## Abstract

Enhanced four-wave-mixing (FWM) effects have been observed with the help of large group-indices near the band edges in one-dimensional (1-D) silicon photonic crystal waveguides (Si PhCWs). A significant increase of the FWM conversion efficiency of about 17 dB was measured near the transmission band edge of the 1-D PhCW through an approximate 3.2 times increase of the group index from 8 to 24 with respect to the central transmission band region despite a large group-velocity dispersion. Numerical analyses based on the coupled-mode equations for the degenerated FWM process describe the experimentally measured results well. Our results indicate that the 1-D PhCWs are good candidates for large group-index enhanced nonlinearity devices even without having any special dispersion engineering.

© 2013 Optical Society of America

## 1. Introduction

Recently, optical nonlinear effects enhanced by slow light in photonic crystal waveguides (PhCWs) with periodic structures have been studied for potential applications to all-optical signal processing and nonlinear signal generation [1–4]. Several approaches demonstrated for slowing light down with periodic structures include uses of coupled resonators [5, 6], one-dimensional (1-D) gratings or holes formed in optical fibers or waveguides [1, 6–14], or two-dimensional (2-D) photonic crystals (PhCs) [1–4]. The interaction between the light field and the waveguide materials is increased due to the reduced group velocity of the light within the periodic structures, and thus enables very short nonlinear optic devices with low power pumping. Among such periodic structures, the 1-D periodic structures are very useful for application in nonlinear optic functional devices because of their simple device structures and enhanced optical nonlinearity. They provide the wavelength-selective properties with high quality factors and also slow light characteristics with large group indices near the band-edge region. Of special note, a 1-D array of small circular holes formed on single-mode linear waveguides shows a wide spectral region of the photonic band-gap prohibiting the light transmission and optical transmission bands beside it. The transmission bands have interference fringe ripples on their spectral profiles, which result from a Fabry-Perot (FP) cavity. The FP cavity is formed by the refractive index difference at the interfaces between the 1-D PhC waveguide section and its input and output strip waveguide sections. Thus, the group-index profile of the 1-D PhCW structures can be determined from the FP interference fringes [6].

Two major properties related to the PhCWs, the group-index dispersion and optical coupling to the input and output waveguides, are very important parameters influencing the efficiency of the third-order nonlinear processes, such as four-wave-mixing (FWM) and third-harmonic generation (THG) [1, 15]. The PhCWs are known to have a large group-index change near the edge of the transmission band, which results in a significant chromatic-dispersion profile in the region. Some dispersion engineering methods have been demonstrated for 2-D PhCWs to achieve the slow-light-enhanced FWM efficiencies [16]. In the 1-D PhCWs, technical approaches to improve the optical coupling efficiency between the input and output strip waveguide sections and the 1-D PhCW section have been reported by using cascaded reduction of the PhC holes near the strip waveguide section or by having a reduced strip waveguide width of the PhC hole section [11]. Furthermore, in bulk-type 1-D PhCs, the optical phase-conjugated condition has been observed through degenerated FWM processes and used to compensate phase mismatching between the pump and signal beams thus delivering an undistorted signal output [17].

In this paper we report on the enhanced FWM efficiency measured near the band-edge of the transmission band of a silicon (Si) 1-D PhCW without any dispersion engineering. In addition, we describe numerical analyses based on the coupled-mode equations for the FWM process in a 1-D PhCW, and the calculated results are compared with the measured results.

## 2. The coupled-mode equations for the FWM process

For the degenerated FWM process, the pump, signal and idler beams, *P _{p}*,

*P*, and

_{s}*P*, propagating along the waveguide follow the coupled-mode equations [18]:

_{i}*α*is the absorption coefficient of the waveguide which is related to the conventional absorption coefficient

*α*of the waveguide material as

_{0}*α*=

*hα*with a proportional parameter

_{0}*h*. The parameter

*h*may be a group-index

*n*dependent parameter. J. F. McMillan,

_{g}*et al*. [19], observed, for two-dimensional PhCWs, that the parameter

*h*was proportional to the ratio of the group-index of the PhCW section with respect to the effective index

*n*of the strip waveguide,

_{eff}*i.e., h*=

*n*/

_{g}*n*. The parameter

_{eff}*γ*is also a group-index-dependent nonlinearity coefficient of the waveguide which is

*γ*=

*γ*(

_{0}*n*/

_{g}*n*)

_{eff}^{2}and

*γ*= 2

_{0}*πn*

_{2}/

*λA*[19, 20].

