |}^{2} A_{i} \\ + γ_{p p s i} A_{s}^{*} A_{p} A_{p} exp (i Δ k x)] . \end{array}

(12)

where the nonlinear coefficient is defined as

γ_{1234} \approx \frac{n_{2} ω}{c A_{eff}^{1234}} τ_{1234}

with ω ≈ ω_p ≈ ω_s ≈ ω_i. Here, we mainly deduce the differential equations for the degenerate FWM in PCW, and these equations are analogous to those in the nonlinear fiber optics. In addition, the deduced process for the nondegenerate FWM is similar to it.

3. Parameters of coupled-mode equations

3.1. Characteristics of slow light engineered PCW

The waveguide considered in this study is the slow light engineered air-bridge Si PCW [39

L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010). [CrossRef]

]. It is with lattice constant a = 410 nm, thickness h = 220 nm and hole radius r = 0.27a. The dispersion is engineered by shifting the first two rows of holes either side of the waveguide as shown in Fig. 2(a), and the displacement is s = −0.127a [15

]. As shown in Fig. 1, the supercell Ω is used in our numerical simulations with a size of

a \times 11 \sqrt{3} a \times 2.15 a

along the x, y, and z axis, respectively. The computational step size along the x, y, and z directions are a/16, a/16, and a/30, respectively. Figure 2(a) shows that the PCW has two TE-like guiding modes located in the frequency bandgap by using 3D PWE method. In order to utilize the even single mode propagation, only the normalized frequency in the thick solid red line of Fig. 2(a) is investigated.

Fig. 2 (a) Dispersion curves of two TE-like line-defect bands in slow light engineered Si PCW. The engineered PCW is obtained by symmetrically displacing the first rows of holes about the waveguide axis. The displacements relative to the unmodified lattice (red lines) are given by s, where shifts toward the waveguide centre are defined to be positive. (b) Group index and propagation loss as a function of the wavelength. (c) β₁ and GVD β₂ as a function of the wavelength.

Although the PCWs are intrinsically lossless, the high refractive contrast and fabrication imperfections such as sidewall roughness and lithographic inaccuracies cause the extrinsic losses including the out-of plane radiation losses and backscattering losses [40

S. Hughes, L. Ramunno, J. Young, and J. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005). [CrossRef] [PubMed]

]. The relationship between extrinsic losses and the group velocity is complex [41

L. O’Faolain, T. White, D. O’Brien, X. Yuan, M. Settle, and T. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express 15, 13129–13138 (2007). [CrossRef]

]. Recently, O’Faolain et al. have developed a theoretical model that can accurately describe the loss spectra of slow light engineered PCWs [39

]. The out-of plane radiation losses depend linearly on the group index, and the backscattering losses depend on its square. As shown in Fig. 2(b), we calculate the propagation losses using the modified MIT Photonics Band (MPB) code [16

S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010). [CrossRef]

]. The losses dramatically increase around wavelength 1605 nm, this will limit the conversion efficiency of FWM process.

The spatial distribution of the refraction index of the PCW is periodic along the x-axis, the Bloch theorem implies that the propagation constant, which is oriented along the x-axis, may be presented by

k = (k_{0} + m) \frac{2 π}{a}

, where m is an integer and represents the high-order Bloch harmonics and k₀ is restricted to the first Brillouin zone −0.5 ≤ k₀ ≤ 0.5 [30

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B 81, 155115 (2010). [CrossRef]

]. For generating forward FWM process, the Bloch wavevector and group velocity of the four waves are required to be parallel to each other [42

E. Centeno and C. Ciracì, “Theory of backward second-harmonic localization in nonlinear left-handed media,” Phys. Rev. B 78, 235101 (2008). [CrossRef]

]. Since the β₁ = ∂k/∂ω = 1/v^g defines the GV of a pulse with frequency ω, we consider the GV and group index to be positive. Thus, the Bloch wavevector may be taken in the range

\frac{π}{a} \leq k \leq \frac{2 π}{a}

, and the pulse GVD (β₂ = ∂²k/∂ω²) is negative. These wavelength dependence of dispersion coefficients (β₁ and β₂), which are calculated for the TE even mode, are shown in Fig. 2(c). There are two low-dispersion regions, one is from 1560 nm to 1580 and the other is from 1595 nm to 1605 nm. The available wavelength range in small group index region is larger than the one in high group index region.

