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An automated self-marking phase-shifting system for measuring the full field phase distribution of interference fringe patterns
Optics & Laser Technology 32 (2000) 225–230
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An automated self-marking phase-shifting system for measuring the full eld phase distribution of interference fringe patterns M.J. Huang ∗, L.Y. Chou Department of Mechanical Engineering, National Chung-Hsing University, Taichung, Taiwan 40227, ROC Received 20 December 1999; received in revised form 18 April 2000; accepted 2 May 2000
Abstract A technique based on the repetition of the interference fringe pattern for the automated marking of the PZT phase shifting interference system is described. A closed-loop control is used to circumvent the inherent hysteresis and nonlinear characteristics of PZT. The whole interference system integrated under LabVIEW graphic programming environment provides users a friendly and ease-of-maintenance interface. An application of the algorithm is used to demonstrate the promotion of the Zygo GPI-LC model interferometer to c 2000 Elsevier Science Ltd. All rights reserved. upgrade its phase resolution ability. Keywords: Phase-shifting technique; PZT; Optical wavefront measurement; Zygo interferometer; PZT calibration
1. Introduction Phase-shifting techniques [1– 4] are important for the interference fringe analysis to transfer interferogram into phase map. Among all the phase-shifting mechanism, the pieozo-electrical transducer (PZT) is the most easy and e ective actuator to do so. Since the movement of PZT phase shifter a ects directly the shifted interference pattern, its precision of movement becomes very critical. There inherently are two fundamental problems associated with the motion of PZT, which are the unknown sensitivity and the nonlinearity of it. Both problems will generate a kind of sinusoidal phase error. Therefore, Cheng and Wyant proposed a calibration-insensitive phase calculation algorithm [5] to circumvent this and later modeled the PZT displacement as a quadratic function [6] to analyze and eliminate the phase error caused by the PZT nonlinearity. Hariharan et al. [7] also proposed a simple error-compensating phase calculation algorithm to overcome the unknown sensitivity of PZT and any tilt introduced across a diameter in the response of the PZT. Creath and Hariharan [8] and Creath et al. [9] also pro∗ Corresponding author. Tel.: +886-4-285-1951; fax: +886-4287-7170. E-mail address: [email protected] (M.J. Huang).
posed algorithm for elimination of errors contributed by the phase-shifting reference wavefront. Besides, there are still problems related to other sources, e.g. the accuracy of the optical set-up angle. An innovative and ecient self-marking method that automatically marks the PZT-driven voltage of 2 phase is presented in this article. The automatic self-marking voltage obtained by this newly developed method has the advantage of not having to know the sensitivity of PZT. Furthermore, for being based on repetition of the interference fringe pattern, the optical set-up angle associated with this technique does not have to accurately align to the exact value as other method does.
2. Linearity control of PZT To overcome the inherent characteristics of the PZT phase shifter a closed-loop control is used in this article. Strain gage sensors are buried in the transducer to monitor the elongation of it and the signals are fed back to the piezoelectric system for linearity control. Several control algorithms have been successfully implemented on this system including PID, Fuzzy, SOC (Self-Organizing Controller) [10], and neutral-network techniques. Therefore, the PZT system in this work can be treated as a quasi-linear transducer.
