1. Introduction

GaAs and other zincblende-structured crystals such as GaP and ZnSe are attractive nonlinear optical materials because of their large nonlinear coefficients and broad transmission ranges. These materials cannot be birefringently phasematched so other means of phasematching are required to achieve efficient nonlinear frequency conversion. Efficient conversion has been demonstrated in GaAs by quasi-phasematching (QPM) [1,2] using periodic domain inversions [3–5] and by Fresnel phasematching [6,7]. QPM in GaAs has also been shown using phase-shifting mirrors [8].

Recently, $\overline{4}$-quasi-phasematching ($\overline{4}$-QPM) in GaAs microdisks and microrings has been proposed [9–11]. $\overline{4}$-QPM allows efficient χ ⁽²⁾ nonlinear optical mixing of the whispering-gallery modes of a GaAs microdisk without using external domain inversions. The $\overline{4}3m$ symmetry of zincblende semiconductors (such as GaAs, GaP and ZnSe) means that a 90° rotation about the <001> axis is equivalent to a domain inversion (see Figs. 1a and 1b). As waves propagate around a <001>-surface-normal microdisk, their crystal environment effectively experiences four 90° rotations, which is equivalent to four domain inversions due to the $\overline{4}$-symmetry (Fig. 1c). The nonlinear tensor for $\overline{4}3m$ materials dictates that one of the three interacting waves in the <001>-surface-normal microdisk must have TM polarization (electric field orthogonal to the disk) while the other two have TE polarization (electric field in the plane of the disk). The QPM effect can be seen by considering the interaction of the fields with the nonlinear tensor [9], as described for $\overline{4}3m$crystals in Appendix A. In additional to microdisks and microrings, $\overline{4}$-QPM can be obtained in other “bent” geometries such as square-shaped resonators [11] and curved waveguides [12]. $\overline{4}$-QPM is also available in other crystals with $\overline{4}$-symmetry such as those of the $\overline{4}2m$ (KH₂PO₄, chalcopyrites, etc.), and $\overline{4}$ crystal classes [11].

 

Fig. 1 Two views of the zincblende crystal structure with (a) inverted relative to (b). $\overline{4}$ symmetry means that a 90° rotation combined with a crystal inversion reproduces the original crystal. Equivalently, a 90° rotation about the $\overline{4}$axis is the same as a crystal inversion. (c) Waves propagating around a <001>-normal GaAs microdisk effectively experience four 90° rotations and hence, four domain inversions.

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The nonlinear conversion efficiency is also increased by the presence of the resonant cavity. External cavities have been used to enhance second-harmonic generation (SHG) in bulk crystals [13]. Resonantly enhanced nonlinear mixing can enable efficient optical-frequency conversion in small volumes, which would be of interest for miniature integrated sources and generation of entangled photons [14]. Cavity-enhanced SHG has been demonstrated in microcavities including LiNbO₃ whispering-gallery-mode resonators [15,16] and GaP photonic-crystal cavities [17]. In nonlinear microcavities where all the interacting waves are on-resonance, 100% conversion can be obtained theoretically with milliwatt-level pumping [18].

In this paper, we explore the interplay between cavity resonances and $\overline{4}$-QPM in enhancing nonlinear-conversion efficiency in a GaAs microdisk. We describe on- and off-resonance nonlinear-optical conversion. As an example, we consider SHG in a GaAs microdisk, but our results can be extended to three-frequency processes such as sum- and difference-frequency generation, and to other $\overline{4}$-symmetry materials. We present a theoretical analysis in Section 2 of $\overline{4}$-QPM in a traveling wave resonator. To maximize conversion, both fundamental and second-harmonic waves should be resonant with the microdisk and the $\overline{4}$-QPM condition should be satisfied. We investigate nonlinear conversion if one of the waves is not resonant with the microdisk cavity and calculate the conversion in these cases. We explore the effects of varying microdisk radius and temperature on the SH conversion spectrum in Section 3, which is important for experimental realization of SHG in a GaAs microdisk. In Section 4, we discuss interpretations of our results including analogies to Fresnel phasematching. We also describe perfectly destructive $\overline{4}$-quasi-phasematching where both the fundamental and second-harmonic waves are on-resonance but SHG is suppressed.

