## Abstract

We have proposed and designed zigzag-folded U-shaped waveguides to realize highly efficient wavelength conversion with an extremely small footprint. Extreme high efficiencies are achieved with a combination of modal phase matching and $\overline{4}$ quasi phase matching in inversion-stacked AlGaAs/Alox waveguides. Numerical simulations reveal that the conversion efficiency of second harmonic generation pumped at 1.55 *μ*m as high as 12000 %W^{−1} can be achieved in an 8.0-mm-long AlGaAs/Alox waveguide that is folded up into a small domain of 0.8×0.6 mm^{2} area. A phase matching signal bandwidth of difference frequency generation, 29 nm, covers 83 % of the telecommunication C band.

© 2017 Optical Society of America

## 1. Introduction

Difference frequency generation (DFG) around 1.55 *μ*m is expected to play important roles for optical cross-connect in the wavelength division multiplexing telecommunication system [1]. In addition, the DFG is a key process for a *χ*^{(2)}-based phase sensitive amplifier [2]. Practical DFG devices need to be fabricated in small footprints comparable to the length of laser diodes (sub mm). 100 % conversion efficiency of DFG pumped with, *e. g.*, 2.5 mW power requires an extremely high normalized DFG conversion efficiency 40000 %W^{−1} (corresponding to 10000 %W^{−1} second-harmonic generation (SHG) efficiency) under no-pump-depletion condition. The highest normalized SHG efficiency of conventional LiNbO_{3} quasi phase matching (QPM) waveguide is 370 %W^{−1}cm^{−2} [3], and thus a 52-mm-long device is needed for the efficiency target. Obviously, new materials and device designs are necessary for coping with both efficiencies and compactness.

AlGaAs is a promising material for the DFG device because of its large quadratic nonlinear optical coefficients (GaAs: 120 pm/V [4] and AlAs: 39 pm/V [4, 5] at 1.53 *μ*m) and mature waveguide-fabrication technologies. Its relatively large refractive indices are advantageous for strong optical confinement. However, since AlGaAs is optically isotropic, it is impossible to achieve birefringence phase matching. As an alternative approach, AlGaAs waveguiding devices with periodic inversion structures have been developed for achieving QPM by utilizing wafer bonding [6, 7] and crystal growth techniques [8–11]. We have developed AlGaAs QPM devices and attained an SHG conversion efficiency of 34 %W^{−1} pumped at 1.55 *μ*m with a 6-mm-long device [12]. The efficiency is limited by relatively large propagation losses at the present stage of our investigation. GaAs microdisk [13, 14] (maximum theoretical efficiency 10^{4} %W^{−1}) or microring [15] (maximum theoretical efficiency 240000 %W^{−1}) resonators, in which the $\overline{4}$ symmetry of zincblende crystal is used to achieve QPM ($\overline{4}$ QPM), have been proposed, and phase matched experimental SHG efficiency of 45 %W^{−1} was reported for a 5-*μ*m-diameter GaAs microdisk pumped at 2.0 *μ*m [14]. SHG efficiency achieved in a telecommunication range was only 0.07 %W^{−1} in an AlGaAs microdisk [16]. The microdisk devices suffer from extremely stringent $\overline{4}-\text{QPM}$ and resonance conditions.