c 2000 Elsevier Science Ltd. All rights reserved. 0030-3992/00/$ - see front matter PII: S 0 0 3 0 - 3 9 9 2 ( 0 0 ) 0 0 0 4 6 - 3
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M.J. Huang, L.Y. Chou / Optics & Laser Technology 32 (2000) 225–230
3. Self-marking system for PZT phase shifter As mentioned above, both the measurement error of incident angle of the reference mirror and the proportional coecient error of PZT driving actuator may induce signi cant error into the calculation of phase angle. Therefore, it becomes very important to precisely calibrate the PZT actuator and to accurately measure the angle of incident light on the reference, PZT-driving, mirror in the eld of phase-shifting technique. In this paper, we propose an innovative algorithm to circumvent this problem and successfully apply it to a phase-shifting system to perform the measurement of full- eld phase distribution. This newly developed system owns the advantage of not having to measure the actual angle between the incident light and the normal of the reference mirror; even not having to know the proportional coecient of PZT phase shifter also. This system can automatically level and mark the voltage needed for PZT actuator to introduce a 2 phase shift into the optical phase-shifting system as soon as the optical system has been set. This algorithm can be easily applied to various kinds of interference systems, like Michelson, Mach Zehnder, Twyman Green, etc., without any modi cation. Following is the explanation of this newly developed algorithm. In interferometry, it is not dicult to nd that the shifted interference fringe pattern will be identical with its original non-shifted fringe pattern if the introduced phase shift is an integer times of 2 phase. Furthermore, the induced phase shift is a linear function of the elongation of pieozo-electric transducer. Therefore, if the elongation of pieozo-electric transducer could be changed linearly and gradually, the induced phase shift of the phase shifter would be changed in correspondence. Owing to this characteristic, if we introduce a small increment of voltage step by step to the PZT phase shifter, it would provide linearly a correspondent phase shift to the interference system also step by step. Whenever PZT shift a small step, the shifted interference fringe pattern has to be recorded and be compared with its original non-shifted interference fringe pattern. When the shifted interference fringe pattern matches identically its original non-shifted interference fringe pattern, the phase change of the PZT phase shifter will de nitely be an integer number times of 2 phase. At this moment, the total voltage we have applied is the exact voltage needed for PZT phase shifter to introduce a 2 phase shift through it into the optical system. By setting the non-shifted interference fringe pattern as an initial fringe pattern for reference, each of the shifted interference fringe patterns associated with di erent amount of PZT shift is compared, respectively, with the original non-shifted fringe pattern. The comparison is carried out in the way of subtracting the intensity of the correspondence pixel of the two, shifted and original, interference fringe patterns rst, then take its root mean square value. The behavior can be represented in the formula
below: f(k) =
1 A
Z Z
k = 1; 2; 3 : : : ;
R
[I0 (x; y) − Ik (x; y)]2 d x dy
1=2
(1)
where R is the region of the interference fringe pattern, A is the total area of the interference fringe pattern for analysis, k is the number of unit voltage applied to the PZT, I0 is the intensity distribution of the original interference fringe pattern, and Ik is the kth interference fringe pattern under PZT driven by k units of unit voltage. Eq. (1) is the expression for comparison between two continuous interference fringe patterns. For experimental data, the interference fringe pattern grabbed from the CCD camera through image grabber card to computer for analyzing is discrete but not continuous. So Eq. (1) has to be modi ed to meet the realistic condition of our experiment. The modi ed equation is given as Eq. (2). )1=2 ( 1 P 2 [I0 (m; n) − Ik (m; n)] ; (2) f(k) = N (m; n)∈R where N is the total number of pixels of the interference fringe pattern, and m; n are the number of pixels of the horizontal and the vertical direction of the interference fringe pattern, respectively. According to the discussion above, the minimum value of the curve of RMS value vs. voltage applied occurs around the point where the introduced phase shift is equal to 2. Marking the voltage applied where the minimum RMS value occurs, therefore this is the voltage that has to be fed to the PZT for a 2 phase shift. When the PZT is under closed-loop control, the linearity of PZT movement is good and acceptable. Therefore, it is not dicult to
Fig. 1. Schematic drawing of the experimental setup for verifying “self-marking phase shifting system for full- eld phase measurement”.
M.J. Huang, L.Y. Chou / Optics & Laser Technology 32 (2000) 225–230
Fig. 2. Schematic diagram of the whole system.
get the applied voltage required for achieving the precise phase shift of PZT. For instance, suppose the applied voltage of the PZT for minimum RMS value is V , then the voltage needed for this PZT to purchase a phase shift will be V=2 and will be V=4 for a =2 phase shift.
