- Email: [email protected]
Two-wavelength phase-shifting interferometry using an electrically addressed liquid crystal spatial light modulator
Optics Communications 242 (2004) 1–6 www.elsevier.com/locate/optcom
Two-wavelength phase-shifting interferometry using an electrically addressed liquid crystal spatial light modulator Youichi Bitou
*
National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 3, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8563, Japan Received 9 April 2004; received in revised form 14 July 2004; accepted 23 July 2004
Abstract A two-wavelength moire´ phase-shifting interferometer using a liquid crystal spatial light modulator is developed. The optical phase shifts for the two wavelengths are simultaneously given by a digital phase shift of the grating displayed on the spatial light modulator. A phase shift of the moire´ fringe is achieved by equal phase shifts with opposite signs in diffracted beams with opposite diffraction orders. A moire´ phase-shifting interferometer with no moving parts and no requirement for calibration of the value of the phase shift was realized. Our experimental result shows measurements of the profile of a step object with a 12.0-lm synthetic wavelength. 2004 Elsevier B.V. All rights reserved. PACS: 07.60.Ly; 42.30.Ms Keywords: Moire´ interferometry; Phase shift; Spatial light modulator
1. Introduction Phase shifting interferometry (PSI) has been widely employed for optical testing [1]. Using this technique, we can obtain the surface profiles of test objects with an accuracy better than 1/100 wavelength. However, the phase distribution across the interferogram is measured modulo 2p in PSI. This problem sets a limit to the phase *
Tel.: +81 298 614 030; fax: +81 298 614 080. E-mail address: [email protected].
measurement range of single-wavelength PSI. The use of a long wavelength K = k1k2/jk1 k2j synthesized from two wavelengths k1 and k2 provides a solution to this ambiguous phase problem. The phase-measurement range is extended to a longer synthetic wavelength K in two-wavelength interferometry, in which there are many kinds of phase-measurement methods. In previous researches, two-wavelength PSI was carried out by subtracting phase measurements made at two wavelengths k1 and k2 [2,3]. The interferograms at each wavelength were sequentially measured
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.07.052
2
Y. Bitou / Optics Communications 242 (2004) 1–6
with the phase shifts given by a piezoelectric transducer (PZT). To directly obtain the phase distribution at the synthetic wavelength K, Polhemus presented a nonlinear processing method on the video signal of two superimposed interferometric fringe patterns taken with two different wavelengths [4]. On the other hand, several types of heterodyne processing methods have been proposed for point measurement systems [5–8]. Two-wavelength moire´ PSI with two-wavelength changes of dual laser diodes (LDs) was developed by Ishii and Onodera [9,10]. The phase shift of two-wavelength moire´ is realized from equal phase shifts in opposite directions by separately varying the currents in two LDs on an unbalanced interferometer. In this method, the phase shifts for two interferometric fringe patterns are simultaneously given and there are no moving parts. The phase distribution at the synthetic wavelength K can be calculated by the same phase-step algorithms as for single-wavelength PSI. However, the value of the phase shift given by the LD depends on the optical path difference of the unbalanced interferometer and wavelength. We have to calibrate or monitor the values of the phase shifts in each optical setup. Furthermore, an appropriate phase-step algorithm is required to compensate the intensity changes in interferograms associated with the current variations in the two LD sources [11]. In this paper, we propose a two-wavelength moire´ PSI using an electrically addressed spatial light modulator (EA-SLM). In the previous work, we developed a single-wavelength PSI using the EA-SLM [12]. In the developed setup, the EASLM displays a binary grating pattern and acts as the phase shifter. The grating pattern displayed on the EA-SLM is controlled by a computer and the optical phase shift in the interferometer is accurately given by the digital phase shift of the displayed grating. By using this phase shifting scheme, unlike in the PSI with the PZT, the PSI with no moving parts can be constructed and the calibration for the value of the optical phase shift is not necessary. In addition, the value of the optical phase shift given by this scheme does not depend on the wavelength of the light source. In this work, this single-wavelength PSI using the
EA-SLM is extended to the two-wavelength moire´ PSI. Equal phase shifts in opposite directions for the two wavelengths can be realized by diffracting the beams into +1 and 1 orders, respectively.
