Real-time normalization apparatus and method of phase generated carrier demodulation in sinusoidal phase modulation interferometer

Abstract

The present invention discloses a real-time normalization apparatus and method of the PGC demodulation in a sinusoidal phase modulation interferometer. An optical setup containing a measuring interferometer and a monitoring interferometer is constructed. An electro-optic phase modulator is placed in the common reference arm of the two interferometers. High-frequency sinusoidal wave modulation and low-frequency triangular wave modulation are applied to the electro-optic phase modulator at the same time. Sinusoidal modulation is used for generating phase carrier, and PGC demodulation is performed to obtain quadrature signals containing the phase information to be measured. Triangular wave modulation makes the quadrature signals change periodically. Ellipse fitting is performed on the Lissajous figure corresponding to the quadrature signals, and real-time normalization of the PGC demodulated quadrature signals is achieved. By calculating the variation of the phase difference between the two interference signals, the measured displacement is obtained, and nanometer scale displacement measurement is achieved.

Claims

1. A real-time normalization apparatus of a phase generated carrier (PGC) demodulation in a sinusoidal phase modulation interferometer, comprising a single frequency laser, a first beam splitter, a second beam splitter, a third beam splitter, a fourth beam splitter, a reference retroreflector, a measuring retroreflector, an electro-optic phase modulator, a first photoelectric detector, a second photoelectric detector, a high voltage amplifier and a signal generator; a linearly polarized beam emitted from the single frequency laser is directed to the first beam splitter and divided into transmitted and reflected beams; the reflected beam from the first beam splitter is modulated by the electro-optic phase modulator and then projected onto the reference retroreflector; the reflected beam from the reference retroreflector is incident on the second beam splitter and divided into transmitted and reflected beams; the transmitted beam from the first beam splitter is incident on the third beam splitter and divided into transmitted and reflected beams; the transmitted beam from the third beam splitter is projected onto the measuring retroreflector; the reflected beam from the measuring retroreflector and the transmitted beam from the second beam splitter are merged at the first beam splitter to form the measurement interference signal, which is received by the first photodetector; the reflected beam from the third beam splitter and the reflected beam from the second beam splitter are merged at the fourth beam splitter to form the reference interference signal, which is received by the second photodetector; the reference retroreflector is fixed, and the measuring retroreflector is mounted on the object to be measured; the electro-optical phase modulator is connected to the signal generator through the high voltage amplifier, and a high-frequency sinusoidal wave voltage and a low-frequency triangular wave voltage output by the signal generator are amplified by the high voltage amplifier and then applied to the electro-optic phase modulator; the polarization direction of the linearly polarized beam emitted from the signal frequency laser is aligned with the direction of the electric field applied to the electro-optic phase modulator.

2. The real-time normalization apparatus of the PGC demodulation in the sinusoidal phase modulation interferometer according to claim 1, wherein the electro-optic phase modulator is placed between the first beam splitter and the reference retroreflector, and is used to modulate the reflected beam from the first beam splitter projected onto the reference retroreflector.

3. The real-time normalization apparatus of the PGC demodulation in the sinusoidal phase modulation interferometer according to claim 1, wherein the first beam splitter, the reference retroreflector and the measuring retroreflector constitute a measuring interferometer; the first beam splitter, the reference retroreflector, the second beam splitter, the third beam splitter and the fourth beam splitter constitute a monitoring interferometer; the first beam splitter, the electro-optic phase modulator, the reference retroreflector and the second beam splitter constitute a common reference arm of the measuring interferometer and the reference interferometer; the periodic high-frequency sinusoidal wave voltage and the low-frequency triangular wave voltage output by the signal generator are applied to the electro-optic phase modulator after being amplified by the high voltage amplifier, and an optical path difference of the common reference arm is modulated by changing a refractive index of electro-optic crystal of the electro-optic phase modulator, a high-frequency sinusoidal phase modulation and a low-frequency triangular wave phase modulation of the measuring interferometer and the monitoring interferometer are realized.

