J. Eur. Opt. Society-Rapid Publ. 21, 32( 2025) 23
Figure 1. Schematic diagram of SPM interferometer, HWP: Half wave plate, P: Polarizer, EOM: Electro-optic modulator, BE: Beam expander, PBS: Polarization beam splitter, QWP: Quarter wave plate, BS: Beam splitter, DM: Deformable mirror, HSC: High-speed camera, PM: Plane mirror with piezoelectric transducer( PZT). DM and PM are exclusively utilized.
The interference signal can be rewritten using Bessel function as
I HSC ðx; y; t Þ ¼ E 2 T þ E 2 R þ 2E T E R cos / ðx; y; t
ÞfJ 0 ðmÞ
þ2 X 1 J 2kðmÞcos ð2kx m tÞg�2E T E R sin / ðx; y; tÞ k¼1
2 X 1 J 2k�1ðmÞsin fð2k � 1Þx m tg; ð4Þ k¼1
where J k ðmÞis the Bessel function of order k with modulation index m. It is revealed that the interference signal is a sum of harmonic waves. It should be noted that the amplitude of even-order harmonics includes cos / ðx; y; tÞ, while the amplitude of odd-order harmonics includes sin / ðx; y; tÞ. The phase / ðx; y; tÞ can be obtained using amplitudes of even and odd order harmonics.
To extract the sin /( x, y, t) and cos /( x, y, t) termsfrom the interference signal, a demodulation( phase-meter) algorithm is implemented in a personal computer( PC) connected to HSC. The division of the interference signal into harmonics from the 0th to 6th order at a given sampling rate is shown in Figure 2.( Note, in Fig. 2, that the amplitude of the harmonics is not proportional to the magnitude of the Bessel function.) In the demodulation algorithm, the interference signal is sampled at 12 equal intervals( 12 sampling points) in 1 modulation period( 2p / x m) synchronizing to the modulation frequency( as shown by black dots). The 0th sampling point is the zero-crossing point at which the first harmonic changes from negative to positive. The values at sampling points 0, 3T m / 12, 6T m / 12 and 9T m / 12 for 2nd harmonic correspond to peaks or valleys. Similarly, values at sampling points T m / 12, 3T m / 12, 5T m / 12, 7T m / 12, 9T m / 12 and 11T m / 12 for 3rd harmonic correspond to peaks or valleys. Therefore, cos /( x, y, t) andsin /( x, y, t) inequation( 4) can be calculated by adding and subtracting the peak and valley values of the second and third harmonics, respectively, using the following equations.
Figure 2. Sampling timing in phase demodulation algorithm. 4pnL
3 16AJ 2 ðmÞcos
¼ I ð0Þ�I k
12 T m
6 þI
12 T 9
m � I 12 T m I cos; ð5Þ
�24AJ 3 ðmÞsin 4pnL k
5 þI 12 T m � I
7 12 T m
þ I
¼ I
1 12 T m
9 12 T m
� I
3 12 T m
11 12 T m I sin:
� I ð6Þ