2. Proposed method Fig. 1 shows the optical setup of the proposed two-wavelength moire´ PSI using the EA-SLM. The basic setup shown in Fig. 1 is similar to that in [12]. Two linearly polarized laser beams with wavelengths k1 and k2 were incident upon Wollaston prisms (WP1 and WP2) which divided each beam into two orthogonally (horizontally and vertically) polarized beams, respectively. The power
λ2 λ/2 WP 2 LCD-coupled PAL-SLM
Gr at
L2
ing
BS1
λ1
λ1
L1
λ/2
λ2 +1 and -1 order diffracted beams
WP 1 L3 L4 L5
PH
CCD camera
L7
L6
A
BS2 Object
PBS Reference mirror
Fig. 1. Schematic setup for the two-wavelength moire´ PSI using the LCD coupled PAL-SLM. WP, Wollaston prism; L, lens; BS, beam splitter; PH, pinhole; PBS, polarizing beam splitter; A, analyzer.
Y. Bitou / Optics Communications 242 (2004) 1–6
ratios between the two orthogonally polarized beams were, respectively, controlled by half-wave (k/2) plates placed in front of the WPs. The spatially separated and orthogonally polarized beams were incident upon the lenses L1 and L2, respectively. After the lenses, the two sets of orthogonally polarized beams were combined on the beam splitter BS1 such that the two vertically polarized beams were spatially overlapped and copropagating and the two horizontally polarized beams were crossed and focused at the position of the EA-SLM. In the setup described here, a liquid crystal display (LCD) coupled parallel-aligned nematic liquid crystal spatial light modulator (PAL-SLM, Hamamatsu X7550) was used as the EA-SLM [13]. In the unit, the electrically addressed SLM (LCD) and the optically addressed SLM (PALSLM) are coupled. The displayed pattern on the LCD is optically written in the PAL-SLM and the readout pattern (reflected pattern) from the PAL-SLM is nonpixelized. The PAL-SLM can be used as the reflection-type phase-only spatial light modulator when the polarization direction of the readout light is parallel to the liquid crystal molecule directions, while there is no modulation effect when the polarization direction of the readout light is perpendicular to the liquid crystal molecule directions. The LCD-coupled PAL-SLM has the programmable capability of an array of 640 · 480 pixels and 256 phase modulation levels. The linearized transfer characteristics of the phase modulation of more than 3p are obtained for 633 nm light in the LCD-coupled PAL-SLM. The LCD-coupled PAL-SLM displayed a binary grating pattern created by a computer. The orientation direction of the liquid crystal molecules was set to be horizontal. Consequently, the PAL-SLM acts as the reflective binary phase grating for the horizontally polarized beam and only as the mirror for the vertically polarized beam. In the setup, the crossing angles between the horizontally and vertically polarized beams were chosen to match the diffraction angles of +1 order for k1 and 1 order for k2, respectively. The crossing angles were adjusted according to the positions of L1 and WP1, and L2 and WP2, respectively. Therefore, after the PAL-SLM, the two reflected
3
vertically polarized beams, the +1 order diffracted horizontally polarized beam with wavelength k1, and the 1 order diffracted horizontally polarized beam with wavelength k2, copropagated. These four beams were collimated by the lenses L3 and L4, and their beam diameters were reduced. Then the four beams were focused onto a pinhole (PH) with a diameter of 25 lm by the objective lens L5 with a numerical aperture of 0.25 and recollimated by lens L6 with a focal length of 20 cm. By use of the PH, other order diffraction beams were eliminated and uniform phase and intensity distributions could be obtained in all four beams. Simultaneously, in the two diffracted horizontally polarized beams, the phase distributions caused by the nonuniformity of the displayed binary phase grating were averaged by the PH. The four recollimated beams were introduced into a polarization interferometer equipped with a polarizing beam splitter (PBS) and beam splitter BS2, as shown in Fig. 1. In the experiment described here, the diffracted horizontally polarized beams were sent to a reference arm and the reflected vertically polarized beams were sent to an object arm. Shifting the phase of the binary phase grating displayed on the PAL-SLM, which was controlled by the computer, digitally shifted the optical phase of the reference beams. The two interference fringe patterns (moire´ pattern) between the reference mirror and the object surface were simultaneously observed after an analyzer A and taken by a charge-coupled device (CCD) camera and a 10-bit frame grabber. The object surface was imaged onto the CCD camera with lens L7. Theoretically, the two interference fringe patterns for k1 and k2 can be, respectively, written as: I 1 ¼ A1 þ B1 cos ½4pW ðx; y Þ=k1 þ U1 þ /G ;
ð1Þ
I 2 ¼ A2 þ B2 cos ½4pW ðx; y Þ=k2 þ U2 /G ;
ð2Þ
where A1 and A2 are the bias intensities, and B1 and B2 are the modulation intensities of the interferograms of k1 and k2, respectively, W(x,y) is the object height distribution, U1 and U2 are the phases due to the optical path differences between the horizontally and vertically polarized beams of k1 and k2, respectively, and /G is the phase given
4
Y. Bitou / Optics Communications 242 (2004) 1–6
by the grating. If the relations a = A1 + A2 and b/2 = B1 = B2 are assumed in Eqs. (1) and (2), respectively, the two-wavelength moire´ pattern i taken by the CCD camera is given as
PAL-SLM used in our experiments was better than k/2.
