Design and analysis of enhanced second harmonic generation in AlGaAs/AlO<sub>x</sub> microring waveguide

Mojtaba Gandomkar; Vahid Ahmadi

doi:10.1364/OE.19.009408

1. Introduction

Phase matching is quite essential in nonlinearities related to wavelength conversion like second harmonic generation (SHG). Several methods have been proposed to provide the required phase matching and among them quasi-phase-matching (QPM) is widely used in waveguides as an effective technique. Recently some efforts have been done to implement QPM in microresonators to enhance SHG. Ilchenko et al demonstrated two artificial poling schemes for enhancement of SHG in LiNbO₃ microresonators [1,2]. Dumeige et al showed that whispering gallery modes in AlGaAs microdisks can be used to obtain QPM without domain inversion for efficient wavelength conversion [3]. Later, Yang et al presented a scheme for achieving enhanced quasi-phase-matched SHG without requiring any artificial variation of χ ⁽²⁾ in microrings. There, QPM was achieved by utilizing the dependence of local mode polarization with respect to the crystal axes on the angle passed on the periphery of the ring [4].

To achieve a design with reasonable dimensions for microring structure, as described later, effective refractive indices for fundamental and second harmonic fields should be close together. One method is matching lowest mode of fundamental wave (FW) with the higher mode of second harmonic field (SH) [4]. This method causes some problems in coupling of different mode shapes which have different coupling factor and different group velocity. Another method is matching the lowest FW mode with the lowest SH mode by utilizing a multilayer AlGaAs/AlO_x microring waveguide that reduces effective refractive index of SH wave but has less effect on the effective refractive index of FW wave [5–7]. Here, we study and design a waveguide based on the latter method. In this method, group velocity dispersion (GVD) is reduced and fields make a better overlap for nonlinear purposes. In addition, small and large radius microrings can be fabricated with the same thickness of layers just by tuning the width of the waveguide. However, another way is using Bragg reflection waveguides (BRWs) [8,9] which makes more complexity in design and fabrication, so it is not studied here.

In addition to the bandwidth of phase matching, resonance of microring has an important effect on the linewidth of SHG. Calculations show that microring can provide a sharper linewidth of SHG than typical straight waveguides [5,10] and larger radius microring provides narrower linewidth.

Loss is another important parameter in SHG that affects the efficiency. The results show that larger radius of microring needs lower input power for efficient SHG in low loss condition. By the same reason, our proposed double microring structure provides higher conversion efficiency with lower input power as compared with a single microring structure. However, in lossy waveguide condition single structure and small radius microring have higher efficiency as compared with double microring structure and small radius microring, respectively.

To calculate refractive index of axisymmetric waveguide of microrings we use quasi 3D finite element method [11,12]. With this method, resonance frequencies and relevant mode shapes are extractable considering azimuthal mode number and all necessary field integration are conceivable. To investigate SHG behaviour, coupled nonlinear Schrödinger equations (C-NLSE) are simplified to a time invariant integrative form. This method is compared to the exact solution of C-NLSE utilizing finite difference time domain (FDTD) method.

2. Theory

Symmetry of a [100]-grown AlGaAs microring results in a periodic modulation of the effective second-order nonlinear coefficient, χ ⁽²⁾. This characteristic can be used to establish QPM and overcome the problem of optical isotropy of this material [3,4,13]. The schematic of a single microring structure is illustrated in Fig. 1 .

Fig. 1 Schematic of a single microring resonator structure.

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Here, we consider an in-plane TE-polarized FW field at frequency ω _p, which couples into the microring from port 1. This field interacts with the material and generates an out-of plane TM-polarized SH field at ω _s = 2ω _p. Both of fields must be at resonance frequencies of the microring. This requires $R k_{p} = m_{p}$ and $R k_{s} = m_{s}$ $(m_{p}, m_{s} \in N)$ where R is the ring radius and k_p and k_s are wave numbers of fundamental and second harmonic fields. On the other hand, QPM condition requires $R k_{s} - 2 R k_{p} = R Δ k = \pm 2$ which results in $m_{s} = 2 m_{p} \pm 2$ .

Having $k_{p} = ω_{p} n_{TE} / c$ and $k_{s} = ω_{s} n_{TM} / c$ , where n _TE and n _TM correspond to the effective refractive indices at FW and SH frequencies, respectively, we obtain the following equation to satisfy the resonance and QPM conditions simultaneously:

n_{TM} = (1 \pm \frac{1}{m_{p}}) n_{TE} .