3.2. Effective interaction areas

From Eq. (10), we calculate

A_{eff}^{p p p p}

as the SPM effective interaction area,

A_{eff}^{p p s s}

as the XPM effective interaction area and

A_{eff}^{p p s i}

as the FWM effective interaction area, where the λ_s = λ_P − Δλ and λ_i = λ_P + Δλ are assumed. Figure 3(d) depicts the calculated effective interaction areas for different nonlinear effects and Δλ. The effective interaction areas for SPM, FWM and XPM with Δλ = 1 nm are about the same. We also calculated the FWM effective interaction areas with Δλ = 3 nm and Δλ = 7 nm. It is found that the

A_{eff}^{p p s i}

agrees with

A_{eff}^{p p p p}

very well because the

A_{eff}^{p p p p}

is between

A_{eff}^{s s s s}

and

A_{eff}^{i i i i}

. However, the XPM effective interaction area

A_{eff}^{p p s s}

changes when the Δλ increases. If the Δλ is large, we can not replace

A_{eff}^{p p s s}

with

A_{eff}^{p p p p}

approximately but to calculate

A_{eff}^{p p s s}

accurately. Especially, it is found that the areas don’t monotonously increase with the increasing wavelength and group index as we expected. In order to explain this phenomenon, we plot three y-component electric fields for different wavelengthes and group index in Fig. 3(a), 3(b) and 3(c). The electric field basically exists in the line-defect of PCW for small group index as shown in Fig. 3(a), and it gradually spreads to two sides when the group index increases as shown in Fig. 3(b). But while the group index is large enough, the electric field gathers tightly in the line-defect and the dielectric among the first row of holes as shown in Fig. 3(c). We also investigated the effective interaction area for a conventional W1 PCW, the result is similar to this. The physical mechanics behind this phenomenon may be depended on the restricted behavior of periodic structure of PCW in the y direction.

Fig. 3 Electric field profiles of Bloch waves in X–Y and Y–Z plane for (a) wavelength 1565 nm with group index 5.11, (b) wavelength 1600 nm with group index 27.72, (c) wavelength 1615 nm with group index 118.14. (d) Effective interaction areas as a function of the pump wavelength λ_p.

In addition, since the energy of electric field mainly exist in the dielectric as shown in Fig. 1, sometimes the dielectric constant may be assumed to be homogeneous for the integral of effective interaction areas. There is an approximate calculation [3

]

A_{eff}^{'} = \frac{1}{a} \frac{{(\int_{Ω} {| e_{1} |}^{2} d V \int_{Ω} {| e_{2} |}^{2} d V \int_{Ω} {| e_{3} |}^{2} d V \int_{Ω} {| e_{4} |}^{2} d V)}^{1 / 2}}{\int_{Ω} e_{1}^{*} e_{2}^{*} e_{3} e_{4} d V} .

(13)

We calculate the SPM approximation of effective interaction area as defined by Eq. (13) in Fig. 3(d). It is seen that the approximation is about twice as larger as our model.

3.3. Slow light enhanced factors and phase mismatches

Based on Eq. (11), the slow light enhanced factors for SPM, XPM and FWM may be represented as τ_pppp, τ_ppss and τ_ppsi, respectively. Figure 4(a) shows the calculated slow light enhanced factors for different nonlinear interactions and Δλ. Similar to the effective interaction areas, the slow light enhanced factors for SPM, XPM and FWM with Δλ = 1 nm are close to each other. The XPM enhanced factor τ_ppss changes and is far away from the SPM enhanced factor τ_pppp as the Δλ increases, while the FWM enhanced factor is consistent with the SPM enhanced factor.

Fig. 4 (a) Slow light enhanced factors τ and (b) phase mismatch Δk as a function of the pump wavelength λ_p.

Figure 4(b) demonstrates the phase mismatch (Δk = k_s + k_i − 2k_p) for different Δλ. The two low-dispersion regions correspond to the small phase mismatch. Meanwhile, the phase mismatch increases following the increasing Δλ. In FWM process, the phase match is very important for the conversion efficiency and conversion bandwidth, so the wavelengths in such two regions is widely used.

