Abstract
The large index contrast and the subwalength tranverse dimensions of nanowires induce strong longitudinal electric field components. We show that these components play an important role for second harmonic generation in III-V wire waveguides. To illustrate this behavior, an efficiency map of nonlinear conversion is determined based on full-vectorial calculations. It reveals that many different waveguide dimensions and directions are suitable for efficient conversion of a fundamental quasi-TE pump mode around the 1550 nm telecommunication wavelength to a higher-order second harmonic mode.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The first demonstration of second harmonic generation (SHG) in 1961 paved the way for decades of research on nonlinear optics [1]. The quest for efficient conversion is still relevant today as key regions of the electromagnetic spectrum lack suitable laser sources. Other applications such as squeezed light generation [2], entangled photon generation [3] or frequency comb stabilization [4] would also benefit from efficient frequency converters.
The advent of integrated photonic platforms the last decade revolutionized frequency conversion. The large nonlinear coefficients as well as the high index contrast inherent to integrated photonics allows for strong nonlinear interaction at low power. Many instances of integrated second harmonics generation have been reported, with novel, low-loss, LiNbO$_3$ on insulator and III-V-on insulator platforms currently holding the record normalized conversion efficiency in nanowires [5–7]. In most theoretical analysis, the light is approximated by a purely transverse mode such that a single incoming polarization state and spatial profile is considered. In practice however, more complex nonlinear wave mixing can be expected because the optical modes in high index contrast waveguides display large longitudinal components. We recently experimentally demonstrated SHG enabled by longitudinal components in III-V wire waveguides [8]. Here we further investigate the impact of strong longitudinal components on SHG. We derive the ordinary differential equations describing the nonlinear coupling of a fundamental wave at $\omega _0$ to its second harmonic (SH) at $2\omega _0$ in a III-V waveguide using full-vectorial calculations [9–15]. We identify ultra-efficient conversion for a wide range of waveguide dimensions and highlight the major role played by the propagation direction.
While we focus on III-V semiconductor wire waveguides [6,7,16], we stress that our analysis can be easily adapted to other platforms. We consider indium gallium phosphide (InGaP) around 1550 nm as the guiding material [8,17]. InGaP displays a $\bar {4}$ symmetry and exhibits very low nonlinear losses at telecom wavelengths in the C-band. The $\chi ^{(2)}_{xyz}$ coefficient, which is the only nonzero tensor element, was measured to be as high as 220 pm/V around 1550 nm [18]. Because of the lack of birefringence in III-V semiconductors, several different approaches have been implemented to achieve phase matching, including form birefringence [19], quasi-phase-matching [20–24] and modal phase matching [6,25]. Our theoretical study is focused on the latter.
The paper is organized as follows. In Section 2, we recall the general formalism used for describing bound modes and the derivation of the first order differential equation modeling their nonlinear coupling. Specifically, we resort to a perturbative method to include second order nonlinearities. In Section 3, we apply the formalism to SHG. In Section 4, we discuss the influence of the propagation direction on the conversion efficiency and present the efficiency map for the specific case of InGaP-on insulator waveguides. Concluding remarks are given in Section 5.
2. General framework
2.1 Linear waveguides
We start by discussing the properties of bound modes in a lossless linear waveguide [26]. We consider a III-V-on-insulator wire waveguide as shown in the inset of Fig. 1. An electromagnetic wave oscillating at a frequency $\omega$ propagating in the waveguide must satisfy the source-free linear Maxwell equations which read in the Fourier domain:
The translational invariance along the propagation axis allows to write the $j^{\mathrm {th}}$ guided mode as a spatial distribution depending on the transversal coordinates of the electric and magnetic fields with a fixed propagation constant. They read:
2.2 Nonlinear coupling
We next derive the expressions for the nonlinear coupling between different forward propagating modes. In this derivation, the nonlinearity is treated as a perturbation to the ideal lossless linear waveguide [9–15]. The perturbed waveguide modes are written in the Fourier domain as:
3. Second-harmonic generation
We apply the formalism from the previous section to the specific case of SHG. For simplicity we only consider type I phase matching. The fundamental wave, with a carrier frequency $\omega _0$, and the second harmonic, with a carrier frequency $2\omega _0$, are each limited to a single spatial mode. The total electric and magnetic fields are:
We define the effective nonlinear coefficient as:
4. Application to III-V-on-insulator wire waveguides
We now focus on the specific case of III-V-on insulator wire waveguides. Because the propagation direction is not fixed in the crystal frame ($xyz$), we introduce the coordinates ($x'y'z'$) to describe the optical wave in the waveguide frame. The propagation Eqs. (24a) and (24b), simply become:
Most III-V wafers are grown along a crystallographic axis. Consequently, we may consider that the light propagates in the $xz$-plane (010) of the crystal. The two coordinate frames are linked through the rotation matrix:The $45^{\circ}$ effective nonlinearity reads:
The $0^{\circ}$ effective nonlinearity, on the other hand, reads:
The main difference between $0^{\circ}$ and $45^{\circ}$ oriented waveguides stems from the profiles of the excited modes. As a reminder there is a single, vertical, symmetry plane. Consequently, only the parity of the profiles along the $x'$ direction matters. In a $45^{\circ}$ waveguide, the $y'$ component of the SH mode must be symmetric as the fundamental components are squared [see Eq. (31)]. Conversely, in $0^{\circ}$ waveguides, only SH modes with an antisymmetric vertical component will be excited because the product of the transverse and longitudinal components of the pump is always antisymmetric. We stress that these considerations are valid for type I SHG, irrespective of the pump mode.