_{eff}*n*is the nonlinear refractive index of the optical waveguide material. We assume that the pump, signal and idler beam wavelengths are close to each other, so that the

_{2}*γ*are the same for all three beams.

*λ*is the wavelength of light, and

*A*is the effective mode area. $\Delta k=k\left({\omega}_{s}\right)+k\left({\omega}_{i}\right)-2k\left({\omega}_{p}\right)$ is the linear phase mismatch among the three beams.

_{eff}*k*(

*ω*

_{s}

*) =*

_{,i,p}*ω*(

_{s,i,p}n_{eff}*ω*)

_{s,i,p}*/c*is the propagation constant of each of the pump, signal and idler lightwaves.

*c*is the speed of light in free space, and

*n*(

_{eff}*ω*) is the effective refractive index of the waveguide at the single beam frequency of

_{s,i,p}*ω*. The maximum power transfer from the pump beam to the signal and idler beams takes place when the phase difference angle

_{s,i,p}*θ*among the three beams equals to π/2.

The 1D-PhCW section has a different average refractive index compared to those of the input and output strip waveguide sections on both its sides, and thus forms a Fabry-Perot cavity between the boundaries. The transmittance at the boundary between either one of the input and output strip waveguide sections and the 1D-PhCW section is related to the Fresnel reflection relationship, which can be written as $T=1-R=1-{{\left({n}_{eff}-{\tilde{n}}_{eff}\right)}^{2}/\left({n}_{eff}+{\tilde{n}}_{eff}\right)}^{2}$_{.} Here, the averaged refractive index of the holes and filled waveguide portions in the 1-D PhCW section ${\tilde{n}}_{eff}$ is written as [21]

*d*is the hole diameter.

## 3. Fabrication and characterization of 1-D PhCW devices

A 1-D PhCW device used in this research was fabricated on a silicon-on-insulator (SOI) wafer with
a 220 nm thick silicon layer and a 3,000 nm thick buried oxide (BOX) layer using 193 nm
deep-ultraviolet lithography (130 nm CMOS process) and dry etching technologies through the
European ePIXfab silicon photonics platform (www.ePIXfab.eu). The main Si waveguide with the
1-D PhC holes had a 220 nm thickness and 520 nm width, and each end of the waveguide was
connected to a wider waveguide of 10 μm width through a 1,000 μm long tapered
waveguide section as shown in Fig. 1. Bragg grating couplers of a 620 nm period, 70 nm etched depth, and 50% filling factor
were formed on each of the wider waveguides for optical input and output coupling from and to
single-mode fibers (SMFs). The details of the fabrication process and structure of the grating
coupler are found in [22]. The PhC holes were formed on
the middle of the total 4,000 μm long plain strip waveguide region. The PhC hole period
and diameter were *a* = 400 nm and *d* = 300 nm, respectively.
There were 169 holes over a span length of 67.6 μm. The effective mode area of the
220x520 nm strip waveguide for the 1/*e* point of the Gaussian profile of the
fundamental mode was calculated to be about 0.106 μm^{2} [23]. This value was used in our numerical analysis for approximated
evaluations of the nonlinear effects even though the actual effective field size causing the
nonlinear effects in a highly nonlinear core can be calculated in various ways [24–26].

In order to determine the optical loss characteristics of the Si strip waveguide section without any PhC hole, as well as that of the Bragg grating coupler section including the tapered waveguide region, several plain Si-strip waveguides of the same core size and the same grating couplers on both ends but with four different lengths of 4, 16, 24 and 32 mm were fabricated on the same wafer. Figure 2(a) shows the measured results of the transmitted output powers from the plain strip waveguides of four different lengths for an optical input power of 0 dBm at a 1,550 nm wavelength. From the slope of this transmitted output power curve, the optical loss of the plain Si-strip waveguide section was determined to be 0.354 dB/mm. The fiber coupling loss, including the combined losses of the Bragg grating section and the tapered waveguide region except the central plain strip waveguide section, was also plotted in Fig. 2(b) as function of wavelength. The amplified spontaneous emission (ASE) beam from an erbium-doped fiber amplifier (EDFA) was used with an optical spectrum analyzer (OSA) for this coupling loss measurement.

