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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 18 — Aug. 29, 2011
  • pp: 16898–16918
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On- and off-resonance second-harmonic generation in GaAs microdisks

Paulina S. Kuo and Glenn S. Solomon  »View Author Affiliations


Optics Express, Vol. 19, Issue 18, pp. 16898-16918 (2011)
http://dx.doi.org/10.1364/OE.19.016898


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Abstract

We present a theoretical description of on- and off-resonance, 4 ¯ -quasi-phasematched, second-harmonic generation (SHG) in microdisks made of GaAs or other materials possessing 4 ¯ symmetry, such as GaP or ZnSe. The theory describes the interplay between quasi-phasematching (QPM) and the cavity-resonance conditions. For optimal conversion, all waves should be resonant with the microdisk and should satisfy the 4 ¯ -QPM condition. We explore χ(2) nonlinear mixing if one of the waves is not resonant with the microdisk cavity and calculate the second-harmonic conversion spectrum. We also describe perfectly destructive 4 ¯ -QPM where both the fundamental and second-harmonic are on-resonance with the cavity but SHG is suppressed.

© 2011 OSA

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

1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light eaves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962). [CrossRef]

,2

2. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992). [CrossRef]

] using periodic domain inversions [3

3. E. Lallier, M. Brevignon, and J. Lehoux, “Efficient second-harmonic generation of a CO2 laser with a quasi-phase-matched GaAs crystal,” Opt. Lett. 23(19), 1511–1513 (1998). [CrossRef] [PubMed]

5

5. L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001). [CrossRef]

] and by Fresnel phasematching [6

6. R. Haidar, N. Forget, P. Kupecek, and E. Rosencher, “Fresnel phase matching for three-wave mixing in isotropic semiconductors,” J. Opt. Soc. Am. B 21, 1522–1534 (2004). [CrossRef]

,7

7. H. Komine, W. H. Long Jr, J. W. Tully, and E. A. Stappaerts, “Quasi-phase-matched second-harmonic generation by use of a total-internal-reflection phase shift in gallium arsenide and zinc selenide plates,” Opt. Lett. 23(9), 661–663 (1998). [CrossRef] [PubMed]

]. QPM in GaAs has also been shown using phase-shifting mirrors [8

8. C. Simonneau, J. P. Debray, J. C. Harmand, P. Vidakovi, D. J. Lovering, and J. A. Levenson, “Second-harmonic generation in a doubly resonant semiconductor microcavity,” Opt. Lett. 22(23), 1775–1777 (1997). [CrossRef] [PubMed]

].

In this paper, we explore the interplay between cavity resonances and 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 4¯-symmetry materials. We present a theoretical analysis in Section 2 of 4¯-QPM in a traveling wave resonator. To maximize conversion, both fundamental and second-harmonic waves should be resonant with the microdisk and the 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 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 z^ surface-normal, SHG occurs for a TE-polarized fundamental (electric field in the plane of the disk, with magnetic field, Hzf) and a TM-polarized second-harmonic (SH) (electric field orthogonal to the disk, EzSH). With coordinates sketched in Fig. 2a
Fig. 2 (a) Microdisk coordinate system. (b) Sketch of coupling between a fiber taper and a disk or ring resonator. ti and κi are the through- and cross-coupling coefficients, respectively; and i represents f or SH.
, the fields can be represented by
Hzf=Af(θ,t)Ψf(r,z)ei(ωftmfθ)EzSH=ASH(θ,t)ΨSH(r,z)ei(ωSHtmSHθ),
(1)
where Ai(θ,t) is the slowly varying amplitude at frequency ωi (the index i refers to the fundamental, f, or the second-harmonic, SH). In this section, Ai(θ,t) is normalized so that |Ai(θ,t)|2 = power. Ψi(r, z) is the mode profile, and mi is the azimuthal number of the mode, which is an integer for a resonant mode. The exp(-imiθ) factors describe the phase accumulated by the waves as they propagate around the disk, analogous to exp(-iki) terms found in linear propagation geometries. The change in the SH amplitude due to nonlinear mixing in the GaAs microdisk is described by [9

9. Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74(6), 063804 (2006). [CrossRef]

,11

11. P. S. Kuo, W. Fang, and G. S. Solomon, “4-quasi-phase-matched interactions in GaAs microdisk cavities,” Opt. Lett. 34(22), 3580–3582 (2009). [CrossRef] [PubMed]

]
ASHθ=Af2(K+ei(Δm+2)θ+Kei(Δm2)θ),
(2)
where Δm=mSH2mf, and K+ and K are the SHG coefficients. Details of the fields inside the microdisk and the derivation of Eq. (2) are discussed in Appendix A. If we assume the fundamental is undepleted (Af is constant), integrating Eq. (2) from θ = 0 to 2π yields
ASH(2π)ASH(0)=Af2K˜,
(3)
where
K˜=2π(K+eiπ(Δm+2)sinc[(Δm+2)π]+Keiπ(Δm2)sinc[(Δm2)π]),
(4)
and sinc(x) = sin(x)/x. Equation (3) describes the change in the amplitude of the second-harmonic wave due to SHG and does not include loss.