i ¼ a þ b cos ½2pW ðx; y Þ=C þ U
4. Experimental results and discussion
cos ½2pW ðx; y Þ=K þ h þ /G ;
ð3Þ
where C is an average wavelength, C = k1k2/ (k1 + k2), U = (U1 + U2)/2, and h = (U1 U2)/2. The condition of B1 = B2 can be achieved by adjusting the powers and the rotation angles of the k/2 plates for k1 and k2, respectively. The argument of the second cosine term of Eq. (3) is a function of the phase shift given by the grating and the synthetic wavelength K, whereas the first cosine term is independent of the phase shift. Therefore, we can calculate the phase distribution of the synthetic wavelength K and then obtain the object surface W(x,y) by the conventional phase-step algorithm.
3. Experimental setup In the experiment, we used a He–Ne laser with an operating wavelength k1 = 632.9 nm and a LD with an operating wavelength k2 = 668.1 nm. Therefore, the synthetic wavelength K is 12.0 lm. The wavelengths of k1 and k2 were measured using an optical wavelength/counter (Anritsu MF9630A). The separation angles of the Wollaston prisms were approximately 0.5 and the focal lengths of lenses L1 and L2 were 5 cm. To achieve a good mode-matching between the two wavelengths by the objective lens and the PH, the mode of the beam k2 was adjusted before the k/2 plate using a variable beam expander. The duty ratio of the displayed binary phase grating on the PAL-SLM was 50% and one period had an 8-pixel width. The spatial frequency of the binary phase grating was approximately 3.1 lp/mm. The sizes of the beam spots on the PAL-SLM were around 1.3 cm, and consequently there were approximately 40 fringes within the beam spots. The diffraction efficiencies of the first order diffraction beams were around 35%. The flatness of the
Figs. 2(a)–(d) show one set consisting of the four moire´ fringe patterns i1, i2, i3, and i4, respectively, observed when the object was a plane mirror. The same four fringe patterns correspond to additional optical phase shifts of 0, p/2, p, and 3p/2, respectively. These p/2 phase steps were achieved by 2-pixel width shifts in the binary phase grating. The accuracies of the p/2 phase steps in single-wavelength PSI were confirmed to be better than 17.4 mrad in [12]. The expected phase shifts of the moire´ fringes can be seen in Fig. 2. In comparing the intensity distributions between i1 and i3, or i2 and i4, we can confirm the p phase differences of the fringes due to the synthetic wavelength K. Fig. 3 shows the measurement result of a step height made by gauge blocks. The surfaces of the gauge blocks were sufficiently flat for the synthetic wavelength K of 12.0 lm and its height was measured to be 5.03 lm with an uncertainty of 0.3 lm by a coordinate measurement machine (CMM). The phase distribution for the synthetic wavelength K can be calculated from arctan[(i4 i2)/ (i1 i3)]. In this method, the phase distribution across the moire´ fringe is measured modulo p and the measurement range is less than K/2. The rms error in the phase distribution of flat parts of the gauge block surface was K/160. The measurement result of the step height was 4.94 lm and its deviation from the measurement result by the CMM is sufficiently smaller than the uncertainty of the CMM. The measurement could be limited by laser speckle generated in the interferometer although the speckles generated in the phase-shifting system are removed by the PH. However, in our experimental results, the speckles did not seem the main limitation of the resolution. Another limitation could be noise in the sensor system. On the other hand, the moire´ technique (MT) using incoherent light source and computer-generated gratings can avoid the speckle noise problem [14,15]. In addition, stability requirements of these MT are far less
Y. Bitou / Optics Communications 242 (2004) 1–6
5
Fig. 2. Four moire´ fringe patterns of the mirror surface observed under the additional phase shifts of: (a) 0, (b) p/2, (c) p, and (d) 3p/2.
