Parallel phase-shifting interferometry based onMichelson-like architecture

Junwei Min,1,2 Baoli Yao,1,* Peng Gao,1 Rongli Guo,1 Juanjuan Zheng,1 and Tong Ye1

1State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics,Chinese Academy of Sciences, Xi’an 710119, China

2Graduate School of the Chinese Academy of Sciences, Beijing 100039, China

*Corresponding author: yaobl@opt.ac.cn

Received 24 August 2010; accepted 25 October 2010;posted 28 October 2010 (Doc. ID 133825); published 24 November 2010

In this paper, we present a new scheme for parallel phase-shifting interferometry that employs aMichelson-like architecture and a simple polarization unit to generate two phase-shifting interferogramswith phase shift of π=2 at a single camera exposure. The parallel phase-shifting unit is built with simpleoptical components, and the distance between the parallel interferograms can be adjusted conveniently.Phase reconstruction is performed by using an algorithm developed for two-step phase-shifting inter-ferometry. The practicability of the proposed configuration and the reconstruction method is demon-strated by experiments. © 2010 Optical Society of AmericaOCIS codes: 050.5080, 090.2880, 260.5430, 180.3170.

1. Introduction

Phase-shifting interferometry is a reliable techniquefor extracting phase information from interfero-grams. This technique has been actively investigatedin many fields, such as high precision optical mea-surements [1,2], digital holography [3,4], specklemetrology [5,6] encryption and information security[7], and other cases [8]. Although the phase-shiftingmethod is effective for eliminating the zero-order andtwin images, it is useless for instantaneous measure-ment of moving objects, since the phase-shiftedfringe patterns are acquired time sequentially.To overcome this drawback, several simultaneousphase-shifting interferometric systems have beenproposed. Smythe and Moore [9], Koliopoulos [10],and Sivakumar et al. [11] employed four CCDs to per-form four-step phase shifting and captured four dif-ferent phase-shifted interferograms simultaneously.However, the employment of multiple imaging sen-sors in an interferometric system brings not only acomplex system configuration, but also potentially

encounters difficulties in accurately mapping pixelpositions between sensors. Another kind of setupusing a single CCD and a micropolarizer array hasbeen established to perform parallel phase-shiftinginterferometry [12–14]. Although the method is lar-gely insensitive to the effects of vibration, the suc-cessful operation of these systems heavily relied onthe effectiveness of the micropolarizer array ele-ment. Chen et al. [15] proposed a simple simulta-neous phase-shifting interferometer that employeda single glass plate to generate parallel phase-shiftedinterferograms. However, the reflection effect of theglass plate influences the fringe’s contrast. Toto-Arellano et al. [16] developed a one-shot phase-shifting interferometry system that employs atwo-dimensional diffraction grating and can acquiremultiple phase-shifted interferograms in one shot,from four interferograms [17] up to nine interfero-grams [18]. However, the distance, i.e., the vacantregion, among the interferograms is not adjustable.Besides, the successful operation of these systemsrelied on the effectiveness of the phase grating.

In this paper, we propose a new configuration forparallel phase-shifting interferometry in which twoparallel interferograms with a relative phase shift

0003-6935/10/346612-05$15.00/0© 2010 Optical Society of America

6612 APPLIED OPTICS / Vol. 49, No. 34 / 1 December 2010

of π=2 are recorded at a single shot. The system isachieved by using a Michelson-like architectureand a polarization unit. The advantages of the pro-posed parallel phase-shifting interferometry arethe possibility to adjust the distance between inter-ferograms and that it is built with simple opticalcomponents. The system layout and the parallelphase-shifting principle are presented in Section 2.Following this, the mathematical principle of thetwo-step phase-shifting algorithm for phase recon-struction is detailed in Section 3. To verify the effec-tiveness of the proposed method, we perform anexperiment in Section 4.

