## 2 Experimental Setup  
The schematic of the Kerr phase-interrogator for GBD measurements is presented in Figure 1. A continuous light-wave (CW) from a tunable laser (TL) operates at wavelength \(\lambda\) and is amplitude-modulated by an electro-optic modulator (EOM). The EOM is driven by a sinusoidal electrical signal oscillating at radio frequency \(f_{m}\) and turns the CW into an SMOS oscillating at a radio frequency \(f_{s}=2f_{m}\). The optical spectrum of the resulting SMOS is comprised of two distinct sidebands separated by \(f_{s}\) and centered at \(v_{0}\). A polarization beam splitter (PBS) splits the SMOS into two orthogonal polarization components. The two orthogonal polarization components propagate through the fast axis and slow axis over equal fiber length, respectively. A polarization beam combiner (PBC) combines the two orthogonal polarization components and sends them into a fiber under test (FUT) through a polarization-maintaining circulator (PMC). Reflected at the far end of the FUT by a polarization-maintaining reflector R, the combined signals are amplified by an erbium-doped fiber amplifier (EFDA) and are injected into a Kerr medium comprised of a 2 km dispersion-shifted fiber (DSF, Fujikura) with waveguide nonlinearity of 2.28 W\({}^{-1}\)/km. The amplitudes of the amplified combined signals are identical, which is realized by finely adjusting the polarization state of the SMOS with a polarization controller (PC) placed before the PBS. In the DSF, the Kerr effect results in the distinct sidebands in the total optical spectrum of the combined signals, and the power of the first-order sideband \(P_{1}\) is expressed as [11; 12]:  
\[P_{1}=P_{10}[1+\cos(2\pi f_{s}t_{d}+\phi_{0})], \tag{1}\]  
where \(P_{10}\) is a constant value, \(\phi_{0}\) is the constant phase and \(t_{d}\) is the DGD representing the round-trip time differential group delay between the slow axis and fast axis of the FUT.  
Using the Hilbert transform and arctan function [12], the relative phase \(\Delta\phi(\lambda)=\phi(\lambda)-\phi_{ref}\) is demodulated from \(P_{1}(\lambda)\), where \(\phi(\lambda)=2\pi f_{s}t_{d}(\lambda)+\phi_{0}\), and \(\phi_{ref}=2\pi f_{s}t_{ref}+\phi_{0}\) with \(t_{ref}\) being the absolute DGD at a reference wavelength \(\lambda_{ref}\). The relative DGD \(\Delta t_{d}(\lambda)=t_{d}(\lambda)-t_{ref}\) is obtained by  
\[\Delta t_{d}(\lambda)=\frac{\Delta\phi(\lambda)}{2\pi f_{s}}. \tag{2}\]To measure \(t_{ref}\), the setup in Figure 1 is modified. Specifically, a 3 GHz optical band-pass filter (OBPF), a photodetector (PD) and an oscilloscope (OSC) are cascaded to record \(P_{1}\) as a function of \(f_{s}\). In the modified setup, the laser wavelength of the two SMOSs \(\lambda\) is set at a reference wavelength \(\lambda_{ref}\) and \(f_{s}\) is linearly scanned, which results in the sinusoidal variation of \(P_{1}(f_{s})\) according to Equation (1). The relative phase \(\Delta\phi(f_{s})=\phi(f_{s})-\phi_{init}\) is demodulated from \(P_{1}(f_{s})\), where \(\phi_{init}=2\pi f_{s-init}t_{ref}+\phi_{0}\) and \(f_{s-init}\) is the initial value of \(f_{s}\). \(t_{ref}\) is calculated from the slope of \(\phi(f_{s})\) with  
\[t_{ref}=\frac{1}{2\pi}\frac{d}{d{f_{s}}}[\Delta\phi(f_{s})]. \tag{3}\]  
\(t_{ref}\), another absolute DGD \(t_{ref2}\) at a wavelength \(\lambda_{ref2}\), needs to be measured and then compared with \(t_{ref}\) to specify the actual sign of \(\Delta t_{d}(\lambda)\). In other words, to verify whether \(t_{d}(\lambda)\) increases or decreases with \(\lambda\).  
The absolute DGD is given by \(t_{d}(\lambda)=t_{ref}+\Delta t_{d}(\lambda)\), resulting in  
\[B(\lambda)=\frac{c}{2L}[t_{d}(\lambda)], \tag{4}\]  
where \(B\), \(c\) and \(L\) are the group birefringence, speed of light in vacuum and the length of the FUT, respectively. Equation (4) leads to the GBD calculated by  
\[D_{B}(\lambda)=\frac{d}{d\lambda}[B(\lambda)], \tag{5}\]  
where \(D_{B}\) represents the GBD.