Strong optical confinement in high-index-contrast waveguides composed of AlGaAs/SiO_{2} and AlGaAs/Alox (AlAs oxides with refractive index of 1.6) enables us to realize extremely high normalized conversion efficiencies. To achieve phase matching, it is indispensable to compensate large modal dispersion in high-index-contrast waveguides. Form-birefringence phase matching (FBPM) [17, 18], high-index-contrast birefringence phase matching (HICBPM) [19, 20] and modal phase matching (MPM) have been investigated. Normalized conversion efficiencies of the FBPM and HICBPM devices are reported to be 24000 %W^{−1}cm^{−2} and 19000 %W^{−1}cm^{−2}, respectively [19]. To accomplish the efficiency target, required devices lengths are 6.5 mm and 7.3 mm, respectively. On the other hand, conversion efficiencies of MPM can be dramatically enhanced by utilizing lateral modulation of nonlinear optical coefficient [21, 22] compensating mode-overlap cancellation. We proposed an Al_{0.5}Ga_{0.5}As/SiO_{2} MPM waveguide with inversion-stacked core structure, where the sign of the nonlinear optical coefficient of AlGaAs is modulated in the transverse direction. For an MPM waveguide utilizing a TE-polarized fundamental wave with the lowest-order mode $({\text{TE}}_{00}^{\omega})$ and a TM-polarized second-harmonic (SH) wave with the first-order mode $({\text{TM}}_{01}^{2\omega})$, the highest normalized SHG efficiency is as high as 45000 %W^{−1}cm^{−2} [23]. AlGaAs inversion-stacked waveguides were successfully fabricated by using surface activation bonding technique [24]. For the efficiency target, the MPM device needs to be 4.7 mm long.

In this paper, we propose and design a novel wavelength conversion device, an Al_{0.5}Ga_{0.5}As/Alox zigzag-folded U-shaped waveguide, which is composed of straight and half-arc (180°) waveguides with the inversion-stacked structure schematically shown in Fig. 1 [23]. MPM between ${\text{TE}}_{00}^{\omega}$ and ${\text{TE}}_{01}^{2\omega}$ is achieved via *d*_{14} in the straight waveguides running along [011] and phase difference between fundamental and SH waves in the half-arc waveguides is adjusted to 2*π* in order to make the SH waves generated in all the waveguide interfere constructively. We reveal by simulation that it is possible to attain > 10^{4} %W^{−1} efficiencies with sub-mm^{2} footprints utilizing optimized zigzag-folded inversion-stacked AlGaAs/Alox waveguides. This paper is composed of 1) performances of the MPM SHG in the straight waveguide, 2) phase matching in half-arc waveguides analyzed using phasor representation, 3) practical design of zigzag folded waveguides, 4) conversion efficiency and phase matching tolerances of the device.

## 2. Modal phase matching in straight waveguides

Our simulation starts with a design of AlGaAs/Alox rib-type straight waveguide phase matchable for SHG at the fundamental wavelength of 1.55 *μ*m. We calculated phase matching thicknesses *t _{g}* of a guiding layer and SHG conversion efficiencies in Al

_{0.5}Ga

_{0.5}As/Alox inversion-stacked rib-type straight waveguides. We assumed the type-I SHG interaction between TE-polarized fundamental and TM-polarized SH waves via nonlinear optical coefficient

*d*

_{14}of AlGaAs. Figure 2(a) shows schematic image of an Al

_{0.5}Ga

_{0.5}As/Alox straight waveguide, where guiding layer is laterally inverted,

*i. e.*, the sign of the nonlinear optical coefficient is inverted as shown in Fig. 2(b), to compensate mode-overlap cancellation for MPM between the TE

_{00}and TM

_{01}modes. The thickness of Alox layers, 0.8

*μ*m, was chosen to avoid the light leakage of the fundamental wave to the substrate. We calculated effective indices

*n*and electric-field distributions of the interacting waves by using full vectorial finite element method (FEMSIM software package developed by Rsoft). We used material dispersion data of AlGaAs and Alox reported by Afromvitz [25] and Knoops

_{eff}*et al.*[26], respectively. We also calculated normalized SHG conversion efficiencies

*η*

_{0}in a lossless waveguide under no depletion of the fundamental wave. The normalized SHG conversion efficiency is given by [27]

*ϵ*

_{0},

*c*,

*λ*, $\tilde{d}$, ${n}_{eff}^{\omega}$, ${n}_{eff}^{2\omega}$, ${\mathcal{E}}_{\omega}$ and ${\mathcal{E}}_{2\omega}$ are permittivity and light speed in vacuum, the wavelength of the fundamental wave, cross-sectional distribution of nonlinear optical coefficient normalized by