The transmission property of the 1-D PhCW device was measured with the OSA under illumination of a tunable laser beam amplified through a high power EDFA. Figure 3(a) shows the measured and simulated transmittance spectra of the waveguide. The black and red solid lines indicate the measured and simulated transmission spectra of the 1-D PhCW, respectively. The measured spectrum shows an interference pattern of a Fabry-Perot (FP) interferometer over the transmission band and a PhC band edge at around 1,530 nm wavelength while the stimulated transmittance spectrum also shows a similar interference pattern. The numerical simulation was carried out with a three-dimensional finite-differential-time-domain method (Lumerical’s FDTD Solutions). The oscillation depth of the measured FP interference is in a range from 3 to 10 dB. The FP interference pattern resulted from the reflectance at the boundary interfaces between the PhC hole array section and the input and output plain strip waveguide sections. The measured transmittance indicates the total transmittance of the entire 1-D PhCW device. The total length of the 1-D PhCW is 4 mm with a 67.5 μm long 1-D PhC section in the middle. The dotted line in Fig. 3(a) is the measured transmission spectrum for a 4 mm long plain strip waveguide without any photonic crystal pattern. This measured loss profile includes the optical losses at the input and output grating coupling sections and the propagation loss along the plain strip waveguide. The transmittance difference between this dotted line and the measured interference fringes peaks corresponds to the total loss caused by the propagation loss in the 1-D PhCW section and by the interface losses between the 1-D PhCW section and the strip waveguides at both sides. Thus, the average optical loss caused by the 1-D PhCW section is about 3.15 dB.

The spectral profile of the group refractive index of the 1-D PhCW can be determined from the FP interference fringes according to the theoretical relationship between the group index and the interference fringes [14],

where${\lambda}_{0}$ is the wavelength of the interference peak, $\Delta \lambda $ is the wavelength separation between two adjacent peaks, and*L*is the length of the FP resonator. The spectral dependency of the group indices of the 1-D PhCW calculated based on Eq. (6) with both the measured and simulated interference peak data of Fig. 3(a) is shown in Fig. 3(b). The black square points are the group indices calculated with the measured interference peak data, while the red circular points are those calculated with the simulated interference peak data. The trends of the spectral profiles of both the measured and the simulated group indices are similar except the measured data show some fluctuation with respect to the simulated results. The group index increases sharply above 20 near the band edge as the wavelength decreases close to 1,530 nm. The group index values below a 1,535 nm wavelength are 3~5 times larger than those above 1,550 nm. At the wavelength of 1,540 nm, the calculated effective index

*n*of the Si-strip waveguide of 220 nm height and 520 nm width is about 2.74, and the average refractive index ${\tilde{n}}_{eff}$ of the 1-D PhCW section is about 1.435 based on Eq. (5). The actual value of this average refractive index ${\tilde{n}}_{eff}$ of the 1-D PhCW section will be a little bit larger than 1.435 because the air gaps are considered empty planar spacers of a thickness corresponding to the hole diameter instead of considering hole-shape blanks. The measured interference fringes are resulted from a Fabry-Perot cavity formed by the optical reflectance at the interface between the 1-D PhCW and the normal strip waveguide sections.

_{eff}## 4. FWM measurement with the 1-D PhCW and numerical analysis

Group-index dependent FWM efficiencies of the 1-D PhCW were measured, and compared with numerically calculated results. Since the group index of the 1-D PhCW varies with the wavelength, the interaction time of the light with the waveguide material also varies with wavelength due to the slow light effect. We measured the dependency of the FWM efficiency on the group indices of the 1-D PhCW at four different wavelengths as shown in Fig. 3 with marks (Points A ~D). Figure 4 shows the experimental setup used for the FWM measurement. Two tunable lasers were used as pump and signal beam sources for the FWM experiment in the 1-D PhCW after being amplified through high-power EDFAs. The ASE noises from the amplified pump and signal beams were filtered out with narrow band-pass filters. The polarizations of the pump and signal beams were aligned to the same polarization direction with polarization controllers (PCs). Then, the two beams were combined with a 3 dB coupler, and their polarizations were aligned to a TE-mode direction of the PhCW. The pump and signal beams were coupled into the 1-D PhCW through a fiber-to-grating coupler with an 8 degree inclined beam injection. The FWM output was measured with an optical spectrum analyzer (OSA). The total coupling loss of an optical beam to the grating-coupler and tapering sections on both sides of the 1-D PhCW excluding the insertion losses of the PhC hole section itself and the plain strip waveguide section of its input and output sides was about 16 dB for each of the pump and signal beams, and that for the idler beam was about 8 dB because its output was measured from only one side.