Let us consider coupling of the microdisk to a waveguide (Fig. 2b). Bn,i and Cn,i are the complex mode amplitudes normalized so that |Bn,i|2, |Cn,i|2 = power. We note that Bn,i and Cn,i differ from the slowly varying amplitudes Ai(θ, t) by phase terms. Following Ref [19

19. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]

], coupling between the resonator and waveguide (in the absence of reflections) is described by

[C1,iC2,i]=[tiκiκi*ti*][B1,iB2,i].
(5)

The coupler is taken to be lossless so that |κi|2 + |ti|2 = 1.

Propagation of the fundamental around the ring produces loss, αf, and phase shift, ϕf,
B2,f=αfexp(iϕf)C2,f.
(6)
At the SH wave, there is loss, phase shift and SHG gain. Equation (3) may be written as

B2,SHαSHexp(iϕSH)C2,SH=|C2,f|2K˜.
(7)

We assume the amplitude of the fundamental wave is constant (Af 2 ≈|C2,f|2 ≈|B2,f|2).

The quantity mi plays a role analogous to the wavevector ki in linear propagation geometries. Both describe the rate of phase accumulation due to propagation and, hence, the effective propagation constant inside the medium. However, mi 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 mi'(λi). This continuous function describes the effective dispersion inside the microdisk. The phase shift from one round trip in the disk is
ϕi=2πmi',
(8)
and the phase mismatch per round trip accumulated between the fundamental and SH waves is
2πΔm=ϕSH2ϕf.
(9)
We can extend the description of SHG in a microdisk to include cases where the fundamental and/or SH waves are not resonant with the cavity by replacing Δm byΔm in Eqs. (2) – (4).

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

19. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]

]
|B2,f|2=αf2(1|tf|2)1+αf2|tf|22αf|tf|cos(ψf+ϕf)|B1,f|2,
(10)
and the transmitted power is
|C1,f|2=αf2+|tf|22αf|tf|cos(ψf+ϕf)1+αf2|tf|22αf|tf|cos(ψf+ϕf)|B1,f|2,
(11)
where

ti=|ti|exp(iψi).
(12)

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 (B1, SH = 0) so that Eq. (5) becomes

C1,SH=κSHB2,SHC2,SH=tSH*B2,SH.
(13)

If we assume that the amplitude of the fundamental wave is unchanged (that is, αf ≈1 and |C2,f|2 ≈|B2,f|2), it follows from Eqs. (7) and (13) that the circulating second-harmonic power is

|B2,SH|2=|B2,f|4|K˜|2αSH21+αSH2|tSH|22αSH|tSH|cos(ψSH+ϕSH)=|B1,f|4|K˜|2αSH21+αSH2|tSH|22αSH|tSH|cos(ψSH+ϕSH)×(αf2(1|tf|2)1+αf2|tf|22αf|tf|cos(ψf+ϕf))2.
(14)

The transmitted SH power is

|C1,SH|2=|B2,SH|2(1|tSH|2).
(15)

Fi=2π2Δ(ψi+ϕi)=παi|ti|1αi|ti|=δωFSRΔωFWHM.
(16)

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

Qi=ω0ΔωFWHM=Fiω0δωFSR=παi|ti|1αi|ti|ω0δωFSR.
(17)

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 λf ≈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 Pfin=|B1,f|2=1mW and Qf0 = Qfc = QSH0 = QSHc = 10000, which corresponds to linewidths Δλf = 2ΔλSH = 0.4 nm. We also assume the phase shifts from the coupler are ψf = ψSH = 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

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 Pfin = 1 mW.

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of Appendix B.

A GaAs disk with radius R = 2.609 μm has both the fundamental and SH waves on-resonance at λf = 2λSH = 1998.7 nm with fundamental wave (mf = 13) TE-polarized and the SH wave (mSH = 28) TM-polarized (satisfying mSH – 2mf = 2). We used finite-element modeling software and GaAs dispersion data [20

20. T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003). [CrossRef]

] to calculate resonance wavelengths for various values of mf and mSH, and linearly interpolated between resonances to calculate mi'(λ) and the phase shifts, ϕi (Eq. (8)). Figure 3
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 λf = 2λSH = 1998.7 nm (where mf = 13, mSH = 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 (2λSH) cavity resonances.
shows the SH conversion efficiency, η = |C 1, SH| 2 /|B 1, f| 2, for this microdisk.

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 = mSH – 2mf = 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 PSHout is roughly proportional to QSHc(Qfc)2.

Another feature of Fig. 3 are the sharp dips in the SH conversion. These dips correspond to wavelengths where Δm=0, ±1, ±3, etc., and there is perfectly destructive 4¯-QPM. These wavelengths need not correspond to cavity resonances; for instance, Δm=0 occurs at λf = 2093.7 nm in Fig. 3 where mf' = 11.87 and mSH' = 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

2. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992). [CrossRef]

].