4. A displacement measurement method based on the real-time normalization of the PGC demodulation in the sinusoidal phase modulation interferometer applied to the apparatus according to claim 1, comprising 1) performing identical real-time normalization of PGC demodulation on the measuring interference signal and the monitoring interference signal detected by the apparatus, and obtaining demodulated phase values of the measuring and the monitoring interference signals respectively; 2) when the measuring retroreflector is moving, the variation of the phase difference between the measuring interference signal and the monitoring interference signal are calculated, and the measured displacement is given using the following equation: $L=\frac{\left(t\right)}{2}\frac{}{2}$ where is a wavelength of the linearly polarized beam emitted from the single frequency laser; and the measured displacement L serves as a movement displacement of the measuring retroreflector.

5. The displacement measurement method based on the real-time normalization of the PGC demodulation in a sinusoidal phase modulation interferometer according to claim 4, wherein in the step 1), the real-time normalization of the PGC demodulation is realized by applying the high-frequency sinusoidal phase modulation to generate a phase carrier signal for performing the PGC demodulation: 1.1) the interference signal is multiplied by a high-frequency sinusoidal modulation signal and its double-frequency signal respectively, passed through two low-pass filters, the outputs of the filters are divided by Bessel function values corresponding to a theoretical value z of phase modulation depth, and then a pair of quadrature signals containing the interference phase information are obtained, 1.2) performing ellipse fitting on a Lissajous figure corresponding to quadrature signals, values of a major axis and a minor axis of an ellipse are measured in real time, which serve as two normalization coefficients of the quadrature signals; and 1.3) normalization the pair of quadrature signals obtained in the step 1.2) by using two normalization coefficients obtained in the step 1.2), the phase of interferometer is obtained after performing division and arctangent operations on the normalized quadrature signals.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is the schematic diagram of the apparatus of the present invention.

(2) FIG. 2 is the schematic diagram of the real-time normalization of the PGC demodulation in the present invention.

(3) In the figures, 1: single frequency laser; 2: first beam splitter; 3: second beam splitter; 4: third beam splitter; 5: fourth beam splitter; 6: reference retroreflector; 7: measuring retroreflector; 8: electro-optic phase modulator; 9: high voltage amplifier; 10: signal generator; 11: first photoelectric detector; 12: second photoelectric detector; 13: interference signal; 14: sinusoidal modulation signal; 15: frequency multiplier; 16: multiplier; 17: multiplier; 18: low-pass filter; 19: low-pass filter; 20: divider; 21: divider; 22: ellipse fitting and normalization coefficient calculating module ; 23: quadrature signals real-time normalization module; 24: divider; and 25: arctangent operation.

DESCRIPTION OF THE EMBODIMENTS

(4) The present invention is further described in details hereinafter with the Figures and Embodiments.

(5) As shown in FIG. 1, the apparatus of the present invention comprises a single frequency laser 1, a first beam splitter 2, a second beam splitter 3, a third beam splitter 4, a fourth beam splitter 5, a reference retroreflector 6, a measuring retroreflector 7, an electro-optic phase modulator 8, a first photoelectric detector 11 and a second photoelectric detector 12. The linearly polarized beam with wavelength emitted from the single frequency laser 1 is incident on the first beam splitter 2 and divided into transmitted and reflected beams. After being modulated by the electro-optic phase modulator 8, the reflected beam from the first beam splitter 2 is projected onto the reference retroreflector 6. Then the reflected beam from the reference retroreflector 6 is incident on the second beam splitter 3 and divided into transmitted and reflected beams. The transmitted beam from the first beam splitter 2 is incident on the third beam splitter 4 and divided into transmitted and reflected beams. The transmitted beam from the third beam splitter 4 is projected onto the measuring retroreflector 7. The reflected beam from the measuring retroreflector 7 and the transmitted beam from the second beam splitter 3 are merged at the first beam splitter 2 to form the measurement interference signal, which is received by the first photodetector 11. The reflected beam from the third beam splitter 4 and the reflected beam from the second beam splitter 3 are merged at the fourth beam splitter 5 to form the reference interference signal, which is received by the second photodetector 12. The reference retroreflector 6 is fixed, and the measuring retroreflector 7 is mounted on the object to be measured. The measuring retroreflector 7 and object move together, and the displacement of the measuring retroreflector 7 represents the displacement of the object.

(6) The high-frequency sinusoidal wave and a low-frequency triangular wave output by the signal generator 10 are amplified by the high voltage amplifier 9 and then applied to the electro-optic phase modulator 8. The electro-optical phase modulator 8 is placed in the common reference arm of the measuring interferometer and a monitoring interferometer. And the polarization direction of the linearly polarized beam emitted from the single frequency laser 1 is aligned with the direction of the electric field applied to the electro-optic phase modulator 8.