_{ω}*d*

_{14}, effective indices and normalized electric fields of the fundamental and SH waves, respectively. Position parameters

*x*and

*y*are parallel to $[0\overline{1}1]$ and [100] AlGaAs crystallographic axes, respectively. We used

*d*

_{14}= 85 pm/V for Al

_{0.5}Ga

_{0.5}As based on the reports by Shoji

*et al.*[4] and Ohashi

*et al.*[5]. Figure 3 shows obtained phase matching thickness

*t*and SHG conversion efficiency dependent on the rib width

_{g}*w*of the straight waveguide. The normalized SHG conversion efficiencies were estimated to be about 20000 %W

_{s}^{−1}cm

^{−2}.

## 3. Adjustment of phase difference in U-shaped waveguides

To investigate the behavior of SH electric fields generated in AlGaAs single U-shaped waveguides, we used phasors of normalized SH amplitudes in the waveguide. The normalized coupled mode equation of SHG process under no depletion of the fundamental wave in a lossless waveguide is given by [28]

where*s*is a position parameter along the waveguide, ${\mathcal{A}}_{2\omega}(s)$ is normalized SH complex amplitude, Δ

*k*(

*s*) is wavevector mismatch, and $\overline{d}(s)$ is effective nonlinear optical coefficient normalized by

*d*

_{14}. Figure 4 is a schematic image of the U-shaped waveguide with the length of the straight waveguides

*l*and the radius of the half-arc waveguide

*R*. In the straight waveguides, $\overline{d}=1$ and Δ

*k*= 0 because MPM is achieved. In contrast, in the half-arc waveguide, $\overline{d}=cos(2s/R)$ [13, 15], which is due to $\overline{4}$ symmetry of AlGaAs crystal, and Δ

*k*≠ 0 in the case that both the thickness and the width are the same with those for the straight waveguides. By solving eq. (2) for the single U-shaped waveguide with

_{ha}*l*= 0.3

*μ*m and

*πR*= 1.0

*μ*m, we obtained phasors of normalized SH amplitudes. The phasors depend on phase difference between interacting waves in the half-arc waveguide Δ

*k*. When Δ

_{ha}πR*k*is the product of non-integer number and

_{ha}πR*π*, the SH wave generated in the second straight waveguide is not in phase with that in the first one. Figure 5 shows phasors for the phase difference Δ

*k*=

_{ha}πR*mπ*, where

*m*is integer. When Δ

*k*= 0, the SH amplitude grows along the imaginary axis in the first straight waveguide, and oscillates with one cycle along the imaginary axis in the half-arc waveguide, and finally reaches twice length of the straight waveguide, 2

_{ha}πR*l*, in the end of the second straight waveguide. When phase difference satisfies Δ

*k*= 4

_{ha}πR*π*, the SH amplitude also reaches 2

*l*because of the phasor oscillation with a full epicycloid curve with two cusps in the half-arc waveguide. In the case of Δ

*k*= 2

_{ha}πR*mπ*where

*m ≠*1, the SH wave generated in the second straight waveguide constructively interferes with that in the first straight waveguide, but that in the half-arc waveguide does not contribute to the total output. On the other hand, when Δ

*k*= 2

_{ha}πR*π*, the SH wave generated in half-arc waveguide grows along the imaginary axis tracing on a cycloid-curve phasor, and the total SH amplitude reaches 2

*l*+

*πR/*2. This clearly shows that the SH wave generated in the half-arc waveguide is constructively added to those in the straight waveguides. Since the wavevector mismatch is compensated by the modulation of the nonlinear optical coefficient by cos(2

*s/R*), QPM is achieved in the half-arc waveguide. The phase difference condition, Δ

*k*= 2

_{ha}πR*π*, is equivalent to the $\overline{4}-\text{QPM}$ condition. Effective nonlinear optical coefficient in this case is 1/2. The obtained effective nonlinear optical coefficient is consistent with the Fourier coefficient of the normalized nonlinear optical coefficient $\overline{d}(s)=cos(2s/R)$ in the half-arc waveguide. In contrast, in case of the phase difference Δ