2. SHG conversion efficiency based on generalized waveguide-microresonator coupling theory

In an air-clad GaAs microdisk with $\widehat{z}$ surface-normal, SHG occurs for a TE-polarized fundamental (electric field in the plane of the disk, with magnetic field, $H_z^f$) and a TM-polarized second-harmonic (SH) (electric field orthogonal to the disk, $E_z^{SH}$). With coordinates sketched in Fig. 2a , the fields can be represented by

 

Fig. 2 (a) Microdisk coordinate system. (b) Sketch of coupling between a fiber taper and a disk or ring resonator. t_i and κ_i are the through- and cross-coupling coefficients, respectively; and i represents f or SH.

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Let us consider coupling of the microdisk to a waveguide (Fig. 2b). B_n,i and C_n,i are the complex mode amplitudes normalized so that |B_n,i|², |C_n,i|² = power. We note that B_n,i and C_n,i differ from the slowly varying amplitudes A_i(θ, t) by phase terms. Following Ref [19], coupling between the resonator and waveguide (in the absence of reflections) is described by

The coupler is taken to be lossless so that |κ_i|² + |t_i|² = 1.

Propagation of the fundamental around the ring produces loss, α_f, and phase shift, ϕ_f,

We assume the amplitude of the fundamental wave is constant (A_f ² ≈|C_2,f|² ≈|B_2,f|²).

The quantity m_i plays a role analogous to the wavevector k_i in linear propagation geometries. Both describe the rate of phase accumulation due to propagation and, hence, the effective propagation constant inside the medium. However, m_i is only well-defined at cavity-resonance wavelengths. Inside the microdisk, it is reasonable to ask what the effective propagation constant is at a wavelength that does not fall at a cavity resonance. We can estimate this effective propagation constant by linearly interpolating between resonances of the same spatial-mode family and constructing a function $m_i\text{'}({\lambda }_i)$. This continuous function describes the effective dispersion inside the microdisk. The phase shift from one round trip in the disk is

Using Eqs. (5) and (6), the circulating fundamental power is [19]

We note that Eqs. (10) and (11) also describe the “passive” behavior of the resonator at the second-harmonic wavelength when the f subscripts are replaced by SH.

For SHG in the microresonator, there is no incoming wave at the second-harmonic (B_{1, SH} = 0) so that Eq. (5) becomes

If we assume that the amplitude of the fundamental wave is unchanged (that is, α_f ≈1 and |C_2,f|² ≈|B_2,f|²), it follows from Eqs. (7) and (13) that the circulating second-harmonic power is

The transmitted SH power is

The parameters α_i and t_i are related to the quality factors of the resonator. From Ref [19], the linewidths of the resonances are calculated by considering the shift in phase ϕ_i where the circulating power (Eq. (10)) is half of its peak value. The finesse at wave i, F_i, is the ratio between the phase shift between adjacent resonances, 2π, and the full-width at half maximum (FWHM) of the resonance. The finesse is also equal to the free spectral range (FSR) divided by the FWHM of the resonance. For a high-finesse resonator where ${\alpha }_i,\left|t_i\right|\approx 1$, the finesse is

The quality factor, Q_i, is ratio between the resonance frequency, ω₀, and the linewidth, Δω_FWHM, and can be calculate from the finesse using

The total (Q_i), intrinsic ($Q_i^0$) and coupling ($Q_i^c$) quality factors are related by

