Operation Principle

The figure.1 shows is a schematic diagram of a Hilbert transformer based on a micro-comb. The micro-comb is produced on a high-Q MRR by pumping. The CW laser is used and the EDFA is used to amplify the MRR whose polarization state is aligned with the TE mode. When the pump wavelength is manually swept across one of the resonances of the MRR and the pump power is large enough to generate sufficient parameter gain, the optical parameter oscillation will occur, and finally a Kerr frequency comb with a spacing equal to the MRR free spectral range will be generated.
The spectral transfer function of a general fractional Hilbert transformer is given by [1, 8]:
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }H_{P}\left(\omega\right)=\left\{\par \begin{matrix}e^{-j\varphi},\ \ if\ 0\leq\omega<\pi\\ e^{\text{jφ}},\ \ if-\pi\leq\omega<0\\ \end{matrix}\right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ }\) (1)
where j = \(\sqrt{-1}\) ,φ = P × π / 2 denotes the phase shift, P is the fractional order (when P = 1, it becomes a standard Hilbert transformer). The corresponding impulse response is given by a continuous hyperbolic function:
\(h_{P}\left(t\right)=\left\{\par \begin{matrix}\frac{1}{\text{πt}},\ t\neq 0\\ \cot\left(\varphi\right),t=0\\ \end{matrix}\right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\)
This hyperbolic function is truncated and sampled in time with discrete taps for digital implementation. The null frequency is given by:
\(f_{c}=1/t\) (3)
where ∆t denotes the sample spacing. The coefficient of the tap at t = 0 can be adjusted to achieve a tunable fractional order [1]. The normalized power of each comb line is:
\(\text{\ \ \ \ }P_{n}=\frac{1}{\pi|n-\frac{N}{2}+0.5|}\) (4)
where N is the number of comb lines, or taps, and n = 0, 1, 2, …,N -1 is the comb index.