In the following, according to these parameters we firstly study the conversion efficiency of degenerate FWM process by using finite difference method, specially for the two low-dispersion regions. Then, the more realistic multiple FWM process has been simulated.

4. Theoretical analysis and numerical simulation

4.1. Degenerate FWM in Si PCW

In Si-based waveguide structures, the losses include the linear propagation loss and the nonlinear losses caused by TPA and TPA-induced free-carrier absorption (FCA). Because the TPA is derived from the imaginary part of third-order nonlinearity

χ_{2222}^{(3)}

while the real part of

χ_{2222}^{(3)}

describes parametric processes [24

N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 257–266 (2010). [CrossRef]

], we can replace the coefficients γ₁₂₃₄ of SPM and XPM with

γ_{1234} + i \frac{β_{T P A} τ_{1234}}{2 A_{eff}^{1234}}

, where β_TPA is the coefficient of TPA. The Eqs. for degenerate FWM in Si PCW are obtained as

\begin{array}{l} \frac{\partial A_{p}}{\partial x} + β_{p}^{1} \frac{\partial A_{p}}{\partial t} + \frac{i}{2} β_{p}^{2} \frac{\partial A_{p}^{2}}{\partial t^{2}} = - \frac{1}{2} (α_{l p} + α_{f c p} + ζ_{p p p p} {| A_{p} |}^{2} + 2 ζ_{p p s s} {| A_{s} |}^{2} + 2 ζ_{p p i i} {| A_{i} |}^{2}) A_{p} \\ + i [δ_{n f c p} A_{p} + γ_{p p p p} {| A_{p} |}^{2} A_{P} + 2 γ_{p p s s} {| A_{s} |}^{2} A_{P} + 2 γ_{p p i i} {| A_{i} |}^{2} A_{p} + 2 γ_{p p s i} A_{p}^{*} A_{s} A_{i} exp (- i Δ k x)], \\ \frac{\partial A_{s}}{\partial x} + β_{s}^{1} \frac{\partial A_{s}}{\partial t} + \frac{i}{2} β_{s}^{2} \frac{\partial A_{s}^{2}}{\partial t^{2}} = - \frac{1}{2} (α_{l s} + α_{f c s} + ζ_{s s s s} {| A_{s} |}^{2} + 2 ζ_{p p s s} {| A_{p} |}^{2} + 2 ζ_{s s i i} {| A_{i} |}^{2}) A_{s} \\ + i [δ_{n f c s} A_{s} + γ_{s s s s} {| A_{s} |}^{2} A_{s} + 2 γ_{p p s s} {| A_{p} |}^{2} A_{s} + 2 γ_{s s i i} {| A_{i} |}^{2} A_{s} + γ_{p p s i} A_{i}^{*} A_{p} A_{p} exp (i Δ k x)], \\ \frac{\partial A_{i}}{\partial x} + β_{i}^{1} \frac{\partial A_{i}}{\partial t} + \frac{i}{2} β_{i}^{2} \frac{\partial A_{i}^{2}}{\partial t^{2}} = - \frac{1}{2} (α_{l i} + α_{f c i} + ζ_{i i i i} {| A_{i} |}^{2} + 2 ζ_{p p i i} {| A_{p} |}^{2} + 2 ζ_{s s i i} {| A_{s} |}^{2}) A_{i} \\ + i [δ_{n f c i} A_{i} + γ_{i i i i} {| A_{i} |}^{2} A_{i} + 2 γ_{p p i i} {| A_{p} |}^{2} A_{i} + 2 γ_{s s i i} {| A_{s} |}^{2} A_{i} + γ_{p p s i} A_{s}^{*} A_{p} A_{p} exp (i Δ k x)] . \end{array}

(14)

where α_lq(q = p, s, i) represents the linear propagation loss for the pump, signal and idler waves.

ζ_{1234} = \frac{β_{T P A} τ_{1234}}{A_{eff}^{1234}}

. The α_fcq = σ_qN_c represents the nonlinear loss from FCA, where σ_q is the FCA cross section and N_c is the free-carrier density. The

δ_{n f c q} = \frac{2 π}{λ_{q}} k_{q c} N_{c}

represents the dispersion from the FC dependence of the Si refractive index, where k_qcN_c is the FC-induced index change.

Under the assumption of strong pump and weak signal, the pump remains undepleted. Meanwhile, the FC-induced loss and index change are proportional to group index [24

N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 257–266 (2010). [CrossRef]

]. These slow light enhanced carrier parameters can be written as [2

, 43

L. Yin and G. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

]

\begin{array}{l} σ_{q} = 1.45 \times 10^{- 9} \sqrt{τ_{p p p p}} {(λ_{q} / 1550)}^{2} μ m^{2}, \\ k_{q c} = - 1.35 \times 10^{- 9} \sqrt{τ_{p p p p}} {(λ_{q} / 1550)}^{2} μ m^{3}, \\ \frac{\partial N_{c} (t)}{\partial t} = \frac{β_{T P A}}{2 h v_{p} {(A_{eff}^{p p p p})}^{2}} {| A_{p} |}^{4} - \frac{N_{c} (t)}{τ_{c}} . \end{array}

(15)

where λ_q (nm) is the wavelength, v_p is the central frequency of the pump. τ_c is the carrier lifetime (typically 100 ps∼1 ns) [14

]. With a CW pump, the steady-state free-carrier density is

N_{c} = τ_{c} \frac{β_{T P A}}{2 h v_{p} {(A_{eff}^{p p p p})}^{2}} {| A_{p} |}^{4}

In order to investigate the FWM over the whole guided-mode wavelength region, we first consider the Δλ = 1 nm. In this case, the differences of effective interaction areas and slow light enhanced factors among the SPM, XPM and FWM can be neglected as shown in Fig. 3(d) and Fig. 4(a). The Kerr and TPA coefficients of Si are n₂ = 6 × 10⁻⁶ μm²/W and β_TPA = 5 × 10⁻⁶ μm/W [43

L. Yin and G. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007). [CrossRef] [PubMed]

]. The carrier lifetime τ_c and input CW signal power are supposed to be 200 ps and 1 mW. We numerically solve the above FWM Eqs. (14) for a CW pump by using the finite difference method, and the FWM conversion efficiency is calculated with

η = 10 {log}_{10} \frac{P_{i} (L)}{P_{s} (0)} = 10 {log}_{10} \frac{{| A_{i} (L) |}^{2}}{{| A_{s} (0) |}^{2}}

From Eqs. (14) and (15), the loss, pump power and waveguide length are important for the FWM conversion efficiency of slow light with large group index. Figure 5(a) shows the conversion efficiency for different pump power with L = 300 μm and Δλ = 1 nm for signal power P_s = 1 mW. It is found that the TPA and FCA may reduce the conversion efficiency seriously when the pump power is high. There is an oscillatory behavior with P_p = 0.5 W when wavelength is larger than 1606 nm. Although the enhanced factor increases with the increasing group index, the loss raises more rapidly. So the loss limits the increase of conversion efficiency and the conversion efficiency will decrease when group index is big enough. Figure 5(b) illustrates the conversion efficiency for different device lengths. With the propagation loss and nonlinear losses, the wavelengths in the inflection points of curve diminish for increasing L. That is, the available wavelength range for FWM is reduced correspondingly.

Fig. 5 The conversion efficiency as a function of the pump wavelength λ_p (a) for fixed waveguide length L = 300 μm and different pump power P_p with or without TPA and FCA, (b) for fixed pump power P_p = 0.3 W, P_s = 1 mW and different device lengthes with or without TPA and FCA.

Next, we focus on the FWM in the two low-dispersion regions. The coupled pump power P_p is assumed to be 100 mW, and it may be reached experimentally [23

J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express 19, 4458–4463 (2011). [CrossRef] [PubMed]

]. In the large group index region, we consider the pump wavelengthes centered in 1599 nm and 1604 nm, with group indices 26.31 and 27.87 and propagation losses 27.21 dB/cm and 41.48 dB/cm, respectively. And in the small group index region, the 1574 nm and 1579 nm pumps are considered. Their group indices are 6.00 and 7.03, and losses are 4.04 dB/cm and 5.46 dB/cm. In low-dispersion regions, the FWM with large Δλ may be achieved. Since the SPM and FWM effective interaction areas and enhanced factors almost are equal, we mainly investigate the XPM effective interaction areas and enhanced factors as shown in Figs. 6(a) and 6(b).