2 µm
1mm 1mm Fig. 3. Three-dimensional phase map of the step-height object.
stringent than in techniques based on interference of coherent light. However, to obtain a high accuracy in the MT, a careful calibration before the measurement is required [14].
5. Conclusions We have developed a two-wavelength moire´ PSI that uses an EA-SLM. Phase shift of the moire´
fringe is achieved by equal phase shifts with opposite signs obtained by diffracting the beams into +1 and 1 orders. The optical phase shifts for the diffracted beams are simultaneously given by a digital phase shift of a grating displayed on the EA-SLM. In the setup, there is no mechanical actuator such as a PZT and the calibration of the value of the phase shift is not required. The experimental results show the p/2 phase steps of two-wavelength moire´ fringe and measurement of the profile of a step object with the 12.0 lm synthetic wavelength using four-step algorithm.
Acknowledgements The author thanks Dr. T. Takatsuji and Dr. T.R. Schibli for useful discussions.
6
Y. Bitou / Optics Communications 242 (2004) 1–6
References [1] K. Creath, in: E. Wolf (Ed.), Progress in Optics, vol. 26, North-Holland, Amsterdam, 1988. [2] Y.-Y. Cheng, J.C. Wyant, Appl. Opt. 23 (1984) 4539. [3] K. Creath, Appl. Opt. 26 (1987) 2810. [4] C. Polhemus, Appl. Opt. 12 (1973) 2071. [5] A.F. Fercher, H.Z. Hu, U. Vry, Appl. Opt. 24 (1985) 2181. [6] R. Dandliker, R. Thalmann, D. Prongue, Opt. Lett. 13 (1988) 339. [7] H.J. Tiziani, A. Rothe, N. Maier, Appl. Opt. 35 (1996) 3525.
[8] S. Yokoyama, J. Ohnishi, S. Iwasaki, K. Seta, H. Matsumoto, N. Suzuki, Meas. Sci. Technol. 10 (1999) 1233. [9] Y. Ishii, R. Onodera, Opt. Lett. 16 (1991) 1523. [10] R. Onodera, Y. Ishii, Opt. Commun. 137 (1997) 27. [11] R. Onodera, Y. Ishii, Appl. Opt. 33 (1994) 5052. [12] Y. Bitou, Opt. Lett. 28 (2003) 1576. [13] Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, T. Hara, Opt. Rev. 6 (1999) 339. [14] R.A. Martinez-Celorio, J.J.J. Dirckx, L. Marti-Lopez, F.G. Pena-Lecona, Rev. Sci. Instrum. 75 (2004) 492. [15] A. Asundi, Opt. Eng. 32 (1992) 107.
Two-wavelength phase-shifting interferometry using an electrically addressed liquid crystal spatial light modulator Youichi Bitou
*
National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 3, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8563, Japan Received 9 April 2004; received in revised form 14 July 2004; accepted 23 July 2004
Abstract A two-wavelength moire´ phase-shifting interferometer using a liquid crystal spatial light modulator is developed. The optical phase shifts for the two wavelengths are simultaneously given by a digital phase shift of the grating displayed on the spatial light modulator. A phase shift of the moire´ fringe is achieved by equal phase shifts with opposite signs in diffracted beams with opposite diffraction orders. A moire´ phase-shifting interferometer with no moving parts and no requirement for calibration of the value of the phase shift was realized. Our experimental result shows measurements of the profile of a step object with a 12.0-lm synthetic wavelength. 2004 Elsevier B.V. All rights reserved. PACS: 07.60.Ly; 42.30.Ms Keywords: Moire´ interferometry; Phase shift; Spatial light modulator
1. Introduction Phase shifting interferometry (PSI) has been widely employed for optical testing [1]. Using this technique, we can obtain the surface profiles of test objects with an accuracy better than 1/100 wavelength. However, the phase distribution across the interferogram is measured modulo 2p in PSI. This problem sets a limit to the phase *
Tel.: +81 298 614 030; fax: +81 298 614 080. E-mail address: [email protected].
