## On- and off-resonance second-harmonic generation in GaAs microdisks |

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, *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

© 2011 OSA

## 1. Introduction

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. 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]

3. E. Lallier, M. Brevignon, and J. Lehoux, “Efficient second-harmonic generation of a CO_{2} laser with a quasi-phase-matched GaAs crystal,” Opt. Lett. **23**(19), 1511–1513 (1998). [CrossRef] [PubMed]

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]

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. 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]

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]

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. 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]

*χ*

^{(2)}nonlinear optical mixing of the whispering-gallery modes of a GaAs microdisk without using external domain inversions. The

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. 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]

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

_{2}PO

_{4}, chalcopyrites, etc.), and

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]

13. W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO_{3} external resonant cavities,” IEEE J. Quantum Electron. **24**(6), 913–919 (1988). [CrossRef]

14. Z. Yang and J. E. Sipe, “Generating entangled photons via enhanced spontaneous parametric downconversion in AlGaAs microring resonators,” Opt. Lett. **32**(22), 3296–3298 (2007). [CrossRef] [PubMed]

_{3}whispering-gallery-mode resonators [15

15. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. **92**(4), 043903 (2004). [CrossRef] [PubMed]

16. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. **104**(15), 153901 (2010). [CrossRef] [PubMed]

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]

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

*A*(

_{i}*θ*,

*t*) is the slowly varying amplitude at frequency

*ω*(the index

_{i}*i*refers to the fundamental,

*f*, or the second-harmonic,

*SH*). In this section,

*A*(

_{i}*θ*,

*t*) is normalized so that |

*A*(

_{i}*θ*,

*t*)|

^{2}= power. Ψ

*(*

_{i}*r*,

*z*) is the mode profile, and

*m*is the azimuthal number of the mode, which is an integer for a resonant mode. The exp(-

_{i}*im*) factors describe the phase accumulated by the waves as they propagate around the disk, analogous to exp(-

_{i}θ*ik*) terms found in linear propagation geometries. The change in the SH amplitude due to nonlinear mixing in the GaAs microdisk is described by [9

_{i}ℓ9. 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]

**34**(22), 3580–3582 (2009). [CrossRef] [PubMed]

*A*is constant), integrating Eq. (2) from

_{f}*θ*= 0 to 2

*π*yieldswhereand 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.

*B*and

_{n,i}*C*are the complex mode amplitudes normalized so that |

_{n,i}*B*|

_{n,i}^{2}, |

*C*|

_{n,i}^{2}= power. We note that

*B*and

_{n,i}*C*differ from the slowly varying amplitudes

_{n,i}*A*(

_{i}*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]

*κ*|

_{i}^{2}+ |

*t*|

_{i}^{2}= 1.

*α*, and phase shift,

_{f}*ϕ*,At the SH wave, there is loss, phase shift and SHG gain. Equation (3) may be written as

_{f}*A*

_{f}^{2}≈|

*C*|

_{2,f}^{2}≈|

*B*|

_{2,f}^{2}).

*m*plays a role analogous to the wavevector

_{i}*k*in linear propagation geometries. Both describe the rate of phase accumulation due to propagation and, hence, the effective propagation constant inside the medium. However,

_{i}*m*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

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

*f*subscripts are replaced by

*SH.*

_{1,}

*= 0) so that Eq. (5) becomes*

_{SH}*α*≈1 and |

_{f}*C*|

_{2,f}^{2}≈|

*B*|

_{2,f}^{2}), it follows from Eqs. (7) and (13) that the circulating second-harmonic power is

*Q*, is ratio between the resonance frequency,

_{i}*ω*, and the linewidth, Δ

_{0}*ω*, and can be calculate from the finesse using

_{FWHM}## 3. On- and off-resonance second-harmonic generation in a GaAs microdisk

*λ*≈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

_{f}*λ*= 2Δ

_{f}*λ*= 0.4 nm. We also assume the phase shifts from the coupler are

_{SH}*ψ*=

_{f}*ψ*= 0 (the main effects of non-zero coupler phase shifts are small wavelength shifts in the resonance locations). Values for

_{SH}*K*

_{+}and

*K*

_{–}are given in Table 1 of Appendix B.