Fig. 6 (a) The XPM effective interaction areas as a function of Δλ. (b)The XPM enhanced factors as a function of Δλ, where Δλ = λ_p − λ_s

Figure 7(a) and 7(c) depict the conversion efficiency versus the waveguide length. It is shown that there exists an optimized waveguide length for the 1599 nm and 1604 nm pumps with Δλ = 3 nm because of the losses and large phase mismatch, and the optimized waveguide length is restricted within 400 μm. This length may be increased through engineering the dispersions of the two regions individually. The waveguide lengths for the two regions are chosen as L = 400 μm and L = 1 mm, respectively. Figure 7(b) and 7(d) show the relationship between conversion efficiency and wavelength detuning (Δλ). In the two regions, the 3-dB bandwidth of Δλ for wavelength with smaller group index is wider than the one with bigger group index. Although the maximum conversion efficiency in high group index region is larger than the small one, the coupled pump power of latter may be higher since its mode field and group index match the ridge access waveguides better [44

Q. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. 95, 061105 (2009). [CrossRef]

Fig. 7 The conversion efficiency as a function of the waveguide length L with Δλ = 1 nm and 3 nm for (a)λ_p = 1604 nm and λ_p = 1599 nm, (c) λ_p = 1579 nm and λ_p = 1574 nm. The conversion efficiency as a function of the Δλ for (b)λ_p = 1604 nm and λ_p = 1599 nm with L = 400 μm, (d) λ_p = 1579 nm and λ_p = 1574 nm with L = 1 mm. Where P_p = 0.1W and P_s = 1 mW are used.

4.2. Numerical Simulation for multiple FWM

In the actual experimental researches of FWM, two idler waves may appear as shown in Fig. 8 [17

,23

J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express 19, 4458–4463 (2011). [CrossRef] [PubMed]

]. The precise simulation of these results have to employ the multiple FWM theory [45

X. Liu, “Theory and experiments for multiple four-wave-mixing processes with multifrequency pumps in optical fibers,” Phys. Rev. A 77, 043818 (2008). [CrossRef]

]. Since |A_p|², |A_s|² ≫ |A_i|², |A_j|², we neglect the terms including |A_i|² and |A_j|². According to Eqs. (14), one can establish the multiple FWM Eqs. as