Figure 4
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 λf – 2λSH = −5.4 nm. The maximum SH conversion occurs at a pumping wavelength of 1987.4 nm with η = 1.4 × 10-3% (Pfin = 1 mW). The top inset shows locations of the fundamental and SH (2λSH) cavity resonances.
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 mf = 13 fundamental resonance occurs at λf = 1987.4 nm and the mSH = 28 SH resonance occurs at 2λSH = 1992.8 nm such that |λf – 2λSH | = 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 Δm = 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 Δmis 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.

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
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 λf = 2λSH = 2186.9 nm where mf = 11, mSH = 20, Δm = −2; and SH conversion is maximized with η = 0.29% (Pfin = 1 mW). The top inset shows locations of the fundamental and SH (2λSH) cavity resonances.
). 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 λf = 2016.7 nm where the fundamental is on-resonance (mf = 13) while the SH is not. At this pumping wavelength, Δm = 1.61 and η = 6.5 × 10-4%.

Figure 6
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 λf = 2λSH = 1998.7 nm in the 2.609-μm-radius disk, and (b) double resonance satisfying Δm = −2 is achieved at λf = 2λSH = 2186.9 nm in the 2.643-μm-radius disk.
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 Δm≈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 (mf = 13) and SH (mSH = 28) resonances become misaligned, resulting in decreased SH conversion. Figure 6b plots the conversion near 2190 nm pumping wavelength where Δm≈-2. Overlapping fundamental and SH resonances are obtained with the R = 2.643 μm disk, resulting in high SH conversion (η = 0.29%).

Maximum SHG is obtained when the fundamental and SH resonances overlap and have azimuthal numbers that satisfy Δm = mSH – 2mf = ±2. As the disk radius, thickness or temperature is changed, the resonances no longer coincide and Δm≠ ±2, which reduces SHG. Figure 7
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 |λf – 2λSH| = 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 |λf – 2λSH| = 0.5 nm and the peak conversion efficiency is seven times smaller than the peak at T = 30 °C.
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 λf = 2λSH = 1998.7 nm (mf = 13, mSH = 28). By changing the radius of the microdisk by 5 nm, the fundamental and SH resonances become detuned by |λf – 2λSH| = 1.3 nm (several times the 0.4 nm linewidth of these Qf = QSH = 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 |λf – 2λSH| = 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%).

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
Fig. 8 Overlapping fundamental and SH resonances (λf = 2λSH) occur in a 2.587-μm-radius, 161-nm-thick GaAs microdisk, but Δm = 4 (mf = 14, mSH = 32), which results in a suppression of SHG due to perfectly destructive 4¯-QPM.
. SHG is suppressed when Δm = 0, ±1, ±3, etc. due to perfectly destructive 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).

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 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 2πR=16μm 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

11. P. S. Kuo, W. Fang, and G. S. Solomon, “4-quasi-phase-matched interactions in GaAs microdisk cavities,” Opt. Lett. 34(22), 3580–3582 (2009). [CrossRef] [PubMed]

] 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

PSHout=|C1,SH|2|B2,f(ϕf)|4×|K˜|2×|B2,SH,passive(ϕSH)|2.
(20)

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 |K˜|2 is broad because it is determined by one roundtrip of nonlinear mixing (essentially, |K˜|2sinc2(Δk/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 PSHout varies as the square of the circulating fundamental power and linearly with the SH power, resonance enhancement at λf has a bigger effect than resonance enhancement at λSH.

There is a strong analogy between 4¯-QPM in a microdisk and Fresnel phasematching in a plate [6

6. R. Haidar, N. Forget, P. Kupecek, and E. Rosencher, “Fresnel phase matching for three-wave mixing in isotropic semiconductors,” J. Opt. Soc. Am. B 21, 1522–1534 (2004). [CrossRef]

,7

7. H. Komine, W. H. Long Jr, J. W. Tully, and E. A. Stappaerts, “Quasi-phase-matched second-harmonic generation by use of a total-internal-reflection phase shift in gallium arsenide and zinc selenide plates,” Opt. Lett. 23(9), 661–663 (1998). [CrossRef] [PubMed]

]. The output SH intensity in Fresnel phasematching is proportion to [6

6. R. Haidar, N. Forget, P. Kupecek, and E. Rosencher, “Fresnel phase matching for three-wave mixing in isotropic semiconductors,” J. Opt. Soc. Am. B 21, 1522–1534 (2004). [CrossRef]

,7

7. H. Komine, W. H. Long Jr, J. W. Tully, and E. A. Stappaerts, “Quasi-phase-matched second-harmonic generation by use of a total-internal-reflection phase shift in gallium arsenide and zinc selenide plates,” Opt. Lett. 23(9), 661–663 (1998). [CrossRef] [PubMed]

]
ISHout[sin(ΔkL/2)ΔkL/2]2[sin(Nε/2)sin(ε/2)]2,
(21)
where L is the distance between zigzag bounces, N is the total number of bounces and ε is the phase error accumulated per zigzag path. The first factor in Eq. (21) represents the gain factor while the second factor represents the resonance condition [6