*k*= (2

_{ha}πR*m*+ 1)

*π*, the SH waves generated in the straight waveguides destructively interfere because of the

*π*phase shift of SH amplitudes between the first and second straight waveguide. It is confirmed that the phasors in the half-arc waveguide turn negative direction along imaginary axis because the phasor shapes in the half-arc waveguide for Δ

*k*=

_{ha}πR*π*and 3

*π*are a half of an astroid curve,

*i.e.*a hypocycloid with 4 cusps, and a half of a epicycloid curve with four cusps, respectively.

## 4. Practical design of U-shaped waveguides

In order to accurately estimate phase difference and power transmission in half-arc waveguides, we simulated time development of electric-fields of both fundamental and SH waves using two-dimensional finite difference time-domain method (2D-FDTD) (Fullwave software package developed by Rsoft). We investigated two types of waveguides with uniform and non-uniform widths. Figure 6 shows computer-aided designs of the two U-shaped waveguides. Width of the half-arc waveguide *w _{ha}* is equal to that of the straight waveguide

*w*in the uniform device shown in Fig. 6(a), while

_{s}*w*≠

_{ha}*w*in the non-uniform device shown in Fig. 6(b). In the non-uniform case, the straight and half-arc waveguides are connected with linearly-tapered waveguides. The lengths of the straight and tapered waveguide are 10

_{s}*μ*m. Refractive indices of guiding segments (red regions) for the fundamental and SH waves are set to effective indices of ${\text{TE}}_{0}^{\omega}$ and ${\text{TM}}_{1}^{2\omega}$ mode, which are analytically calculated for Alox/Al

_{0.5}Ga

_{0.5}As/Alox 3-layer-slab waveguides with phase matching thicknesses

*t*corresponding to

_{g}*w*. The lowest-order modes of the interacting waves were launched from the left-side end faces of the waveguides. The phase difference from position A to B was calculated, and the power transmittance that is defined by the power at position B transmitting to the second straight waveguide divided by the power at position A coupled from the first straight waveguide was also calculated.

_{s}Figure 7 shows phase difference and SH transmittance in a uniform U-shaped waveguide as functions of the radius of the half-arc waveguide calculated for various *w _{s}*. The phase difference is dominated by the increase of the propagation constant of the SH wave which travels outer side of the bent waveguide. Thus, the phase difference is larger for smaller

*R*or larger

*w*. SH transmission reduction for smaller

_{ha}*R*is mainly due to radiation losses except for

*w*= 1.2

_{ha}*μ*m and

*R*= 20

*μ*m for which accidental mode matching at the point B increases the transmittance (large variation of the transmittance due to meandering propagation of the SH wave in the

*π*arc will be discussed below). The phase difference can be adjusted to 2

*π*with the radius of 11

*μ*m for

*w*= 1.0

_{ha}*μ*m and of 24

*μ*m for

*w*= 1.2

_{ha}*μ*m, respectively. Although the calculated power transmittances for the fundamental wave are larger than 0.99, those for the SH wave with the phase matching radii are smaller than 0.7 as shown in Fig 7(b). This means that only 4 % (= 0.7

^{9}) of SH power generated in the first straight waveguide survives through the nine-folded U-shaped waveguide. Therefore, we can not simultaneously realize high power transmittance and phase matching using U-shaped waveguides with uniform widths.

