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Polarization shifting interferometric profilometer
ARTICLE IN PRESS
Optics and Lasers in Engineering 46 (2008) 203–210 www.elsevier.com/locate/optlaseng
Polarization shifting interferometric profilometer Hui-Kang Tenga,, Kuo-Chen Langb a
Graduate Institute of Electrical Engineering, Nan-Kai Institute of Technology, N568, Chung Chang Road, Nan-Tou, Taiwan 542, ROC Department of Computer and Communication Engineering, Nan-Kai Institute of Technology, N568, Chung Chang Road, Nan-Tou, Taiwan 542, ROC
b
Received 11 March 2007; received in revised form 7 October 2007; accepted 1 November 2007
Abstract A phase sensitive Michelson interferometer based on interference microscope configuration with a polarization adjustment approach is proposed to determine the two-dimensional (2-D) surface profile of optical grating with real time capability. In the proposed method, nonlinear behavior of a PZT phase shifter is avoided by use of polarization stepping and a phase map is developed with the four-bucket algorithm. The phase map is unwrapped to give the true surface profile of the sample. A close agreement of measurements is found between the measured result determined by the proposed method and that determined by an atomic force microscope (AFM). We also analyzed the estimated uncertainty of measurement in the nanometer range for the random fluctuations of the experimental parameters. r 2007 Elsevier Ltd. All rights reserved. Keywords: Interference microscope; Polarization shifting; Profilometry
1. Introduction Surface profile measurement is an important technique in mass information storage device [1], the semiconductor industry [2] and the manufacturing of optical components [3]. As the manufacturing tolerances become tighter, quality control and performance testing for surface properties are also becoming a part of the manufacturing process [3]. Thus, the measurement of surface profile for characterizing the manufacturing process or determining the quality of machining becomes more important than ever. The determination of the surface profile by an optical interferometric approach is generally recognized as a high resolution, noninvasive and relatively inexpensive technology [4] in comparison with an atomic force microscope (AFM), or scanning electronic microscope (SEM), because of the high sensitivity of phase detection. The measurement of AFM and SEM need longer times for imaging, and the latter is limited by depth resolution into the nanometer range. With the optical interferometric approach, the surface profile can be recovered from the phase informaCorresponding author. Tel.: +886 49 2563489; fax: +886 49 2563489.
E-mail addresses: [email protected] (H.-K. Teng), [email protected] (K.-C. Lang). 0143-8166/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2007.11.001
tion by scanning with the heterodyne technique to determine the surface characteristics [5,6], or by a twodimensional (2-D) imaging method with the phase shifting technique for characterizing the crystal surface [7], fiber tip profile [8], or determining the contour of a binary microlens array [9], to name a few. Among these methods, the phase shifting interferometry provides significant contributions on full field profile measurements with the advantages of high accuracy, noise and parasitic fringes reduction, and the independence of intensity variation across the pupil. However, great care should be taken on the mechanical movement of the reference mirror for phase shifting by a piezoelectric (PZT) phase shifter because of the measurement error from the nonlinear behavior and the hysteresis of PZT. This drawback had been analyzed by many reports [10–13]. There are many algorithms developed to deal with this problem, providing a compromise between accuracy and computational tasks. To avoid the errors that arise from nonlinear behavior and hysteresis of the PZT phase shifter, Onuma et al. [14] proposed a threebucket algorithm without a PZT phase shifter to monitor the growth of a crystal surface, in which three interference images with encoded p/2 phase difference from one another are generated simultaneously. However, this technique needs three CCD cameras for recording individual interference
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image at the same time, which leads to a difficult alignment for superposing the images from the three CCD cameras and the calibration task for image intensity recorded by each CCD camera. Recently, Helen et al. [15] proposed a white light interferometer for surface profiling by using a rotation polarizer as a phase shifter. Since the polarizer is located in front of the interferometer, the propagation direction of beams may be deviated slightly during the rotation. This deviation usually results in the perturbation of optical alignment which, in turn, contaminates the raw image data so that the resolution of the measurement is degraded. Phase shifting interferometry is developed based on the fact that the optical phase is changed by changing the distance of optical path. The optical phase of a light wave, however, can also be varied by modifying the polarization state of composite components of the original light wave. This phenomenon can be described, for example, by Berry [16] using Poincare´ sphere. By virtue of this fact, we present in this report a polarization shifting approach based on a Michelson interference microscope configuration [17] for characterizing the surface profile of interest. The phase shifting is achieved by adjusting the polarization state of the output beams with a rotating quarter wave plate (QWP), which is located at the beam path where the reference and probe beams propagate collinearly after recombination. Therefore, the alignment of reference and probe beams is not disturbed by rotating the QWP. Since the rotation can be precisely controlled and free from nonlinear behavior and hysteresis, the high resolution in surface profile measurement by the proposed technique can be achieved. In addition, the non-uniform laser intensity distribution has no effect on the measurement result. We used an electronically controlled stage for the rotation of the QWP, thus making the in situ measurement possible. To demonstrate the capability of the proposed technique, we took an aluminum coated blazed grating as the test sample for a full field surface profile measurement. The raw phase map that represents the phase difference between reference and probe beams is first generated by a fourbucket algorithm on a pixel-by-pixel basis. Because the raw phase map was modulo 2p due to the inherent nature of the arctangent function, it was then unwrapped to obtain the grating profile. The profile of the same blazed grating was also investigated by an AFM for comparison. Close agreement was found between the measured results from the AFM and that from the proposed technique. Finally, an estimated uncertainty of measurement was given by considering the random fluctuations of incident intensity and component imperfection of the interferometer. 2. System description We used a linearly polarized monochromatic plane wave, split into a horizontal linearly polarized (P) and vertical linearly polarized (S) waves by a polarized beam splitter (PBS) as shown in Fig. 1. We refer to the P (S) wave as the
M Q1 SF
HWP
Q2
PBS
He-Ne Laser BX Q3
L1
G
LP (PSU)
FG and PC
CCD
Fig. 1. The microscopic interferometer in Michelson configuration of proposed technique.