4. Experiment and results The validity of the method has been examined in experiments using Zygo GPI-LC interferometer [11], which has been proved to be stable and reliable for optical at examination. The experimental set up of the interferometer is schematically shown in Fig. 1. In the interferometer a 4 in diameter, expanded and collimated low-power HeNe laser beam is used as the light source. A transmission element with =20 accuracy, mounted in front of the aperture, re ects some of the laser light back into the interferometer, thus creating a reference wavefront. The remainder of the laser light passes through the transmission element to the component – optical at being tested and is referred to as the measurement wavefront. The measurement wavefront re ects back to the interferometer. The phase di erences between the two wavefronts result in an image of light and dark fringe patterns that is a direct indication of the quality of the optical at being tested. The interference pattern is converted to electric signals by a solid-state video camera (CCD), and is grabbed through the image grabber card to a computer for analysis. The optical component being tested is a Newport 1 1 in diameter, 1=10 accuracy front mirror, model number 10D20BD.1. Due to its high re ectance an extra attenuation lter is added between the transmission
component and the test component to attenuate the light intensity. The Zygo GPI-LC interferometer used in this experiment is a Fizeau interferometer with ease of use and great accuracy characteristics. But the model is not the full version. This model can only display the interference pattern on the video monitor and without any phase resolution capacity. The readout of the interference fringe pattern depends strongly on the experience of the engineer. From automation and result-repetition point of view, this has to be avoided. For this reason and further to verify this newly developed algorithm, the phase-shifting capacity mentioned above has been added to a Zygo GPI-LC model interferometer to upgrade its phase resolution ability and to verify the validity of the method. The promotion of the Zygo GPI-LC model interferometer has been carried out as follows. First, link the image output terminal of the Zygo GPI-LC interferometer with a frame grabber card, model NI 2 IMAQ PCI-1408, to enable the digital image acquisition ability of the interference system. Then, add a phase-shifting mechanism to the Zygo GPI-LC interferometer to enhance its phase resolution capacity. In our experiment, we attach a computer-controlled PZT, model PI 3 P-821.10, to the tested mirror to act as the phase-shifting mechanism. The movement of the PZT is controlled by the computer-driven-signal; this signal is converted by a 16 bit D=A card, model NI (see footnote 2) PCI-MIO16XE-10, to the analog signal and fed to the PZT for precise phase shift. The whole phase-shifting hardware system was integrated systematically under the LabVIEW 4 graphic programming software from driving the phase-shifting mechanism, marking the voltage required for a 2 phase shift of PZT, acquiring the phase-shifted interference fringe pattern, solving the wrapped phase, unwrapping the wrapped phase map, analyzing the results, to even plotting the 3D phase distribution. The whole schematic diagram of the integrated system is shown in Fig. 2. To verify our self-marking algorithm of PZT, a discrete voltage series with 2.5 mV increment each was fed to the PZT gradually through the D=A card. Each time an increment of voltage has been sent to the PZT, the shifted interference fringe pattern after being stable was grabbed from image grabber to the computer and its RMS value was calculated. The result was plotted in Fig. 3(a) with the RMS value of the shifted interference fringe pattern as vertical axis and the number of unit voltage applied as horizontal axis. From the experimental data, except from the null-voltage applied point – the non-shifted original interference fringe pattern, the total voltage applied where the minimum RMS value occurred is the voltage needed for the PZT to achieve a 2 phase shift of the
2
National Instruments Corporation Austin, TX, USA. Physik Instrumente (PI) GmbH & Co Germany. 4 LabVIEW, Graphical programming National Instruments Corporation Austin, TX, USA. 3
1
Newport Corporation, CA, USA.