We calculated the phase differences and power transmittances in non-uniform-width U-shaped waveguides whose guiding thicknesses are kept at phase matching thickness *t _{g}* for

*w*in all the waveguide portions. We have chosen

_{s}*R*= 30

*μ*m, and

*w*≃ 1.0

_{s}, w_{ha}*μ*m in order to ensure high SH power transmittances. Figure 8(a) shows obtained phase differences dependent on

*w*in the non-uniform waveguides. For a certain

_{ha}*w*smaller than

_{ha}*w*, the phase differences can be adjusted to 2

_{s}*π*. Figure 8(b) represents SH transmittance as a function of

*w*in the non-uniform U-shaped waveguides. SH transmittance is strongly dependent on

_{ha}*w*because of meandering propagation of the SH wave in the bent waveguide. Since the SH wave propagates in a meandering path in the arc portion, very high coupling at the junction between the arc and tapered waveguides takes place when the transverse mode distribution maximum coincides with the center of the tapered/straight waveguide as shown in Fig. 8(c), while SH transmission is remarkably reduced when the SH mode deviates from the center at the end of the meandering path in the arc as shown in Fig. 8(d). Obtained SH power transmittances in the phase-adjusted waveguides are 0.8456, 0.9957, and 0.7670 for the

_{ha}*w*=1.00, 1.04 and 1.08

_{s}*μ*m, respectively. Based on this simulation, we conclude that phase matching and high transmittance can be achieved in the non-uniform waveguide with

*w*= 1.007

_{ha}*μ*m and

*w*= 1.04

_{s}*μ*m. The effective propagation losses are

*α*= 0.16 cm

_{ω}^{−1}(corresponding to the fundamental power transmittance of 0.9982) and

*α*

_{2}

*= 0.38 cm*

_{ω}^{−1}(corresponding to the SH transmittance of 0.9957).

## 5. SHG conversion efficiency of a zigzag-folded U-shaped waveguide

To evaluate the effects of propagation losses and the phase variations in the tapered waveguides, we calculated phasor of the normalized SH amplitude generated in a non-uniform U-shaped waveguide by solving the coupled wave equation taking account of propagation losses in the U-shaped portions. We assumed the effective losses calculated above are uniformly distributed over the whole length of the U-shaped waveguide. The coupled-wave equation in a lossy waveguide is given by [28]

*k*(

_{t}*s*)) was assumed to be proportional to

*s*, so that Δ

*k*(

_{t}*s*) changes linearly from 0 to $4\pi ({n}_{eff}^{2\omega}({w}_{ha})-{n}_{eff}^{\omega}({w}_{ha}))/{\lambda}_{\omega}$, where

*n*(

_{eff}*w*) denote effective indices of AlGaAs/Alox rib-type waveguide with the width of

_{ha}*w*and with the phase matching thickness

_{ha}*t*. Since the total phase difference ($=\mathrm{\Delta}{k}_{ha}\pi R+2{\displaystyle {\int}_{0}^{{l}_{t}}\mathrm{\Delta}{k}_{t}(s)ds}$, where

_{g}*l*is the length of the tapered waveguide) is adjusted to 2

_{t}*π*, the phase difference in a half-arc waveguide is $\mathrm{\Delta}{k}_{ha}=2/R-2{\displaystyle {\int}_{0}^{{l}_{t}}\mathrm{\Delta}{k}_{t}(s)ds/\pi R}$. Figure 9 shows calculated phasor for a non-uniform U-shaped waveguide with

*t*= 0.3904

_{g}*μ*m,

*w*= 1.04

_{s}*μ*m,

*w*= 1.007

_{ha}*μ*m,

*R*= 30

*μ*m,

*l*= 10

_{t}*μ*m,

*α*= 0.16 cm

_{ω}^{−1}and

*α*

_{2}

*= 0.38 cm*

_{ω}^{−1}. The asymptotical approach of the phasor to the imaginary axis indicates that SH waves generated in the tapered and half-arc waveguides coherently interfere with that in the preceding straight waveguide. Effective nonlinear optical coefficient was estimated to be 0.57 (= 64.6

*μ*m/114

*μ*m, the ratio between the length of the phasor and the total interaction length). The obtained coefficient is reduced slightly by 3.6 % compared to the ideal value ((

*πR/*2 + 2

*l*)/(

_{t}*πR*+ 2

*l*) = 0.59) owing to the small phase mismatch in the tapered waveguides.