*R*= 2.609 μm has both the fundamental and SH waves on-resonance at

*λ*= 2

_{f}*λ*= 1998.7 nm with fundamental wave (

_{SH}*m*= 13) TE-polarized and the SH wave (

_{f}*m*= 28) TM-polarized (satisfying

_{SH}*m*– 2

_{SH}*m*= 2). We used finite-element modeling software and GaAs dispersion data [20

_{f}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]

*m*and

_{f}*m*, and linearly interpolated between resonances to calculate

_{SH}*ϕ*(Eq. (8)). Figure 3 shows the SH conversion efficiency,

_{i}*η*= |

*C*

_{1,}

_{SH}|^{2}/|

*B*

_{1,}

_{f}|^{2}, for this microdisk.

*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 Δ

*m*=

*m*– 2

_{SH}*m*= 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

_{f}*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

*λ*= 2093.7 nm in Fig. 3 where

_{f}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]

*R*= 2.587 μm,

*h*= 161 nm) where the fundamental and SH resonances near 1990 nm do not overlap. For this disk, the

*m*= 13 fundamental resonance occurs at

_{f}*λ*= 1987.4 nm and the

_{f}*m*= 28 SH resonance occurs at 2

_{SH}*λ*= 1992.8 nm such that |

_{SH}*λ*– 2

_{f}*λ*| = 5.4 nm. A maximum conversion of

_{SH}*η*= 1.4×10

^{-3}% is obtained at a pumping wavelength of 1987.4 nm (where only the fundamental is on-resonance and

*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

*m*= −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*= 2016.7 nm where the fundamental is on-resonance (

_{f}*m*= 13) while the SH is not. At this pumping wavelength,

_{f}*η*= 6.5 × 10

^{-4}%.

*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*= 13) and SH (

_{f}*m*= 28) resonances become misaligned, resulting in decreased SH conversion. Figure 6b plots the conversion near 2190 nm pumping wavelength where

_{SH}*R*= 2.643 μm disk, resulting in high SH conversion (

*η*= 0.29%).

*m*=

*m*– 2

_{SH}*m*= ±2. As the disk radius, thickness or temperature is changed, the resonances no longer coincide and

_{f}*R*

_{0}= 2.609 μm, and supports doubly resonant SHG at

*λ*= 2

_{f}*λ*= 1998.7 nm (

_{SH}*m*= 13,

_{f}*m*= 28). By changing the radius of the microdisk by 5 nm, the fundamental and SH resonances become detuned by |

_{SH}*λ*– 2

_{f}*λ*| = 1.3 nm (several times the 0.4 nm linewidth of these

_{SH}*Q*=

_{f}*Q*= 5000 resonances), and the maximum SH conversion efficiency drops from 1.2% to 2.8×10

_{SH}^{-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*– 2

_{f}*λ*| = 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

_{SH}*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%).

*m*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 Δ

*m*= 0, ±1, ±3, etc. due to perfectly destructive

*m*values of the resonances, but the presence of the dip clearly indicates that Δ

*m*≠ ±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

*χ*

^{(2)}conversion in a GaAs microdisk — resonance enhancement at the fundamental wave, resonance enhancement at the second-harmonic wave, and

*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

**34**(22), 3580–3582 (2009). [CrossRef] [PubMed]

*k*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

*λ*has a bigger effect than resonance enhancement at

_{f}*λ*.

_{SH}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. 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]

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. 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]

*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

**21**, 1522–1534 (2004). [CrossRef]

*ε*, the second factor can be written as [7

**23**(9), 661–663 (1998). [CrossRef] [PubMed]

*N*) lead to more SHG while narrowing the generated spectrum in the same way that higher

*Q*and

_{f}*Q*lead to more roundtrips in the cavity, higher circulating powers and narrower spectra. In both Fresnel phasematching and

_{SH}**21**, 1522–1534 (2004). [CrossRef]

*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.

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

*λ*will be high. This light can initiate second-harmonic generation at the entrance of the cavity. The second-harmonic will increase due to

_{f}*λ*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

_{SH}*λ*/2 matches a cavity resonance, then there will still be an enhancement of SHG in the cavity. Since

_{f}*Q*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.

_{f}*m*= ±2. It would be interesting to explore square [11

**34**(22), 3580–3582 (2009). [CrossRef] [PubMed]

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]

*m*= 0, ±1, ±3, etc. in a circular cavity, SHG is suppressed due to perfectly destructive

*η*∝ sinc

^{2}(Δ

*kL*/2) = 0, which occurs when

*ΔkL*/2 = ±

*π*, ± 2

*π*, etc. We can derive an expression for the width of the destructive

*λ*= 2

_{f}*λ*) and Δ

_{SH}*m*= 0, ±1, ±3, etc. In the vicinity of the dip,

^{2}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*and

_{f}*w*, respectively (in wavelength units). If

_{SH}*w*= 2

_{f}*w*(same total quality factors for both waves), then the width of the destructive

_{SH}*w*. The sharpness of a measured, destructive

_{f}*m*≠ ± 2 can be used to characterize the circularity and ideality of a microdisk.