227
228
M.J. Huang, L.Y. Chou / Optics & Laser Technology 32 (2000) 225–230
Fig. 3. Averaged RMS phase error vs. voltage applied to PZT for phase stepping (each with 2.5 mV increment) (a) – (c). The same experiment has been repeated for three times, the repetition is good and the minimum point corresponds almost around the same voltage applied.
interference system. The same experiment was repeated at least for three times. The results are shown in Fig. 3(a) – (c). It is evident that the repetition of the experiment is good and the algorithm can be really treated as an easy and powerful way to nd the voltage required for a 2 phase shift of the PZT phase-shifting system. Since the error associated with ve-frame technology is less serious in comparison with three- and four-frame technology, ve-frame technology is used here. Therefore, ve voltages of PZT associated, respectively, with 0, 90, 180, 270 ◦ and 360 phase shift were automatically calculated by our system and fed also automatically to the PZT to achieve
the phase shifting. After phase stepping, the associated ve PZT-driven phase shifting interference fringe patterns were grabbed from image grabber to the analyzing computer. The wrapped phase map is then calculated and the unwrapped phase map by cellular-automata method [12] are shown in Fig. 4. The unwrapped phase map is then curve tted by a plane in least-square means. The results, shown in Fig. 5, are expressed in the form of di erences between the unwrapped phase distribution and the curve- tted plane with optical wavelength (632:8 nm) as the unit. The maximum and minimum height between the measured and the tted plane is 0.036 and −0:024,
M.J. Huang, L.Y. Chou / Optics & Laser Technology 32 (2000) 225–230
229
Fig. 6. 3D plot of the experimental wavefront.
Fig. 4. Wrapped and unwrapped phase map of the experiment. Fig. 7. Contour plot of the experimental result with 0:01 increment each.
5. Conclusions and discussion
Fig. 5. Unwrapped phase map expressed in di erences with a least-squares tted plane in unit .
that is to say, the total peak-to-valley of the mirror being tested is 0:06. The 3D distribution of the tested mirror is plotted in Fig. 6 and its contour plot is also shown in Fig. 7 with each of 0:01 increment one from another.
An easy, precise, e ective, and automatic technique for marking the PZT phase-shifting system has been successfully developed and veri ed. The voltage required, for the PZT, to achieve a 2 phase shift to the associated interference system is implemented by locating the point of the minimum RMS value vs. voltage applied. The newly proposed PZT self-marking algorithm circumvents the determination of the sensitivity of PZT material as well as the angle formed between lights of PZT incident on and re ected from it. The algorithm can be applied directly to other kind of interference system without much modi cation. Experimental results show that this method is e ective as an accurate and precise calibration of the PZT phase-shifting system. The ultimate resolution of the method is limited by the spatial resolution of the CCD camera as well as by the numbers of bit of the D=A card that used to drive the PZT phase shifter. The technique is especially suitable for PZT under closed-loop control for its hysteresis and nonlinearity
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repression characteristics. The repetition of the experimental results depend strongly on the precise control of the environmental conditions such as temperature, pressure, humidity, vibration, air ow, etc., all parameters that are related directly or indirectly with the refractive index, as the same demands for all other interference systems. Acknowledgements This research was performed at the Holographic Laboratory of National Chung-Hsing University, supported by the ROC National Science Council Contract No. NSC87-2212E-005-019 and NSC88-2212-E-005-006. References [1] Bruning JH, Herriott DR, Gallagher JE, Rosenfeld DP, White AD, Brangaccio DJ. Digital wavefront measuring interferometer for testing optical surfaces and lenses. Appl Opt 1974;13:2693–703. [2] Wyant JC. Use of an AC heterrodyne lateral shear interferometer with real-time wavefront corection systems. Appl Opt 1975;14: 2622–6.