_{t}SHG conversion efficiency in a zigzag folded U-shaped device shown in Fig. 1 was calculated based on the phasor analysis described above. The device is composed of 0.7-mm-long straight waveguides (*w _{s}* = 1.04

*μ*m and

*t*= 0.3904

_{g}*μ*m) and half-arc waveguides (

*R*= 30

*μ*m,

*w*= 1.007

_{ha}*μ*m, and

*t*= 0.3904

_{g}*μ*m) connected with 10-

*μ*m-long linearly-tapered waveguides (

*t*= 0.3904

_{g}*μ*m). Uniformly distributed propagation losses

*α*= 0.16 cm

_{ω}^{−1}and

*α*

_{2}

*= 0.38 cm*

_{ω}^{−1}are taken in account. We used a normalized conversion efficiency 21000 %W

^{−1}cm

^{−2}obtained for a straight waveguide with

*w*= 1.04

_{s}*μ*m. SHG conversion efficiency of nine-time-folded waveguide is as high as 12000 %W

^{−1}as shown in Fig. 10. In the device, the total interaction length of 8.0 mm is folded up into a small footprint with a 0.8 × 0.6 mm

^{2}area. In the present simulation, we have not taken into account the propagation losses due to waveguide imperfections, for example, side-wall roughness. It should be noted that the conversion efficiency is reduced to 2300 %W

^{−1}when typical propagation losses of 1.5 cm

^{−1}for the fundamental and 4.0 cm

^{−1}for the SH waves in a GaAs/Alox waveguide [29] are assumed to be uniformly distributed over the whole interaction length.

We calculated the wavelength tolerances for DFG and fabrication tolerances of the nine-time-folded device shown in Fig. 1. Figure 11(a) shows signal wavelength tolerance for the DFG pumped at 0.775 *μ*m. The full width at half maximum (FWHM) is 29 nm, which covers 83 % of the telecommunication C band (1530–1565 nm). The pump wavelength tolerance for the fixed signal wavelength of 1.550 *μ*m is 0.05 nm as shown in Fig. 11(b). This is comparable to the linewidth of external cavity lasers oscillating around 0.775 *μ*m. The stringent pump bandwidth is due to the relatively long interaction length and large index dispersion of Al_{0.5}Ga_{0.5}As. In order to ease the restriction, we need to shorten the interaction length or to make use of AlGaAs alloys with higher Al compositions exhibiting smaller dispersions. Propagation losses discussed above also make the bandwidth broader. We have to optimize the trade-off between the efficiency and the tolerance for practical applications.

Fabrication tolerances were estimated based on SHG conversion efficiency calculations. The tolerances of the widths of half-arc and straight waveguides are 4.0 nm and 0.6 nm, respectively. The estimated fabrication tolerance of 0.6 nm on the straight waveguide is very challenging. However, recent progress in fabrication technologies (for example, the alignment accuracy of EB lithography is improved down to 0.68 nm [30]) would make this device feasible in the near future.

## 6. Conclusion

We have proposed a novel ultra-compact and highly-efficient wavelength conversion device composed of zigzag folded high-index contrast AlGaAs/Alox rib waveguide with inversion-stacked core structure. We have shown using numerical simulations that ~ 5 × 10^{4} %W^{−1} DFG conversion efficiency is attainable in the telecommunication wavelength band with a 0.8 × 0.6 mm^{2}-footprint device where an 8-mm-long waveguide is folded using half-arc waveguides connecting straight waveguides via tapered ones. MPM interactions in the straight waveguide are made efficient by the lateral sign reversal of the nonlinear optical coefficient in the inversion-stacked structure, and the half-arc waveguides are designed to efficiently contribute to the constructive interaction utilizing $\overline{4}$ symmetry of AlGaAs (equivalent to $\overline{4}$ QPM). We believe that this will open up a new route to high-performance wavelength conversion devices compatible with laser diodes.

## Funding

Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 26706019.

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