\begin{array}{l} \frac{\partial A_{p}}{\partial x} + β_{p}^{1} \frac{\partial A_{p}}{\partial t} + \frac{i}{2} β_{p}^{2} \frac{\partial A_{p}^{2}}{\partial t^{2}} = - \frac{1}{2} (α_{l p} + α_{f c p} + ζ_{p p p p} {| A_{p} |}^{2} + 2 ζ_{p p s s} {| A_{s} |}^{2}) A_{p} \\ + i [δ_{n f c p} A_{p} + γ_{p p p p} {| A_{p} |}^{2} A_{P} + 2 γ_{p p s s} {| A_{s} |}^{2} A_{P} + 2 γ_{p p s i} A_{p}^{*} A_{s} A_{i} exp (i Δ k_{p p s i} x) \\ + γ_{s s p j} A_{j}^{*} A_{s} A_{s} exp (- i Δ k_{s s p j} x) + 2 γ_{s p i j} A_{s}^{*} A_{i} A_{j} exp (i Δ k_{s p i j} x)], \\ \frac{\partial A_{s}}{\partial x} + β_{s}^{1} \frac{\partial A_{s}}{\partial t} + \frac{i}{2} β_{s}^{2} \frac{\partial A_{s}^{2}}{\partial t^{2}} = - \frac{1}{2} (α_{l s} + α_{f c s} + ζ_{s s s s} {| A_{s} |}^{2} + 2 ζ_{p p s s} {| A_{p} |}^{2}) A_{s} \\ + i [δ_{n f c s} A_{s} + γ_{s s s s} {| A_{s} |}^{2} A_{s} + 2 γ_{p p s s} {| A_{p} |}^{2} A_{s} + 2 γ_{s s p j} A_{s}^{*} A_{p} A_{j} exp (i Δ k_{s s p j} x) \\ + γ_{p p s i} A_{i}^{*} A_{p} A_{p} exp (- i Δ k_{p p s i} x) + 2 γ_{s p i j} A_{p}^{*} A_{i} A_{j} exp (i Δ k_{s p i j} x)], \\ \frac{\partial A_{i}}{\partial x} + β_{i}^{1} \frac{\partial A_{i}}{\partial t} + \frac{i}{2} β_{i}^{2} \frac{\partial A_{i}^{2}}{\partial t^{2}} = - \frac{1}{2} (α_{l i} + α_{f c i} + 2 ζ_{p p i i} {| A_{p} |}^{2} + 2 ζ_{s s i i} {| A_{s} |}^{2}) A_{i} \\ + i [δ_{n f c i} A_{i} + 2 γ_{p p i i} {| A_{p} |}^{2} A_{i} + 2 γ_{s s i i} {| A_{s} |}^{2} A_{i} + γ_{p p s i} A_{s}^{*} A_{p} A_{p} exp (- i Δ k_{p p s i} x) \\ + 2 γ_{s p i j} A_{j}^{*} A_{s} A_{p} exp (- i Δ k_{s p i j} x)] . \\ \frac{\partial A_{j}}{\partial x} + β_{j}^{1} \frac{\partial A_{j}}{\partial t} + \frac{i}{2} β_{j}^{2} \frac{\partial A_{j}^{2}}{\partial t^{2}} = - \frac{1}{2} (α_{l i} + α_{f c j} + 2 ζ_{p p j j} {| A_{p} |}^{2} + 2 ζ_{s s j j} {| A_{s} |}^{2}) A_{j} \\ + i [δ_{n f c j} A_{j} + 2 γ_{p p j j} {| A_{p} |}^{2} A_{j} + 2 γ_{s s j j} {| A_{s} |}^{2} A_{j} + γ_{s s p j} A_{s}^{*} A_{s} A_{p} exp (- i Δ k_{s s p j} x) \\ + 2 γ_{s p i j} A_{i}^{*} A_{s} A_{p} exp (- i Δ k_{s p i j} x)] . \end{array}

(16)

where Δk_ppsi = k_s + k_i − 2k_p, Δk_sspj = k_p + k_j − 2k_s and Δk_spij = k_i + k_j − k_p − k_s. The TPA and FCA parameters are from Eqs. (15). The FC lifetime τ_c is long enough and the time between two pulses is long compared to τ_c, we can neglect FC recombination within the picosecond duration of the probe pulses. The N_c can be calculated by integrating the TPA-induced FC generation term over the pulse duration.

Fig. 8 Scheme of multiple FWM process.

In fact, the multiple FWM process mainly includes two degenerate FWM processes and one nondegenerate FWM process. The two degenerate FWM processes are the ω_i = 2ω_p − ω_s and ω_j = 2ω_s − ω_p processes. The nondegenerate FWM process is the ω_j + ω_i = ω_p + ω_s. Since the generated two idler wave power are very low, the nondegenerate FWM process is not important. We numerically solve the above multiple FWM Eqs. for a 100 ps wide Gaussian pulse pump centered at 1604 nm and a CW probe signal at 1602 nm also by using the finite difference method. The corresponding λ_i = 1606 nm and λ_j = 1600 nm. To clearly display the pulse evolution within the waveguide, the pump, signal and two idler waves propagating along the waveguide are simulated in Fig. 9. From Fig. 9(b), the CW signal wave gradually evolves into a pulsed wave and the pulse is decreased owing to large propagation loss. Figures 9(c) and 9(d) display the two idler waves evolutions in the multiple FWM process. It is found that there is an optimized waveguide length with loss for the wavelength conversion in the FWM process. The L = 400 μm is considered in the rest of the discussion.