[3] Frantz LM, Sawchuk AA, von der Ohe W. Optical phase measurement in real time. Appl Opt 1979;18:3301–6. [4] Hariharan P, Oreb BF, Brown N. Digital phase-measurement system for real-time holographic interferometry. Opt Commun 1982;41:393–6. [5] Cheng YY, Wyant JC. Phase shifter calibration in phase-shifting interferometry. Appl Opt 1985;24:3049–52. [6] Ai C, Wyant JC. E ect of piezoelectric transducer nonlinearity on phase shift interferometry. Appl Opt 1987;26:1112–6. [7] Hariharan P, Oreb BF, Eiju T. Digital phase-shifting interferometry: a simple error-compensating phase calculation. Appl Opt 1987;26:2504–5. [8] Creath K, Hariharan P. Phase-shifting errors in interferometric tests with high-numerical-aperture reference surfaces. Appl Opt 1994;33:24–5. [9] Creath K, Gilliand YA, Hariharan P. Interferometric testing of high-numerical-aperture convex surfaces. Appl Opt 1994;33: 2585–8. [10] Procky TJ, Mammdani EH. Self-organizing process controller. Automatica 1979;15:15–30. [11] Zygo GPI-LC manual. Zgyo Corporation Middle eld, CT, 064550448 September 1997. [12] Ghiglia DC, Mastin GA, Romero LA. Cellular-automata method for phase unwrapping. J Opt Soc Am A 1987;4:267–80.
www.elsevier.com/locate/optlastec
An automated self-marking phase-shifting system for measuring the full eld phase distribution of interference fringe patterns M.J. Huang ∗, L.Y. Chou Department of Mechanical Engineering, National Chung-Hsing University, Taichung, Taiwan 40227, ROC Received 20 December 1999; received in revised form 18 April 2000; accepted 2 May 2000
Abstract A technique based on the repetition of the interference fringe pattern for the automated marking of the PZT phase shifting interference system is described. A closed-loop control is used to circumvent the inherent hysteresis and nonlinear characteristics of PZT. The whole interference system integrated under LabVIEW graphic programming environment provides users a friendly and ease-of-maintenance interface. An application of the algorithm is used to demonstrate the promotion of the Zygo GPI-LC model interferometer to c 2000 Elsevier Science Ltd. All rights reserved. upgrade its phase resolution ability. Keywords: Phase-shifting technique; PZT; Optical wavefront measurement; Zygo interferometer; PZT calibration
1. Introduction Phase-shifting techniques [1– 4] are important for the interference fringe analysis to transfer interferogram into phase map. Among all the phase-shifting mechanism, the pieozo-electrical transducer (PZT) is the most easy and e ective actuator to do so. Since the movement of PZT phase shifter a ects directly the shifted interference pattern, its precision of movement becomes very critical. There inherently are two fundamental problems associated with the motion of PZT, which are the unknown sensitivity and the nonlinearity of it. Both problems will generate a kind of sinusoidal phase error. Therefore, Cheng and Wyant proposed a calibration-insensitive phase calculation algorithm [5] to circumvent this and later modeled the PZT displacement as a quadratic function [6] to analyze and eliminate the phase error caused by the PZT nonlinearity. Hariharan et al. [7] also proposed a simple error-compensating phase calculation algorithm to overcome the unknown sensitivity of PZT and any tilt introduced across a diameter in the response of the PZT. Creath and Hariharan [8] and Creath et al. [9] also pro∗ Corresponding author. Tel.: +886-4-285-1951; fax: +886-4287-7170. E-mail address: [email protected] (M.J. Huang).
posed algorithm for elimination of errors contributed by the phase-shifting reference wavefront. Besides, there are still problems related to other sources, e.g. the accuracy of the optical set-up angle. An innovative and ecient self-marking method that automatically marks the PZT-driven voltage of 2 phase is presented in this article. The automatic self-marking voltage obtained by this newly developed method has the advantage of not having to know the sensitivity of PZT. Furthermore, for being based on repetition of the interference fringe pattern, the optical set-up angle associated with this technique does not have to accurately align to the exact value as other method does.