This shows that we have to design n _TE and n _TM close together to design a microring with reasonable m_p and radius. The schematic of our proposed structure to compensate material and waveguide dispersion of AlGaAs is illustrated in Fig. 2 . A similar structure has been fabricated for straight waveguide in [6]. The waveguide structure consists of a multilayer core 110 nm AlGaAs /90 nm AlO_x /110 nm AlGaAs with thick AlO_x cladding. The thin AlO_x layer does not significantly affect the TE mode profile at the FW frequency but generates large discontinuities in the electric-field distribution of the TM mode at the SH frequency. The effective refractive index of TM mode is thus decreased enough to achieve the required phase matching. By adjusting the width of the waveguide (typically 700–1000 nm), we achieve the desired QPM at telecommunication wavelength (typically 1550 nm).

Fig. 2 Schematic of multilayer AlGaAs/AlO_x waveguide.

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Second harmonic generation can be modelled by coupled nonlinear Schrödinger equations (C-NLSE). These equations in lossy medium are

{\begin{cases} (\frac{1}{v_{s}} \frac{\partial}{\partial t} + \frac{\partial}{\partial ξ}) S = - \frac{α_{1 s}}{2} S + i η ν (θ) F^{2} e^{- i Δ k ξ} \\ (\frac{1}{v_{p}} \frac{\partial}{\partial t} + \frac{\partial}{\partial ξ}) F = - \frac{α_{1 p}}{2} F + i η^{*} ν (θ) S F^{*} e^{i Δ k ξ} \end{cases},

where S and F represent the slowly varying envelope of FW and SH fields. v_p and v_s are group velocities. α ₁ represents the loss of ring waveguide, which is due to bending loss, roughness or optical absorption. v(θ) = sinθ cosθ determines the variation of nonlinearity with θ, which arises because of the variation of local propagation direction with respect to the crystal axes. In addition, ζ = Rθ, Δk = k_s - 2k_p define the phase mismatch between two fields [4]. Nonlinear coupling parameter is

η = {(P A)}^{- 1 / 2}

where P and A are

P = 4 ε_{0} {\bar{n}}^{6} v_{p}^{2} v_{s} {(χ^{(2)})}^{- 2} ω_{p}^{- 1} ω_{s}^{- 1},

A = {| ε_{0}^{3 / 2} {\bar{n}}^{3} \int \frac{1}{2} e_{T E} (\vec{ρ}) e_{T M} {(\vec{ρ})}^{*} d \vec{ρ} |}^{- 2},

where

\bar{n}

is a reference refractive index for nonlinear material and

\vec{ρ}

is perpendicular to the direction of propagation in nonlinear material.

e_{T E} (\vec{ρ})

and

e_{T M} (\vec{ρ})

are radial and axial electric field component of TE (FW) and TM (SH) mode shapes, respectively, which are normalized as

ε_{0} \int {\bar{n}}^{2} (ω_{F, S}, \vec{ρ}) {| e_{T E, T M} (\vec{ρ}) |}^{2} d \vec{ρ} = 1.

The coupling between the straight waveguide and the ring waveguide is described by

[\begin{matrix} G_{2} \\ G_{3} \end{matrix}] = [\begin{matrix} τ_{p, s} & i κ_{p, s} \\ i κ_{p, s} & τ_{p, s} \end{matrix}] [\begin{matrix} G_{1} \\ G_{4} \exp (i k_{s, p} L) \end{matrix}],

where G represents S or F. κ_p,s and τ_p,s are coupling and transmission coefficients of the microring for FW and SH frequencies.

These equations are general to consider transient and steady state behaviour of the structure. However, it is appropriate to simplify them into time invariant form assuming $\partial / \partial t = 0$ . Then an integrative and iterative form of equations can be rewritten for each microring as [14]

{\begin{cases} S_{4}^{n + 1} = S_{3}^{n} + \int_{0}^{L} (- \frac{α_{1 s}}{2} S (ξ) + i η ν (θ) F^{2} (ξ) e^{- i Δ k ξ}) d ξ \\ F_{4}^{n + 1} = F_{3}^{n} + \int_{0}^{L} (- \frac{α_{1 p}}{2} F (ξ) + i η^{*} ν (θ) S (ξ) F^{*} (ξ) e^{i Δ k ξ}) d ξ \end{cases},

where the subscript “n” denotes the number of round-trips inside the microring and L = 2πR is the periphery of the ring. Keeping the data for any round-trip in an array provides a good estimation for transient behaviour of the structure even with the time invariant form of the equations. According to the coupling of straight and ring waveguides,

