# A New Method to Verify the Measurement Speed and Accuracy of Frequency Modulated Interferometers
## 3 Experiments and Results  
A photograph of the experimental setup is shown in Figure 1b. The laser source was an LD (HL6312G, Thorlabs Inc., Newton, NJ, USA), and it was frequency modulated with a sine-wave signal (a frequency modulation of 20 MHz and modulation excursion of 570 MHz) by modulating the injection current. To produce a high-speed virtual motion, the 4002 EOM with V\({}_{\pi}\) = \(\sim\)125 V at \(\lambda\) = 635 nm was used. The EOM was phase changed using a 20 kHz sine-wave signal V(t), which had V\({}_{0}\) = 2 V, by utilizing a digital function generator (Moku:labs, Liquid Instruments), and it was then amplified 30 times by a high-voltage amplifier. \(\Delta\Phi_{EOM}\) was produced, and the 20 kHz sine-wave virtual displacement all could thus be attained with a peak-to-peak (p-p) amplitude of \(\sim\)152 nm (=635 nm \(\times\) 2 V \(\times\) 30/2/125 V). The l(\(\tau\), t) was detected using an avalanche photodetector (DET08 CFC/M, Thorlabs). To extract I\({}_{2\omega}\) (2\(\omega\) = 40 MHz) and I\({}_{3\omega}\) (2\(\omega\) = 60 MHz), we used two digital LIAs (Moku:labs, Liquid Instruments), which were synchronized with each other and had a cutoff frequency of 200 kHz. The I\({}_{2\omega}\) and I\({}_{3\omega}\) were then recorded using an analog-to-digital converter (AD16-16U(PCI)EV, Contec Co., Osaka, JP) at a sampling frequency of 715 kHz to eliminate MHz-order noises in the harmonics. In this experiment, the modulation index was m \(\approx\) 2.4. This led to J(2\(\omega\)) = 2.15 \(\times\) J(3\(\omega\)). In addition, because (L-l)n\({}_{\text{air}}\) + n\({}_{\text{el}}\)\(\approx\) n\({}_{\text{air}}\)L \(\approx\) 0.2 m. Equation (13) is rewritten as  
\[\Delta l=\frac{\lambda}{4\pi}\left\{\arctan\!\left(-\frac{1}{2.15}\frac{I_{3 \omega}}{I_{2\omega}}\right)\right\}-Ln_{air}. \tag{14}\]  
The experiments were performed under the conditions in Table 1. The second and third harmonics that were clearly detected by LIAs are shown in Figure 2a. A Lissajous diagram using the two harmonics is shown in Figure 2b. A sine-wave displacement at 20 kHz and a p-p amplitude of \(\sim\)152 nm was obtained over 1 ms, as shown in Figure 3a. This led to a measured speed of 6.08 mm/s (=2\(\Delta l\)\(\times\)\(f_{EOM}\) = 2 \(\times\) 152 nm \(\times\) 20 kHz). The result obtained by the interferometer was compared with a calculated displacement function \(F\) using Equation (11), \(F=A+B\times\sin(2\pi\times 20\,\text{kHz}\times t)\), where \(A=-0.152\) um and \(B=-0.076\) um. The difference between the measured result and \(F\) is shown in Figure 3b. A displacement difference of \(\sim\)30 nm and a standard deviation of \(\sim\)6 nm were attained. The displacement noise floor analyzed in Figure 3b is depicted in Figure 3c. Above 2 kHz, a displacement noise floor of less than 100 pm/\(\surd\)Hz was achieved. A dominated noise peak at 20 kHz can be seen. This led to the cycle error of the sine-wave displacement in Figure 3a, and this noise level was \(\sim\)100 pm/\(\surd\)Hz. The result showed that our method could generate a stable and high-precision displacement that could be used as a reference. Even though unwanted noises caused by vibration and air disturbance still existed in the measurement, the method was promising for high-speed measurements. \begin{table}
\begin{tabular}{l l} \hline Modulation frequency \(\omega\) for LD & 2\(\pi\times\) 20 MHz \\ Frequency modulation excursion \(\Delta\omega\) & 2\(\pi\times\) 570 MHz \\ Cutoff frequency of LIAs & 200 kHz \\ \(V_{\pi}\) at 536 nm & 125 V \\ Applied angle frequency for \(\omega_{EOM}\) & 2\(\pi\times\) 20 kHz \\ Unbalanced length \(L\) & 0.2 m \\ Working sampling frequency & 715 kHz \\ \hline \end{tabular}
\end{table} 
Table 1: The experimental conditions for the virtual displacement measurement. Figure 2: Harmonic detection: (**a**) second and third harmonics; (**b**) Lissajous diagram using \(I_{2\omega}\) and \(I_{3\omega}\) harmonics.