The FWM measurement was carried out with the selected pump and signal wavelengths close to the
interference peaks shown in Fig. 3 in order to have the
generated idler signal appear close to one of the interference peaks. The spectra of the
measured FWM output of the 1-D PhCW at two points, B and D, as well as the idler generation
efficiency compared to the input signal, were plotted in Fig.
5(a). The wavelength separation between the pump and signal beam was about 2.42 nm ( =
Δ*λ*) at Point B and 2.38 nm at Point D. The FWM idler beam
generation efficiency was calculated by taking the idler beam output power divided by the input
signal power, i.e. *η* =
*P _{idler}*(

*L*)/

_{total}*P*(0). The input signal power

_{signal}*P*(0) was taken as the signal power at the beginning of the input plain strip waveguide connected to the 1-D PhC hole section, and

_{signal}*P*(

_{idler}*L*) was taken as the idler beam power at the output end of the output plain strip waveguide, both by excluding the spectral profile of the coupling losses shown in Fig. 2(b). The measured FWM conversion efficiency (

_{total}*η*) at the pump wavelength of

*λ*= 1534.77 nm (Point B) was −35.5 dB for its group index (

_{p}*n*

_{g}) of 21.3, and that at Point D (

*λ*= 1551.9 nm) was −42.7 dB for

_{p}*n*= 7.6. From comparison of these two points, B and D, the efficiency was increased about 8.2 dB while the group index was increased from 7.6 to 21.3. At Point A (

_{g}*λ*= 1533.73 nm),

_{p}*η*= −25.83 dB,

*n*= 24.4, and Δ

_{g}*λ*= 0.32 nm, while

*η*= −40.51 dB,

*n*= 14.0, and Δ

_{g}*λ*= 1.61 nm at Point C (

*λ*= 1540.07 nm). The parameters used for the FWM measurements, and the measured and calculated FWM efficiencies at the four points are summarized in Table 1.

_{p}Figure 5(b) shows the measured FWM conversion efficiencies (black squares) at the four points (A~D) illustrated in Fig. 3 along with the group-index profile. The FWM efficiency in the logarithmic scale increases almost linearly with the group index. The group index change is significant near the band edge, which thus causes a large group-velocity dispersion profile, since the dispersion is related to the group-velocity as ${\beta}_{2}=-\left({\lambda}^{2}/2\pi {c}^{2}\right)\left(d{n}_{g}/d\lambda \right)$. It is known that the large group-velocity dispersion causes the phase mismatch among the pump, signal and idler beams in the FWM process and deteriorates the signal-to-idler conversion efficiency. We may expect that an efficient FWM efficiency is difficult due to the large dispersion at the high index side since the group-index curve has a steep slope in the wavelength region below 1536 nm. However, between the two points A and D, the measured FWM conversion efficiency increases by 16.9 dB from – 42.7 dB to −25.8 dB for a 3.2 fold increase of the group index from 7.6 to 24.4 toward the band edge even though the group-velocity dispersion increases by 7.1 times from 1.26 ps^{2}/mm to 8.98 ps^{2}/mm.

A numerical analysis was performed with the coupled-mode equations, Eqs. (1)-(4), to explain the
above measured FWM results. In the coupled mode equations, the group-index-dependent absorption
parameter *h* was determined by solving the coupled-mode equations for three
different values of the parameter, and by comparing the calculated results with the measured
ones as shown in Figs. 5(b) and Fig. 7. The calculated results were the best fit to the measured results when
the parameter value *h* was set to be
(*n _{g}*/

*n*)

_{eff}^{1.5}, so that

*α*=

*hα*= (

_{0}*n*/

_{g}*n*)

_{eff}^{1.5}

*α*. The absorption coefficient

_{0}*α*of the Si-strip waveguide was taken to be 0.354 dB/mm from Fig. 2(a). The nonlinear refractive index of the silicon was set to be

_{0}*n*

_{2}= 4.5 × 10

^{−18}m

^{2}/W [19]. By accounting for the coupling loss at the input grating coupler as shown in Fig. 2(b), the initial input pump and signal powers,

*P*(0) and

_{pump}*P*(0), were taken at the very beginning of the input strip-waveguide section right after the tapered section of the input grating coupler. Figure 6 shows the calculated pump-beam power and FWM-signal-conversion efficiency along the 1-D PhCW for the group-index profile of point A. The FWM efficiency drops marked with

_{signal}*T*and

_{in}*T*indicate the transmission losses due to the 10% Fresnel reflections at the input and output boundaries of the 1-D PhC section, respectively. The results imply that a significant FWM conversion takes place within the 1-D PhC section.

