24
J. Eur. Opt. Society-Rapid Publ. 21, 32( 2025)
Table 1. Mixing into I cos and I sin from harmonics, when m ¼ 3:14 rad.
J k ðmÞ ð Þ % J k ðmÞ
Harmonic order k |
cos or sin |
J 2 m |
J 3 ðmÞ % |
Mixing into equation( 5) |
Mixing into equation( 6) |
0 |
cos 0 |
�62.576 |
�91.175 |
0 |
0 |
1 |
sin x m t |
58.756 |
85.609 |
0 |
0 |
2 |
cos 2x m t |
100.000 |
145.703 |
1 |
0 |
3 |
sin 3x m t |
68.633 |
100.000 |
0 |
1 |
4 |
cos 4x m t |
31.145 |
45.379 |
0 |
0 |
5 |
sin 5x m t |
10.718 |
15.616 |
0 |
0 |
6 |
cos 6x m t |
2.988 |
4.354 |
1 |
0 |
7 |
sin 7x m t |
0.702 |
1.023 |
0 |
0 |
8 |
cos 8x m t |
0.143 |
0.208 |
0 |
0 |
9 |
sin 9x m t |
0.026 |
0.037 |
0 |
�1 |
10 |
cos 10x m t |
0.004 |
0.006 |
1 |
0 |
In equations( 5) and( 6), only 2nd and 3rd harmonics( Bessel functions) are considered. If the modulation index m is larger than 3 rad, higher harmonics( higher Bessel functions) may be included in cos /( x, y, t) andsin /( x, y, t) terms. Table 1 shows which harmonics are mixed in the calculations of equations( 5) and( 6) at the modulation index m = 3.14 rad. In the table, from the left column, the harmonic order k, the target harmonic( sine or cosine), J k( m)/ J 2( m)( ratio of kth-order Bessel function to 2ndorder Bessel function), J k( m)/ J 3( m)( ratio of kth-order Bessel function to 3rd-order Bessel function), the harmonic mixedinequation( 5)( I cos)( 1: in-phase, 0: not mixed), and harmonics mixed in equation( 6)( I sin)( 0: not mixed, �1: in-phase). Table 1 shows that harmonics of the 0th, 1st, 4th, 5th, 7th, and 8th orders have no effect on the calculations of equations( 5) and( 6) for cos /( x, y, t) and sin /( x, y, t). In the cos /( x, y, t), the 2nd, 6th, and 10th harmonics are mixed in phase, and the ratio of the 6th and 2nd magnitudes is J 6( m)/ J 2( m) 2.988 %, which is not negligible. 10th harmonics are negligible with a similar ratio of J 10( m)/ J 2( m) 0.004 %. In the calculation of sin /( x, y, t) in equation( 6), the 9th harmonic is introduced in the opposite phase, but the same ratio of J 9( m)/ J 3( m) 0.037 % is negligible. Based on the above discussion, rewriting equation( 5) for a modulation index of m = 3.14 rad, we obtain
3 12 T 6
m þ I
12 T m
¼ 16AfJ 2 ðmÞþJ 6 ðmÞgcos
I cos ¼ I ð0Þ�I
� I
4pnL k
9 12 T m
; ð7Þ
When the modulation index m = 3.14 rad, we can obtain 16fJ 2 ðmÞþJ 6 ðmÞg ¼ 24J 3 ðmÞ, so that jI cos j ¼ jI sin j. Then we can illustrate a Lissajous diagram of a perfect circle using I cos and I sin as lateral and vertical axes. The phase / ðx; y; tÞ ¼ 4pnL can be calculated by arctangent k
of I cos and I sin. In the case of 2-D in-plane displacement measurement, interference signals are recorded for all pixels of HSC. However, due to the huge amount of image data, it is currently impossible to calculate and output the displacement of each pixel in real time. Most current HSCs store the
Figure 3. Block diagram of the comb filter.
captured video data in the camera’ s internal memory and transfer the video data to a PC or FPGA when the image capture is completed. A linear image sensor can transfer the output signals of each pixel to an analog-to-digital converter incorporated in a PC or an FPGA in real time [ 21 ], and phase( displacement) measurement with the linear image sensor is considered possible in real time. Similarly, if the output signal of each pixel of a HSC can be transferred in real time to an FPGA capable of large-scale, high-speed computation via a high-speed line such as an optical fiber [ 22 ], real-time measurement may be possible with the HSC. The calculations of the 2-D in-plane displacement of all pixels of the HSC used in the experiments are performed in post-processing.
However, it is possible to apply a pre-filter to the interference signal before calculations of equations( 6) and( 7), albeit in post-processing. To improve the resolution, we introduce the feedback type comb filter for pre-filtering the interference signal. Since the output of a SPM interferometer contains only the harmonics of the modulation signal, a comb filter that transmits only its harmonic components is employed as a prefilter. The block diagram of the comb filter is shown in Figure 3. Its Bode diagram with D = 12, K i = 0.9 and the cut-off frequency = 145 Hz is shown in Figure 4. The feedback type comb filter has sharp pass bands at 5 kHz, 10 kHz, 15 kHz,.... Dueto matching the pass bands and harmonics, we can decrease the noise included in the interference signal.
3 Experiment
The experimental setup is illustrated in Figure 5. A He-Ne laser( Spectra-Physics, 117A) is used as a light source. With