3. On- and off-resonance second-harmonic generation in a GaAs microdisk

Using Eqs. (14) and (15), we can understand SHG in a microdisk for the cases when both the fundamental and SH are resonant with the microdisk cavity, and for the cases when one or more of the waves is not resonant. We consider SHG with λ_f ≈2 μm in 161-nm thick GaAs microdisks where both fundamental and SH modes are the lowest-order vertical and lowest-order radial modes. The disk thickness is chosen so that doubly resonant, quasi-phasematched SHG can be supported with disk radii near 2.6 μm. Throughout this section, we take the incident fundamental power (in the fiber waveguide) as $P_f^{in}={\left|B_{1,f}\right|}^2=1$mW and $Q_f^0$ = $Q_f^c$ = $Q_{SH}^0$ = $Q_{SH}^c$ = $10000$, which corresponds to linewidths Δλ_f = 2Δλ_SH = 0.4 nm. We also assume the phase shifts from the coupler are ψ_f = ψ_SH = 0 (the main effects of non-zero coupler phase shifts are small wavelength shifts in the resonance locations). Values for K ₊ and K _– are given in Table 1 of Appendix B.

Table 1. Comparison between coupled-mode theory (CMT), which utilizes the energy normalization and Eq. (62), and on-resonance generalized waveguide-microresonator theory (GWMT) discussed in Section 2, which utilizes the power normalization and Eq. (63). The incident fundamental power P_fⁱⁿ = 1 mW.

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A GaAs disk with radius R = 2.609 μm has both the fundamental and SH waves on-resonance at λ_f = 2λ_SH = 1998.7 nm with fundamental wave (m_f = 13) TE-polarized and the SH wave (m_SH = 28) TM-polarized (satisfying m_SH – 2m_f = 2). We used finite-element modeling software and GaAs dispersion data [20] to calculate resonance wavelengths for various values of m_f and m_SH, and linearly interpolated between resonances to calculate $m_i\text{'}(\lambda )$ and the phase shifts, ϕ_i (Eq. (8)). Figure 3 shows the SH conversion efficiency, η = |C _{1, SH}| ² /|B _{1, f}| ², for this microdisk.

 

Fig. 3 SH conversion efficiency for various pumping wavelengths in a GaAs microdisk (R = 2.609 μm, h = 161nm) that supports doubly resonant SHG at λ_f = 2λ_SH = 1998.7 nm (where m_f = 13, m_SH = 28, Δm = 2). The maximum conversion efficiency is η = 1.2% for 1 mW of incident fundamental power. The top inset shows locations of the fundamental and SH (2λ_SH) cavity resonances.

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A maximum conversion of η = 1.2% is obtained at a pumping wavelength of 1998.7 nm. At this wavelength, both the fundamental and SH waves are resonant with the microdisk and Δm = m_SH – 2m_f = 2. Figure 3 also shows the SH conversion efficiency at other wavelengths that do not necessarily correspond to cavity resonances. When the fundamental is on-resonance but the SH is not, there is a local maximum in η, but its value is more than four orders of magnitude smaller than the maximum conversion found at 1998.7 nm. There are also local maxima when only the SH is on-resonance and the fundamental is not, but these peaks are even weaker than the fundamental-only peaks. The difference in relative peak sizes confirms that resonance-enhancement at the fundamental wave is a larger contributor to increased SHG than resonance-enhancement at the SH wave, which can also be seen in the expression for SHG calculated by coupled-mode theory (Eq. (62) in Appendix B) where $P_{SH}^{out}$ is roughly proportional to $Q_{SH}^c{(Q_f^c)}^2$.

Another feature of Fig. 3 are the sharp dips in the SH conversion. These dips correspond to wavelengths where $\Delta m^{\prime }=0,$ ±1, ±3, etc., and there is perfectly destructive $\overline{4}$-QPM. These wavelengths need not correspond to cavity resonances; for instance, $\Delta m^{\prime }=0$ occurs at λ_f = 2093.7 nm in Fig. 3 where $m_f\text{'}$ = 11.87 and $m_{SH}\text{'}$ = 23.75. SHG is suppressed at these wavelengths in the same way that nonlinear conversion is suppressed in even-order QPM gratings having 50% duty cycle [2].