2. Linearity control of PZT To overcome the inherent characteristics of the PZT phase shifter a closed-loop control is used in this article. Strain gage sensors are buried in the transducer to monitor the elongation of it and the signals are fed back to the piezoelectric system for linearity control. Several control algorithms have been successfully implemented on this system including PID, Fuzzy, SOC (Self-Organizing Controller) [10], and neutral-network techniques. Therefore, the PZT system in this work can be treated as a quasi-linear transducer.
c 2000 Elsevier Science Ltd. All rights reserved. 0030-3992/00/$ - see front matter PII: S 0 0 3 0 - 3 9 9 2 ( 0 0 ) 0 0 0 4 6 - 3
226
M.J. Huang, L.Y. Chou / Optics & Laser Technology 32 (2000) 225–230
3. Self-marking system for PZT phase shifter As mentioned above, both the measurement error of incident angle of the reference mirror and the proportional coecient error of PZT driving actuator may induce signi cant error into the calculation of phase angle. Therefore, it becomes very important to precisely calibrate the PZT actuator and to accurately measure the angle of incident light on the reference, PZT-driving, mirror in the eld of phase-shifting technique. In this paper, we propose an innovative algorithm to circumvent this problem and successfully apply it to a phase-shifting system to perform the measurement of full- eld phase distribution. This newly developed system owns the advantage of not having to measure the actual angle between the incident light and the normal of the reference mirror; even not having to know the proportional coecient of PZT phase shifter also. This system can automatically level and mark the voltage needed for PZT actuator to introduce a 2 phase shift into the optical phase-shifting system as soon as the optical system has been set. This algorithm can be easily applied to various kinds of interference systems, like Michelson, Mach Zehnder, Twyman Green, etc., without any modi cation. Following is the explanation of this newly developed algorithm. In interferometry, it is not dicult to nd that the shifted interference fringe pattern will be identical with its original non-shifted fringe pattern if the introduced phase shift is an integer times of 2 phase. Furthermore, the induced phase shift is a linear function of the elongation of pieozo-electric transducer. Therefore, if the elongation of pieozo-electric transducer could be changed linearly and gradually, the induced phase shift of the phase shifter would be changed in correspondence. Owing to this characteristic, if we introduce a small increment of voltage step by step to the PZT phase shifter, it would provide linearly a correspondent phase shift to the interference system also step by step. Whenever PZT shift a small step, the shifted interference fringe pattern has to be recorded and be compared with its original non-shifted interference fringe pattern. When the shifted interference fringe pattern matches identically its original non-shifted interference fringe pattern, the phase change of the PZT phase shifter will de nitely be an integer number times of 2 phase. At this moment, the total voltage we have applied is the exact voltage needed for PZT phase shifter to introduce a 2 phase shift through it into the optical system. By setting the non-shifted interference fringe pattern as an initial fringe pattern for reference, each of the shifted interference fringe patterns associated with di erent amount of PZT shift is compared, respectively, with the original non-shifted fringe pattern. The comparison is carried out in the way of subtracting the intensity of the correspondence pixel of the two, shifted and original, interference fringe patterns rst, then take its root mean square value. The behavior can be represented in the formula
below: f(k) =
1 A
Z Z
k = 1; 2; 3 : : : ;
R
[I0 (x; y) − Ik (x; y)]2 d x dy
1=2
(1)
where R is the region of the interference fringe pattern, A is the total area of the interference fringe pattern for analysis, k is the number of unit voltage applied to the PZT, I0 is the intensity distribution of the original interference fringe pattern, and Ik is the kth interference fringe pattern under PZT driven by k units of unit voltage. Eq. (1) is the expression for comparison between two continuous interference fringe patterns. For experimental data, the interference fringe pattern grabbed from the CCD camera through image grabber card to computer for analyzing is discrete but not continuous. So Eq. (1) has to be modi ed to meet the realistic condition of our experiment. The modi ed equation is given as Eq. (2). )1=2 ( 1 P 2 [I0 (m; n) − Ik (m; n)] ; (2) f(k) = N (m; n)∈R where N is the total number of pixels of the interference fringe pattern, and m; n are the number of pixels of the horizontal and the vertical direction of the interference fringe pattern, respectively. According to the discussion above, the minimum value of the curve of RMS value vs. voltage applied occurs around the point where the introduced phase shift is equal to 2. Marking the voltage applied where the minimum RMS value occurs, therefore this is the voltage that has to be fed to the PZT for a 2 phase shift. When the PZT is under closed-loop control, the linearity of PZT movement is good and acceptable. Therefore, it is not dicult to
Fig. 1. Schematic drawing of the experimental setup for verifying “self-marking phase shifting system for full- eld phase measurement”.