(7) The specific implementation for the present invention includes the following steps:

(8) The modulation voltage signal is applied to the electro-optic phase modulator 8:
V(t)=(V.sub..sub.1(t)+V.sub..sub.2(t))=[A cos .sub.1t+Btri(.sub.2t)](1)
where is the amplification factor of the high voltage amplifier 9, V.sub..sub.1(t)=A cos .sub.1t and V.sub..sub.2(t)=Btri(.sub.2t) are the sinusoidal wave and the triangular wave output from the signal generator 10, respectively; A and .sub.1 are the amplitude and the angle frequency of the sinusoidal modulation signal, respectively; B and .sub.2 are the amplitude and the angle frequency of the triangular wave modulation signal, respectively; and t is time.

(9) tri(.sub.2t) is a unit triangular wave function, which is expressed as:

(10) $\begin{array}{cc}\mathrm{tri}\left({}_{2}t\right)=1-2\frac{.Math.{}_{2}t-2n.Math.}{},-\frac{}{{}_2}+\frac{2n}{{}_2}t\frac{}{{}_2}+\frac{2n}{{}_2},nZ& \left(2\right)\end{array}$
where n is the period number of the triangular wave, and Z represents the set of integers.

(11) Then the phase change .sub.EOM caused by the electro-optic phase modulator 8 is:

(12) $\begin{array}{cc}{}_{\mathrm{EOM}}=\frac{A}{V_{}}\cos {}_{1}t+\frac{B}{V_{}}\mathrm{tri}\left({}_{2}t\right)& \left(3\right)\end{array}$
where V.sub. is the half-wave voltage of the electro-optic phase modulator 8.

(13) The linearly polarized beam with wavelength emitted from the single frequency laser 1 is projected onto the measuring interferometer composed of the first beam splitter 2, the reference retroreflector 6 and the measuring retroreflector 7 and the monitoring interferometer composed of the first beam splitter 2, the reference retroreflector 6, the second beam splitter 3, the third beam splitter 4 and the fourth beam splitter 5. The measuring interference signal and the reference interference signal are respectively formed, and are respectively received by the first photoelectric detector 11 and the second photoelectric detector 12.

(14) Denoting the interferometer composed of the first beam splitter 2, the reference retroreflector 6 and the measuring retroreflector 7 as the measuring interferometer, denoting the interferometer composed of the first beam splitter 2, the reference retroreflector 6, the second beam splitter 3, the third beam splitter 4 and the fourth beam splitter 5 as the monitoring interferometer, and denoting l.sub.1 and l.sub.2 as the initial optical path differences between the two arms of the measuring interferometer and the monitoring interferometer before the electro-optical phase modulator 8 is applied to voltage modulation, respectively, the measuring interference signal S.sub.1(t) and a monitoring interference signal S.sub.2(t) received by the first photoelectric detector 11 and the second photoelectric detector 12 are respectively shown as follows:

(15) $\begin{array}{cc}\begin{array}{c}{S}_1\left(t\right)=S_{01}+S_{11}\cos \left\{\frac{A}{V_{}}\cos \left({}_{1}t-{}_1\right)+\frac{B}{V_{}}\mathrm{tri}\left({}_{2}t\right)+\frac{2l_1}{}\right\}\\ =S_{01}+S_{11}\cos \left[z\cos \left({}_{1}t-{}_1\right)+{}_1\left(t\right)\right]\end{array}& \left(4\right)\\ \begin{array}{c}{S}_2\left(t\right)=S_{02}+S_{12}\cos \left\{\frac{A}{V_{}}\cos \left({}_{1}t-{}_2\right)+\frac{B}{V_{}}\mathrm{tri}\left({}_{2}t\right)+\frac{2l_2}{}\right\}\\ =S_{02}+S_{12}\cos \left[z\cos \left({}_{1}t-{}_2\right)+{}_2\left(t\right)\right]\end{array}& \left(5\right)\end{array}$
where S.sub.01 and S.sub.02 are the amplitudes of the direct current components of the measuring interference signal and the reference interference signal, respectively; S.sub.11 and S.sub.12 are the amplitudes of alternating current components of the measuring interference signal and the reference interference signal, respectively; is the wavelength of the linearly polarized beam emitted from the single frequency laser 1; z is the sinusoidal phase modulation depth and z=A/V.sub.; .sub.1 and .sub.2 are the carrier phase delays corresponding to the measuring interferometer and the monitoring interferometer, respectively; and .sub.1(t) and .sub.2(t) are the phases to be demodulated corresponding to the measuring interferometer and the monitoring interferometer, respectively.