6. R. Haidar, N. Forget, P. Kupecek, and E. Rosencher, “Fresnel phase matching for three-wave mixing in isotropic semiconductors,” J. Opt. Soc. Am. B 21, 1522–1534 (2004). [CrossRef]

]. For small ε, the second factor can be written as [7

7. H. Komine, W. H. Long Jr, J. W. Tully, and E. A. Stappaerts, “Quasi-phase-matched second-harmonic generation by use of a total-internal-reflection phase shift in gallium arsenide and zinc selenide plates,” Opt. Lett. 23(9), 661–663 (1998). [CrossRef] [PubMed]

]

[sin(Nε/2)sin(ε/2)]2=N2sinc2(Nε/2).
(22)

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 Qf and QSH lead to more roundtrips in the cavity, higher circulating powers and narrower spectra. In both Fresnel phasematching and 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

6. R. Haidar, N. Forget, P. Kupecek, and E. Rosencher, “Fresnel phase matching for three-wave mixing in isotropic semiconductors,” J. Opt. Soc. Am. B 21, 1522–1534 (2004). [CrossRef]

].

A big difference between Fresnel phasematching and 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 4¯-QPM conversion. Curved waveguide structures have also been proposed for 4¯-QPM [12

12. R. T. Horn and G. Weihs, “Directional Quasi-Phase Matching in Curved Waveguides,” http://arXiv.org/abs/1008.2190v1.

]. 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.

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

11. P. S. Kuo, W. Fang, and G. S. Solomon, “4-quasi-phase-matched interactions in GaAs microdisk cavities,” Opt. Lett. 34(22), 3580–3582 (2009). [CrossRef] [PubMed]

] or deformed cavities [21

21. J. U. Nöckel, A. D. Stone, and R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19(21), 1693–1695 (1994). [CrossRef] [PubMed]

,22

22. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998). [CrossRef] [PubMed]

] 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 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 ∝ sinc2kL/2) = 0, which occurs when ΔkL/2 = ± π, ± 2π, etc. We can derive an expression for the width of the destructive 4¯-QPM dip when the fundamental and SH resonances overlap (λf = 2λSH) and Δm = 0, ±1, ±3, etc. In the vicinity of the dip, |K˜|2(λ-λ0)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 wf and wSH, respectively (in wavelength units). If wf = 2wSH (same total quality factors for both waves), then the width of the destructive 4¯-QPM dip is 0.31wf. The sharpness of a measured, destructive 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 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 mf = 13, mSH = 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 λf = 1998.7 nm external fundamental light (assuming critical coupling and Qf = QSH = 5000). Our theory also describes SHG when the fundamental and SH resonances no longer overlap (λf –2λSH ≠ 0), and SHG is no longer doubly resonant. When |λf – 2λSH| = 1.3 nm, we expect a maximum conversion efficiency of η = 2.8 × 10-2% and when |λf – 2λSH| = 5.4 nm, we expect η = 1.4×10-3% (for Qf = QSH = 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, |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 (λf = 2λSH). Using a series of lower-quality-factor, waveguide-coupled GaAs microdisks would allow higher nonlinear conversion with broader spectral bandwidth [23

23. Y. Dumeige, “Quasi-phase-matching and second-harmonic generation enhancement in a semiconductor microresonator array using slow-light effects,” Phys. Rev. A 83(4), 045802 (2011). [CrossRef]

]. We also identify the process of perfectly destructive 4¯-QPM; SHG is suppressed in a circular GaAs microdisk at the wavelengths where Δm = 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.

Appendix A. Derivation of second-harmonic generation coefficients

A.1 Stationary eigenmodes

The eigenmodes of a microdisk can be approximated by the analytical expressions presented in Ref [25

25. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13(5), 1515–1530 (2005). [CrossRef] [PubMed]

], which we will summarize here. In a thin microdisk with surface normal z^, the fields naturally decouple into transverse electric (TE, {Hz, Er, Eθ}) and transverse magnetic (TM, {Ez, Hr, Hθ}) polarizations. By using Maxwell’s equations, the radial and azimuthal components can be derived from the z-components (Hz or Ez)

TM={Ez,Hr,Hθ}TE={Hz,Er,Eθ}Hr=mrμ0ωEzEr=mrε0n2ωHz.Hθ=1iμ0ωEzrEθ=iε0n2ωHzr
(23)

The field Fz (where Fz = Hz or Ez) is separable and can be written as

Fzexp(iωt)=A(θ)ψ(r)Z(z)exp[i(ωtmθ)].
(24)

Using Eq. (24), Maxwell’s wave equation in cylindrical coordinates becomes three differential equations
2Zz2+k02(n2n¯2)Z=02ψr2+1rψr+k02n¯2ψ=l2r2ψ,2Aθ22imAθm2A=l2A
(25)
where k 0 = ω/c is the vacuum wavevector and n = n(r) is the refractive index. If we take the slowly varying envelope approximation (SVEA) so that ∂2 A/∂θ 2 = ∂A/∂θ = 0, then the last equation implies l = m and A(θ) is approximately constant.