Figure 4 plots the SH conversion for a slightly smaller GaAs microdisk (R = 2.587 μm, h = 161 nm) where the fundamental and SH resonances near 1990 nm do not overlap. For this disk, the m_f = 13 fundamental resonance occurs at λ_f = 1987.4 nm and the m_SH = 28 SH resonance occurs at 2λ_SH = 1992.8 nm such that |λ_f – 2λ_SH | = 5.4 nm. A maximum conversion of η = 1.4×10^-3% is obtained at a pumping wavelength of 1987.4 nm (where only the fundamental is on-resonance and $\Delta m^{\prime }$ = 2.24). As in Fig. 3, there are local maxima in η when either the fundamental or the SH are resonant with the microdisk cavity. SH conversion is maximized at 1987.4 nm since the value of $\Delta m^{\prime }$is close to 2, but conversion at this wavelength is only one or two orders of magnitude larger than other peaks. In contrast, the global maximum in conversion efficiency when the resonances overlap (Fig. 3) can be four or more orders of magnitude larger than other peaks.

 

Fig. 4 SH conversion efficiency for different pumping wavelengths in a GaAs microdisk (R = 2.587 μm, h = 161nm). Near 1990 nm where Δm’ ≈2, the fundamental and SH resonances do not overlap and are separated by λ_f – 2λ_SH = −5.4 nm. The maximum SH conversion occurs at a pumping wavelength of 1987.4 nm with η = 1.4 × 10^-3% (P_fⁱⁿ = 1 mW). The top inset shows locations of the fundamental and SH (2λ_SH) cavity resonances.

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By increasing the microdisk radius to 2.643 μm, doubly resonant SHG can be achieved at a pumping wavelength of 2186.9 nm with Δm = −2 (Fig. 5 ). At this wavelength, η = 0.29%. The difference in maximum η between this case and the R = 2.609 μm case is due to different values of K ₊ and K _- (see Table 1 in Appendix B). Other peaks in SH conversion are produced at wavelengths where only the fundamental or only the SH wave is on-resonance. For instance, a local maximum occurs at λ_f = 2016.7 nm where the fundamental is on-resonance (m_f = 13) while the SH is not. At this pumping wavelength, $\Delta m^{\prime }$ = 1.61 and η = 6.5 × 10^-4%.

 

Fig. 5 SH conversion efficiency for various pumping wavelengths in a GaAs microdisk (R = 2.643 μm, h = 161nm). Doubly resonant SHG is achieved at λ_f = 2λ_SH = 2186.9 nm where m_f = 11, m_SH = 20, Δm = −2; and SH conversion is maximized with η = 0.29% (P_fⁱⁿ = 1 mW). The top inset shows locations of the fundamental and SH (2λ_SH) cavity resonances.

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Figure 6 shows details of the SH conversion spectra for these three microdisk sizes in two different pumping-wavelength ranges. Figure 6a plots conversion efficiency near a pump wavelength of 2000 nm where $\Delta m^{\prime }$≈2. For the R = 2.609 μm disk, the fundamental and SH resonances are aligned and very large SH conversion is achieved (η = 1.2%). For the other disk sizes, the fundamental (m_f = 13) and SH (m_SH = 28) resonances become misaligned, resulting in decreased SH conversion. Figure 6b plots the conversion near 2190 nm pumping wavelength where $\Delta m^{\prime }$≈-2. Overlapping fundamental and SH resonances are obtained with the R = 2.643 μm disk, resulting in high SH conversion (η = 0.29%).

 

Fig. 6 Detailed pump-wavelength dependence of SH conversion efficiency for three different GaAs microdisk sizes (h = 161nm). (a) Double resonance satisfying Δm = 2 is achieved at λ_f = 2λ_SH = 1998.7 nm in the 2.609-μm-radius disk, and (b) double resonance satisfying Δm = −2 is achieved at λ_f = 2λ_SH = 2186.9 nm in the 2.643-μm-radius disk.

