## Abstract

We propose a novel phase-matching scheme in GaP whispering-gallery-mode microdisks grown on Si substrate combining modal and $\overline{4}$ -quasi-phase-matching for second-harmonic-generation. The technique consists in unlocking parity-forbidden processes by tailoring the antiphase domain distribution in the GaP layer. Our proposal can be used to overcome the limitations of form birefringence phase-matching and $\overline{4}$ -quasi-phase-matching using high order whispering-gallery-modes. The high frequency conversion efficiency of this new scheme demonstrates the competitiveness of nonlinear photonic devices monolithically integrated on silicon.

© 2016 Optical Society of America

## 1. Introduction

III–V semiconductor materials have large nonlinear susceptibilities [1] and good optical confining properties due to their high refractive indices. Moreover, their technology is mature and a lot of III–V integrated optoelectronic devices are already available. These assets can be used to obtain integrated photonic devices for second-order nonlinear optics applications. Furthermore III–V semiconductors present a broad transmission range and can be operated from ultraviolet to infrared [2, 3]. Unfortunately these materials are isotropic and birefringence cannot be used to obtain the phase-matching (PM) condition strictly required to reach high nonlinear conversion efficiencies. Periodical poling usually used to achieve quasi-phase-matching (QPM) by electric field application cannot be implemented since III–V materials are not ferroelectric. Alternative methods have been developed to obtain the QPM in the mid-infrared such as Fresnel PM [4] and reversed domain plate bonding. [3] However, in the near infrared domain, these methods cannot be used since the coherence length becomes too short. Consequently other PM scenarios such as form birefringence [5], modal PM [6–8], phase shift in Bragg mirrors [9], coupled resonator PM [10] or band-edge PM [11] have been proposed and demonstrated in semiconductor waveguides, Bragg reflection waveguides [12] or microcavities [13]. QPM can also be implemented by using complex growth techniques on orientation patterned substrates [14–17] or periodic quantum well intermixing [18–20]. It has been shown recently that the $\overline{4}$ -symmetry of zincblende crystals can be used in whispering-gallery-mode (WGM) microdisks to obtain a natural $\overline{4}$ -QPM [21–23]. Moreover, this method takes advantage of electric field enhancement due to the high quality (Q) factor and low mode volume of WGM resonators [24, 25]. Several second-order optical nonlinear functions such as frequency conversion, entangled photon sources or THz wave generation benefiting from natural $\overline{4}$ -QPM have then been proposed [26–28]. From an experimental point of view, second-harmonic generation (SHG) based on this particular QPM has been demonstrated in GaAs, AlGaAs and GaP WGM microcavities [29–31]. However, due to the very strong dispersion of III–V semiconductors, the $\overline{4}$ -natural QPM technique must be associated to other PM techniques like form birefringence [32] or modal PM using radial low-order fundamental (F) modes and high-order second-harmonic (SH) modes. In the second case, the overlap between the SH mode and the nonlinear polarization can be very weak and a sub-wavelength patterning of the WGM microcavity then becomes mandatory to keep SHG efficiency significant [33, 34].

Meanwhile, the monolithic integration of GaP on Si substrates has been studied for decades thanks to the very low lattice mismatch between these two semiconductors [35–38]. GaP thus holds great promises for the development of advanced photonic functionalities on silicon [39]. However material issues due to the growth of polar crystals on nonpolar silicon seemed to so far prevent any practical development of GaP-on-Si nonlinear photonic devices [40]. Indeed, in any heterogeneous epitaxial system with polar growth on nonpolar substrate, defects called antiphase domains (APDs) can be generated in the polar crystal. At the crossing of the boundary (called APB) between two APDs the lattice continuity is maintained but an exchange occurs between gallium and phosphorus atoms on the crystal sites. Because the GaP polarity and thus the *χ*^{(2)} susceptibility are locally reversed, the APDs were often considered as detrimental for the implementation of second order nonlinear devices unless a full control over the APD spatial distribution was obtained in order to fabricate orientation patterned crystals [15, 41].

In this paper we demonstrate how thickness-controlled natural distributions of APDs can be used to unlock parity-forbidden $\overline{4}$
-QPM second order process in GaP microdisks monolithically grown on Si. The efficiency of this new scheme is compared to more conventional experimental approaches of second order processes in microdisks, unveiling the competitiveness of this method for future practical development of GaP-based nonlinear functionalities on Si. Such a parity tuning of the nonlinear susceptibility of a system was already proposed in multiple quantum well structures and angled gratings to achieve new schemes of modal PM and QPM respectively [42] [43]. This paper is organized as follows. We first briefly review basics of structural and optical properties of polar crystals grown on nonpolar substrate. In particular we discuss the impact of APD on crystal *χ*^{(2)} properties. Then, we describe the basic principle of the modal PM optimization. Finally, conversion efficiency calculations are shown and discussed.