M.J. Huang, L.Y. Chou / Optics & Laser Technology 32 (2000) 225–230
Fig. 2. Schematic diagram of the whole system.
get the applied voltage required for achieving the precise phase shift of PZT. For instance, suppose the applied voltage of the PZT for minimum RMS value is V , then the voltage needed for this PZT to purchase a phase shift will be V=2 and will be V=4 for a =2 phase shift.
4. Experiment and results The validity of the method has been examined in experiments using Zygo GPI-LC interferometer [11], which has been proved to be stable and reliable for optical at examination. The experimental set up of the interferometer is schematically shown in Fig. 1. In the interferometer a 4 in diameter, expanded and collimated low-power HeNe laser beam is used as the light source. A transmission element with =20 accuracy, mounted in front of the aperture, re ects some of the laser light back into the interferometer, thus creating a reference wavefront. The remainder of the laser light passes through the transmission element to the component – optical at being tested and is referred to as the measurement wavefront. The measurement wavefront re ects back to the interferometer. The phase di erences between the two wavefronts result in an image of light and dark fringe patterns that is a direct indication of the quality of the optical at being tested. The interference pattern is converted to electric signals by a solid-state video camera (CCD), and is grabbed through the image grabber card to a computer for analysis. The optical component being tested is a Newport 1 1 in diameter, 1=10 accuracy front mirror, model number 10D20BD.1. Due to its high re ectance an extra attenuation lter is added between the transmission
component and the test component to attenuate the light intensity. The Zygo GPI-LC interferometer used in this experiment is a Fizeau interferometer with ease of use and great accuracy characteristics. But the model is not the full version. This model can only display the interference pattern on the video monitor and without any phase resolution capacity. The readout of the interference fringe pattern depends strongly on the experience of the engineer. From automation and result-repetition point of view, this has to be avoided. For this reason and further to verify this newly developed algorithm, the phase-shifting capacity mentioned above has been added to a Zygo GPI-LC model interferometer to upgrade its phase resolution ability and to verify the validity of the method. The promotion of the Zygo GPI-LC model interferometer has been carried out as follows. First, link the image output terminal of the Zygo GPI-LC interferometer with a frame grabber card, model NI 2 IMAQ PCI-1408, to enable the digital image acquisition ability of the interference system. Then, add a phase-shifting mechanism to the Zygo GPI-LC interferometer to enhance its phase resolution capacity. In our experiment, we attach a computer-controlled PZT, model PI 3 P-821.10, to the tested mirror to act as the phase-shifting mechanism. The movement of the PZT is controlled by the computer-driven-signal; this signal is converted by a 16 bit D=A card, model NI (see footnote 2) PCI-MIO16XE-10, to the analog signal and fed to the PZT for precise phase shift. The whole phase-shifting hardware system was integrated systematically under the LabVIEW 4 graphic programming software from driving the phase-shifting mechanism, marking the voltage required for a 2 phase shift of PZT, acquiring the phase-shifted interference fringe pattern, solving the wrapped phase, unwrapping the wrapped phase map, analyzing the results, to even plotting the 3D phase distribution. The whole schematic diagram of the integrated system is shown in Fig. 2. To verify our self-marking algorithm of PZT, a discrete voltage series with 2.5 mV increment each was fed to the PZT gradually through the D=A card. Each time an increment of voltage has been sent to the PZT, the shifted interference fringe pattern after being stable was grabbed from image grabber to the computer and its RMS value was calculated. The result was plotted in Fig. 3(a) with the RMS value of the shifted interference fringe pattern as vertical axis and the number of unit voltage applied as horizontal axis. From the experimental data, except from the null-voltage applied point – the non-shifted original interference fringe pattern, the total voltage applied where the minimum RMS value occurred is the voltage needed for the PZT to achieve a 2 phase shift of the
2
National Instruments Corporation Austin, TX, USA. Physik Instrumente (PI) GmbH & Co Germany. 4 LabVIEW, Graphical programming National Instruments Corporation Austin, TX, USA. 3
1
Newport Corporation, CA, USA.