linearly polarized waves that polarized parallel (perpendicular) to the incident plane on the interface of PBS. The S wave reflected by a plane mirror (M) acts as a reference beam while the P wave impinged on a test sample (G) acts as a probe beam. Due to the change of direction of linear polarization by the QWP (Q1 and Q2) on the measurement and reference paths respectively, the probe and reference beams recombine at PBS and propagate along the same path toward the subsequent optical components. These two coherent monochromatic plane waves after a round trip can be described by using Jones matrix as: ! rR E R expðiaR Þ E1 ¼ . (1) rG E G expðiaG Þ We used Jones matrices in Eq. (1) and in the subsequent analysis, because it allows the phase of the transverse optical field emerging from a polarizing component to be calculated. The amplitude of electric fields of reference and probe beams in Eq. (1) is not normalized, this clearly demonstrates the individual phase shift and amplitude change of reference and probe beams. In Eq. (1), rR, rG are real coefficients responsible for the polarization-independent amplitude reflection from the reference mirror and isotropic sample, respectively. ER and EG are the amplitudes of reference and probe beams. aR and aG are the phase shifts accumulated along the path of the reference and probe beams, respectively, before recombination. For a plane wave of incident beam, the phase shift aR is uniform distributed across the transverse plane of the reference beam if a plane reference mirror is employed. The phase shift aG on the wave front of the probe beam is dependent on the surface profile of the sample. Therefore, the resultant phase difference ad ¼ aRaG to be determined by the proposed method carries the profile information of the sample. The 2-D raw phase map is then built up through pixel-by-pixel calculation of ad which will be described in the subsequent paragraph. However,
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due to the inherent nature of the arctangent function, only the principal value of ad is obtained, thus the raw phase map may be wrapped if the optical phase difference is larger than 2p. Once the raw phase map is unwrapped to reflect the true value of phase difference aT, the surface profile h(x, y) and aT can be related by hðx; yÞ ¼
laT , 4p
(2)
where (x, y) denotes the position of image plane where aT is calculated. To adjust the polarization state of the output beam, the recombined light wave then passes through a polarization-shifting unit (PSU), composed of a QWP (Q3) and a linear polarizer (LP), thus the output optical field EOUT becomes: E OUT ¼ QLP QQWP E 1 ,
(3)
where 0 QQWP
cos2 ðbÞ expðij1 Þ
B B þsin2 ðbÞ expðij2 Þ ¼B B 1 sinð2bÞ½expðij Þ @2 1 expðij2 Þ
1 2 sinð2bÞ½expðij1 Þ
expðij2 Þ sin2 ðbÞ expðij1 Þ 2
þcos ðbÞ expðij2 Þ
1
QLP ¼
sinðgÞ cosðgÞ sin2 ðgÞ
where Iin ¼ |Eo|2. When b is rotated sequentially at p/8, p/4, 3p/8 and 3p/4, the intensities at these angles are measured as: " # pffiffiffi I in 3 2 1 2 1 2 I1 ¼ r þ r þ rG rR cosðad Þ þ rG rR sinðad Þ ; 2 4 R 4 G 2 2
b¼
p 8
(8)
I in 1 2 1 2 r þ r þ rG rR sinðad Þ ; I2 ¼ 2 2 R 2 G
b¼
p 4
" # pffiffiffi I in 3 2 1 2 1 2 r þ r rG rR cosðad Þ þ rG rR sinðad Þ ; I3 ¼ 2 2 4 R 4 G 2
I4 ¼
and sinðgÞ cosðgÞ
expressed by I in 2 1 2 rR ðrR r2G Þ sin2 ð2bÞ Iðx; yÞ ¼ 2 2 1 þ rR rG ½sinð4bÞ cosðad Þ þ 2 sinð2bÞ sinðad Þ , ð7Þ 2
(9)
b¼
3p 8
(10)
C C C C A (4)
cos2 ðgÞ
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! (5)
are the Jones matrices of QWP and LP, respectively. The angle b is the orientation of the optical axis of Q3, and g is the angle of transmission axis of LP. Both angles are measured from the x-axis. j1 and j2 are the phase shifts with respect to the wave linearly polarized along the slow and fast axes of Q3. If the orientation of the linear polarization of the pffiffiffi incident beam is set at y ¼ p/4 so that E R ¼ E G ¼ E o = 2 where Eo is the amplitude of the laser source, and the transmission axis of LP is fixed at g ¼ 01, the electric field EOUT of laser beam towards CCD becomes: Eo E OUT ¼ pffiffiffi rR cos2 ðbÞ exp½iðj1 þ aR Þ 2 1 þ rR sin2 ðbÞ exp½iðj2 þ aR Þ þ rG sinð2bÞ 2 1 exp½iðj1 þ aG Þ rG sinð2bÞ exp½iðj2 þ aG Þ . 2 ð6Þ Eq. (6) clearly shows that four plane waves propagate in a common path after passing through the PSU, each of which carries different amplitude and phase but presents the same polarization state. If the phase retardation Dj ¼ j1j2 ¼ p/2 of Q3 is assumed, the intensity I(x, y) of the interferogram captured by the CCD can be
I in 1 2 1 2 rR þ rG rG rR sinðad Þ ; 2 2 2
b¼
3p 4
(11)
Thus, the wrapped phase map can be obtained in terms of the image intensities by I2 I4 ad ¼ arctan . (12) 2ðI 1 I 3 Þ It is worth noting that ad calculated by Eq. (12) is regardless of the amplitude reflectance factors rR, rG, and the non-uniform distribution of laser intensity. This leads to the precise determination of the surface profile h(x, y) of the sample of interest in a pixel-by-pixel base by use of Eq. (2). 3. Optical setup and experimental results Fig. 1 shows the optical arrangement of the proposed method. A He–Ne laser launches a linearly polarized beam at wavelength 0.6328 mm to the interferometer. A beam expander (BX) composed of a 10 objective and an achromatic lens with focal length 50 mm expands and collimates the laser beam to a 5.6 mm diameter with a spatial filter (SF) of 20 mm diameter in between. The orientation of polarization state of incident laser beam is fixed at p/4 to the x-axis by a half wave plate (HWP). The recombined reference and probe beams pass through the PSU, and then toward CCD. The QWP (Q3) of the PSU is mounted on a DC motor-driven rotation stage while the transmission axis of LP is set at 01. The intensity detected by the CCD is converted to a 10-bit digital signal by a frame grabber (FG) and stored in a personal computer (PC) for further processing. All the orientations of optic axis of polarization components are carefully calibrated by the previous developed techniques [18,19]. The probe beam is focused onto a grating G of 300 grooves/mm blazed at
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a
b
β = π/4
β = π/8
c
d
β = 3π/8
β = 3π/4
Fig. 2. Interference fringes of reference and probe beams at four given b of Q3. The status of alignment is indicated by the straight fringes. The fringe position is shifted in accordance with the polarization adjustment.
