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Fig. 1. Schematic of the core functional link for Si 3 N 4-TFLN heterogeneous integrated device with phase modulation and frequency doubling.
the intensity of the electric field applied to TFLN waveguide, with the unit of volt( V); x m denotes the modulation angular frequency, corresponding to the modulation bandwidth f m = x m / 2p, with the unit of gigahertz( GHz); / 0 is the initial phase of the RF modulation signal, with the unit of radian( rad). According to the electrooptic effect, the refractive index change Dn of the LN material is proportional to the applied electric field. Combining the antisymmetric characteristic relationship of the phases between the two arms of the DD-MZM, the phases of the optical signals output from the two arms of the DD- MZM can be derived as follows [ 27 ]:
/ 1 ðÞ¼ t / stat þ V m cos ðx m t þ / 0 Þ; ð2Þ
/ 2 ðÞ¼ t / stat � V m cos ðx m t þ / 0 Þ: ð3Þ
/ 1( t) and / 2( t) represent the time-varying optical phases of the two arms of the DD-MZM, with the unit of radian( rad). / stat is the inherent phase difference generated by the propagation of optical signals in the DD-MZM arms without external modulation signals. It is the static phase reference of the device itself, measured in radians( rad). The electro-optic response coefficient c of LN is defined as c =( pr 33 n eff L)/( kV p). Among the parameters in this expression, n eff denotes the effective refractive index of the LN waveguide; r 33 is the electro-optic coefficient of TFLN with the unit of picometer per volt( pm / V) and a value of 30.8 pm / V; L stands for the length of the DD-MZM modulation arm with the unit of meter( m); k is the operating optical wavelength with the unit of meter( m). V p refers to the half-wave voltage of the DD-MZM, with the unit of volt( V). Additionally, d represents the spacing between the T-shaped track electrodes with the unit of meter( m). Other parameters are consistent with equation( 1). After the modulated light is combined via MZM interference, the total phase of the output modulated light is determined by the phase difference between the two arms. Considering the constructive interference condition, the total phase can be simplified as:
/ mod ðÞ¼x t 0 t þ / dynamic ðÞ t; ð4Þ
where D / dynamic( t)= 2cV 0 cos( x m t) is the dynamic phase shift introduced by modulation, which contains the optical carrier x 0 and the upper / lower sideband( x 0 ± x m). These components correspond to the fundamental frequency optical signals required for frequency doubling.
2.1.2 Resonance phase expression of the MRR
The resonance condition of the MRR requires that the phase change of light propagating round-trip within the cavity is an integer multiple of 2p. Considering static perturbations such as temperature and stress, the resonance phase of the MRR can be expressed as [ 28, 29 ]:
2pL ring
/ res ðxÞ ¼ n eff; MRR kx ð Þ þ / static; ð5Þ
where n eff, MRR is the effective refractive index of the MRR waveguide, L ring is the perimeter of the MRR. The optical wavelength k of the incident light and its angular frequency x satisfy the relationship x = 2pc / k, where c is the speed of light in vacuum. / static is the static phase offset-typically introduced by process errors or environmental perturbations [ 30 ].
2.1.3 Phase matching adjustment
To enable the fundamental frequency x 0 and sideband( x 0 ± x m) of the modulated light to couple efficiently into the MRR and excite frequency doubling, the matching condition between the dynamic phase and the resonant phase must be satisfied.
Fundamental frequency optical matching: