J. Eur. Opt. Society-Rapid Publ. 21, 32( 2025) 29
9 Pisani M, et al., A portable picometer reference actuator with 100 lm range, picometer resolution, subnanometer accuracy and submicroradian tip-tilt error for the characterization of measuring instruments at the nanoscale, Metrologia 55, 541( 2018). https:// doi. org / 10.1088 / 1681-7575 / aaca6f.
10 Hori Y, et al., Periodic error evaluation system for linear encoders using a homodyne laser interferometer with 10 picometer uncertainty, Precis. Eng. 51, 388( 2018). https:// doi. org / 10.1016 / j. precisioneng. 2017.09.009.
11 Dandridge A, et al., Homodyne demodulation scheme for fiber optic sensors using phase generated carrier, IEEE J. Quantum Electron 18, 1647( 1982). https:// doi. org / 10.1109 / TMTT. 1982.1131302.
12 Sasaki O, et al., Sinusoidal phase modulating interferometry for surface profile measurement, Appl. Opt. 25, 3137( 1986). https:// doi. org / 10.1364 / AO. 25.003137.
13 Higuchi M, et al., Development of band-limitless demodulation method for sinusoidal phase modulation interferometry( evaluation of noise floors, maximum measurement speeds and resolutions), J. Jpn. Soc. Precis. Eng. 90, 153( 2024)( Written in Japanese). https:// doi. org / 10.2493 / jjspe. 90.153.
14 Platt CB, et al. History and principles of Shack-Hartmann wavefront sensing, J. Refract. Surg. 17, S573( 2001). https:// doi. org / 10.3928 / 1081-597x-20010901-13.
15 Shack-Hartmann wavefront sensor WFS200 and WFS21, Thorlabs, https:// www. thorlabs. co. jp / newgrouppage9. cfm? objectgroup _ id = 5287.
16 Wang X., et al., Sinusoidal phase-modulating Fizeau interferometer using a self-pumped phase conjugator for surface profile measurements, Opt. Eng. 33, 8( 1994). https:// doi. org / 10.1117 / 12.173590.
17 Sasaki O., et al., Sinusoidal phase modulating Fizeau interferometry, Appl. Opt. 29, 4( 1990). https:// doi. org / 10.1364 / AO. 29.000512.
18 A Yariv, et al., Photonics: optical electronics in modern communications, 6th edn,( Oxford Univ Pr on Demand, 2007).
19 Minoni U, et al., A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: development and evaluation, Rev. Sci. Instrum. 62, 2579( 1991). https:// doi. org / 10.1063 / 1.1142233.
20 Heinzel G, et al., Deep phase modulation interferometry, Opt. Express 18, 19076( 2010). https:// doi. org / 10.1364 / OE. 18.019076.
21 Aranchuk V, et al., Laser Doppler multi-beam differential vibration sensor based on a line-scan CMOS camera for realtime buried objects detection, Opt. Express 31, 1( 2022). https:// doi. org / 10.1364 / OE. 477115.
22 S711 high-speed machine vision camera product information, Vision Research. Available at https:// www. phantomhighspeed. com / products / cameras / machinevision / s711.
Appendix
The approximation in modulation index m calculation is explained in this section. The SPM interferometer described below uses EOM for phase modulation. The phase of the outgoing light from the EOM varies depending on the crystal orientation of the EOM, the direction of the applied voltage, and the polarization of the incident light. Figure A1 shows the considered arrangement. The E EOM is the electrical field applied to EOM along with y axis( thickness direction). Due to the polarizer with 45-degree inclination, the electrical field of the incident beam to the EOM is written as
E 1 ðz; tÞ ¼ E
x
¼ 1 E y 1 e ikz�xtþ ð / 0Þ
where, E x, E y t, k, x and / 0 are electric fields along x and y axes time, wave number, angular frequency of the beam and initial phase, respectively. EOM shifts the phase with refractive index of EOM, which efficiency are different in the axis direction. Amounts of the phase shift with applying voltage V are written as
/ x ¼ 2pl k
/ y ¼ 2pl k
n 1 � n 1 3 2 c 13
n 3 � n 3 3 2 c 33
V EOM d
V EOM d
A1
; A2
; A3
where, l, k, n 1, n 3, c 13, c 33 and V EOM are the length of the EOM crystal, the wavelength, the refractive index of the EOM, the electro-optic coefficient of the EOM crystal and applied voltage to EOM, respectively. The refractive index n 1 and n 3 are not same in LiNbO3. The electrical field of the transmitted beam is written as
E 2 ðz; t
Þ ¼ 1 pffiffi
2 ei / x e i / y
e ikz�xtþ ð / 0 Þ: ðA4Þ
If the y axis polarization beam goes to the reference arm and the x axis polarization beam goes to the target arm, the combined electrical field at HSC is written as
E 3 ðz; t
Þ ¼ p 1 ffiffi 8 ei ð 2kL Rþ / x
Þ þ e i ð 2kLTþ / y
0
! Þ e i ð �xtþ / 0 Þ: ðA5Þ
The interference signal I could be written as I ¼ jE 3 ðz; tÞj 2 ¼ 1
4 þ 1 4 cos 2k L �
ð R � L T Þþ / x � / y: ðA6Þ
In case of L R < L T, the beam along with the x axis polarization delays t ¼ 2nLT�L ð RÞ compared to the beam along with the y axis c polarization. The phase difference / x � / y by applying sinusoidal voltage V EOM sin x m t is rewritten as
u y ðÞ�u t x ðt þ tÞ ¼ 2pl n 1 � n
1 3 k 2 c V EOM sin x m t
13 d
� n 3 � n
3 3
2 c V EOM sin x m ðt þ tÞ
33 g d
¼ 2pl ðn 1 � n 3 Þþ V
EOM
� n 3 1 c k
2d 13 sin x m t � n 3 3 c 33 sin x m ðt þ tÞ
:
Using qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
the relationship of a sin h � b sin ðh þ Þ ¼ ða � b cos Þ 2 þ ðb sin Þ 2 sin �b sin h þ atan, the phase a�b cos difference is rewritten as u y ðÞ�u t x ðt þ tÞ ¼ 2pl ð k n 1 � n 3 Þþm sin fx m t þ hg; ðA7Þ where, m ¼ p k
V EOM d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 13 c 13 � n 33 c 33 cos x m tÞ 2 þ ðn 33 c 33 sin x m tÞ 2;
ðA8Þ