Importantly, we find that III-V nanowaveguides are suitable for efficient conversion in both propagation orientations.
Next we study the impact of the waveguide dimensions on the SHG efficiency when considering a fundamental mode wavelength around 1550 nm. Specifically, we vary the width and height of the III-V section and look for phase matching between a fundamental quasi-TE mode and a SH higher-order mode in a 10 nm window around 1550 nm. For each instance of phase matching, we compute the effective nonlinearity for different propagation directions and store only the maximum coefficient. Due to the symmetry of the crystal, we limit ourselves to the first quadrant. We investigate waveguides with a width between 600 nm and 1000 nm and a height between 50 nm and 350 nm. To limit the computational time, we use a resolution of 5 nm. Every phase matching point is shown as a marker in Fig. 2. The marker color codes the strength of the coupling and its shape indicates the angle between the waveguide and the crystal axis that maximizes the interaction. Squares are used for the $0^{\circ}$ waveguides while diamonds represent $45^{\circ}$ waveguides. To highlight similar interactions, we evaluate the overlap between neighboring markers via Eq. (5). We define a threshold at 70%, beyond which we infer it is the same mode and connect the two markers with a line. We identify 8 independent modes. Their Poynting vector distribution is shown in Fig. 2. The maximum effective nonlinearity [$\kappa = 2816\;\mathrm {(\sqrt {W}m)^{-1}}$] is found for a waveguide with a width of 810 nm and a height of 110 nm, directed at $45^{\circ}$. The SH propagates in a TM$_{00}$ mode and the corresponding conversion efficiency is as high as 79300 %/(Wcm$^2)$. This interaction is well-known as it is mostly due to the mixing of the transverse components of the modes [6]. More interesting are the many square markers indicating coupling enabled by longitudinal components. The maximum conversion efficiency [52350 %/(Wcm$^2$)], found for a TM$_{10}$ mode, is predicted to be almost as efficient as the conversion to a TM$_{00}$ mode. Moreover, we find that the phase matching is more sensitive to width variations in the latter. In waveguides with a thickness of 200 nm or more, it is the $0^{\circ}$ configuration that is the most efficient. These thicknesses are commonly used as they lead to lower propagation losses than in thinner layers [31,32]. Importantly, these highly efficient geometries could not be predicted using a scalar approach.
We stress that not all possible couplings are shown in Fig. 2. We use a 5 nm resolution and wave vectors are very sensitive to waveguide dimensions. Also, quasi-phase-matching can be used to efficiently couple two modes with different effective indices [33]. Yet, many of the novel nonlinear couplings we show here are predicted to be very efficient. We expect them to play a significant role in future integrated wavelength converters.
5. Conclusion
We have theoretically investigated SHG in III-V semiconductor wire waveguides. By using a full-vectorial model we found many instances of efficient conversion between a fundamental pump mode and a higher-order SH mode. Our results highlight the crucial role played by the longitudinal component of the electric field. When propagating along the crystal axis, only wave mixing involving different components is permitted by the single nondiagonal $\chi ^{(2)}_{xyz}$ element. Due the high index contrast, the longitudinal electric field component can be almost as large as its transverse counterpart [34] making this configuration very efficient.
Funding
European Research Council (757800, 759483); Conseil Régional des Pays de la Loire (Paris-Scientifique "Nano-Light").
Acknowledgements
This work was supported by funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement Nos 759483 & 757800) and by the Fonds de la Recherche Fondamentale Collective (grant agreement No PDR.T.0185.18). CC acknowledges the financial support from Région Pays de la Loire through the grant Paris Scientifique "Nano-Light."
Disclosures
The authors declare no conflicts of interest.
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