21340 by a 10 objective L1 in which the diameter of focal point on the grating about 16 mm is obtained. The nominal period of this grating was 3.33 mm and the nominal depth of groove was 145 nm. The calibration of numerical aperture (NA) effects in our case is not necessary since the NA of the objective is smaller than 0.5. However, for NAX0.5, the spacing between fringes increased significantly such that a procedure should be employed to determine the actual height change across the fringes [20]. Noting that the plane wave front of the beam waist should impinges onto the surface of the grating, this can be checked against the interference fringe patterns as depicted in Fig. 2 where straight fringes are observed to indicate the status of alignment of the wave front of the reference and probe beams. Fig. 2 also demonstrates the shifting of fringe position in accordance with the polarization state shifted by PSU. In addition, the blazed grating is slightly tilted so that the zero order beam is retro-reflected from the grating. To reduce the random noise of the image data, 10 frames of the interferogram at each given b are captured and averaged to give the image data of I1 to I4. The raw phase map shown in Fig. 3 is then determined by pixel-by-pixel calculations according to Eq. (12) with these image data. The raw phase map is unwrapped using the Gold-stein’s algorithm [21] which was developed for studying the image from the synthetic-aperture radar signal. Similar algorithms are also developed, for example, for studying the local stress distribution [22]. We use Gold-stein’s algorithm to isolate and connect the residues which indicate phase
Fig. 3. Raw phase map of the blazed grating.
inconsistency in the raw phase map as boundary for the region where path-independent integration of the phase value is determined to generate the unwrapped phase map. The optical system must be carefully aligned so that the wave front reflected from the sample coincides with that
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from reference mirror, which results in fewer residues occurring at phase jumps in raw phase map. The resultant 2-D of the blazed grating is then obtained and is shown in Fig. 4(a) where a step profile is clearly evident. The line scan along x-axis at yE5 mm of Fig. 4(a) is shown in Fig. 4(b) where the step high at 142 nm and the grating period at 3.36 mm are determined from the image data. The same grating is also investigated by AFM, the 2-D image is shown in Fig. 5(a) where the step profile is clearly observed. The line scan of this blazed grating with the AFM is shown in Fig. 5(b) where peak to valley distances 140.05 and 139.42 nm and grating period 3.372 mm are measured. From Figs. 4 and 5, the close agreement of profiling the same grating can be observed between the two methods. 4. The estimated uncertainty of measurements For the full field interferometric profile imaging either by phase shifting interferometry or by the proposed polarization shifting microscopic interferometry, the phase map is determined based on calculating the phase difference ad in single pixel with respect to the intensities of the
Fig. 5. Profile image of the same blazed grating determined by AFM (a), while the line profile is shown in (b). The depth of groove at 140.05 and 139.42 nm and grating spacing 3.372 mm are determined by AFM.
500 450 400 350 nm
300 250 142.3 nm
200 150
3.36 um
100 50 0
0
2
4
6
um
8
10
12
Fig. 4. Surface profile of the blazed grating determined by proposed method in (a), while the line profile along x-axis at yE5 mm is displayed in (b). The depth of groove at about 142 nm and grating period about 3.36 mm can be determined from the image data.
interferogram. The measurement error introduced by the system errors, such as the defect on components, imperfect alignment, or nonlinear response of the detector, can be calibrated to reduce the effects on the measurement. On the other hand, the resolution of the proposed technique is determined by the uncertainty on the measurement of ad [23] that results from the random fluctuation of interference intensities. In view of the interference intensities from Eqs. (8)–(11), the random fluctuation is introduced by the random deviation db of azimuth angle b and the random fluctuation dIin of incident intensity, these two fluctuations are assumed to be statistically independent. The quantization error arising from the analog to digital conversion of the CCD is neglected if the converted digital image spans the whole digits where the number of digits is not smaller than 10 [24]. Hence the uncertainty of the wrapped phase d(ad) can be considered as the square root of the sum of each random fluctuation squared for the image intensity at each b by following equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 X qad 2 X qad dðad Þ ¼ db þ dI in : (13) qb qI in n n In Eq. (13), the index n runs from 1 to 4 due to the four given b values at p/8, p/4, 3p/8 and 3p/4, respectively.
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By using the relation: qad qad qI n ¼ ; qb qI n qb
qad qad qI n ¼ , qI in qI n qI in
(14)
where qI n ¼ qb
qI ; qb n
qI n ¼ qI in
qI qI in
,
(15)
n
are the derivatives of Eq. (7) with b and Iin respectively at given b. After substituting above equations into Eq. (13), it follows that "(
) 2 pffiffiffi <1 2 sinðad Þ þ 8 cos4 ðad Þ ðdbÞ2 < ( " # pffiffiffi 1 3< þ 1 2 1 2 3< þ 1 2 2 þ sin ðad Þ þ ð1 þ 2 sin ðad ÞÞ þ sinðad Þ 8 < 2 < 2 " #) #1=2 1 <þ1 2 1 dI in 2 þcos2 ðad Þ þ sin2 ðad Þ , ð16Þ I in 16 < 4
dðad Þ ¼
2 sin2 ðad Þ
where R ¼ rR/rG is the ratio of amplitude reflectance coefficients. From Eq. (16), the ratio of amplitude reflective coefficients R becomes a factor to the measurement uncertainty. This is because the interference visibility deteriorates due to unequal amplitude reflectance coefficients from the reference mirror and the sample, respectively. The derivation of Eq. (16) is based on: (a) The random deviation of rotation for shifting the azimuth angle of Q3 is independent of b and equals 0.011 from the data sheet provided by the manufacturer. p Thus, ffiffiffiffiffi the averaged random fluctuation becomes db ¼ 0:01 N where N ¼ 10 is the number of image frames to be averaged at rotation step n. (b) The relative random fluctuation of incident intensity is smaller than 0.02% according to the data sheet provided by the manufacturer. This relative random fluctuation is also independent of the rotation. Thus, pffiffiffiffiffiafter averaging at each rotation step, dI in =I in p0:0002= N . By substituting the values of db and dIin/Iin into Eq. (16), the estimated uncertainty of measurement in nanometer scale using Eq. (2) at l ¼ 0.6328 mm is depicted in Fig. 6. The uncertainty of measurement reveals a sinusoidal-like variation as the phase difference ad varied from 0 to 2p. This uncertainty is smaller than 0.02 nm when the ratio RE1 and approaches to 0.1 nm as R approaches 10. Fig. 6 also shows the uncertainty of the proposed method monotonically increasing when the ratio R ¼ rR/rG deviates from unity. This may give low resolution for measuring the profile of a sample with low reflectivity such as a biological tissue or ceramic material. Under this circumstance, the optical axis y of HWP that is located after BX in Fig. 1 can be adjusted so that more power is assigned to the probe beam to compensate for the low reflection. Eq. (12) for determining the phase difference ad still remains unchanged. However, the ratio R becomes: rR cosðyÞ <¼ . rG sinðyÞ
(17)
Fig. 6. Estimated uncertainty of measurement as function of ad and R. The uncertainty is smaller than 0.02 nm as R ¼ 1 and approaches 0.1 nm as R approaches 10.