## 5. Conclusions

*m*= 13,

_{f}*m*= 28 and Δ

_{SH}*m*= 2 in a 2.609-μm-radius, 161-nm-thick microdisk, the theory predicts 1.2% conversion efficiency with 1 mW of

*λ*= 1998.7 nm external fundamental light (assuming critical coupling and

_{f}*Q*=

_{f}*Q*= 5000). Our theory also describes SHG when the fundamental and SH resonances no longer overlap (

_{SH}*λ*–2

_{f}*λ*≠ 0), and SHG is no longer doubly resonant. When |

_{SH}*λ*– 2

_{f}*λ*| = 1.3 nm, we expect a maximum conversion efficiency of

_{SH}*η*= 2.8 × 10

^{-2}% and when |

*λ*– 2

_{f}*λ*| = 5.4 nm, we expect

_{SH}*η*= 1.4×10

^{-3}% (for

*Q*=

_{f}*Q*= 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,

_{SH}*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*= 2

_{f}*λ*). Using a series of lower-quality-factor, waveguide-coupled GaAs microdisks would allow higher nonlinear conversion with broader spectral bandwidth [23

_{SH}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]

*m*≠ ± 2 and to evaluate the circularity and ideality of the microdisk.

## Appendix A. Derivation of second-harmonic generation coefficients

*A*(

_{i}*θ*)|

^{2}represents the circulating power inside the microdisk. The fields can also be normalized to represent stored energy, which is used in coupled-mode theory [11

**34**(22), 3580–3582 (2009). [CrossRef] [PubMed]

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

### A.1 Stationary eigenmodes

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]

*H*}) and transverse magnetic (TM, {

_{z}, E_{r}, E_{θ}*E*}) polarizations. By using Maxwell’s equations, the radial and azimuthal components can be derived from the

_{z}, H_{r}, H_{θ}*z*-components (

*H*or

_{z}*E*)

_{z}*F*(where

_{z}*F*=

_{z}*H*or

_{z}*E*) is separable and can be written as

_{z}*k*

_{0}=

*ω/c*is the vacuum wavevector and

*n*=

*n*(

**) is the refractive index. If we take the slowly varying envelope approximation (SVEA) so that ∂**

*r*^{2}

*A*/∂

*θ*

^{2}= ∂

*A*/∂

*θ*= 0, then the last equation implies

*l*=

*m*and

*A*(

*θ*) is approximately constant.

*Z*(

*z*), can be solved by considering a slab-waveguide model [26], which yields an effective index,

*q*= 1, 2,… that counts the number of vertical antinodes. We should therefore write the effective index as

*Z*(

*z*) involves real functions: exponentials, sines and cosines [26].

*R*is the disk radius,

*k*

_{0}(

^{2}–

*n*

^{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]

*r*>

*R*is an approximation for the actual solution, which is the Hankel function of the second kind:

**74**(6), 063804 (2006). [CrossRef]

*ψ*(

*r*) is continuous at

*r*=

*R*(i. e.,

*H*and

_{z}*E*are continuous at the disk boundary). The other boundary condition is that the tangential fields are continuous; that is [25

_{z}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]

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

### A.2 Normalization of eigenmodes

*P*, of the microdisk is

_{circ}*P*= |

_{circ}*A*(

*θ*)|

^{2}, which implies

*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*) and

*ψ*(

_{p}*r*) constructed in section A.1 are real, so the absolute value in Eq. (32) is not needed.

### A.3 Nonlinear optical coupling

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

*μ*=

*μ*(for a non-magnetic material),

_{0}*ε*=

*ε*

_{0}n^{2}inside the microdisk, and

*ε*=

*ε*outside the disk. We can introduce a perturbing polarization source arising from the nonlinear interaction,

_{0}**P**

*, which produces a perturbed wave equation*

^{NL}*E*, can be written as a sum over the TM-polarized SH eigenmodes that solved the unperturbed equation:

_{z}*z*-component of the nonlinear polarization:

*ω*so that ∂

_{SH}^{2}

*P*/∂

_{z}^{NL}*t*

^{2}≈ -

*ω*. We can now utilize the orthogonality of the eigenmodes [28

_{SH}^{2}P_{z}^{NL}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]

