# A New Method to Verify the Measurement Speed and Accuracy of Frequency Modulated Interferometers
## 2 Methodology  
Figure 1a shows the schematic of the frequency modulated interferometer. The laser source was a laser diode (LD), in which the frequency was modulated with a sinusoidal signal by a modulation current injection. The angle frequency \(\omega(t)\) of the LD is expressed as  
\[\omega(t)=\omega_{0}+\Delta\omega\sin(\omega t) \tag{1}\]where \(\omega_{0}\) (=2\(\pi\)\(f_{0}\)), \(\omega\), and \(\Delta\omega\) are the initial angle frequency of the LD, acting as the carrier angle frequency, the angle modulation frequency, and the angle modulation frequency excursion of \(\omega\), respectively. An isolator was used to prevent unwanted lights returning to the LD. A half-wave plate (HWP) was employed to rotate the polarization plane of the incident light at 45\({}^{\circ}\). A beam splitter (BS) divided the incident light from the isolator into two beams: a reflected beam and a transmitted beam. The reflected beam was sent to a mirror (M1) in the reference arm. In the measurement arm, the second beam was sent to another mirror (M2) along a polarizer (P) and an EOM. The polarizer was placed in front of the EOM to ensure the correct polarization of the input light. After reflection from the mirrors, the reflected lights of the two arms recombined with each other at the BS to generate interference. The intensity \(I(\tau,t)\) of the interference signal [13,14] is written as  
\[ \begin{split}& I(\tau,t)=E^{2}{}_{01}+E^{2}{}_{02}+2E_{01}E_{02} \cos(\omega_{0}\tau)J_{0}(m)+4E_{01}E_{02}\cos(\omega_{0}\tau)\sum\limits_{n=1} ^{\infty}J_{2n}(m)\cos(2n\omega t)\\ &-4E_{01}E_{02}\sin(\omega_{0}\tau)\sum\limits_{n=1}^{\infty}J_{2 n-1}(m)\cos[(2n-1)\omega t],\end{split}  \tag{2}\]  
where \(\tau\), \(m\), \(E_{01}\) and \(E_{02}\), \(n\), \(J_{0}(m)\), \(J_{2n}(m)\), and \(J_{2n-1}(m)\) are the changes in time between the two arms of the interferometer, the modulation index, the amplitudes of the electric fields in the reference and measurement arms, an integer, and the Bessel functions, respectively. Here,  
\[m=\frac{\Delta\omega}{\omega}\sin\left(\frac{\omega\tau}{2}\right)\approx\frac {2\pi\Delta fn_{air}L}{c} \tag{3}\]  
where \(\Delta f\) (\(\Delta\omega=2\pi\Delta f\)), \(n_{air}\), \(L\), and \(c\) are the frequency modulation excursion, the refractive index of air, the unbalance length of the interferometer, and the speed of light in a vacuum. A divider split \(I(\tau,t)\) into two parts, which are coupled with the two purely sinusoidal signals of \(2\omega\) and \(3\omega\) from the function generator. By using the two LIAs [13,14], the intensities \(I_{2\omega}\) and \(I_{3\omega}\) of the \(2\omega\) and \(3\omega\) harmonics of \(I(\tau,t)\), respectively, are produced as follows:  
\[ \begin{split}& I_{2\omega}=2E_{01}E_{02}\cos(\omega_{0}\tau)J_{2}(m),\\ & I_{3\omega}=-2E_{01}E_{02}\sin(\omega_{0}\tau)J_{3}(m).\end{split}  \tag{4}\]  
Figure 1: Virtual displacement-measuring interferometer using a frequency-modulated laser source. **(a)** Schematic design and **(b)** the experimental setup. LD: laser diode; HWP: half-wave plate; BS: beam splitter; M: mirror; P: polarizer; EOM: electro-optic modulator; EO crystal: electro-optic crystal; NF: neutral density filter; HVA: high-voltage amplifier; FG: function generator; APD: avalanche photodetector; LIA: lock-in amplifier; ADC: analog-to-digital converter. Here, \(J_{2}(m)\) and \(J_{3}(m)\) are the second- and third-order Bessel functions, respectively. By using the Lissajous diagram, \(\omega_{0}\tau\) is determined. The total phase difference \(\Phi\) between the arms is  
\[\Phi=\omega_{0}\tau=\arctan\biggl{(}-\frac{I_{3\omega}}{I_{2\omega}}\cdot\frac{ J_{2}(m)}{J_{3}(m)}\biggr{)}. \tag{5}\]  
To generate the virtual displacement, the EOM was applied using a sinusoidal voltage \(V(t)\)  
\[V(t)=V_{EOM}\sin(\omega_{EOM}t), \tag{6}\]  
where \(V_{EOM}\) is the amplitude of \(V(t)\).