(16) $\begin{array}{cc}{}_1\left(t\right)=\frac{B}{V_{}}\mathrm{tri}\left({}_{2}t\right)+\frac{2l_1}{}& \left(6\right)\\ {}_2\left(t\right)=\frac{B}{V_{}}\mathrm{tri}\left({}_{2}t\right)+\frac{2l_2}{}& \left(7\right)\end{array}$

(17) As can be seen from the Eqs. (6) and (7), the phases of the two interferometers periodically change with the modulation of the triangular wave signal tri(.sub.2t).

(18) In order to demodulate the phases .sub.1(t) and .sub.2(t) corresponding to the measuring interferometer and the monitoring interferometer, the interference signal S.sub.i(t) (i=1, 2, wherein i=1 represents the measuring interference signal, and i=2 represents the monitoring interference signal) is expanded into the following form:

(19) $\begin{array}{cc}{S}_i\left(t\right)=S_{0i}+S_{1i}\cos {}_i\left(t\right)\left[J_0\left(z\right)+2\underset{m=1}{\overset{}{.Math.}}{\left(-1\right)}^m{J}_{2m}\left(z\right)\cos 2m\left({}_{1}t-{}_i\right)\right]+S_{1i}\sin {}_i\left(t\right)\left[-2\underset{m=1}{\overset{}{.Math.}}{\left(-1\right)}^m{J}_{2m-1}\left(z\right)\cos \left(2m-1\right)\left({}_{1}t-{}_i\right)\right],& \left(8\right)\end{array}$
where S.sub.0i is the amplitude of the direct current component of the measuring interference signal or the reference interference signal; S.sub.1i is the amplitude of the alternating current component of the measuring interference signal or the reference interference signal; J.sub.(2m1)(z) and J.sub.2m(z) denote the odd- and even-order Bessel functions, respectively; and J.sub.0(z) is the zero order Bessel function.

(20) The principle of the real-time normalization of the PGC demodulation is shown in FIG. 2. With the multiplier 16 and the multiplier 17, the interference signal 13 is respectively multiplied by the sinusoidal modulation signal 14 and its double-frequency signal output by the frequency multiplier 15. The outputs of multiplier 16 and multiplier 17 are respectively filtered with two lower-pass filters 18 and 19. The outputs of filters 18 and 19 are respectively divided by the first-order Bessel function value J.sub.1(z) and the second-order Bessel function value J.sub.2(z) through the divider 20 and the divider 21. The J.sub.1(z) and J.sub.2(z) are calculated according to the theoretical value z of the phase modulation depth. Then, a pair of quadrature signals containing phase information of the interferometer are obtained as:

(21) $\begin{array}{cc}{P}_{1i}\left(t\right)=\frac{\mathrm{LPF}\left[S_i\left(t\right).Math.V_{{}_1}\left(t\right)\right]}{J_1\left(z^{}\right)}=-\frac{J_1\left(z\right)}{J_1\left(z^{}\right)}{K}_{1i}{S}_{1i}A\cos {}_i\sin {}_i\left(t\right)& \left(9\right)\\ P_{2i}\left(t\right)=\frac{\mathrm{LPF}\left[S_i\left(t\right).Math.V_{2{}_1}\left(t\right)\right]}{J_2\left(z^{}\right)}=-\frac{J_2\left(z\right)}{J_2\left(z^{}\right)}{K}_{2i}{S}_{1i}A\cos 2{}_i\cos {}_i\left(t\right)& \left(10\right)\end{array}$
where P.sub.1i(t) and P.sub.2i(t) are respectively the sine item and the cosine item of the quadrature signals, K.sub.1i is the total gain of the multiplier 16 and the low-pass filer 18, K.sub.2i (i=1,2) is the total gain of the multiplier 17 and the low-pass filter 19, LPF[ ] means low-pass filtering, V.sub.1(t) is the sinusoidal modulation signal, V.sub.2(t) is the double-frequency signal of the sinusoidal modulation signal, and S.sub.1i (i=1,2, wherein i=1 represents the measuring interference signal, and i=2 represents the monitoring interference signal) shows the alternating current component amplitude of the interference signal.