Fig. 9 Optical pulse evolutions along the waveguide with 100 ps pulse input pump with peak power 100 mW and 1 mW CW input signal, (a) pump wave power P_p, (b) signal wave power P_s, (c) idler wave power P_i, (d) idler wave power P_j.

The parametric gain is calculated as the ratio of output signal relative to the input CW signal. Figure 10(a) gives the output signal gain dependence on pump peak pulse powers. The gain is 3.8 dB which is achieved with an input peak pump power of 900 mW. Figure 10(b) illustrates the output pump and two idler pulses. The output pump has a width of 100 ps and is identical to the input pump pulse. The pulsed λ_j idler wave has the same pulse width to pump pulse when the λ_i idler wave exhibits pulse narrowing of about 30% to 70 ps. The two idler wave widths depend on the two degenerate FWM processes. The λ_i idler wave generation is determined by the time dependent pump pulse power, and gain is not a constant and will compress the λ_i idler pulse. On the other hand, the λ_j idler wave generation is approximate constant in time domain owing to the CW signal, so the temporal waveform is basically unchanged.

Fig. 10 (a) Output signal power gain for various coupled input peak pump powers. (b) Output pump and two idler pulses with input pump peak power 100 mW and signal power 1mW.

The effect of multiple FWM on the conversion efficiency is studied for different signal powers. Figure 11(a) and 11(b) display the idler peak powers and peak conversion efficiency as a function of the input signal power. From Fig. 11(a), the power of λ_j increase more rapidly than λ_i power. As shown in Fig. 11(b), the peak conversion efficiency for λ_j increases with the increasing input signal power when the one for λ_i keep changelessly. It seems that the generation of λ_j has nothing to do with the wavelength conversion of λ_i for small signal multiple FWM process. This phenomenons can also be understood through the two degenerate FWM processes.

Fig. 11 (a) The peak powers and (b) conversion efficiencies of two idler waves as a function of the input signal power with pump peak power 100 mW.

5. Conclusion

Based on the modal theory, we present a derive of the couple-mode Eqs. for third-order nonlinear effects in PCWs. The Eqs. are similar to that in nonlinear fiber optics, and may be conveniently used to solve the nonlinear processes. Considering a slow light engineered Si PCW, we analyze the waveguide properties include group index, GVD and optical propagation loss. Especially, the effective interaction area, slow light enhanced factor and phase mismatch for FWM process have been calculated exactly. According to these accurate parameters, we systematically study the degenerate FWM conversion efficiency and conversion bandwidth for two low-dispersion regions with TPA and FC effects. Furthermore, we numerically simulate the more realistic multiple FWM process. The efficient parametric interaction and multiple wavelength conversion have been discussed detailedly. The wavelength conversion process of another idler wave has nothing to do with the conversion efficiency of FWM process. It should be noted that the theoretical model presented here can be easily extended to other periodic photonic structures, in which case one can investigate the effects such as stimulated Raman scattering, mode switching, or spontaneous FWM.

Acknowledgment

This work was supported by the National Natural Science Foundations of China under Grant Nos. 60977044 and 11105049, the Natural Science Foundation of Hubei Province under Grant No. 2011CDC010, the Foundation for Youths of Hubei normal university under Grant No. 2010C22 and the Hubei Province Science Foundation for Youths under Grant No. Q20112501.

References and links

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OCIS Codes
(160.6000) Materials : Semiconductor materials
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics
(190.4223) Nonlinear optics : Nonlinear wave mixing
(130.5296) Integrated optics : Photonic crystal waveguides

ToC Category:
Nonlinear Optics

History
Original Manuscript: March 23, 2012
Revised Manuscript: May 13, 2012
Manuscript Accepted: August 6, 2012
Published: August 17, 2012

Citation
Tao Chen, Junqiang Sun, and Linsen Li, "Modal theory of slow light enhanced third-order nonlinear effects in photonic crystal waveguides," Opt. Express 20, 20043-20058 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20043