measurement range of single-wavelength PSI. The use of a long wavelength K = k1k2/jk1 k2j synthesized from two wavelengths k1 and k2 provides a solution to this ambiguous phase problem. The phase-measurement range is extended to a longer synthetic wavelength K in two-wavelength interferometry, in which there are many kinds of phase-measurement methods. In previous researches, two-wavelength PSI was carried out by subtracting phase measurements made at two wavelengths k1 and k2 [2,3]. The interferograms at each wavelength were sequentially measured
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.07.052
2
Y. Bitou / Optics Communications 242 (2004) 1–6
with the phase shifts given by a piezoelectric transducer (PZT). To directly obtain the phase distribution at the synthetic wavelength K, Polhemus presented a nonlinear processing method on the video signal of two superimposed interferometric fringe patterns taken with two different wavelengths [4]. On the other hand, several types of heterodyne processing methods have been proposed for point measurement systems [5–8]. Two-wavelength moire´ PSI with two-wavelength changes of dual laser diodes (LDs) was developed by Ishii and Onodera [9,10]. The phase shift of two-wavelength moire´ is realized from equal phase shifts in opposite directions by separately varying the currents in two LDs on an unbalanced interferometer. In this method, the phase shifts for two interferometric fringe patterns are simultaneously given and there are no moving parts. The phase distribution at the synthetic wavelength K can be calculated by the same phase-step algorithms as for single-wavelength PSI. However, the value of the phase shift given by the LD depends on the optical path difference of the unbalanced interferometer and wavelength. We have to calibrate or monitor the values of the phase shifts in each optical setup. Furthermore, an appropriate phase-step algorithm is required to compensate the intensity changes in interferograms associated with the current variations in the two LD sources [11]. In this paper, we propose a two-wavelength moire´ PSI using an electrically addressed spatial light modulator (EA-SLM). In the previous work, we developed a single-wavelength PSI using the EA-SLM [12]. In the developed setup, the EASLM displays a binary grating pattern and acts as the phase shifter. The grating pattern displayed on the EA-SLM is controlled by a computer and the optical phase shift in the interferometer is accurately given by the digital phase shift of the displayed grating. By using this phase shifting scheme, unlike in the PSI with the PZT, the PSI with no moving parts can be constructed and the calibration for the value of the optical phase shift is not necessary. In addition, the value of the optical phase shift given by this scheme does not depend on the wavelength of the light source. In this work, this single-wavelength PSI using the
EA-SLM is extended to the two-wavelength moire´ PSI. Equal phase shifts in opposite directions for the two wavelengths can be realized by diffracting the beams into +1 and 1 orders, respectively.
2. Proposed method Fig. 1 shows the optical setup of the proposed two-wavelength moire´ PSI using the EA-SLM. The basic setup shown in Fig. 1 is similar to that in [12]. Two linearly polarized laser beams with wavelengths k1 and k2 were incident upon Wollaston prisms (WP1 and WP2) which divided each beam into two orthogonally (horizontally and vertically) polarized beams, respectively. The power
λ2 λ/2 WP 2 LCD-coupled PAL-SLM
Gr at
L2
ing
BS1
λ1
λ1
L1
λ/2
λ2 +1 and -1 order diffracted beams
WP 1 L3 L4 L5
PH
CCD camera
L7
L6
A
BS2 Object
PBS Reference mirror
Fig. 1. Schematic setup for the two-wavelength moire´ PSI using the LCD coupled PAL-SLM. WP, Wollaston prism; L, lens; BS, beam splitter; PH, pinhole; PBS, polarizing beam splitter; A, analyzer.