Thus, R closes to unity by adjusting the optical axis y of HWP. To determine the angle of optical axis, additional interference intensity I0 at b ¼ 0 is suggested at y ¼ p/4 where I in 2 r . (18) 2 R In combination with Eqs. (9) and (11), the ratio R can be determined by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rR I0 <¼ , (19) ¼ arctan rG I2 þ I4 I0
I0 ¼
which gives the azimuth angle y of HWP from Eq. (17). In the proposed method, the interference intensity is varied as Q3 is rotated at given angles. It is useful to estimate the error as a result from the imperfections of Q3. The theoretical uncertainties dad,q introduced by the phase retardation uncertainty d(Dj) defined by d(Dj) ¼ p/2Dj of Q3 and d(Dj)5Dj can be calculated by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X @ad 2 2 dad;q ¼ (20) dðDjÞ . @Dj n n It is assumed that the random fluctuations of interference intensities at each step of n due to phase retardation uncertainty are statistically independent. Going back to Eq. (6) and following a similar expansion in Eqs. (14) and (15), the estimated error of the measurements due to imperfect QWP can be written as: ( 2 pffiffiffi 1 1 2 dad;q ¼ 2 sin2 ðad Þ < þ 2 cos2 ðad Þ 2 < " #) 2 1 1 < þ 2 cos2 ðad Þ dðDjÞ2 . þcos2 ðad Þ 2 < ð21Þ From Eq. (21), it is found that dad,q is also a function of the ratio R ¼ rR/rG, and becomes minimal when R ¼ 1. The estimated result is shown in Fig. 7 where the imperfect
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Fig. 7. The estimated uncertainty of measurement as function of ad,q and R. The imperfect phase retardation of Q3 at d(Dj) ¼ 0.51 is assumed. The estimated uncertainty also shows cyclic variations as function of ad,q, and smaller than 0.5 nm as R ¼ 1 and larger than 1 nm as R approaches 10.
of Q3 with d(Dj) ¼ 0.51 is assumed for a typical commercially available QWP. The estimated uncertainty of measurement translated in nanometer scale shows cyclic variation as function of measured ad. The theoretical uncertainty approaches 0.5 nm as RE1 and becomes larger when R deviates from unit. By comparing the theoretical uncertainties depicted in Fig. 6 with that shown in Fig. 7, it is found that the imperfect of Q3 may be more critical to the uncertainty of proposed measurement method. 5. Discussions and conclusions In the proposed method, the polarization state of light waves reflected from the reference mirror and the sample of interest are adjusted simultaneously by PSU at four steps whereas the interference intensity at each adjustment was captured by a CCD. Since this adjustment is performed after the combination of reference and probe beams at PBS, the optical alignment of the Michelson interferometer remains unaffected during the measurement. Another advantage of polarization shifting is that the nonlinear behavior of PZT that is usually employed as a phase shifter can be avoided. A blazed grating can be employed as, for example, a beam steering device in the liquid crystal display [25], or a coupler in surface emitting lasers [26]. The grating period in our case was larger than the wavelength of light source so that it was not necessary to take into consideration of the polarizationdependent reflection from the grating. This is evident from the experimental result. Once the grating period is smaller than the wavelength of the probe beam, the polarizationdependent reflection should be considered [27]. As far as the time consuming is concerned, it takes 20 s to complete the experiment under PC-controlled rotation stage and image recording procedure. This time for completing the measurement depends on the rotation speed of the Q3 and the number of frames captured at each adjustment for reducing random noise. The real time measurement can be improved
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if an electronically controlled polarization adjustment device, such as liquid crystal wave plate [28], is employed. However, from the estimated error analysis in Eq. (21), we conclude the critical requirement on a programmable wave plate using liquid crystal for real time measurement is unavoidable. The phase shifting microscopic interferometry is a powerful, versatile and noninvasive method for 2-D surface profile measurement. We present a polarization shifting alternative to determine the 2-D surface profile while avoiding the disadvantage of using PZT as a phase shifter in phase shifting interferometry. A single frequency linearly polarized He–Ne laser is employed due to the high resolution of measurement that can be achieved with a phase sensitive approach. An estimated error analysis is given based on the randomness from interference intensities and from the rotation of Q3. The compensation for the degradation of measurement uncertainty due to unequal reflectance coefficients from the reference mirror and the sample respectively is possible by simply rotated the HWP, in which a method to determine the ratio R is also suggested. We also found that the imperfect phase retardation of Q3 in PSU is more critical to the error of measurement. This suggests the careful use of an electronically rotatable wave plate for real time measurement. Acknowledgment This work is supported by National Science Council of Taiwan, ROC, through contract number NSC93-2215-E252-001. References [1] Bartoli A, Poggi P, Quercioli F, Tiribilli B. Fast one-dimensional profilometer with a compack disc pickup. Appl Opt 2001;40:1044–8. [2] Klein EJ, Ramirez WF, Hall JL. A common-path heterodyne interferometer for surface profiling in microelectronic fabrication. Rev Sci Instrum 2001;72:2455–67. [3] White DJ. Surface metrology. Meas Sci Technol 1997;8:955–72. [4] Dorrio BV, Fernandez JL. Phase-evaluation methods in whole-field optical measurement techniques. Meas Sci Technol 1999;10:R33–55. [5] Vatsya SR, Li C, Nikumb SK. Surface profile of material ablated with high-power lasers in ambient air medium. J Appl Phys 2005;97: 034912. [6] Chou C, Lyu CW, Peng LC. Polarized differential-phase laser scanning microscope. Appl Opt 2001;40:95–9. [7] Booth NA, Stanojev B, Chemov AA, Vekilov PG. Differential phaseshifting interferometry for in situ surface characterization during solution growth of crystals. Rev Sci Instrum 2002;73:3540–5. [8] Wang S, Quan C, Tay CJ, Reading I, Fong Z. Measurement of a fiber-end surface profile by use of phase-shifting laser interferometry. Appl Opt 2004;43:49–56. [9] Liang P, Ding J, Jin Z, Guo CS, Wang HT. Two-dimensional wavefront reconstruction from lateral shearing interferograms. Opt Express 2006;14:625–34. [10] Wyant JC. Effect of piezoelectric transducer nonlinearity on phase shift interferometery. Appl Opt 1987;26:1112–6. [11] Creath K. Temporal phase measurement methods. In: Robinson DW, Reid GT, editors. Interferogram analysis. New York: Wiley; 1993.