_{out}The calculated FWM-efficiency values at the four points (A) to (D) are compared to the measured values in Table 1 and Fig. 5(b). Figure 5(b) shows the calculated values plotted with open diamonds, squares, and circles, with each symbol representing the case of the group-index dependent absorption parameter *h* = 1 (*i.e*., *α* = α), *n _{g}*/

*n*(

_{eff}*i.e*.,

*α*= α

*n*/

_{g}*n*) and (

_{eff}*n*/

_{g}*n*)

_{eff}^{1.5}[

*i.e*.,

*α*= α (

*n*/

_{g}*n*)

_{eff}^{1.5}], respectively. The black squares are the measured FWM efficiencies. The calculated FWM efficiency values are the best fit to the measured results when

*h*= (

*n*/

_{g}*n*)

_{eff}^{1.5}[

*i.e*.,

*α*= α (

*n*/

_{g}*n*)

_{eff}^{1.5}], except the large group-index case very near the bandedge. The measured FWM conversion efficiencies follow a similar trend with the group-index profile. The FWM-conversion efficiency increases with the increasing group index as the pump wavelength approaches to the band-edge wavelength. Some discrepancy between the calculated and measured efficiency values near point A may have resulted from ignorance of the nonlinear absorption, such as two photon absorption (TPA) and free-carrier absorption (FCA), free-carrier index (FCI) changes, and nonlinear-effect-induced effective mode area (NEI-EMA) in our coupled-mode analysis [27, 28]. The effects of nonlinear absorption in the FWM processing in the silicon waveguides have been described by Lin

*et al.*[27]. The details of the group-index dependent nonlinear absorption effects in the FWM processing should be further investigated in a separate research. Except for the large group-index region near the band edge, the calculated FWM-conversion efficiencies are close to the measured ones. The inset in Fig. 5(a) shows the generated idler power at the end of the waveguide versus the coupled input signal power at point B. The linear dependence of the generated idler power with respect to the input signal power indicates that there are no significant TPA, FCA, FCI and NEI-EMA effects as reported in [14]. The closed diamonds in Fig. 5(b) indicate the case when no dispersion effect exists but

*α*=

*α*(

*n*/

_{g}*n*)

_{eff}^{1.5}. The plot is compared to the normal case having a chromatic dispersion with the group-index-dependent absorption parameter

*α*=

*α*(

*n*/

_{g}*n*)

_{eff}^{1.5}. Based on our experimental measurement and numerical analyses, we can state that the FWM efficiency is affected more dominantly by the group-index-dependent absorption parameter than by the dispersion effect. In addition, the FWM process is enhanced by the large group-index profile near the band edge of the 1-D PhCW without any dispersion engineering even when the large dispersion profile exists over the band-edge region. The FWM efficiency depends not only on the dispersion effect but also on the group-index-dependent loss parameter.

Figure 7 shows the measured and calculated idler beam
powers for an input signal power of −6 dBm through the FWM process at points B and D as
functions of the pump powers. The calculated values using the coupled-mode Eqs. (1)-(4) match well with the measured ones when the group-index-dependent absorption
parameter *h* =
(*n _{g}*/

*n*)

_{eff}^{1.5}[

*i.e*.,

*α*=

*α*(

*n*/

_{g}*n*)

_{eff}^{1.5}]. The generated idler-beam power increases almost linearly with the increasing of the pump power.

## 5. Conclusion

The signal-to-idler conversion efficiency of a 1-D PhCW through the FWM process has been measured and compared with its group-index profile. The measured conversion efficiency followed a similar profile to the group index, and thus the large group index near the band edge of the transmission band of the 1-D PhCW provided an efficient FWM conversion despite a significant dispersion profile. The FWM conversion efficiency was about −25.83 dB near its band edge where the group index was 24.4, while it was −40.51 dB and −43.67 dB for group indices of 12 and 8, respectively, at the other central region of its transmission band. The increase of the FWM-conversion efficiency at the large group-index region even with a significant dispersion profile was compared with the numerically calculated results with the coupled-mode equations. We have found that the FWM efficiency depends on the group-index-dependent absorption parameter significantly rather than the dispersion effect. Our research indicates that the 1-D PhCWs are very useful devices for high-efficient FWM signal conversion in a simple device structure even without any dispersion engineering. Future works are left to have a further increase of the FWM efficiency by optimizing the waveguide and coupling losses of the 1-D PhCWs, and to identify the detailed roles of the dispersion effect and group-index-dependent absorption parameter in the FWM efficiency by having a variety of the 1-D PhCW samples. Further theoretical analysis on the effects of the two-photon absorption, free-carriers, and accurate effective mode area are also needed to predict accurate FWM efficiencies.

## Acknowledgments

The authors acknowledge that this work was supported in part by the Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Korean Ministry of Education, Science and Technology under Grants 2009-0084514 and 2009-0079527 and in part by the Ministry of Science, ICT & Future Planning under the Grant number 2013R1A1A2012409.

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