The vertical dependence, Z(z), can be solved by considering a slab-waveguide model [26

26. C. R. Pollock, Fundamentals of Optoelectronics (Irwin, 1995).

], which yields an effective index, n¯. The slab can support multiple modes, which are indexed by an integer q = 1, 2,… that counts the number of vertical antinodes. We should therefore write the effective index as n¯q. The functional form of Z(z) involves real functions: exponentials, sines and cosines [26

26. C. R. Pollock, Fundamentals of Optoelectronics (Irwin, 1995).

].

The unnormalized radial dependence is approximated by
ψ(r)={Jm(k0n¯qr)r<RJm(k0n¯qR)exp(α˜(rR))r>R,
(26)
where R is the disk radius, Jm(k0n¯qr) is the Bessel function of the first kind and α˜ = k 0(n¯q 2n 2)1/2 [25

25. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13(5), 1515–1530 (2005). [CrossRef] [PubMed]

]. The decaying exponential for r>R is an approximation for the actual solution, which is the Hankel function of the second kind: Hm(2)(k0n¯qr) [9

9. Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74(6), 063804 (2006). [CrossRef]

]. We match boundary conditions to solve for the resonant wavelength where Eq. (25) is a valid solution. Equation (26) incorporates the first boundary condition that ψ(r) is continuous at r = R (i. e., Hz and Ez are continuous at the disk boundary). The other boundary condition is that the tangential fields are continuous; that is [25

25. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13(5), 1515–1530 (2005). [CrossRef] [PubMed]

]

TM:HθiscontinuousTE:EθiscontinuousEzr|r<R=Ezr|r>R1n2Hzr|r<R=Hzr|r>R.
(27)

We assume that n = 1 outside the microdisk. These boundary conditions allow us to find the resonant wavelengths by solving the following transcendental equations

TM:Jm(k0n¯qR)[mR+α˜]=k0n¯qJm+1(k0n¯qR)TE:Jm(k0n¯qR)[mR+α˜(n¯n)2]=k0n¯qJm+1(k0n¯qR).
(28)

There are multiple solutions to Eq. (28) indexed by the number of radial antinodes, p, in ψ(r).

Solving Eq. (28) gives us the wavelengths of the resonant modes, λmpq. Both Zq(z) and ψp(r) are not normalized. They can be inserted into expressions for Hz and Ez (Eq. (24)), which can then be normalized to determine the constant A(θ).

A.2 Normalization of eigenmodes

Z˜q(z)=dqZq(z)ψ˜p(r)=cpψp(r).
(29)

The circulating power, Pcirc, of the microdisk is

Pcirc=Pθ^=12(E×H*)θ^drdz.
(30)

The integration is performed over the cross-sectional area of the microdisk and includes the evanescent field extending slightly outside the disk. For the power normalization, Pcirc = |A(θ)|2, which implies

TM:Pcirc=12EzHr*drdz=|A(θ)|2m2μ0ω1r|ψ˜p(r)Z˜q(z)|2drdz=1TE:Pcirc=12Hz*Erdrdz=|A(θ)|2m2ε0n2ω1r|ψ˜p(r)Z˜q(z)|2drdz=1.
(31)

The double integral over r and z separates into two normalization equations. We can set the integral over z to unity and obtain the following conditions

-|Z˜q(z)|2dz=1TM:m2μ0ω01r|ψ˜p(r)|2dr=1TE:m2ε0n2ω01r|ψ˜p(r)|2dr=1.
(32)

We note that Zq(z) and ψp(r) constructed in section A.1 are real, so the absolute value in Eq. (32) is not needed.

A.3 Nonlinear optical coupling

Nonlinear interactions between the modes of the microdisk can be described using a perturbative approach [27

27. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973). [CrossRef]

]. The eigenmodes, as described in Section A.1, obey the unperturbed wave equation
2Eμε2Et2=0,
(33)
where μ = μ0 (for a non-magnetic material), ε = ε0n 2 inside the microdisk, and ε = ε0 outside the disk. We can introduce a perturbing polarization source arising from the nonlinear interaction, P NL, which produces a perturbed wave equation

2Eμε2Et2=μ02PNLt2.
(34)

The eigenmodes that solve the unperturbed wave equation (Eq. (33)) form a complete, mutually orthogonal set, and therefore the solution to Eq. (34) can be written as a linear combination of these eigenmodes.

Let us consider SHG with a TE-polarized fundamental and a TM-polarized second harmonic. The SH field that satisfies Eq. (34), Ez, can be written as a sum over the TM-polarized SH eigenmodes that solved the unperturbed equation:

Ez=m'SH,p'SH,q'SHAm'SHp'SHq'SH(θ)ψ˜m'SHp'SHq'SH(r)Z˜q'SH(z)ei(ωSHtm'SHθ)+radiationmodes.
(35)

The terms Am'SHp'SHq'SH(θ)are slowly-varying envelope functions and act like the weighting factors. We have explicitly written out the indices associated with the constituent functions. For this analysis, we neglect the effect of the radiation modes since we are more interested in the guided modes of the microdisk.