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References

T. Baba, “Slow light in photonic crystals,” Nat. Photonics2, 465–473 (2008). [CrossRef]
C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express17, 2944–2953 (2009). [CrossRef] [PubMed]
C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010). [CrossRef]
A. Baron, A. Ryasnyanskiy, N. Dubreuil, P. Delaye, Q. Vy Tran, S. Combrié, A. De Rossi, R. Frey, and G. Roosen, “Light localization induced enhancement of third order nonlinearities in a GaAs photonic crystal waveguide,” Opt. Express17, 552–557 (2009). [CrossRef] [PubMed]
S. Combrié, Q. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett.95, 221108 (2009). [CrossRef]
K. Inoue, H. Oda, N. Ikeda, and K. Asakawa, “Enhanced third-order nonlinear effects in slow-light photonic-crystal slab waveguides of line-defect,” Opt. Express17, 7206–7216 (2009). [CrossRef] [PubMed]
J. McMillan, M. Yu, D. Kwong, and C. Wong, “Observation of spontaneous Raman scattering in silicon slow-light photonic crystal waveguides,” Appl. Phys. Lett.93, 251105 (2008). [CrossRef]
X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308 (2010). [CrossRef]
B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009). [CrossRef]
C. Monat, C. Grillet, B. Corcoran, D. J. Moss, B. J. Eggleton, T. P. White, and T. F. Krauss, “Investigation of phase matching for third-harmonic generation in silicon slow light photonic crystal waveguides using Fourier optics,” Opt. Express18, 6831–6840 (2010). [CrossRef] [PubMed]
J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express18, 15484–15497 (2010). [CrossRef] [PubMed]
C. Xiong, C. Monat, A. Clark, C. Grillet, G. Marshall, M. Steel, J. Li, L. O’Faolain, T. Krauss, J. Rarity, and B. J. Eggleton, “Slow-light enhanced correlated photon pair generation in a silicon photonic crystal waveguide,” Opt. Lett.36, 3413–3415 (2011). [CrossRef] [PubMed]
P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010). [CrossRef]
M. Ebnali-Heidari, C. Monat, C. Grillet, and M. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express17, 18340–18353 (2009). [CrossRef] [PubMed]
J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express16, 6227–6232 (2008). [CrossRef] [PubMed]
S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt.12, 104004 (2010). [CrossRef]
V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett.35, 1440–1442 (2010). [CrossRef] [PubMed]
I. Cestier, A. Willinger, V. Eckhouse, G. Eisenstein, S. CombriÚ, P. Colman, G. Lehoucq, and A. De Rossi, “Time domain switching/demultiplexing using four wave mixing in GaInP photonic crystal waveguides,” Opt. Express19, 6093–6099 (2011). [CrossRef] [PubMed]
I. Cestier, A. Willinger, P. Colman, S. Combrié, G. Lehoucq, A. De Rossi, and G. Eisenstein, “Efficient parametric interactions in a low loss GaInP photonic crystal waveguide,” Opt. Lett.36, 3936–3938 (2011). [CrossRef] [PubMed]
P. Colman, I. Cestier, A. Willinger, S. Combrié, G. Lehoucq, G. Eisenstein, and A. De Rossi, “Observation of parametric gain due to four-wave mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett.36, 2629–2631 (2011). [CrossRef] [PubMed]
C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express18, 22915–22927 (2010). [CrossRef] [PubMed]
B. Corcoran, M. D. Pelusi, C. Monat, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Ultracompact 160Gbaud all-optical demultiplexing exploiting slow light in an engineered silicon photonic crystal waveguide,” Opt. Lett.36, 1728–1730 (2011). [CrossRef] [PubMed]
J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express19, 4458–4463 (2011). [CrossRef] [PubMed]
N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 257–266 (2010). [CrossRef]
M. Santagiustina, C. Someda, G. Vadala, S. Combrie, and A. De Rossi, “Theory of slow light enhanced four-wave mixing in photonic crystal waveguides,” Opt. Express18, 21024–21029 (2010). [CrossRef] [PubMed]
S. Roy, M. Santagiustina, P. Colman, S. Combrie, and A. De Rossi, “Modeling the Dispersion of the Nonlinearity in Slow Mode Photonic Crystal Waveguides,” IEEE Photon. J.4, 224–233 (2012). [CrossRef]
G. Agrawal, Nonlinear Fiber Optics and Applications of Nonlinear Fiber Optics, 4th ed. (Elsevier Science, New York, 2007).
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