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Optics and Lasers in Engineering 46 (2008) 203–210 www.elsevier.com/locate/optlaseng
Polarization shifting interferometric profilometer Hui-Kang Tenga,, Kuo-Chen Langb a
Graduate Institute of Electrical Engineering, Nan-Kai Institute of Technology, N568, Chung Chang Road, Nan-Tou, Taiwan 542, ROC Department of Computer and Communication Engineering, Nan-Kai Institute of Technology, N568, Chung Chang Road, Nan-Tou, Taiwan 542, ROC
b
Received 11 March 2007; received in revised form 7 October 2007; accepted 1 November 2007
Abstract A phase sensitive Michelson interferometer based on interference microscope configuration with a polarization adjustment approach is proposed to determine the two-dimensional (2-D) surface profile of optical grating with real time capability. In the proposed method, nonlinear behavior of a PZT phase shifter is avoided by use of polarization stepping and a phase map is developed with the four-bucket algorithm. The phase map is unwrapped to give the true surface profile of the sample. A close agreement of measurements is found between the measured result determined by the proposed method and that determined by an atomic force microscope (AFM). We also analyzed the estimated uncertainty of measurement in the nanometer range for the random fluctuations of the experimental parameters. r 2007 Elsevier Ltd. All rights reserved. Keywords: Interference microscope; Polarization shifting; Profilometry
1. Introduction Surface profile measurement is an important technique in mass information storage device [1], the semiconductor industry [2] and the manufacturing of optical components [3]. As the manufacturing tolerances become tighter, quality control and performance testing for surface properties are also becoming a part of the manufacturing process [3]. Thus, the measurement of surface profile for characterizing the manufacturing process or determining the quality of machining becomes more important than ever. The determination of the surface profile by an optical interferometric approach is generally recognized as a high resolution, noninvasive and relatively inexpensive technology [4] in comparison with an atomic force microscope (AFM), or scanning electronic microscope (SEM), because of the high sensitivity of phase detection. The measurement of AFM and SEM need longer times for imaging, and the latter is limited by depth resolution into the nanometer range. With the optical interferometric approach, the surface profile can be recovered from the phase informaCorresponding author. Tel.: +886 49 2563489; fax: +886 49 2563489.
E-mail addresses: [email protected] (H.-K. Teng), [email protected] (K.-C. Lang). 0143-8166/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2007.11.001
tion by scanning with the heterodyne technique to determine the surface characteristics [5,6], or by a twodimensional (2-D) imaging method with the phase shifting technique for characterizing the crystal surface [7], fiber tip profile [8], or determining the contour of a binary microlens array [9], to name a few. Among these methods, the phase shifting interferometry provides significant contributions on full field profile measurements with the advantages of high accuracy, noise and parasitic fringes reduction, and the independence of intensity variation across the pupil. However, great care should be taken on the mechanical movement of the reference mirror for phase shifting by a piezoelectric (PZT) phase shifter because of the measurement error from the nonlinear behavior and the hysteresis of PZT. This drawback had been analyzed by many reports [10–13]. There are many algorithms developed to deal with this problem, providing a compromise between accuracy and computational tasks. To avoid the errors that arise from nonlinear behavior and hysteresis of the PZT phase shifter, Onuma et al. [14] proposed a threebucket algorithm without a PZT phase shifter to monitor the growth of a crystal surface, in which three interference images with encoded p/2 phase difference from one another are generated simultaneously. However, this technique needs three CCD cameras for recording individual interference
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image at the same time, which leads to a difficult alignment for superposing the images from the three CCD cameras and the calibration task for image intensity recorded by each CCD camera. Recently, Helen et al. [15] proposed a white light interferometer for surface profiling by using a rotation polarizer as a phase shifter. Since the polarizer is located in front of the interferometer, the propagation direction of beams may be deviated slightly during the rotation. This deviation usually results in the perturbation of optical alignment which, in turn, contaminates the raw image data so that the resolution of the measurement is degraded. Phase shifting interferometry is developed based on the fact that the optical phase is changed by changing the distance of optical path. The optical phase of a light wave, however, can also be varied by modifying the polarization state of composite components of the original light wave. This phenomenon can be described, for example, by Berry [16] using Poincare´ sphere. By virtue of this fact, we present in this report a polarization shifting approach based on a Michelson interference microscope configuration [17] for characterizing the surface profile of interest. The phase shifting is achieved by adjusting the polarization state of the output beams with a rotating quarter wave plate (QWP), which is located at the beam path where the reference and probe beams propagate collinearly after recombination. Therefore, the alignment of reference and probe beams is not disturbed by rotating the QWP. Since the rotation can be precisely controlled and free from nonlinear behavior and hysteresis, the high resolution in surface profile measurement by the proposed technique can be achieved. In addition, the non-uniform laser intensity distribution has no effect on the measurement result. We used an electronically controlled stage for the rotation of the QWP, thus making the in situ measurement possible. To demonstrate the capability of the proposed technique, we took an aluminum coated blazed grating as the test sample for a full field surface profile measurement. The raw phase map that represents the phase difference between reference and probe beams is first generated by a fourbucket algorithm on a pixel-by-pixel basis. Because the raw phase map was modulo 2p due to the inherent nature of the arctangent function, it was then unwrapped to obtain the grating profile. The profile of the same blazed grating was also investigated by an AFM for comparison. Close agreement was found between the measured results from the AFM and that from the proposed technique. Finally, an estimated uncertainty of measurement was given by considering the random fluctuations of incident intensity and component imperfection of the interferometer. 2. System description We used a linearly polarized monochromatic plane wave, split into a horizontal linearly polarized (P) and vertical linearly polarized (S) waves by a polarized beam splitter (PBS) as shown in Fig. 1. We refer to the P (S) wave as the
M Q1 SF
HWP
Q2
PBS
He-Ne Laser BX Q3
L1
G
LP (PSU)
FG and PC
CCD
Fig. 1. The microscopic interferometer in Michelson configuration of proposed technique.
