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.