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Fig. 2. Phase-matching relationship and key tuning parameter working range.( a) Variation of fundamental / sideband light phase with modulation signal amplitude V 0.( b) Variation of MRR resonant phase with temperature tuning amount DT.
through the efficiency calculation model, and defines the optimal working interval of the modulation bandwidth combined with the visual curve.
2.2.1 Core influencing factors of frequency doubling efficiency
Based on the second-order nonlinear effect of LN, the frequency doubling efficiency can be expressed as [ 31, 32 ]:
g SHG / x2 0 d2 33j
E fundj 2 Q 2 L ring n 3 eff; MRR c3 a
: ð15Þ
In this formula, d 33 denotes the second-order nonlinear coefficient of LN, | E fund | is the fundamental frequency optical field intensity within the MRR, Q is the quality-factor of the MRR, and a represents the round-trip loss of the MRR.
Given d 33 of TFLN is 30.8 pm / V, the fundamental frequency optical field intensity within the MRR is:
� jE fund j¼P found = A v g: ð16Þ
The fundamental optical power P found is 100 mw, and the spot area A is taken as the effective area of the mode field of the MRR waveguide, which is derived based on the waveguide structural parameters. The physical cross- sectional area of the waveguide is 1.2 lm 0.3 lm = 3.6 10 �13 m 2. Considering the penetration of the optical field into the cladding, the effective area of the TE 00 mode field obtained through COMSOL Multiphysics simulation is A = 4.2 10 �13 m 2. The group velocity is v g = c / n eff 1.4 10 8 m / s, calculated | E fund | 1.98
10 6 V / m. The MRR quality factor Q is 3.2 10 5, with a roundtrip = 0.3 dB, loss consisting of propagation loss of 0.15 dB / cm and interface scattering loss of 0.2 dB / cm. The round-trip length is 1.147 mm, and the total loss is approximately 0.3 dB, calculated as( 0.15 + 0.2) 1.147 10 �1 0.3 dB. Substituting into formula( 15),
g SHG /( 30.8 10 �12) 2( 1.98 10 6)( 3.2 10 5) 2 / 0.3 1.53 10 �3. Combining this with the experimental calibration coefficient K = 10 2, we ultimately obtain g SHG 15.3 %.
Combined with the modulation characteristics of the DD-MZM, the relationship between the fundamental frequency optical field intensity | E fund | and the modulation bandwidth f m is:
jE fund j /
V 0 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ð17Þ 1 þ ðf m = f 3dB Þ 2
where f 3dB is the 3 dB modulation bandwidth of the DD- MZM. When f m f 3dB,| E fund | is approximately constant; when f m approaches f 3dB, | E fund | attenuates as f m increases.
2.2.2 Bandwidth compatibility constraint
The FSR of the MRR should be greater than twice the modulation bandwidth, i. e., FSR > 2 f m. This prevents overlapping interference between different sideband components within the MRR [ 31 ]. Combining the expression for FSR:
� FSR MMR ¼ c = 2n eff; MRR L ring: ð18Þ
The bandwidth constraint condition is obtained as: c f m <:
4n eff; MRR L ring ð19Þ
Combined with the FSR expression of the racetrack MRR, the bandwidth constraint condition of the system is obtained. For the Si 3 N 4 racetrack MRR designed in this paper, n eff, MRR = 1.99768, L ring = 1.147 mm, and the maximum allowable modulation bandwidth is about 75 GHz.
The variation trend of frequency doubling efficiency with modulation bandwidth is shown in Figure 3, and the key parameter range and optimal working interval of the system are defined based on the curve characteristic.
3 dB modulation bandwidth of DD-MZM: f 3dB = 72 GHz. This parameter is determined by the RC delay of the T-shaped track electrode and the microwave-optical velocity matching, which is the upper limit of the effective modulation bandwidth of the device.