Equation (35) can be substituted into Eq. (34), and many of the terms will sum to zero because the constituent functions are all solutions to the homogeneous equation, Eq. (33). The only remaining terms are those that involve the derivatives of Am'SHp'SHq'SH(θ) and the z-component of the nonlinear polarization:

m'SH,p'SH,q'SH1r2(2Am'SHp'SHq'SHθ22im'SHAm'SHp'SHq'SHθ)×ψ˜m'SHp'SHq'SH(r)Z˜q'SH(z)ei(ωSHtm'SHθ)=μ0ωSH2PzNL.
(36)

Since |2Am'SH/θ2|«|2mSHAm'SH/θ| (SVEA), the first term in the parentheses can be neglected. We also assumed the nonlinear polarization oscillates at frequency ωSH so that ∂2 PzNL/∂t 2 ≈ -ωSH2 PzNL. We can now utilize the orthogonality of the eigenmodes [28

28. K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1-3), 37–61 (2005). [CrossRef]

]
02π0Fz,mpqFz,m'p'q'rdrdθdzδmm'δpp'δqq',
(37)
where Fz = Hz or Ez, and “project out” the coefficients associated with (mSH, pSH, qSH) by multiplying both sides of Eq. (36) by rψ˜mSHpSHqSH(r)Z˜qSH(z)eimSHθand integrating over all θ, r and z:

02π0m'SH,p'SH,q'SH(2im'SHAm'SHθ)(ψ˜m'SHp'SHq'SH(r)ψ˜mSHpSHqSH(r)r)×Z˜q'SH(z)Z˜qSH(z)ei(mSHm'SH)θdrdθdzeiωSHt=μ0ωSH202π0PzNLrψ˜mSHpSHqSH(r)Z˜qSH(z)eimSHθdrdθdz.
(38)

Since Am'SHp'SHq'SHis the slowly varying envelope, its derivative Am'SHp'SHq'SH/θ is essentially a constant with respect to θ, so the 02πei(mSHm'SH)θdθ integral acts like the Kronecker delta function, δm'SHmSH. The integral over θ on the left-hand side of Eq. (38) is

m'SH02πAm'SHp'SHq'SHθei(mSHm'SH)θdθ2πAmSHp'SHq'SHθ.
(39)

The r and z integrals on the left-hand side of Eq. (38) select out the terms pSH = pSH and qSH = qSH. The integral can be simplified using the normalization expressions. Using Eq. (32) so that |AmSHpSHqSH|2 represent power, Eq. (38) becomes

4πimSHAmSHpSHqSHθ(2μ0ωSHmSH)eiωSHt=μ0ωSH202π0PzNLrψ˜mSHpSHqSH(r)Z˜qSH(z)eimSHθdrdθdz.
(40)

In GaAs and other crystals with 4¯3mpoint-group symmetry, the only non-zero nonlinear-susceptibility-tensor elements are d 14 = d 25 = d 36 = dzxy so the z-component of P NL is

PzNL=2ε0d14ExfEyf.
(41)

In terms of Er and Eθ,
Ex=ErcosθEθsinθEy=Ersinθ+Eθcosθ,
(42)
so that

PzNL=2ε0d14[cos2θErfEθf+12sin2θ(Erf2Eθf2)].
(43)

From Eq. (23), Erf and Eθfare proportional to Hzfand Hzf/r, respectively, which are both proportional to eimfθ. The sine and cosine terms in Eq. (43) produce factors of ei2θand ei2θ, so PzNL is proportional to ei(2mf+2)θ and ei(2mf2)θ.

If we let ψ˜SH=ψ˜mSHpSHqSH(r), then combining and rearranging Eqs. (23), (24), (40) and (43), we see that
2πAmSHpSHqSHθ=d142ε0ωSH(Amfpfqfnf2)2h/2h/2Z˜qSH(z)Z˜qf2(z)dz×02π[0Rei(Δm+2)θrψ˜SH(mfrψ˜f+ψ˜fr)2dr0Rei(Δm2)θrψ˜SH(mfrψ˜fψ˜fr)2dr]dθ
(44)
where Δm=mSH2mf. Note that the limits of integration are [0,R] for r and [-h/2, h/2] for z (where R and h are the radius and height of the microdisk) since the nonlinearity, d 14, is only non-zero inside the disk. For good vertical mode overlap, Z˜qf2(z)Z˜qSH(z) (i.e., the shape of Z˜qf2(z) should match that of Z˜qSH(z)).

We had approximated that AmSHpSHqSH/θis a constant with respect to θ, so the left-hand side is actually 02π(AmSHpSHqSH/θ)dθ. Terms on the right-hand side of Eq. (44) can be collected and we identify
AmSHpSHqSHθ=Amfpfqf2(K+ei(Δm+2)θ+Kei(Δm2)θ),
(45)
where

K+pow=d142ε0ωSHnf4h/2h/2Z˜qSH(z)Z˜qf2(z)dz0Rrψ˜SH(mfrψ˜f+ψ˜fr)2drKpow=d142ε0ωSHnf4h/2h/2Z˜qSH(z)Z˜qf2(z)dz0Rrψ˜SH(mfrψ˜fψ˜fr)2dr.
(46)

K±pow in Eq. (46) are labeled by the pow superscript to indicate that they are calculated using the power normalization of the fields.