F_{3}^{n}

and

S_{3}^{n}

can be calculated as

\begin{array}{l} S_{3}^{n} = i κ_{s} S_{1}^{n} + τ_{s} S_{4}^{n} \exp (i k_{s} L), \\ F_{3}^{n} = i κ_{p} F_{1}^{n} + τ_{p} F_{4}^{n} \exp (i k_{p} L), \end{array}

where,

F_{1}^{n}

is the fundamental input field and

S_{1}^{n}

is the second harmonic input field for n^th round-trip, which is supposed to be zero for the first microring. κ and τ are coupling and transmission coefficients of the microring. We can perform our simulation several times faster than FDTD by this formulation.

We define two kinds of efficiency; The total efficiency, η _tot = Power_SH-out/Power_FW-in and the external efficiency, η _ext = Power_SH-out/(Power_SH-out + Power_FW-out). In lossless microrings, efficiency of SHG depends on the power of input fundamental field. By increment of the input power, total efficiency increases and at a certain power reaches to 100%, then decreases slowly. Maximum efficiency occurs at input power P _c that [4]

P_{c} = 16 P A κ_{p}^{4} {(1 - τ_{s})}^{2} κ_{s}^{- 2} L^{- 2} .

2.1 Linewidth of SHG in microrings

We define λ ₀ as the wavelength in which FW field is in resonance and λ as input wavelength which is detuned from λ ₀ by Δλ. To study the linewidth of SHG in microrings, off-resonance condition of the circulating fields must be considered so that exp(ik_s,p(λ)L) is not ignorable in Eqs. (6) and (8). In addition, phase mismatch of the structure at input wavelength is rewritten as

R Δ k (λ) = \frac{2 π R}{c . λ} (n_{e f f}^{s} (λ) - 2 n_{e f f}^{p} (λ)),

where

n_{e f f}^{p}

and

n_{e f f}^{s}

are effective refractive index of TE and TM fields, respectively, which are dispersive by λ and may be approximated by n _TE and n _TM at input wavelength close to λ ₀ to have

R Δ k (λ) = \frac{2 π R}{c . λ} (n_{T M} - 2 n_{T E}) .

Accuracy of Eq. (10) and (11) will be compared later. If at λ ₀, n _TE and n _TM satisfy the Eq. (1), SH field will be in resonance at λ ₀/2 and QPM condition leads to RΔk(λ ₀) = 2. However, experimentally, Eq. (1) is difficult to be satisfied at λ ₀ and there will be a phase mismatch so that RΔk(λ ₀) ≠ 2. This rudimentary phase mismatch should be considered to investigate the linewidth of SHG in microrings.

2.2 Double microring structure

To enhance the efficiency SHG in microrings we propose a double microring structure whose schematic structure is illustrated in Fig. 3 . In double microring resonator, the incident fundamental power is converted partially to SH in the first microring. The remaining fundamental field and the generated second harmonic field are coupled into the second microring simultaneously.

Fig. 3 Schematic of a double microring resonator structure.

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The first microring acts as if it is a single one. However, in the second microring we must consider the effect of the second harmonic field, which is coupled into it. This field must be at a proper phase difference with FW field, Δφ, to be constructively added to the second harmonic field, which will be generated in the second microring. When each field travels a length of L_b in the straight waveguide, its phase varies by k_pbL_b for fundamental field and k_sbL_b for second harmonic field where k_pb and k_sb are wave numbers of FW and SH fields in the straight waveguide, respectively. Therefore, the mentioned proper phase difference can be designed by choosing the appropriate length of the straight waveguide between two microrings.

3. Results and discussion

Resonance and QPM conditions can be satisfied by designing an appropriate width of waveguide. In Fig. 4 dependence of resonance wavelength of FW and SH waves on w, width of waveguide is illustrated for R = 10 μm. This figure shows that at w = 830.8 nm, resonance wavelength of m_p = 98 (λ ₀ = 1.5495 μm) is exactly matched with resonance wavelength of m_s = 198 at λ ₀/2. For higher azimuthal modes we can see that m_p = 99 and m_s = 200 are matched at w = 838.4 nm and λ ₀ = 1.5393 μm and azimuthal modes m_p = 100 and m_s = 202 are matched at w = 848.5 nm and λ ₀ = 1.5293 μm. In addition, this structure provides the possibility of perfect phase matching of azimuthal modes m_p = 100 and m_s = 200 at w = 940.4 nm.