227
228
M.J. Huang, L.Y. Chou / Optics & Laser Technology 32 (2000) 225–230
Fig. 3. Averaged RMS phase error vs. voltage applied to PZT for phase stepping (each with 2.5 mV increment) (a) – (c). The same experiment has been repeated for three times, the repetition is good and the minimum point corresponds almost around the same voltage applied.
interference system. The same experiment was repeated at least for three times. The results are shown in Fig. 3(a) – (c). It is evident that the repetition of the experiment is good and the algorithm can be really treated as an easy and powerful way to nd the voltage required for a 2 phase shift of the PZT phase-shifting system. Since the error associated with ve-frame technology is less serious in comparison with three- and four-frame technology, ve-frame technology is used here. Therefore, ve voltages of PZT associated, respectively, with 0, 90, 180, 270 ◦ and 360 phase shift were automatically calculated by our system and fed also automatically to the PZT to achieve
the phase shifting. After phase stepping, the associated ve PZT-driven phase shifting interference fringe patterns were grabbed from image grabber to the analyzing computer. The wrapped phase map is then calculated and the unwrapped phase map by cellular-automata method [12] are shown in Fig. 4. The unwrapped phase map is then curve tted by a plane in least-square means. The results, shown in Fig. 5, are expressed in the form of di erences between the unwrapped phase distribution and the curve- tted plane with optical wavelength (632:8 nm) as the unit. The maximum and minimum height between the measured and the tted plane is 0.036 and −0:024,
M.J. Huang, L.Y. Chou / Optics & Laser Technology 32 (2000) 225–230
229
Fig. 6. 3D plot of the experimental wavefront.
Fig. 4. Wrapped and unwrapped phase map of the experiment. Fig. 7. Contour plot of the experimental result with 0:01 increment each.
5. Conclusions and discussion
Fig. 5. Unwrapped phase map expressed in di erences with a least-squares tted plane in unit .
that is to say, the total peak-to-valley of the mirror being tested is 0:06. The 3D distribution of the tested mirror is plotted in Fig. 6 and its contour plot is also shown in Fig. 7 with each of 0:01 increment one from another.
An easy, precise, e ective, and automatic technique for marking the PZT phase-shifting system has been successfully developed and veri ed. The voltage required, for the PZT, to achieve a 2 phase shift to the associated interference system is implemented by locating the point of the minimum RMS value vs. voltage applied. The newly proposed PZT self-marking algorithm circumvents the determination of the sensitivity of PZT material as well as the angle formed between lights of PZT incident on and re ected from it. The algorithm can be applied directly to other kind of interference system without much modi cation. Experimental results show that this method is e ective as an accurate and precise calibration of the PZT phase-shifting system. The ultimate resolution of the method is limited by the spatial resolution of the CCD camera as well as by the numbers of bit of the D=A card that used to drive the PZT phase shifter. The technique is especially suitable for PZT under closed-loop control for its hysteresis and nonlinearity
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repression characteristics. The repetition of the experimental results depend strongly on the precise control of the environmental conditions such as temperature, pressure, humidity, vibration, air ow, etc., all parameters that are related directly or indirectly with the refractive index, as the same demands for all other interference systems. Acknowledgements This research was performed at the Holographic Laboratory of National Chung-Hsing University, supported by the ROC National Science Council Contract No. NSC87-2212E-005-019 and NSC88-2212-E-005-006. References [1] Bruning JH, Herriott DR, Gallagher JE, Rosenfeld DP, White AD, Brangaccio DJ. Digital wavefront measuring interferometer for testing optical surfaces and lenses. Appl Opt 1974;13:2693–703. [2] Wyant JC. Use of an AC heterrodyne lateral shear interferometer with real-time wavefront corection systems. Appl Opt 1975;14: 2622–6.
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