linearly polarized waves that polarized parallel (perpendicular) to the incident plane on the interface of PBS. The S wave reflected by a plane mirror (M) acts as a reference beam while the P wave impinged on a test sample (G) acts as a probe beam. Due to the change of direction of linear polarization by the QWP (Q1 and Q2) on the measurement and reference paths respectively, the probe and reference beams recombine at PBS and propagate along the same path toward the subsequent optical components. These two coherent monochromatic plane waves after a round trip can be described by using Jones matrix as: ! rR E R expðiaR Þ E1 ¼ . (1) rG E G expðiaG Þ We used Jones matrices in Eq. (1) and in the subsequent analysis, because it allows the phase of the transverse optical field emerging from a polarizing component to be calculated. The amplitude of electric fields of reference and probe beams in Eq. (1) is not normalized, this clearly demonstrates the individual phase shift and amplitude change of reference and probe beams. In Eq. (1), rR, rG are real coefficients responsible for the polarization-independent amplitude reflection from the reference mirror and isotropic sample, respectively. ER and EG are the amplitudes of reference and probe beams. aR and aG are the phase shifts accumulated along the path of the reference and probe beams, respectively, before recombination. For a plane wave of incident beam, the phase shift aR is uniform distributed across the transverse plane of the reference beam if a plane reference mirror is employed. The phase shift aG on the wave front of the probe beam is dependent on the surface profile of the sample. Therefore, the resultant phase difference ad ¼ aRaG to be determined by the proposed method carries the profile information of the sample. The 2-D raw phase map is then built up through pixel-by-pixel calculation of ad which will be described in the subsequent paragraph. However,
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due to the inherent nature of the arctangent function, only the principal value of ad is obtained, thus the raw phase map may be wrapped if the optical phase difference is larger than 2p. Once the raw phase map is unwrapped to reflect the true value of phase difference aT, the surface profile h(x, y) and aT can be related by hðx; yÞ ¼
laT , 4p
(2)
where (x, y) denotes the position of image plane where aT is calculated. To adjust the polarization state of the output beam, the recombined light wave then passes through a polarization-shifting unit (PSU), composed of a QWP (Q3) and a linear polarizer (LP), thus the output optical field EOUT becomes: E OUT ¼ QLP QQWP E 1 ,
(3)
where 0 QQWP
cos2 ðbÞ expðij1 Þ
B B þsin2 ðbÞ expðij2 Þ ¼B B 1 sinð2bÞ½expðij Þ @2 1 expðij2 Þ
1 2 sinð2bÞ½expðij1 Þ
expðij2 Þ sin2 ðbÞ expðij1 Þ 2
þcos ðbÞ expðij2 Þ
1
QLP ¼
sinðgÞ cosðgÞ sin2 ðgÞ
where Iin ¼ |Eo|2. When b is rotated sequentially at p/8, p/4, 3p/8 and 3p/4, the intensities at these angles are measured as: " # pffiffiffi I in 3 2 1 2 1 2 I1 ¼ r þ r þ rG rR cosðad Þ þ rG rR sinðad Þ ; 2 4 R 4 G 2 2
b¼
p 8
(8)
I in 1 2 1 2 r þ r þ rG rR sinðad Þ ; I2 ¼ 2 2 R 2 G
b¼
p 4
" # pffiffiffi I in 3 2 1 2 1 2 r þ r rG rR cosðad Þ þ rG rR sinðad Þ ; I3 ¼ 2 2 4 R 4 G 2
I4 ¼
and sinðgÞ cosðgÞ
expressed by I in 2 1 2 rR ðrR r2G Þ sin2 ð2bÞ Iðx; yÞ ¼ 2 2 1 þ rR rG ½sinð4bÞ cosðad Þ þ 2 sinð2bÞ sinðad Þ , ð7Þ 2
(9)
b¼
3p 8
(10)
C C C C A (4)
cos2 ðgÞ
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! (5)
are the Jones matrices of QWP and LP, respectively. The angle b is the orientation of the optical axis of Q3, and g is the angle of transmission axis of LP. Both angles are measured from the x-axis. j1 and j2 are the phase shifts with respect to the wave linearly polarized along the slow and fast axes of Q3. If the orientation of the linear polarization of the pffiffiffi incident beam is set at y ¼ p/4 so that E R ¼ E G ¼ E o = 2 where Eo is the amplitude of the laser source, and the transmission axis of LP is fixed at g ¼ 01, the electric field EOUT of laser beam towards CCD becomes: Eo E OUT ¼ pffiffiffi rR cos2 ðbÞ exp½iðj1 þ aR Þ 2 1 þ rR sin2 ðbÞ exp½iðj2 þ aR Þ þ rG sinð2bÞ 2 1 exp½iðj1 þ aG Þ rG sinð2bÞ exp½iðj2 þ aG Þ . 2 ð6Þ Eq. (6) clearly shows that four plane waves propagate in a common path after passing through the PSU, each of which carries different amplitude and phase but presents the same polarization state. If the phase retardation Dj ¼ j1j2 ¼ p/2 of Q3 is assumed, the intensity I(x, y) of the interferogram captured by the CCD can be
I in 1 2 1 2 rR þ rG rG rR sinðad Þ ; 2 2 2
b¼
3p 4
(11)
Thus, the wrapped phase map can be obtained in terms of the image intensities by I2 I4 ad ¼ arctan . (12) 2ðI 1 I 3 Þ It is worth noting that ad calculated by Eq. (12) is regardless of the amplitude reflectance factors rR, rG, and the non-uniform distribution of laser intensity. This leads to the precise determination of the surface profile h(x, y) of the sample of interest in a pixel-by-pixel base by use of Eq. (2). 3. Optical setup and experimental results Fig. 1 shows the optical arrangement of the proposed method. A He–Ne laser launches a linearly polarized beam at wavelength 0.6328 mm to the interferometer. A beam expander (BX) composed of a 10 objective and an achromatic lens with focal length 50 mm expands and collimates the laser beam to a 5.6 mm diameter with a spatial filter (SF) of 20 mm diameter in between. The orientation of polarization state of incident laser beam is fixed at p/4 to the x-axis by a half wave plate (HWP). The recombined reference and probe beams pass through the PSU, and then toward CCD. The QWP (Q3) of the PSU is mounted on a DC motor-driven rotation stage while the transmission axis of LP is set at 01. The intensity detected by the CCD is converted to a 10-bit digital signal by a frame grabber (FG) and stored in a personal computer (PC) for further processing. All the orientations of optic axis of polarization components are carefully calibrated by the previous developed techniques [18,19]. The probe beam is focused onto a grating G of 300 grooves/mm blazed at
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a
b
β = π/4
β = π/8
c
d
β = 3π/8
β = 3π/4
Fig. 2. Interference fringes of reference and probe beams at four given b of Q3. The status of alignment is indicated by the straight fringes. The fringe position is shifted in accordance with the polarization adjustment.