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Maximum SHG is obtained when the fundamental and SH resonances overlap and have azimuthal numbers that satisfy Δm = m_SH – 2m_f = ±2. As the disk radius, thickness or temperature is changed, the resonances no longer coincide and $\Delta m^{\prime }$≠ ±2, which reduces SHG. Figure 7 illustrates the effect of radius and temperature tuning on SH conversion. The reference microdisk has 161-nm thickness and R ₀ = 2.609 μm, and supports doubly resonant SHG at λ_f = 2λ_SH = 1998.7 nm (m_f = 13, m_SH = 28). By changing the radius of the microdisk by 5 nm, the fundamental and SH resonances become detuned by |λ_f – 2λ_SH| = 1.3 nm (several times the 0.4 nm linewidth of these Q_f = Q_SH = 5000 resonances), and the maximum SH conversion efficiency drops from 1.2% to 2.8×10^-2% (see Fig. 7a). Temperature can be used for fine tuning, as shown in Fig. 7b. Maximum conversion is obtained at T = 30 °C while at T = 10 °C or 50 °C, the fundamental and SH resonances are detuned by |λ_f – 2λ_SH| = 0.5 nm, resulting in a sevenfold decrease in peak SH conversion. The curves become noticeably asymmetric as the fundamental and SH resonances only partially overlap. Varying the geometry or temperature also changes the strength of the interaction through K ₊ and K _–, but the SH conversion is more strongly affected by the detuning of the resonances (for this example, changing the microdisk radius by 5 nm changes the magnitudes of K ₊ and K _– by only 0.2%).

 

Fig. 7 (a) Varying disk radius around R ₀ = 2.609 μm for fixed disk thickness of 161 nm. When the radius is changed by ± 5 nm, the resonances become detuned by |λ_f – 2λ_SH| = 1.3 nm (linewidths of passive cavity resonances are 0.4 nm), which results in a 41-fold reduction in maximum conversion efficiency. (b) Temperature tuning in a R ₀ = 2.609 μm, h = 161 nm GaAs microdisk. At T = 10 °C and 50 °C, fundamental and SH resonances are separated by |λ_f – 2λ_SH| = 0.5 nm and the peak conversion efficiency is seven times smaller than the peak at T = 30 °C.

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It is possible for the fundamental and SH resonances of a microdisk to be aligned but Δm is not equal to ±2 but another integer instead. This situation occurs in a 2.587-μm radius, 161-nm-thick GaAs microdisk at a pumping wavelength of 1903.4 nm, shown in Fig. 4 with a detailed plot in Fig. 8 . SHG is suppressed when Δm = 0, ±1, ±3, etc. due to perfectly destructive $\overline{4}$-quasi-phasematching. The width of the dip is comparable to the linewidths of the passive cavity resonances (discussed in Section 4). This feature is interesting from an experimental standpoint. It is difficult to measure directly the m values of the resonances, but the presence of the dip clearly indicates that Δm ≠ ±2. Since the dip arises from perfectly balanced destructive interference of the SH wave, the observed sharpness and depth of the dip can be a measure of the quality of the microdisk (its circularity, uniformity and loss).

 

Fig. 8 Overlapping fundamental and SH resonances (λ_f = 2λ_SH) occur in a 2.587-μm-radius, 161-nm-thick GaAs microdisk, but Δm = 4 (m_f = 14, m_SH = 32), which results in a suppression of SHG due to perfectly destructive $\overline{4}$-QPM.

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4. Discussion

Three effects enhance χ ⁽²⁾ conversion in a GaAs microdisk — resonance enhancement at the fundamental wave, resonance enhancement at the second-harmonic wave, and $\overline{4}$-quasi-phasematching. SHG is significantly enhanced when all three resonance conditions are satisfied. In the examples discussed in Section 3, up to 1.2% conversion efficiency can be obtained using 1 mW of external fundamental power. Resonantly enhanced SHG in a microdisk is far more efficient than single-pass SHG in a GaAs crystal of comparable size. Confocally focused SHG in a GaAs sample of length $2\pi R=16$μm only produces η ≈10^-6% with 1mW of incident power. As a side note, Section 3 discusses mixing of lowest-order radial modes, which produces larger η than mixing a fundamental with one radial antinode and a SH with two [11] due to better radial-mode overlap.