## 2. Antiphase domains

The heteroepitaxial growth of GaP on Si is mainly studied in the aim of integrating light sources on Si substrate [44] and for photovoltaic applications [45]. In the framework of second order nonlinear optics, a strict definition of the GaP local polarity is required to assess the effects of APDs on the nonlinear properties. As mentioned previously, APDs consist in local inversion of Ga and P atoms in the crystal lattice. In the growth plane, this spatial inversion operation of the zincblende cell appears as a 90° rotation around the [001] direction [46].

Figure 1.a) is a dark field cross-sectional transmission electron microscopy imaging of a GaP/Si sample, where APDs have been voluntarily annihilated in the first tens of nanometers of the III–V growth. In this image, main crystal phase and antiphase are evidenced by the bright/dark contrast in the GaP layers. Fig. 1.b) shows a plan-view thresholded TEM image of a GaP/Si sample where that APBs were left free to emerge. Thresholded TEM images as shown as in Fig. 1.b) allow for a good experimental evaluation the geometrical parameters of the APD distribution. Assigning +1(−1) local polarity to white (black) domains in this image, the mean polarity of the GaP epilayer *P _{APD}* can be defined as the spatial average of the local polarity distribution. The APD correlation length, characterizing the mean size of the domains is measured through the lorentzian fit of the thresholded image Fourier transform. In the present case, the mean polarity is found to be equal to 0.3 and the correlation length of the APD distribution is in the range of 10 nm. For efficient second-order optical nonlinear interactions and in particular for SHG, single orientation (

*P*= ±1) is generally wanted. APDs are detrimental since they reduce the mean polarity of the crystal [15]. A deep experimental investigation of APD nucleation conditions in GaP-on-Si epilayers in several research groups now allows us to consider that a relative control over several key parameters of the APD distribution is achievable. First, the results of Supplie

_{APD}*et al.*[47] suggest that the mean polarity of a GaP APD distribution can be tuned through the careful preparation of the Si substrate. Second, as shown in Fig. 1.a), the reproducible annihilation of most APDs (more than 80% reduction within 10 nm) was observed at the position of a thin AlGaP layer (in dark gray in Fig. 1.a) embedded into the GaP matrix. When such a layer is not used, APDs propagate vertically towards the sample surface [48]. The detailed growth conditions leading to such an annihilation of APDs are provided in [38]. Structural investigations (not presented here) unambiguously indicate that the AlGaP marker is at the origin of APB annihilation, and therefore suggests that height at which such an annihilation occurs can be controlled at will. However, tuning the correlation length of APDs is so far mainly correlated to their annihilation either through the aforementioned technique or through the reorientation of antiphase boundaries [49]. Natural APD in-plane spatial distributions generally feature correlation length in the range of a few to a few tens of nanometers as depicted in Fig. 1.b).

## 3. Modal PM optimization using APD in GaP microdisks

#### 3.1. Structural description

Figure 2(a) is a sketch of the studied configuration, it consists of a GaP WGM microdisk (of radius R) grown on a Si (001) substrate. The vertical confinement of the electric field is obtained thanks to a lateral etching of the Si substrate into a Si pedestal for the microdisk. The epitaxial GaP is assumed to present APDs over a thickness *e _{APD}*. Subsequent epilayers are considered composed of purely monodomain GaP. The overall GaP thickness of the device is noted

*e*in the following. The relative height of the APDs is thus defined by

*ξ*=

*e*. The refractive index of GaP at fundamental and SH frequencies is

_{APD}/e*n*with

_{i}*i*∈ {

*f*,

*SF*}.

*Z̃*(

_{i}*z*) gives the vertical dependences of the electric fields of both fundamental and SH modes and can be calculated considering a symmetrical Air/GaP slab-waveguide model [50]. The modes of this slab waveguide are indexed by

*q*, and their effective indices are used to obtain the WGM assuming a cylinder surrounded by air [50].

_{i}*m*are the azimuthal numbers of the fundamental and SH WGM. The radial dependences of fields are given by normalized Bessel functions noted

_{i}*ψ*(

_{i}*r*); these functions have

*p*antinodes representing the radial mode orders. Practically, the fundamental field can be coupled inside the microdisk using a tapered fiber which is also used to extract the SH field as shown in Fig. 2(b) [31]. The power coupling efficiency is characterized by the coupling coefficient 1 − |

_{i}*t*|

_{i}^{2}where

*t*is the transmission coefficient through the coupling tapered fiber for the fundamental or SH mode. It is is related to the coupling quality (Q) factor

_{i}*Q*by ${Q}_{c,i}=\pi {m}_{i}\sqrt{\left|{t}_{i}\right|}/\left(1-\left|{t}_{i}\right|\right)$.