Fig. 4 Resonance wavelength of FW (TE) and SH (TM) fields versus width of the waveguide. Resonance wavelength of SH field is multiplied by 2 to be comparable with the wavelength of FW field.

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For the following simulations, we calculate design parameters mentioned in Table 1 for three microrings with R = 6, 10 and 20 μm using FEM. We assume χ ⁽²⁾ = 100 pm/V [15], refractive index of AlO_x to be 1.7 and refractive index of AlGaAs to be 3.4206 and 3.5723 for free space wavelengths 1.55 and 0.775 μm, respectively. Coupling coefficients of FW and SH fields are assumed to be the same everywhere.

Table 1. Design Parameters of the Structure

View Table

To show the capability of our proposed integrative method a comparison between solution of Eq. (2) using FDTD method and solution of Eq. (7) with integrative method is illustrated in Fig. 5 . A lossless double microring structure is studied here with R = 10 μm and Δφ = 180°. Input fundamental power is 13 mW (P_c of this structure) and κ_p = κ_s = κ = 0.2. This input power causes the best efficiency for this structure. Although the group velocity of fields is not taken into account in the integrative method, we can estimate the transient response by selecting an approximate proper round trip time for every field. Curves corresponding to integrative method are fitted to real time FDTD curves so that group velocities of FW and SH fields seem to be 88 and 84 µm/ps, respectively. These values are close to the exact group velocities (91.9 and 83.9 µm/ps for FW and SH, respectively). The difference between these curves is mostly due to the dissimilarity between group velocities of the fields. Despite the small difference in amplitudes, the overall behaviors are rather similar and the steady state response is exactly the same for two methods. Integrative method simulates the transient behavior of SHG in microrings more than two orders of magnitude faster than FDTD method and simulates the steady state response several orders faster. Calculation of steady state response in following simulations is performed by integrative method.

Fig. 5 Comparison between FDTD method and integrative method. Results of integrative method are fitted on time axes with an approximate round trip time. Inset shows more details for short time response.

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3.1 SHG Efficiency

Figure 6 compares the total efficiency of SHG in a single microring with that of proposed double microring structure. The structure is supposed to be lossless, R = 10 μm and κ = 0.2. Here we see that P _c decreases by a coefficient of 1/4 when two microrings are cascaded. However, for higher input powers, the efficiency decreases. Here, conversion of SH power to FW is also possible due to difference frequency generation.

Fig. 6 Comparison between total efficiency of SHG in single and double microring structures (R = 10 μm).

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Comparison between total efficiency of SHG in single microrings with different radii is illustrated in Fig. 7 . Here, P_c is calculated to be 14, 57, 162 mW for R = 20, 10 and 6 μm, respectively.

Fig. 7 Comparison of total efficiency of SHG as a function of input power for various ring radii of a single microring structure.

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Efficiency of SHG in microring is sensitive to the loss. We assume that both fields practice the same loss. Figure 8 shows the SHG efficiency of a single microring structure versus loss for 57 mW input power (which is equivalent to P _c of this structure for κ = 0.2) with coupling coefficient as a parameter. It can be observed that total efficiency decreases exponentially with the loss. Figure 8 also shows that in low loss microrings, total efficiency and external efficiency are higher for κ = 0.2 than that of κ = 0.3 and 0.4, while for high loss microrings, κ = 0.4 shows higher efficiency as compared with κ = 0.2. In addition, external efficiency can reach 100% for a lossy microring by choosing appropriate input power and κ.

Fig. 8 Efficiency of SHG in a single microring versus loss, with R = 10 μm and coupling coefficient as parameter.

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Figure 9 shows total efficiency of SHG normalized to input power versus loss with input power and radius as a parameter. It is observed that lossy microring with R = 10 μm can provide higher efficiency than the microring with R = 20 μm even when input power is 14 mW (equivalent to P_c for R = 20 μm). A double microring structure has much better efficiency in low loss condition though its efficiency is deteriorated when loss increases.

Fig. 9 Total efficiency of SHG normalized to input power versus loss with input power and radius as parameter.

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Total efficiency of a double microring structure versus phase difference of SH at the second microring with FW field, Δφ is illustrated in Fig. 10 with loss as parameter. Here, microrings are assumed to be identical, input FW power is 13 mW and κ equals 0.2. It is obvious that in low loss condition, the highest efficiency occurs at Δφ = 180° but in lossy waveguide, efficiency is maximum when Δφ = 0°. This is because, for low loss structures, output FW field of the first microring has 180° phase difference with its input FW but when the loss increases, output FW field of the first microring observes no phase shift.