It is possible to observe SHG in a GaAs microdisk even if the resonance conditions are not all satisfied. Comparing Eqs. (10), (14), and (15), we see the spectrum of the SHG output has three factors

The first factor is the square of the circulating fundamental spectrum, and the last factor is the circulating SH spectrum of the passive cavity. The middle factor in Eq. (20) is the contribution from the nonlinear mixing, whose spectral dependence is described by Eq. (4). The bandwidth of $|\tilde{K}{|}^2$ is broad because it is determined by one roundtrip of nonlinear mixing (essentially, $|\tilde{K}{|}^2\propto {\text{sinc}}^2(\Delta k\ell /2)$ where Δk is the wavevector mismatch and λ the length of one cavity roundtrip). The relative spectral overlap between the three factors in Eq. (20) determine the total SH conversion. Since $P_{SH}^{out}$ varies as the square of the circulating fundamental power and linearly with the SH power, resonance enhancement at λ_f has a bigger effect than resonance enhancement at λ_SH.

There is a strong analogy between $\overline{4}$-QPM in a microdisk and Fresnel phasematching in a plate [6,7]. The output SH intensity in Fresnel phasematching is proportion to [6,7]

This factor is analogous to the passive-cavity-spectra factors in Eq. (20); more bounces in the plate (i. e., larger N) lead to more SHG while narrowing the generated spectrum in the same way that higher Q_f and Q_SH lead to more roundtrips in the cavity, higher circulating powers and narrower spectra. In both Fresnel phasematching and $\overline{4}$-QPM in a microdisk, the gain factors (represented by the first factor in Eq. (21) and the second factor in Eq. (20)) are multiplied by the cavity-resonance factors, much in the same way that a Fabry-Perot cavity resonance interacts with a gain medium [6].

A big difference between Fresnel phasematching and $\overline{4}$-QPM in a microdisk is the presence of the resonant cavity for the latter. Firstly, the microdisk cavity allows for a much more compact device compared to a Fresnel-phasematched device. Secondly, the cavity enforces high periodicity for long, total path-length. Variations in plate thickness (from roughness or wedge) can lead to large phase error, ε. If the phase error is too large, then the useful net path-length (that is, useful N) is reduced. Large phase errors also reduce the wavelength-acceptance bandwidth.

The high periodicity of the effective domain inversions provided by the microdisk cavity makes microcavity structures attractive for efficient $\overline{4}$-QPM conversion. Curved waveguide structures have also been proposed for $\overline{4}$-QPM [12]. These structures are challenging to fabricate since the length of each domain inversion can vary due to fabrication errors. The domain-length errors are similar to phase errors in Fresnel phasematching; both types of errors will reduce the net conversion efficiency and reduce the wavelength-acceptance bandwidth.

Equations (14) and (15) describe very well $\overline{4}$-quasi-phasematched SHG in a microdisk for cases where either the fundamental or SH wave is not resonant with the cavity. We can also give an intuitive description for this process. If light is not resonant with a cavity, then there is destructive interference between the incident and circulating waves, which results in suppression of light circulating inside the cavity. However, due to finite quality factors, the amount of light inside the cavity is non-zero even if the wave is off-resonance. If the fundamental wave is resonant with the cavity, then the circulating intensity inside the cavity at λ_f will be high. This light can initiate second-harmonic generation at the entrance of the cavity. The second-harmonic will increase due to $\overline{4}$-QPM as it propagates around the cavity and if the SH is resonant with the cavity, then the SH light from previous round trips will add constructively to subsequent round trips. However, if the SH light is not resonant with the cavity, then some amount of SH is coupled out, and the remaining SH inside the cavity interferes destructively with light from previous round trips. In essence, when the SH wave is off-resonance with the cavity, there is negligible λ_SH light at the entrance of the cavity, and the net amount of SH generated is essentially the amount produced on one round trip times the out-coupling coefficient. Having the fundamental wave resonant with the cavity will significantly increase SHG since the driving fields will be large (due to resonance enhancement). Conversely, if the fundamental is not resonant with the cavity, but λ_f/2 matches a cavity resonance, then there will still be an enhancement of SHG in the cavity. Since Q_f is finite, there will be some small amount of fundamental light in the cavity and this light can initiate SHG. The generated second-harmonic light is resonant with the cavity and will experience build-up as SH light from subsequent round-trips add constructively.