21340 by a 10 objective L1 in which the diameter of focal point on the grating about 16 mm is obtained. The nominal period of this grating was 3.33 mm and the nominal depth of groove was 145 nm. The calibration of numerical aperture (NA) effects in our case is not necessary since the NA of the objective is smaller than 0.5. However, for NAX0.5, the spacing between fringes increased significantly such that a procedure should be employed to determine the actual height change across the fringes [20]. Noting that the plane wave front of the beam waist should impinges onto the surface of the grating, this can be checked against the interference fringe patterns as depicted in Fig. 2 where straight fringes are observed to indicate the status of alignment of the wave front of the reference and probe beams. Fig. 2 also demonstrates the shifting of fringe position in accordance with the polarization state shifted by PSU. In addition, the blazed grating is slightly tilted so that the zero order beam is retro-reflected from the grating. To reduce the random noise of the image data, 10 frames of the interferogram at each given b are captured and averaged to give the image data of I1 to I4. The raw phase map shown in Fig. 3 is then determined by pixel-by-pixel calculations according to Eq. (12) with these image data. The raw phase map is unwrapped using the Gold-stein’s algorithm [21] which was developed for studying the image from the synthetic-aperture radar signal. Similar algorithms are also developed, for example, for studying the local stress distribution [22]. We use Gold-stein’s algorithm to isolate and connect the residues which indicate phase
Fig. 3. Raw phase map of the blazed grating.
inconsistency in the raw phase map as boundary for the region where path-independent integration of the phase value is determined to generate the unwrapped phase map. The optical system must be carefully aligned so that the wave front reflected from the sample coincides with that
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from reference mirror, which results in fewer residues occurring at phase jumps in raw phase map. The resultant 2-D of the blazed grating is then obtained and is shown in Fig. 4(a) where a step profile is clearly evident. The line scan along x-axis at yE5 mm of Fig. 4(a) is shown in Fig. 4(b) where the step high at 142 nm and the grating period at 3.36 mm are determined from the image data. The same grating is also investigated by AFM, the 2-D image is shown in Fig. 5(a) where the step profile is clearly observed. The line scan of this blazed grating with the AFM is shown in Fig. 5(b) where peak to valley distances 140.05 and 139.42 nm and grating period 3.372 mm are measured. From Figs. 4 and 5, the close agreement of profiling the same grating can be observed between the two methods. 4. The estimated uncertainty of measurements For the full field interferometric profile imaging either by phase shifting interferometry or by the proposed polarization shifting microscopic interferometry, the phase map is determined based on calculating the phase difference ad in single pixel with respect to the intensities of the
Fig. 5. Profile image of the same blazed grating determined by AFM (a), while the line profile is shown in (b). The depth of groove at 140.05 and 139.42 nm and grating spacing 3.372 mm are determined by AFM.
500 450 400 350 nm
300 250 142.3 nm
200 150
3.36 um
100 50 0
0
2
4
6
um
8
10
12
Fig. 4. Surface profile of the blazed grating determined by proposed method in (a), while the line profile along x-axis at yE5 mm is displayed in (b). The depth of groove at about 142 nm and grating period about 3.36 mm can be determined from the image data.
interferogram. The measurement error introduced by the system errors, such as the defect on components, imperfect alignment, or nonlinear response of the detector, can be calibrated to reduce the effects on the measurement. On the other hand, the resolution of the proposed technique is determined by the uncertainty on the measurement of ad [23] that results from the random fluctuation of interference intensities. In view of the interference intensities from Eqs. (8)–(11), the random fluctuation is introduced by the random deviation db of azimuth angle b and the random fluctuation dIin of incident intensity, these two fluctuations are assumed to be statistically independent. The quantization error arising from the analog to digital conversion of the CCD is neglected if the converted digital image spans the whole digits where the number of digits is not smaller than 10 [24]. Hence the uncertainty of the wrapped phase d(ad) can be considered as the square root of the sum of each random fluctuation squared for the image intensity at each b by following equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 X qad 2 X qad dðad Þ ¼ db þ dI in : (13) qb qI in n n In Eq. (13), the index n runs from 1 to 4 due to the four given b values at p/8, p/4, 3p/8 and 3p/4, respectively.
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By using the relation: qad qad qI n ¼ ; qb qI n qb
qad qad qI n ¼ , qI in qI n qI in
(14)
where qI n ¼ qb
qI ; qb n
qI n ¼ qI in
qI qI in
,
(15)
n
are the derivatives of Eq. (7) with b and Iin respectively at given b. After substituting above equations into Eq. (13), it follows that "(
) 2 pffiffiffi <1 2 sinðad Þ þ 8 cos4 ðad Þ ðdbÞ2 < ( " # pffiffiffi 1 3< þ 1 2 1 2 3< þ 1 2 2 þ sin ðad Þ þ ð1 þ 2 sin ðad ÞÞ þ sinðad Þ 8 < 2 < 2 " #) #1=2 1 <þ1 2 1 dI in 2 þcos2 ðad Þ þ sin2 ðad Þ , ð16Þ I in 16 < 4
dðad Þ ¼
2 sin2 ðad Þ
where R ¼ rR/rG is the ratio of amplitude reflectance coefficients. From Eq. (16), the ratio of amplitude reflective coefficients R becomes a factor to the measurement uncertainty. This is because the interference visibility deteriorates due to unequal amplitude reflectance coefficients from the reference mirror and the sample, respectively. The derivation of Eq. (16) is based on: (a) The random deviation of rotation for shifting the azimuth angle of Q3 is independent of b and equals 0.011 from the data sheet provided by the manufacturer. p Thus, ffiffiffiffiffi the averaged random fluctuation becomes db ¼ 0:01 N where N ¼ 10 is the number of image frames to be averaged at rotation step n. (b) The relative random fluctuation of incident intensity is smaller than 0.02% according to the data sheet provided by the manufacturer. This relative random fluctuation is also independent of the rotation. Thus, pffiffiffiffiffiafter averaging at each rotation step, dI in =I in p0:0002= N . By substituting the values of db and dIin/Iin into Eq. (16), the estimated uncertainty of measurement in nanometer scale using Eq. (2) at l ¼ 0.6328 mm is depicted in Fig. 6. The uncertainty of measurement reveals a sinusoidal-like variation as the phase difference ad varied from 0 to 2p. This uncertainty is smaller than 0.02 nm when the ratio RE1 and approaches to 0.1 nm as R approaches 10. Fig. 6 also shows the uncertainty of the proposed method monotonically increasing when the ratio R ¼ rR/rG deviates from unity. This may give low resolution for measuring the profile of a sample with low reflectivity such as a biological tissue or ceramic material. Under this circumstance, the optical axis y of HWP that is located after BX in Fig. 1 can be adjusted so that more power is assigned to the probe beam to compensate for the low reflection. Eq. (12) for determining the phase difference ad still remains unchanged. However, the ratio R becomes: rR cosðyÞ <¼ . rG sinðyÞ
(17)
Fig. 6. Estimated uncertainty of measurement as function of ad and R. The uncertainty is smaller than 0.02 nm as R ¼ 1 and approaches 0.1 nm as R approaches 10.