_{c,i}#### 3.2. SHG model in an inhomogeneous WGM microdisk

In the undepleted fundamental field approximation, for fundamental and SH fields fulfilling simultaneously the WGM resonance condition, the conversion efficiency *η* is given by [21,50]:

*P*is the fundamental input power. The attenuation parameter

_{in}*α*describes the propagation optical losses over a microdisk round-trip. In SHG experiments, it can by the way be extracted to calculate the intrinsic Q-factor

_{i}*Q*

_{0,i}by ${Q}_{0,i}=\pi {m}_{i}\sqrt{{\alpha}_{i}}/(1-{\alpha}_{i})$. The overlap between the fundamental field polarized in the microdisk plane and the nonlinear polarization oriented along

*z*-axis is taken into account by

*K̃*

_{±}which reads [50]:

*ω*is the SH angular frequency and Δ

_{SH}*m*=

*m*− 2

_{SH}*m*. The function

_{f}*d*(

*z*,

*r*,

*θ*) takes into account the inhomogeneities of the polarity of the GaP layer:

*d*

_{14}describes the spatial second-order nonlinear susceptibility sign fluctuations induced by the reversal of the crystal in the APD. Finally we have: where

*f*(

*z*) is a step function. Its value

*f*

_{1}in the upper part of the microdisk (

*z*> −

*e*/2 +

*e*) is necessarily larger than the value

_{APD}*f*

_{2}in the layer including APD. The value

*f*

_{2}does not result from a mere average of the polarity distribution but takes into account the APD disorder as shown by the (

*r*,

*θ*) dependence of Eq. (3). It can drop to zero for balanced APD distributions (

*P*= 0). We consider now that we want to fulfil the $\overline{4}$ -QPM condition Δ

_{APD}*m*= 2 for

*p*= 1 and

_{f}*p*= 1 which gives the best radial function overlap [21]. Without recourse to APDs, the only solution is to use form birefringence and thus very thin GaP layers which makes the WGM very sensitive to surface defects. On the contrary, if an APD distribution with tailored vertical extent can be included in the geometry, modal PM in the

_{SH}*z*direction can be used by choosing

*q*= 1 and

_{f}*q*= 2 (see Fig. 2(c). As shown by Eq. (4) such a process is parity-forbidden in a homogeneous GaP layer (

_{SH}*f*

_{1}=

*f*

_{2}): the field overlap

*K̃*

_{+}is equal to zero as shown in Fig. 2(d) since the fundamental mode is symmetric whereas the SH mode is antisymmetric. If we consider now the ideal case of an inhomogeneous GaP layer with

*e*=

_{APD}*e*/2 and

*f*

_{2}= 0 the field overlap can be greatly enhanced since the symmetry of the structure is broken (see Fig. 2(e)).

## 4. Simulation results

The semi-analytic model described in section 3.2 is applied to estimate the conversion efficiency of GaP microdisks grown on Si substrate. The refractive index dispersion of GaP is taken from [51] for wavelengths between 500 nm and 800 nm and from [52] between 800 nm and 2.3 *μ*m. The value of the nonlinear coefficient *d*_{14} = 28.5 pm/V at *λ _{f}* ≈ 1.5

*μ*m is extrapolated from measurements reported in [1]. We first consider a device with a radius

*R*= 2.242

*μ*m and a total thickness

*e*= 400 nm. The double resonance and the $\overline{4}$ -QPM condition Δ

*m*= 2 is found for a fundamental wavelength

*λ*= 1.536

_{f}*μ*m and WGMs corresponding to (

*p*= 1,

_{f}*q*= 1) and (

_{f}*p*= 1,

_{SH}*q*= 2). We assumed a critical coupling

_{SH}*Q*=

_{c,i}*Q*

_{0,i}and an overall Q-factor

*Q*defined by ${Q}_{i}^{-1}={Q}_{c,i}^{-1}+{Q}_{0,i}^{-1}$ equal to 5 × 10

_{i}^{3}both for the fundamental and SH fields. The emergence of APBs leads to typical surface roughness in the 1 – 3 nm range [53]. For semiconductor microdisks, such roughness values certainly degrade the mode quality but experiments show that Q-factors above 10

^{5}can be conserved [54,55]. This strengthens our choice of overall Q-factors for the simulations one order of magnitude below state of the art GaP microdisks grown on GaP substrates [56]. To analyze the role of the mean polarity

*P*of the APD layer, we numerically generated artificial APD distributions mimicking thresholded distributions obtained from real patterns like that shown in Fig. 1(b). Actually, the overlap between the fundamental and the SH modes is different from zero only on the 20 % external part of the microdisk (shown with bright color on Fig 2(b)), thus we estimate