(22) It can be seen that due to the difference (zz) between the theoretical value z and the actual value z of the phase modulation depth, unequal gain or gain change (K.sub.1iK.sub.2i) of the multiplier and the low-pass filter and the carrier phase delay .sub.i, the amplitude values of the sinusoidal signals are unequal. This will result in a nonlinear error during phase demodulation.

(23) As P.sub.1i(t) and P.sub.2i(t) change periodically with triangular wave modulation, ellipse fitting is performed on Lissajous figures of P.sub.1i(t) and P.sub.2i(t) through the ellipse fitting and normalization coefficient calculating module 22, the major axis .sub.1i=J.sub.1(z)K.sub.1iS.sub.1iA cos .sub.i/J.sub.1(z) and the minor axis .sub.2i=J.sub.2(z)K.sub.2iS.sub.1iA cos 2.sub.i/J.sub.2(z) of the ellipse can be obtained, and .sub.1i and .sub.2i serve as the normalization coefficients of the quadrature signals in PGC demodulation.

(24) The quadrature signals real-time normalization module 23 is utilized for performing normalization correction on P.sub.1i(t) and P.sub.2i(t) to obtain:

(25) $\begin{array}{cc}{Q}_{1i}\left(t\right)=\frac{P_{1i}\left(t\right)}{{}_{1i}}=\sin {}_i\left(t\right)& \left(11\right)\\ Q_{2i}\left(t\right)=\frac{P_{2i}\left(t\right)}{{}_{2i}}=\cos {}_i\left(t\right)& \left(12\right)\end{array}$

(26) It can be seen that, the Lissajous figure corresponding to the corrected quadrature signals is a unit circle, and the nonlinear error of phase demodulation is eliminated. By using the divider 24 and the arctangent operation 25, the phases of the measuring and reference interferometers are obtained with Q.sub.1i(t) and Q.sub.2i(t):

(27) $\begin{array}{cc}{}_i\left(t\right)=\arctan \frac{Q_{1i}\left(t\right)}{Q_{2i}\left(t\right)}& \left(13\right)\end{array}$

(28) According to Eqs. (6) and (7), the initial phase difference (t) between the two interference signals is:

(29) 0$\begin{array}{cc}\left(t\right)={}_1\left(t\right)-{}_2\left(t\right)=\frac{2}{}\left(l_1-l_2\right)& \left(14\right)\end{array}$

(30) As can be seen from the Eq. (14), the phase modulation of the interference signals caused by triangular wave modulation is counteracted, and the phase difference between the two interference signals is only relevant to the initial optical path difference between the two arms of the two interferometers.

(31) When the movement displacement of the measuring retroreflector 7 is L, l.sub.1 is changed into l.sub.1+2L. The phase difference (t) between the two interference signals becomes:

(32) $\begin{array}{cc}{}^{}\left(t\right)={}_1\left(t\right)-{}_2\left(t\right)=\frac{2}{}\left(l_1+2L-l_2\right)& \left(15\right)\end{array}$

(33) Denoting (t)=(t)(t), according to the Eqs. (14) and (15), the movement displacement L of the measuring retroreflector 7 is obtained as:

(34) $\begin{array}{cc}L=\frac{\left(t\right)}{2}\frac{}{2}& \left(16\right)\end{array}$

(35) In the embodiment of the present invention, the laser source is the single frequency HeNe stabilized laser with the model of XL80 made by Renishaw Company from England, which emits a linearly polarized beam. And the laser wavelength is typically 632.990577 nm. The accuracy of PGC phase demodulation can reach 0.2 by performing the real-time normalization. Thereby, by substituting these typical values into Eq. (16), the obtained displacement measuring accuracy is 0.18 nm.

(36) As can be seen from the embodiment, the present invention realizes real-time normalization of the PGC demodulation. The nonlinear error of phase demodulation caused by changes of the phase modulation depth and the phase delay, unequal gain or gain change of the multiplier and the filter, etc. can be eliminated and the sub-nanometer displacement measurement accuracy has realized. In addition, the present invention is applicable to both the relative displacement measurement and the absolute distance measurement. The present invention has the advantages of wide application range, simple optical configuration, convenient to use and remarkable technical effects.