The quasi-phasematching condition can be seen in Eq. (45): the sinusoidally varying portion of AmSHpSHqSH/θ vanishes if Δm = mSH - 2mf = ±2. Equation (45) differs from Eq. (2) in Ref [11

11. P. S. Kuo, W. Fang, and G. S. Solomon, “4-quasi-phase-matched interactions in GaAs microdisk cavities,” Opt. Lett. 34(22), 3580–3582 (2009). [CrossRef] [PubMed]

]. by a factor i since here, PzNL is shown to be imaginary.

Appendix B. Conversion efficiency and coupled-mode theory

In this appendix, we derive the conversion efficiency for SHG in microdisks based on coupled-mode theory (CMT) [29

29. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

]. In CMT, |A(θ)|2 represents the stored energy inside the resonator rather than the circulating power. We describe energy normalization of the fields and its effect on the SHG coefficients (K+ and K). We calculate the conversion efficiency using CMT and show it agrees with results from Section 2 that were based on the generalized waveguide-microresonator theory (GWMT).

B.1 Energy normalization and SHG coefficients

To have |A(θ)|2 represent the stored energy, W, we can relate Pcirc to W by [29

29. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

]
Pcirc=Wvg=|A|2vg.
(48)
is the length of the resonator and vg is the group velocity. The group velocity of waves in a microdisk is [30

30. M. Borselli, Ph.D. thesis (California Institute of Technology, 2006).

]
vg=rδωFSR,
(49)
where r is the radial coordinate inside the microdisk, and δωFSR is the angular-frequency separation between adjacent modes or the free-spectral range (FSR). Equation (49) implies that the group velocity of the wave depends on its radial location in the disk. However, also depends on r through = 2πr. Thus the ratio vg/ is independent of r and
Pcirc=|A|2δfFSR,
(50)
where δfFSR = δωFSR/2π is the FSR in frequency units. Combining Eqs. (30) and (50), we obtain the energy normalization:

-|Z˜q(z)|2dz=1TM:m2μ0ω01r|ψ˜pen(r)|2dr=δfFSR.TE:m2ε0n2ω01r|ψ˜pen(r)|2dr=δfFSR
(51)

Here, the normalized radial functions, ψ˜pen(r), are given the en superscript since they are calculated using the energy normalization of the fields.

Following the procedure outlined in Section A.3 (most notably using Eq. (51) instead of (32) to derive Eq. (40)), the SHG coefficients using the energy normalization are

K+en=d142ε0δfFSR,SHωSHnf4h/2h/2Z˜qSH(z)Z˜qf2(z)dz0Rrψ˜SHen(mfrψ˜fen+ψ˜fenr)2drKen=d142ε0δfFSR,SHωSHnf4h/2h/2Z˜qSH(z)Z˜qf2(z)dz0Rrψ˜SHen(mfrψ˜fenψ˜fenr)2dr.
(52)

Numerically, K±en can be calculated from K±pow (the coefficients calculated using the power normalization, Eq. (46)) using K±en=K±powδfFSR,f/δfFSR,SH.

B.2 Coupled-mode theory

ai=Aiexp(iωit).
(54)

Ref [29

29. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

]. presents the nonlinear source term, sNL, as an overlap integral between the interacting waves and the nonlinear tensor. A more intuitive picture is to cast the nonlinear source in terms of an effective gain time, τg, defined by

sNL=aSHτg.
(55)

Pgain=2αgPcirc.
(57)

Combining Eqs. (48), (56) and (57), we obtain
τg=1αgvg,
(58)
where vg is the group velocity in the microdisk (Eq. (49)). In terms of field amplitudes,
αg=1ASHdASHrdθ,
(59)
where we abbreviate ASH=AmSHpSHqSH. We can combine Eqs. (45), (49), (54) (55), (58) and (59) to find
sNL=af2δωFSR,SHK±en.
(60)
K±en=K+en if Δm = –2, andK±en=Ken if Δm = +2 since the dominant contribution to Eq. (45) is from the phasematched component.

The circulating powers and SHG conversion efficiency can be calculated by looking for steady-state solutions to Eq. (53). Utilizing Eq. (50), we find

Pfcirc=|af|2δfFSR,f=δfFSR,f4Qfcωf1(1+Qfc/Qc0)2Pfin.
(61)

The second-harmonic generated is

PSHout=2τSHc|aSH|2=4QSHcωSH(1+QSHc/QSH0)2(4Qfcωf(1+Qfc/Qf0)2Pfin2πδfFSR,SH|K±en|)2.
(62)

Equation (62) can be compared to results of the GWMT discussed in Section 2. If both the fundamental and SH are resonant with the cavity (with Δm = 2 or −2) and the phase shift from the coupler can be neglected (ψf=ψSH=0), thenK˜(φf,φSH)=2πK±pow, and Eq. (15) becomes
PSHout=(Pfin)2(1|tSH|2)|2πK±pow|2αSH2(1αSH|tSH|)2(αf2(1|tf|2)(1αf|tf|)2)2,
(63)
where PSHout=|C1,SH|2 and Pfin=|B1,f|2. The parameters αi and ti can be written in terms of the quality factors using Eq. (19).