_{APD}*P*on this area. Figure 3(a) shows the conversion efficiency for several values of the APD layer average polarity as a function of

_{APD}*ξ*. For small values of

*ξ*, the conversion efficiency is almost zero since these cases correspond to a single domain GaP layer as sketched in Fig. 2(d). For

*ξ*≈ 1, the nm-scale spatial reversal of the polarity induced by the APD distribution along the whole microdisk height leads to very small values of the conversion

*η*. Between these two bounds, the cases where

*ξ*≈ 0.5 exhibit possible high efficiencies thanks to the symmetry breaking induced by the mean polarity contrast between the lower and upper half of the microdisk which prevents the two opposite sign parts of ${\tilde{Z}}_{\mathit{SH}}(z){\tilde{Z}}_{f}^{2}(z)$ from cancelling each others. The optimal configuration is obtained for

*P*= 0; in this particular situation, the APD layer does not contribute to the SHG which maximizes the field overlap as sketched in Fig. 2(e). Note that

_{APD}*P*= 0.286 corresponds to the typical APD distribution shown in Fig. 1(b) which was deduced from TEM observations. Figure 3(b) shows the maximal conversion efficiency of this process (

_{APD}*ξ*= 0.5) for a large set of APD distributions featuring different

*P*(between 0 and 1) and correlation lengths (varying from 10 to 500 nm). The influence of

_{APD}*P*on the process efficiency appears here clearly while the simulation points dispersion reveals the influence of the APD disorder (especially for large correlation lengths) on the conversion efficiency. The tolerance of the QPM scheme to variations in the APD distribution (along the growth axis for example due to the non-verticality of the APD walls) can thus be estimated: In the worst case, if the effective APD distribution polarity varies from 0 to 0.1, the decrease of the conversion efficiency can in no case exceed 40%. In Fig. 4 we compare several PM configurations in WGM GaP microdisks assuming

_{APD}*Q*=

_{f}*Q*= 5 × 10

_{SH}^{3}and Δ

*m*= 2. In all the simulations we considered that

*p*= 1 and

_{f}*q*= 1. First we calculated

_{f}*η*for homogeneous GaP layers. The optimal configuration (

*p*= 1,

_{SH}*q*= 1) is shown in blue. Due to the strong GaP dispersion, in addition to natural QPM, form birefringence must be used to reach full PM operation. This is obtained through very thin microdisks. For example to reach the PM around

_{SH}*λ*= 1.5

_{f}*μ*m, a GaP layer of

*e*= 130 nm is required. From an experimental point of view such a thin microdisk would be very sensitive to surface roughness and defects which would limit its Q-factor especially for the fundamental mode which is less confined than the SH mode. To obtain the PM in thicker GaP microdisks, modal PM must be used. SHG with a

*p*= 2 and

_{SH}*q*= 1 WGM is shown in yellow in Fig. 4. Double resonance condition is now found at

_{SH}*λ*≈ 1.5

_{f}*μ*m for a disk thickness of

*e*= 150 nm at the expense of the conversion efficiency which is decreased by two orders of magnitude. This issue can be circumvented by using the present proposition for modal PM with inhomogeneous GaP layers (

*p*= 1,

_{SH}*q*= 2). In the grey area of Fig. 4, we plotted the conversion efficiency for

_{SH}*ξ*= 0.5 and

*P*= 0. In this case the PM is obtained for

_{APD}*e*= 360 nm with a conversion yield up to 0.5 %/mW which is almost equivalent to what is calculated for (

*p*= 1,

_{SH}*n*= 1).

_{SH}## 5. Conclusion

We have proposed a method to authorize modal PM in WGM GaP microdisks grown on Si. This technique relies on the reduction of the microdisk nonlinear susceptibility symmetry in order to break parity selection rules in the nonlinear process by means of APD engineering. We have shown that SHG using this modal PM leads to conversion efficiency at the same level than other PM schemes using only the lowest order modes both at fundamental and SH frequencies. The advantage of our approach is that it enables the use of thicker microdisks, less sensitive to WGM scattering losses. Moreover this method broadens the spectral range of phase-matched nonlinear interactions using GaP WGM. This unique property combined with the large bandgap of GaP (2.26 eV) leads to efficient frequency conversion with fundamental wavelengths as low as *λ _{f}* = 1.35

*μ*m. From the simulation point of view, further calculations could take into account a vertical density gradient in the PD distribution. This proposal paves the way towards the development of fully integrated low-power second-order nonlinear optics on Si substrate.

## Acknowledgments

This research was supported by the Labex CominLabs through the ”3D Optical Many Cores” project, the French National Research Agency Project ANTIPODE (Grant No. 14-CE26-0014-01), the Région Bretagne, Rennes Métropole and the Institut Universitaire de France (IUF).

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