22
J. Eur. Opt. Society-Rapid Publ. 21, 32( 2025)
as temporal and spatial variations of the refractive index, which caused dynamic wavefront aberration leading dynamic 2-dimensional( 2-D) in-plane displacement. Precise observation of air fluctuation in such situation has been strongly necessary to improve the resolution of the displacement measurement in air.
Shack-Hartmann sensor is popular to measure wavefront aberration of optical paths [ 14 ]. It measures the wavefront by focusing the incident beam on a camera using a micro lens array. The resolution and measurable range depend on the pixel size and effective focal length of the lens, and the measurement speed depends on the camera frame rate. Some Shack-Hartmann sensors can measure at around 1 kfps or a resolution of k = 200 in real-time [ 15 ]. However, Shack-Hartmann sensors are not capable of measuring the wavefront( i. e., 2-D in-plane displacement) of every single camera pixel. Surface profile interferometers such as Fizeau interferometers are considered capable of measuring the wavefront(= 2-D in-plane displacement) at each pixel of the camera at a hundred nano meter range [ 16, 17 ]. Although the surface profile measurement resolution with Fizeau interferometers has reached sub-nm, the sampling rate is slow, at most several 10 Hz. For the air fluctuation measurement with Shack-Hartmann sensors or Fizeau interferometers, both the measurement resolution and the measurement speed are not enough. The authors believe that a 2-D in-plane displacement measurement system with a sampling rate of several kHz or higher and a resolution of sub-nm or less displacement per camera pixel is needed to study air fluctuations for precision interferometry and astronomical observations.
In this paper, to solve the above problem, we propose a SPM interferometer for 2-D in-plane displacement measurement with sub-nm resolution and fast speed sampling rate of 5 kHz. The interferometer consists of a Michelson type interferometer incorporating an electric-optic modulator( EOM) with a modulation frequency of 5 kHz and a highspeed camera( HSC) synchronized to a clock signal at a frequency of 60 kHz, 12 times the modulation frequency. Phase demodulation of all pixels in the camera is performed by acquiring the interference signal to that pixel synchronously with the sampling signal and performing a specific addition and subtraction between the signals obtained synchronously. By applying this procedure to all pixels in the camera, 2-D in-plane displacements can be obtained. In this paper, we report on the measurement equipment, the demodulation principle, the pre-filter for noise reduction and experimental results.
2 Principle
The schematic diagram of the SPM interferometer with an external EOM configuration is shown in Figure 1. A light source outputs a single frequency and a linear polarized beam. The polarization angle is controlled to 45 ° against to p-polarization axis( along with x-axis) using a half wave plate( HWP) and a polarizer( P). The beam passes through an EOM which modulates phases of the beam in a sinusoidal way. It is noticed that the EOM modulates phases in both p-polarizing beam and s-polarizing beam with different modulation efficiencies. If p- and s-polarizations are 100 % and is 31 %, respectively, because the electro-optic coefficients of r 13 and r 33 are 9.6 fm / V and 30.9 fm / V in a LiNbO3 crystal used in the paper, respectively [ 18 ]. The difference results in phase modulation of 69 %, thus this configuration has a lower phase modulation index than conventionally used configuration in which the EOM is inserted reference arm [ 19, 20 ].
The beam diameter is around 2 mm at the EOM aperture. To measure a mirror deformation with the area around ten-millimeter diameter, the 10 times beam expander( BE) is inserted. We use unbalanced arm length Michelson interferometer with polarizing optics. The beam from the BE is split into a reference optical path L R and a target optical path L T by the PBS. A reference mirror( RM) is mechanically fixed to base frame. A deformable mirror( DM) or a plane mirror( PM) with piezoelectric transducer( PZT) is employed as a target mirror. Beams which are split by the polarizing beam splitter( PBS) are reflected from these mirrors, then go to HSC and interfere with each other. The phase modulation and the mirror deformation with DM( or 2-D in-plane displacement with PM) generate a periodic interference fringe change like a beat fringe in heterodyne interferometer. We represent the spatial phase difference / ðx; y; tÞ to be measured as equation( 1),
/ ðx; y; t
Þ ¼ 4pn k fL R ðx; y; t Þ�L T ðx; y; t
Þg
¼ 4p k nx ð; y; tÞLðx; y; tÞ; ð1Þ
where x, y, t, k, nðx; y; tÞ Lðx; y; tÞ are pixel positions along x- and y axes at sensor of camera, time, vacuum wavelength, air refractive index and 2-D in-plane displacement to be measured which corresponds to the path difference out of flat plane, respectively. It is noted that the lateral resolution corresponded to the pixel pitch of the camera is 20 lm in following experiments. The interference fringe I HSC ðx; y; tÞ at HSC is represented by equation( 2).
I HSC ðx; y; tÞ ¼ E 2 T þ E 2 R þ 2E T E R cos f/ ðx; y; tÞþm sin x m tg; ð2Þ
where E T, E R, x m and m are complex amplitudes in target and reference paths, a modulation angular frequency and a modulation index. The modulation index m is represented as( see Appendix),
m ¼ pL EOM
� n 3 1 c kd 13 � n 3 3 c 33 V EOM K EOM ðtÞ; ð3Þ
EOM
where L EOM, d EOM, V EOM, K EOM and Dt =| L R � L T |/ c, c are a length and a thickness of the EOM crystal, an applied voltage to the EOM, a modified term due to optical path difference, delay time and speed of light in vacuum, respectively. The modified term K EOM ðtÞ ffi 1, when the optical path difference is within 1 m in case of the modulation frequency is 5 kHz( see Appendix). The modulation index m can be fixed by determining V EOM.