Y. Bitou / Optics Communications 242 (2004) 1–6
ratios between the two orthogonally polarized beams were, respectively, controlled by half-wave (k/2) plates placed in front of the WPs. The spatially separated and orthogonally polarized beams were incident upon the lenses L1 and L2, respectively. After the lenses, the two sets of orthogonally polarized beams were combined on the beam splitter BS1 such that the two vertically polarized beams were spatially overlapped and copropagating and the two horizontally polarized beams were crossed and focused at the position of the EA-SLM. In the setup described here, a liquid crystal display (LCD) coupled parallel-aligned nematic liquid crystal spatial light modulator (PAL-SLM, Hamamatsu X7550) was used as the EA-SLM [13]. In the unit, the electrically addressed SLM (LCD) and the optically addressed SLM (PALSLM) are coupled. The displayed pattern on the LCD is optically written in the PAL-SLM and the readout pattern (reflected pattern) from the PAL-SLM is nonpixelized. The PAL-SLM can be used as the reflection-type phase-only spatial light modulator when the polarization direction of the readout light is parallel to the liquid crystal molecule directions, while there is no modulation effect when the polarization direction of the readout light is perpendicular to the liquid crystal molecule directions. The LCD-coupled PAL-SLM has the programmable capability of an array of 640 · 480 pixels and 256 phase modulation levels. The linearized transfer characteristics of the phase modulation of more than 3p are obtained for 633 nm light in the LCD-coupled PAL-SLM. The LCD-coupled PAL-SLM displayed a binary grating pattern created by a computer. The orientation direction of the liquid crystal molecules was set to be horizontal. Consequently, the PAL-SLM acts as the reflective binary phase grating for the horizontally polarized beam and only as the mirror for the vertically polarized beam. In the setup, the crossing angles between the horizontally and vertically polarized beams were chosen to match the diffraction angles of +1 order for k1 and 1 order for k2, respectively. The crossing angles were adjusted according to the positions of L1 and WP1, and L2 and WP2, respectively. Therefore, after the PAL-SLM, the two reflected
3
vertically polarized beams, the +1 order diffracted horizontally polarized beam with wavelength k1, and the 1 order diffracted horizontally polarized beam with wavelength k2, copropagated. These four beams were collimated by the lenses L3 and L4, and their beam diameters were reduced. Then the four beams were focused onto a pinhole (PH) with a diameter of 25 lm by the objective lens L5 with a numerical aperture of 0.25 and recollimated by lens L6 with a focal length of 20 cm. By use of the PH, other order diffraction beams were eliminated and uniform phase and intensity distributions could be obtained in all four beams. Simultaneously, in the two diffracted horizontally polarized beams, the phase distributions caused by the nonuniformity of the displayed binary phase grating were averaged by the PH. The four recollimated beams were introduced into a polarization interferometer equipped with a polarizing beam splitter (PBS) and beam splitter BS2, as shown in Fig. 1. In the experiment described here, the diffracted horizontally polarized beams were sent to a reference arm and the reflected vertically polarized beams were sent to an object arm. Shifting the phase of the binary phase grating displayed on the PAL-SLM, which was controlled by the computer, digitally shifted the optical phase of the reference beams. The two interference fringe patterns (moire´ pattern) between the reference mirror and the object surface were simultaneously observed after an analyzer A and taken by a charge-coupled device (CCD) camera and a 10-bit frame grabber. The object surface was imaged onto the CCD camera with lens L7. Theoretically, the two interference fringe patterns for k1 and k2 can be, respectively, written as: I 1 ¼ A1 þ B1 cos ½4pW ðx; y Þ=k1 þ U1 þ /G ;
ð1Þ
I 2 ¼ A2 þ B2 cos ½4pW ðx; y Þ=k2 þ U2 /G ;
ð2Þ
where A1 and A2 are the bias intensities, and B1 and B2 are the modulation intensities of the interferograms of k1 and k2, respectively, W(x,y) is the object height distribution, U1 and U2 are the phases due to the optical path differences between the horizontally and vertically polarized beams of k1 and k2, respectively, and /G is the phase given
4
Y. Bitou / Optics Communications 242 (2004) 1–6
by the grating. If the relations a = A1 + A2 and b/2 = B1 = B2 are assumed in Eqs. (1) and (2), respectively, the two-wavelength moire´ pattern i taken by the CCD camera is given as
PAL-SLM used in our experiments was better than k/2.
i ¼ a þ b cos ½2pW ðx; y Þ=C þ U
4. Experimental results and discussion
cos ½2pW ðx; y Þ=K þ h þ /G ;
ð3Þ
where C is an average wavelength, C = k1k2/ (k1 + k2), U = (U1 + U2)/2, and h = (U1 U2)/2. The condition of B1 = B2 can be achieved by adjusting the powers and the rotation angles of the k/2 plates for k1 and k2, respectively. The argument of the second cosine term of Eq. (3) is a function of the phase shift given by the grating and the synthetic wavelength K, whereas the first cosine term is independent of the phase shift. Therefore, we can calculate the phase distribution of the synthetic wavelength K and then obtain the object surface W(x,y) by the conventional phase-step algorithm.