Table 1 compares the SH conversion efficiency calculated using coupled-mode theory Eq. (62), and η calculated using the generalized waveguide-microresonator theory Eq. (63). The GaAs microdisk has R = 2.609 μm and h = 161 nm, and supports doubly resonant SHG with λf = 2λSH = 1998.7 nm with TE-polarized fundamental wave (mf = 13) and TM-polarized SH wave (mSH = 28) where Δm = mSH – 2mf = 2 is satisfied (pf = pSH = qf = qSH = 1). The free-spectral ranges for the waves are δfFSR,f=6.3×1012Hz and δfFSR,SH=3.4×1012Hz. We use d 14 = 94 pm/V [31

31. T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002). [CrossRef] [PubMed]

] for GaAs. CMT predicts a SH conversion efficiency of 1.23% while the GWMT predicts η = 1.16% for 1 mW of incident fundamental power. There is good agreement between the two theories.

Acknowledgements

We thank Kartik Srinivasan for helpful discussions. P. S. Kuo acknowledges support from the National Research Council Postdoctoral Research Associate program. We also acknowledge support through the NSF Physics Frontier Center at the Joint Quantum Institute and the Zeno-based Optoelectronics program at DARPA.

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K. Rivoire, Z. Lin, F. Hatami, W. T. Masselink, and J. Vucković, “Second harmonic generation in gallium phosphide photonic crystal nanocavities with ultralow continuous wave pump power,” Opt. Express 17(25), 22609–22615 (2009). [CrossRef] [PubMed]

18.

A. Rodriguez, M. Soljačić, J. D. Joannopoulos, and S. G. Johnson, “χ((2)) and χ((3)) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15(12), 7303–7318 (2007). [CrossRef] [PubMed]

19.

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]

20.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003). [CrossRef]

21.

J. U. Nöckel, A. D. Stone, and R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19(21), 1693–1695 (1994). [CrossRef] [PubMed]

22.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998). [CrossRef] [PubMed]

23.

Y. Dumeige, “Quasi-phase-matching and second-harmonic generation enhancement in a semiconductor microresonator array using slow-light effects,” Phys. Rev. A 83(4), 045802 (2011). [CrossRef]

24.

A. Andronico, I. Favero, and G. Leo, “Difference frequency generation in GaAs microdisks,” Opt. Lett. 33(18), 2026–2028 (2008). [CrossRef] [PubMed]

25.

M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13(5), 1515–1530 (2005). [CrossRef] [PubMed]

26.

C. R. Pollock, Fundamentals of Optoelectronics (Irwin, 1995).

27.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973). [CrossRef]

28.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1-3), 37–61 (2005). [CrossRef]

29.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

30.

M. Borselli, Ph.D. thesis (California Institute of Technology, 2006).

31.

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002). [CrossRef] [PubMed]

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 20, 2011
Revised Manuscript: July 25, 2011
Manuscript Accepted: July 25, 2011
Published: August 15, 2011

Citation
Paulina S. Kuo and Glenn S. Solomon, "On- and off-resonance second-harmonic generation in GaAs microdisks," Opt. Express 19, 16898-16918 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16898


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References

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  17. K. Rivoire, Z. Lin, F. Hatami, W. T. Masselink, and J. Vucković, “Second harmonic generation in gallium phosphide photonic crystal nanocavities with ultralow continuous wave pump power,” Opt. Express 17(25), 22609–22615 (2009). [CrossRef] [PubMed]
  18. A. Rodriguez, M. Soljačić, J. D. Joannopoulos, and S. G. Johnson, “χ((2)) and χ((3)) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15(12), 7303–7318 (2007). [CrossRef] [PubMed]
  19. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]
  20. T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003). [CrossRef]
  21. J. U. Nöckel, A. D. Stone, and R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19(21), 1693–1695 (1994). [CrossRef] [PubMed]
  22. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998). [CrossRef] [PubMed]
  23. Y. Dumeige, “Quasi-phase-matching and second-harmonic generation enhancement in a semiconductor microresonator array using slow-light effects,” Phys. Rev. A 83(4), 045802 (2011). [CrossRef]
  24. A. Andronico, I. Favero, and G. Leo, “Difference frequency generation in GaAs microdisks,” Opt. Lett. 33(18), 2026–2028 (2008). [CrossRef] [PubMed]
  25. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13(5), 1515–1530 (2005). [CrossRef] [PubMed]
  26. C. R. Pollock, Fundamentals of Optoelectronics (Irwin, 1995).
  27. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973). [CrossRef]
  28. K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1-3), 37–61 (2005). [CrossRef]
  29. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  30. M. Borselli, Ph.D. thesis (California Institute of Technology, 2006).
  31. T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002). [CrossRef] [PubMed]

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