Fig. 10 Total efficiency of a double microring structure versus Δφ with loss as parameter (R = 10 μm).

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3.2 SHG Linewidth

Figure 11 shows wavelength response of SHG in single microring structure with R = 10 μm for input power equal to its P _c. It reveals that dependence of n_eff of fields on the input wavelength can reduce the linewidth of SHG. In this condition Eq. (11) has lower precision than Eq. (10). Here, λ ₀ = 1.5495 μm and there is no phase mismatch between FW and SH fields at wavelength detuning Δλ = λ - λ ₀ = 0.

Fig. 11 Effect of refractive index dispersion on total efficiency of SHG in single microring structure versus wavelength detuning. R = 10 μm, λ ₀ = 1.5495 μm.

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Figure 12 shows linewidth of SHG in single microring structure for various phase mismatches. Here, n_eff is supposed to be dispersive and λ ₀ = 1.5495 μm. Phase mismatch is supposed to be due to deviation of n _TM from Eq. (1). We see that a deviation of order 10⁻⁴ in n _TM can cause an obvious decrease in the efficiency (e.g. Δn _TM = 1.8 × 10⁻⁴, 3.6 × 10⁻⁴ and 6.1 × 10⁻⁴ correspond to RΔk(λ ₀) = 2.015, 2.03 and 2.05, respectively). Linewidth of SHG in microring is less than 0.2 nm, which is comparable to the bandwidth of filtering response of microring. In addition, when phase mismatch increases, wavelength response splits with two peaks. The higher peak is at the resonance wavelength of FW field and the next peak is at the resonance of SH field. Therefore, filtering response of microring plays a major role in SHG by increasing the power density of fields.

Fig. 12 Total efficiency of SHG in single microring structure versus wavelength detuning for various phase mismatches. R = 10 μm, λ ₀ = 1.5495 μm.

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Total efficiency of SHG versus wavelength detuning for different radii is shown in Fig. 13 . Increasing the radius of microring decreases the linewidth of SHG since larger microring has narrower filtering response. FWHM of SHG in microring reduces from 0.1 nm for R = 10 μm to 0.05 nm for R = 20 μm.

Fig. 13 Total efficiency of SHG in single microring structure versus wavelength detuning for R = 10 and R = 20 μm.

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4. Conclusions

In this paper, a multilayer AlGaAs/AlO_x waveguide is presented and has been investigated for efficient SHG in microring structure. To perform the simulations a fast integrative method has been proposed to study the steady state and transient second harmonic generation in microrings and has been compared to the FDTD method. FEM was utilized to extract TE and TM eigen-modes and eigen-frequencies of arbitrary waveguide structures.

Wavelength response of SHG in microring structures has been modelled and studied here. Simulations show that waveguide dispersion is not ignorable in calculation of wavelength response of SHG. Linewidth of SHG in microring decreases to less than 0.1 nm by filtering response of microring. Large radius microring provides narrower linewidth.

It has been shown that the total efficiency decreases exponentially with the loss and the external efficiency can reach 100% for a proper coupling coefficient and input power. In low loss double microring structure, 100% total efficiency is obtained with lower input power as compared with a single microring structure. Although double microring structure is more efficient in low loss condition, it is more sensitive to the loss than a single microring. In addition, it has been shown that higher radius of microring can provide higher efficiency of SHG, but considering loss, smaller microring structures yield better efficiency.

Acknowledgements

This work was supported in part by Iran Telecommunication Research Center (ITRC).

References and links

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Parameter	m _p	m _s	w (nm)	λ ₀ (μm)	v _p (μm/ps)	v _s (μm/ps)	P (MW)	A (μm²)	n _TE (FW)	n _TM (SH)
R = 6 μm	59	120	831.0	1.5473	91.5	82.4	1.27	0.666	2.4215	2.4625
R = 10 μm	98	198	830.8	1.5495	91.9	83.9	1.31	0.629	2.4168	2.4414
R = 20 μm	196	394	844.9	1.5515	92.3	84.7	1.34	0.593	2.4199	2.4322

Design and analysis of enhanced second harmonic generation in AlGaAs/AlO_x microring waveguide

Abstract

1. Introduction

2. Theory

2.1 Linewidth of SHG in microrings

2.2 Double microring structure

3. Results and discussion

3.1 SHG Efficiency

3.2 SHG Linewidth

4. Conclusions

Acknowledgements

References and links

Cited By

Figures (13)

Tables (1)

Equations (11)

Optics Express