3. Experimental setup In the experiment, we used a He–Ne laser with an operating wavelength k1 = 632.9 nm and a LD with an operating wavelength k2 = 668.1 nm. Therefore, the synthetic wavelength K is 12.0 lm. The wavelengths of k1 and k2 were measured using an optical wavelength/counter (Anritsu MF9630A). The separation angles of the Wollaston prisms were approximately 0.5 and the focal lengths of lenses L1 and L2 were 5 cm. To achieve a good mode-matching between the two wavelengths by the objective lens and the PH, the mode of the beam k2 was adjusted before the k/2 plate using a variable beam expander. The duty ratio of the displayed binary phase grating on the PAL-SLM was 50% and one period had an 8-pixel width. The spatial frequency of the binary phase grating was approximately 3.1 lp/mm. The sizes of the beam spots on the PAL-SLM were around 1.3 cm, and consequently there were approximately 40 fringes within the beam spots. The diffraction efficiencies of the first order diffraction beams were around 35%. The flatness of the
Figs. 2(a)–(d) show one set consisting of the four moire´ fringe patterns i1, i2, i3, and i4, respectively, observed when the object was a plane mirror. The same four fringe patterns correspond to additional optical phase shifts of 0, p/2, p, and 3p/2, respectively. These p/2 phase steps were achieved by 2-pixel width shifts in the binary phase grating. The accuracies of the p/2 phase steps in single-wavelength PSI were confirmed to be better than 17.4 mrad in [12]. The expected phase shifts of the moire´ fringes can be seen in Fig. 2. In comparing the intensity distributions between i1 and i3, or i2 and i4, we can confirm the p phase differences of the fringes due to the synthetic wavelength K. Fig. 3 shows the measurement result of a step height made by gauge blocks. The surfaces of the gauge blocks were sufficiently flat for the synthetic wavelength K of 12.0 lm and its height was measured to be 5.03 lm with an uncertainty of 0.3 lm by a coordinate measurement machine (CMM). The phase distribution for the synthetic wavelength K can be calculated from arctan[(i4 i2)/ (i1 i3)]. In this method, the phase distribution across the moire´ fringe is measured modulo p and the measurement range is less than K/2. The rms error in the phase distribution of flat parts of the gauge block surface was K/160. The measurement result of the step height was 4.94 lm and its deviation from the measurement result by the CMM is sufficiently smaller than the uncertainty of the CMM. The measurement could be limited by laser speckle generated in the interferometer although the speckles generated in the phase-shifting system are removed by the PH. However, in our experimental results, the speckles did not seem the main limitation of the resolution. Another limitation could be noise in the sensor system. On the other hand, the moire´ technique (MT) using incoherent light source and computer-generated gratings can avoid the speckle noise problem [14,15]. In addition, stability requirements of these MT are far less
Y. Bitou / Optics Communications 242 (2004) 1–6
5
Fig. 2. Four moire´ fringe patterns of the mirror surface observed under the additional phase shifts of: (a) 0, (b) p/2, (c) p, and (d) 3p/2.
2 µm
1mm 1mm Fig. 3. Three-dimensional phase map of the step-height object.
stringent than in techniques based on interference of coherent light. However, to obtain a high accuracy in the MT, a careful calibration before the measurement is required [14].
5. Conclusions We have developed a two-wavelength moire´ PSI that uses an EA-SLM. Phase shift of the moire´
fringe is achieved by equal phase shifts with opposite signs obtained by diffracting the beams into +1 and 1 orders. The optical phase shifts for the diffracted beams are simultaneously given by a digital phase shift of a grating displayed on the EA-SLM. In the setup, there is no mechanical actuator such as a PZT and the calibration of the value of the phase shift is not required. The experimental results show the p/2 phase steps of two-wavelength moire´ fringe and measurement of the profile of a step object with the 12.0 lm synthetic wavelength using four-step algorithm.
Acknowledgements The author thanks Dr. T. Takatsuji and Dr. T.R. Schibli for useful discussions.
6
Y. Bitou / Optics Communications 242 (2004) 1–6
References [1] K. Creath, in: E. Wolf (Ed.), Progress in Optics, vol. 26, North-Holland, Amsterdam, 1988. [2] Y.-Y. Cheng, J.C. Wyant, Appl. Opt. 23 (1984) 4539. [3] K. Creath, Appl. Opt. 26 (1987) 2810. [4] C. Polhemus, Appl. Opt. 12 (1973) 2071. [5] A.F. Fercher, H.Z. Hu, U. Vry, Appl. Opt. 24 (1985) 2181. [6] R. Dandliker, R. Thalmann, D. Prongue, Opt. Lett. 13 (1988) 339. [7] H.J. Tiziani, A. Rothe, N. Maier, Appl. Opt. 35 (1996) 3525.
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