Equation (2) implies that a perfectly circular cavity has only two components that can contribute to quasi-phasematching: Δm = ±2. It would be interesting to explore square [11] or deformed cavities [21,22] that would provide resonance enhancement of the circulating waves and allow higher-order QPM. The deformed cavities would have interesting applications with free-space couplers because of their directional output.

When Δm = 0, ±1, ±3, etc. in a circular cavity, SHG is suppressed due to perfectly destructive $\overline{4}$-QPM (Fig. 8). In this process, each round trip in the cavity produces no net SH, similar to the way there is no net SH is produced after exactly two coherence lengths in a non-phasematched crystal. This suppression is analogous to operating a nonlinear crystal at η ∝ sinc²(ΔkL/2) = 0, which occurs when ΔkL/2 = ± π, ± 2π, etc. We can derive an expression for the width of the destructive $\overline{4}$-QPM dip when the fundamental and SH resonances overlap (λ_f = 2λ_SH) and Δm = 0, ±1, ±3, etc. In the vicinity of the dip, $|\tilde{K}{|}^2\propto {\text{(}\lambda \text{-}{\lambda }_{\text{0}}\text{)}}^{\text{2}}$ since the sinc² function varies quadratically near its zeros. The circulating power spectrum of the fundamental and SH resonances (Eq. (10)) can be approximated by Lorentzian lineshapes with FWHM widths w_f and w_SH, respectively (in wavelength units). If w_f = 2w_SH (same total quality factors for both waves), then the width of the destructive $\overline{4}$-QPM dip is 0.31w_f. The sharpness of a measured, destructive $\overline{4}$-QPM dip where Δm ≠ ± 2 can be used to characterize the circularity and ideality of a microdisk.

5. Conclusions

We have presented a theory of $\overline{4}$-quasi-phasematched second-harmonic generation in a GaAs microdisk that describes both on- and off-resonance conversion. For doubly resonant mixing of lowest-order vertical and radial modes satisfying m_f = 13, m_SH = 28 and Δm = 2 in a 2.609-μm-radius, 161-nm-thick microdisk, the theory predicts 1.2% conversion efficiency with 1 mW of λ_f = 1998.7 nm external fundamental light (assuming critical coupling and Q_f = Q_SH = 5000). Our theory also describes SHG when the fundamental and SH resonances no longer overlap (λ_f –2λ_SH ≠ 0), and SHG is no longer doubly resonant. When |λ_f – 2λ_SH| = 1.3 nm, we expect a maximum conversion efficiency of η = 2.8 × 10^-2% and when |λ_f – 2λ_SH| = 5.4 nm, we expect η = 1.4×10^-3% (for Q_f = Q_SH = 5000, the linewidths of the passive-cavity resonances are 0.4 nm). We show that the SH conversion spectrum is a product of the circulating-power cavity spectra at the fundamental and SH wavelengths, and the nonlinear gain spectrum, $|\tilde{K}{|}^2$. In analogy to Fresnel phasematching, narrowing the passive cavity spectra (through higher Q) is associated with longer total interaction length and higher conversion. Higher quality factors also lead to tighter fabrication tolerances for achieving the double-resonance condition (λ_f = 2λ_SH). Using a series of lower-quality-factor, waveguide-coupled GaAs microdisks would allow higher nonlinear conversion with broader spectral bandwidth [23]. We also identify the process of perfectly destructive $\overline{4}$-QPM; SHG is suppressed in a circular GaAs microdisk at the wavelengths where $\Delta m^{\prime }$ = 0, ±1, ±3, etc. In this process, there is destructive interference of the second-harmonic wave, which result in no net SHG per cavity round-trip and a dip in the SH conversion spectrum. This dip can be used to confirm Δm ≠ ± 2 and to evaluate the circularity and ideality of the microdisk.