Thus, R closes to unity by adjusting the optical axis y of HWP. To determine the angle of optical axis, additional interference intensity I0 at b ¼ 0 is suggested at y ¼ p/4 where I in 2 r . (18) 2 R In combination with Eqs. (9) and (11), the ratio R can be determined by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rR I0 <¼ , (19) ¼ arctan rG I2 þ I4 I0
I0 ¼
which gives the azimuth angle y of HWP from Eq. (17). In the proposed method, the interference intensity is varied as Q3 is rotated at given angles. It is useful to estimate the error as a result from the imperfections of Q3. The theoretical uncertainties dad,q introduced by the phase retardation uncertainty d(Dj) defined by d(Dj) ¼ p/2Dj of Q3 and d(Dj)5Dj can be calculated by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X @ad 2 2 dad;q ¼ (20) dðDjÞ . @Dj n n It is assumed that the random fluctuations of interference intensities at each step of n due to phase retardation uncertainty are statistically independent. Going back to Eq. (6) and following a similar expansion in Eqs. (14) and (15), the estimated error of the measurements due to imperfect QWP can be written as: ( 2 pffiffiffi 1 1 2 dad;q ¼ 2 sin2 ðad Þ < þ 2 cos2 ðad Þ 2 < " #) 2 1 1 < þ 2 cos2 ðad Þ dðDjÞ2 . þcos2 ðad Þ 2 < ð21Þ From Eq. (21), it is found that dad,q is also a function of the ratio R ¼ rR/rG, and becomes minimal when R ¼ 1. The estimated result is shown in Fig. 7 where the imperfect
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Fig. 7. The estimated uncertainty of measurement as function of ad,q and R. The imperfect phase retardation of Q3 at d(Dj) ¼ 0.51 is assumed. The estimated uncertainty also shows cyclic variations as function of ad,q, and smaller than 0.5 nm as R ¼ 1 and larger than 1 nm as R approaches 10.
of Q3 with d(Dj) ¼ 0.51 is assumed for a typical commercially available QWP. The estimated uncertainty of measurement translated in nanometer scale shows cyclic variation as function of measured ad. The theoretical uncertainty approaches 0.5 nm as RE1 and becomes larger when R deviates from unit. By comparing the theoretical uncertainties depicted in Fig. 6 with that shown in Fig. 7, it is found that the imperfect of Q3 may be more critical to the uncertainty of proposed measurement method. 5. Discussions and conclusions In the proposed method, the polarization state of light waves reflected from the reference mirror and the sample of interest are adjusted simultaneously by PSU at four steps whereas the interference intensity at each adjustment was captured by a CCD. Since this adjustment is performed after the combination of reference and probe beams at PBS, the optical alignment of the Michelson interferometer remains unaffected during the measurement. Another advantage of polarization shifting is that the nonlinear behavior of PZT that is usually employed as a phase shifter can be avoided. A blazed grating can be employed as, for example, a beam steering device in the liquid crystal display [25], or a coupler in surface emitting lasers [26]. The grating period in our case was larger than the wavelength of light source so that it was not necessary to take into consideration of the polarizationdependent reflection from the grating. This is evident from the experimental result. Once the grating period is smaller than the wavelength of the probe beam, the polarizationdependent reflection should be considered [27]. As far as the time consuming is concerned, it takes 20 s to complete the experiment under PC-controlled rotation stage and image recording procedure. This time for completing the measurement depends on the rotation speed of the Q3 and the number of frames captured at each adjustment for reducing random noise. The real time measurement can be improved
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if an electronically controlled polarization adjustment device, such as liquid crystal wave plate [28], is employed. However, from the estimated error analysis in Eq. (21), we conclude the critical requirement on a programmable wave plate using liquid crystal for real time measurement is unavoidable. The phase shifting microscopic interferometry is a powerful, versatile and noninvasive method for 2-D surface profile measurement. We present a polarization shifting alternative to determine the 2-D surface profile while avoiding the disadvantage of using PZT as a phase shifter in phase shifting interferometry. A single frequency linearly polarized He–Ne laser is employed due to the high resolution of measurement that can be achieved with a phase sensitive approach. An estimated error analysis is given based on the randomness from interference intensities and from the rotation of Q3. The compensation for the degradation of measurement uncertainty due to unequal reflectance coefficients from the reference mirror and the sample respectively is possible by simply rotated the HWP, in which a method to determine the ratio R is also suggested. We also found that the imperfect phase retardation of Q3 in PSU is more critical to the error of measurement. This suggests the careful use of an electronically rotatable wave plate for real time measurement. Acknowledgment This work is supported by National Science Council of Taiwan, ROC, through contract number NSC93-2215-E252-001. References [1] Bartoli A, Poggi P, Quercioli F, Tiribilli B. Fast one-dimensional profilometer with a compack disc pickup. Appl Opt 2001;40:1044–8. [2] Klein EJ, Ramirez WF, Hall JL. A common-path heterodyne interferometer for surface profiling in microelectronic fabrication. Rev Sci Instrum 2001;72:2455–67. [3] White DJ. Surface metrology. Meas Sci Technol 1997;8:955–72. [4] Dorrio BV, Fernandez JL. Phase-evaluation methods in whole-field optical measurement techniques. Meas Sci Technol 1999;10:R33–55. [5] Vatsya SR, Li C, Nikumb SK. Surface profile of material ablated with high-power lasers in ambient air medium. J Appl Phys 2005;97: 034912. [6] Chou C, Lyu CW, Peng LC. Polarized differential-phase laser scanning microscope. Appl Opt 2001;40:95–9. [7] Booth NA, Stanojev B, Chemov AA, Vekilov PG. Differential phaseshifting interferometry for in situ surface characterization during solution growth of crystals. Rev Sci Instrum 2002;73:3540–5. [8] Wang S, Quan C, Tay CJ, Reading I, Fong Z. Measurement of a fiber-end surface profile by use of phase-shifting laser interferometry. Appl Opt 2004;43:49–56. [9] Liang P, Ding J, Jin Z, Guo CS, Wang HT. Two-dimensional wavefront reconstruction from lateral shearing interferograms. Opt Express 2006;14:625–34. [10] Wyant JC. Effect of piezoelectric transducer nonlinearity on phase shift interferometery. Appl Opt 1987;26:1112–6. [11] Creath K. Temporal phase measurement methods. In: Robinson DW, Reid GT, editors. Interferogram analysis. New York: Wiley; 1993.
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