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Nonlinear optics in gallium phosphide cavities: simultaneous second and third harmonic generation

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Abstract

We demonstrate the simultaneous generation of second and third harmonic signals from a telecom wavelength pump in a gallium phosphide (GaP) microdisk. Using analysis of the power scaling of both the second and third harmonic outputs and calculations of nonlinear cavity mode coupling factors, we study contributions to the third harmonic signal from direct and cascaded sum frequency generation processes. We find that despite the relatively high material absorption in GaP at the third harmonic wavelength, both of these processes can be significant, with relative magnitudes that depend closely on the detuning between the second harmonic wavelengths of the cavity modes.

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Supplementary Material (1)

NameDescription
Supplement 1       Derivations of equations used in main text.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (4)

Fig. 1.
Fig. 1. (a) Energy level diagram of SHG, CSFG, and THG processes. Solid lines indicate WGM resonant modes, while dashed lines represent virtual energy levels. (b) Image of FDTD simulated normalized absolute field magnitude of the ${{\rm TE}_{{23,1}}}$ microdisk mode. White outline corresponds to the edge of the microdisk. (c) Image of FDTD simulated normalized absolute field magnitude of the ${{\rm TM}_{{48,2}}}$ microdisk mode.
Fig. 2.
Fig. 2. Absolute values of second-order (outlined in red circles) and third-order (outlined in green squares) nonlinear coupling factors with the ${{\rm TE}_{{23,1}}}$ first harmonic mode. Groups of phase matched microdisk modes are plotted by the frequency and azimuthal number of the highest harmonic mode in the wave mixing process.
Fig. 3.
Fig. 3. (a) Transmission spectrum of GaP microdisk pump mode around 1557 nm for 8.7 mW pump power. The Fano-like line shape of the observed resonance is attributed to a phase mismatch between the coupled fiber taper modes. (b) Second and (c) third harmonic output signal powers for several pump wavelengths and input powers. Top axes show the observed peak harmonic wavelength measured in the spectrometer.
Fig. 4.
Fig. 4. On-resonance output power for second (orange) and third harmonic (green) signals. Solid lines are fits to the saturated resonance theory. The dashed line corresponds to quadratic power scaling predicted from zero depletion.

Equations (19)

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SHG: m 2 = 2 m 1 ± 2 ,
CSFG: m 3 = m 1 + m 2 ± 2 ,
DTHG: m 3 = 3 m 1 .
d d t a k = ( i ω k κ k 2 ) a k + κ k , e x 2 s + ,
s = s + κ k , e x 2 a k ,
κ k = κ k , i + p + κ k , e x .
d d t a 1 = ( i ω 1 κ 1 2 ) a 1 + κ 1 , e x 2 s + ,
d d t a 2 = ( i ω 2 κ 2 2 ) a 2 i ω 2 β C S F G a 1 a 3 + i ω 2 β S H G a 1 2 ,
d d t a 3 = ( i ω 3 κ 3 2 ) a 3 + i ω 3 β C S F G a 1 a 2 + i ω 3 β D T H G a 1 3 ,
P 2 = η S H G P i n 2 ( 1 + ζ P i n ) 2 ,
P 3 = η D T H G P i n 3 + η C S F G P i n 3 ( 1 + ζ P i n ) 2 ,
η S H G = 32 ω 1 2 | β S H G | 2 κ 1 , e x 2 κ 2 , e x κ 1 4 κ 2 2 ,
η D T H G = 144 ω 1 2 | β D T H G | 2 κ 1 , e x 3 κ 3 , e x κ 1 6 κ 3 2 ,
η C S F G = 2304 ω 1 4 | β S H G | 2 | β C S F G | 2 κ 1 , e x 3 κ 3 , e x κ 1 6 κ 2 2 κ 3 2 ,
ζ = 96 ω 1 2 | β C S F G | 2 κ 1 , e x κ 1 2 κ 2 κ 3 ,
δ ω k ω k = 1 2 ε E δ P k d 3 x ε | E | 2 d 3 x ,
β S H G = 1 4 d 3 x ijk ε χ ijk ( 2 ) [ E 1 i E 2 j E 1 k + E 1 i E 1 j E 2 k ] d 3 x ε | E 1 | 2 ( d 3 x ε | E 2 | 2 ) 1 / 2 ,
β C S F G = 1 4 d 3 x ijk ε χ ijk ( 2 ) [ E 2 i E 3 j E 1 k + E 2 i E 1 j E 3 k ] ( d 3 x ε | E 1 | 2 ) 1 / 2 ( d 3 x ε | E 2 | 2 ) 1 / 2 ( d 3 x ε | E 3 | 2 ) 1 / 2 ,
β D T H G = 3 8 d 3 x ε χ ( 3 ) ( E 1 E 1 ) ( E 1 E 3 ) ( d 3 x ε | E 1 | 2 ) 1 / 2 ( d 3 x ε | E 3 | 2 ) 3 / 2 ,
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