J. Eur. Opt. Society-Rapid Publ. 22, 16( 2026) 151
/ mod ðx 0 Þ ¼ / res ðx 0 Þþ2kp; ðk 2 ZÞ: ð6Þ
This fundamental frequency optical matching condition ensures the fundamental frequency light is resonantly enhanced within the MRR, providing sufficient optical field energy for nonlinear frequency doubling. Sideband optical matching:
/ mod ðx 0 x m Þ ¼ / res ðx 0 x m Þþ2kp; ðk 2 ZÞ: ð7Þ
It prevents the sidebands from being filtered out by the MRR due to phase mismatch, thus retaining the sideband components required for frequency doubling.
Correspondingly, by combining equations( 6) and( 7) and neglecting the dynamic fluctuations of / static, thefinal phase matching formula is derived as:
x 0 L ring x 0 t þ 2V 0 cos ðx m tÞ ¼ n eff; MRR þ / c static þ 2kp; ðk 2 ZÞ: ð8Þ
This condition indicates that phase matching between the two can be achieved by adjusting the amplitude V 0 of the modulation signal or tuning n eff, MRR of the MRR, laying the foundation for efficient frequency doubling.
To verify the feasibility of the phase matching condition, the core parameters of this design are substituted for quantitative verification. The known parameters include the angular frequency of the input optical carrier:
x 0 ¼ 2pc = k ¼ 2p 3 10 8 = 1550 10 �9 1:21 10 15 rad = s:
Assuming f m = 70 GHz, the modulation angular frequency x m = 2pf m, we have x m 4.40 10 11 rad / s. The MRR is made of Si 3 N 4 material, and its effective refractive index is n eff, MRR = 1.99768. The perimeter of the MRR is:
L ring ¼ 2pR avg þ 2l straight ¼ 2p 80 10 �6 þ 2 280 10 �6 1:147 mm:
Static phase shift is D / static = 0.02 rad. Fundamental frequency light matching verification:
x 0 L ring n eff; MRR = c ¼ 1:21 10 15 1:147 10 �3
satisfied
1:99768 = 3 10 8 912:3 rad; ð9Þ
ð10Þ
ð11Þ
912:3 rad ¼ 2kp þ / 0: ð12Þ
With k = 145.2 145 p + 0.02 912.88 rad, with an error of less than 0.02 %, resonant enhancement can be achieved with fundamental frequency light. Sideband light matching verification: ðx 0 þ x m ÞL ring n eff; MRR = c ¼ð1:21 10 15 þ 4:40 10 11 Þ1:147 10 �3 satisfied
1:99768 = 3 10 8 912:7 rad; ð13Þ
912:7 rad ¼ 2ðk þ 1Þp þ / 0: ð14Þ
With k = 144.2 144 p + 0.02 904.78 rad, after adjusting the MRR temperature tuning amount DT = 0.8 ℃, thechangeinn eff, MRR is 0.0005, with a final error of less than 0.1 %, and ensures no filtering loss of sideband light.
The phase-matching relationship between the modulated light and the MRR, as well as the influence law of key tuning parameters, is visually shown in Figure 2. The red dashed line in the figure is the phase-matching threshold; the intersection of the phase curve and the threshold is the optimal working point of the system, and the parameter range near the working point is the theoretical range for the normal operation of the device.
Theoretical working range of V 0: 1.5 – 4.0 V. When V 0 < 1.5 V, the phase offset of the modulated light is insufficient and cannot reach the matching threshold; when V 0 > 4.0 V, the electric field intensity exceeds the breakdown threshold of the TFLN waveguide, which causes irreversible damage to the device.
Theoretical working range of DT: 0.5 – 1.2 ℃. When DT < 0.5 ℃, the sideband light is filtered by the MRR with a loss of more than 3 dB; when DT > 1.2 ℃, theFSRofthe MRR drifts by more than 5 GHz, which destroys the bandwidth compatibility constraint.
Optimal working point: V 0 = 2.8V, DT = 0.8 ℃. At this point, the fundamental frequency / sideband light phase and the MRR resonant phase are exactly matched, the resonance loss of the MRR is the lowest(< 0.1 dB), and the frequency doubling efficiency reaches the theoretical maximum value.
The linear change trend of the phase with V 0 and DT in Figure 2 is consistent with the electro-optic effect of TFLN and the temperature refractive index characteristic of Si 3 N 4, which verifies the feasibility of adjusting the phase matching state of the system through electrical and thermal tuning.
2.2 Correlation model between frequency doubling efficiency and modulation bandwidth
The frequency doubling efficiency of the device is jointly determined by the optical field energy accumulation in the MRR and the bandwidth compatibility between DD- MZM and MRR. There is a synergistic constraint relationship between the modulation bandwidth f m and the frequency doubling efficiency g SHG: the increase of the modulation bandwidth will lead to the attenuation of the fundamental frequency optical field intensity in the MRR, and then reduce the frequency doubling efficiency. This section clarifies the quantitative relationship between the two