*F*=

_{z}*H*or

_{z}*E*, and “project out” the coefficients associated with (

_{z}*m*,

_{SH}*p*,

_{SH}*q*) by multiplying both sides of Eq. (36) by

_{SH}*θ*,

*r*and

*z*:

*θ*, so the

*θ*on the left-hand side of Eq. (38) is

*r*and

*z*integrals on the left-hand side of Eq. (38) select out the terms

*p*′

*=*

_{SH}*p*and

_{SH}*q*′

*=*

_{SH}*q*. The integral can be simplified using the normalization expressions. Using Eq. (32) so that

_{SH}*d*

_{14}=

*d*

_{25}=

*d*

_{36}=

*d*so the

_{zxy}*z*-component of

**P**

*is*

^{NL}*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,

*θ*, so the left-hand side is actually

*pow*superscript to indicate that they are calculated using the power normalization of the fields.

*m*=

*m*- 2

_{SH}*m*= ±2. Equation (45) differs from Eq. (2) in Ref [11

_{f}**34**(22), 3580–3582 (2009). [CrossRef] [PubMed]

*i*since here,

## Appendix B. Conversion efficiency and coupled-mode theory

*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 (

### B.1 Energy normalization and SHG coefficients

*A*(

*θ*)|

^{2}represent the stored energy,

*W*, we can relate

*P*to

_{circ}*W*by [29]

*ℓ*is the length of the resonator and

*v*is the group velocity. The group velocity of waves in a microdisk is [30]where

_{g}*r*is the radial coordinate inside the microdisk, and

*δω*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,

_{FSR}*ℓ*also depends on

*r*through

*ℓ*= 2

*πr*. Thus the ratio

*v*/

_{g}*ℓ*is independent of

*r*andwhere

*δf*=

_{FSR}*δω*/2

_{FSR}*π*is the FSR in frequency units. Combining Eqs. (30) and (50), we obtain the energy normalization:

*en*superscript since they are calculated using the energy normalization of the fields.

### B.2 Coupled-mode theory

*s*, 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,

^{NL}*v*is the group velocity in the microdisk (Eq. (49)). In terms of field amplitudes,where we abbreviate

_{g}*m*= –2, and

*m*= +2 since the dominant contribution to Eq. (45) is from the phasematched component.

*m*= 2 or −2) and the phase shift from the coupler can be neglected (

*α*and

_{i}*t*can be written in terms of the quality factors using Eq. (19).

_{i}*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

*λ*= 2

_{f}*λ*= 1998.7 nm with TE-polarized fundamental wave (

_{SH}*m*= 13) and TM-polarized SH wave (

_{f}*m*= 28) where Δ

_{SH}*m*=

*m*– 2

_{SH}*m*= 2 is satisfied (

_{f}*p*=

_{f}*p*=

_{SH}*q*=

_{f}*q*= 1). The free-spectral ranges for the waves are

_{SH}*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]

*η*= 1.16% for 1 mW of incident fundamental power. There is good agreement between the two theories.

## Acknowledgements

## References and links

1. | J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light eaves in a nonlinear dielectric,” Phys. Rev. |

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. |

3. | E. Lallier, M. Brevignon, and J. Lehoux, “Efficient second-harmonic generation of a CO |

4. | S. Koh, T. Kondo, Y. Shiraki, and R. Ito, “GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices,” J. Cryst. Growth |

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. |

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 |

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. |

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. |

9. | Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A |

10. | Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett. |

11. | P. S. Kuo, W. Fang, and G. S. Solomon, “4-quasi-phase-matched interactions in GaAs microdisk cavities,” Opt. Lett. |

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

13. | W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO |

14. | Z. Yang and J. E. Sipe, “Generating entangled photons via enhanced spontaneous parametric downconversion in AlGaAs microring resonators,” Opt. Lett. |

15. | V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. |

16. | J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. |

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 |

18. | A. Rodriguez, M. Soljačić, J. D. Joannopoulos, and S. G. Johnson, “ |

19. | A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. |

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. |

21. | J. U. Nöckel, A. D. Stone, and R. K. Chang, “ |

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 |

23. | Y. Dumeige, “Quasi-phase-matching and second-harmonic generation enhancement in a semiconductor microresonator array using slow-light effects,” Phys. Rev. A |

24. | A. Andronico, I. Favero, and G. Leo, “Difference frequency generation in GaAs microdisks,” Opt. Lett. |

25. | M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high- |

26. | C. R. Pollock, |

27. | A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. |

28. | K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. |

29. | H. A. Haus, |

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. |

**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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