2. Principle of Parallel Phase Shiftingand Optical Setup

The proposed parallel two-step phase-shifting inter-ferometry configuration is shown in Fig. 1(a). Basedon the polarization phase-shifting principle [19], per-pendicularly linearly polarized object and referencewaves are used to generate the phase-shifting inter-ferograms. The proposed configuration is based on aMichelson-like architecture, incorporated with a po-larization unit. Differently from the systems that em-ployed a phase grating in the Fourier plane [16–18],where the distance between the interferograms is notadjustable since it is determined by the period ofphase grating, in the proposed configuration, the dis-tance between the two interferograms can be ad-justed by changing the angle of reflection. It isseen from the ray trajectory in Fig. 1(a) that thecombined beam (of perpendicularly linearly polar-ized object and reference waves) is split by thenonpolarizing cube beam splitter (NPBS) into a re-flection beam and a transmission beam. The twobeams are reflected by mirrors M1 and M2 at theirfocal points S1 and S2, respectively. The normals ofmirrors M1 and M2 have a small angle with respectto the optical axial direction, thus the reflectedbeams are in directions with angular offsets α andβ to the optical axis, respectively. The two split beams

can be seen as being emitted from the same point S1and S0

2, and S02 is the image of point S2. Thus, in the

Fourier plane, ðξ; ηÞ of lens L1, the frequency spectraGrðξ; ηÞ and Gtðξ; ηÞ of the two split beams can bewritten as

Grðξ; ηÞ ¼ FTfOþ Rg expð−i2πξ sin α=λÞ;Gtðξ; ηÞ ¼ FTfOþ Rg expði2πη sin β=λÞ;

ð1Þ

where FTfOþ Rg denotes the Fourier transform ofthe object and reference waves. ξ and η denote thecoordinates in the frequency domain. After passingthrough lens L2, the complex amplitude distributionsof the two split beams are given by

Urðx; yÞ ¼ Oðxþ f sin α; yÞ þ Rðxþ f sin α; yÞ;Utðx; yÞ ¼ Oðx − f sin β; yÞ þ Rðx − f sin β; yÞ;

ð2Þ

where f denotes the focal length of lens L2. FromEq. (2), we know that the split beams have a distanceof X1 ¼ f sin α and X2 ¼ f sin β with respect to the op-tical axis in the image plane, respectively. Thus thedistance d ¼ f ðsin αþ sin βÞ between the two splitbeams can be controlled by changing the angle α,β. The combination of the quarter-wave plate, QW,and the polarizer array is used for obtaining desiredphase shift. Behind the quarter-wave plate, whichhas its fast axis at π=4 with respect to the polariza-tions of O and R, the linearly polarized object and re-ference waves in the two split beams becomeperpendicularly circularly polarized. The polarizer-array device, which is comprised of two polarizerswith their polarizations crossed at a π=4 angle toeach other, is located in the paths of the two splitbeams. According to the polarization phase-shiftingprinciple [19], the two parallel interferograms havea phase shift of π=2 between each other, as is shownin Fig. 1(b).

The experimental setup based on a micro-scopic Mach–Zehnder interferometer is illustrated

Fig. 1. (Color online) Parallel two-step phase-shifting interferometry. (a) Schematic of parallel two-step phase-shifting interferometry, (b)obtained phase-shifting interferograms with phase shift π=2.O andR, object wave and reference wave; L1, L2, lenses; NPBS, nonpolarizingbeam splitter; M1, M2, mirrors; QW, quarter-wave plate with its principal axis having the angle π=4 with respect to the polarizations of Oand R; the polarizations of O and R are linearly perpendicular. The polarizer array is comprised of two polarizers with their polarizationscrossed at π=4 angle.

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in Fig. 2. A randomly polarized He–Ne laser is usedas a light source. A variable neutral density filter isused for the adjustment of the intensity of the inter-ferometer. After passing through linear polarizer P,the laser beam becomes a linearly polarized wave,which is split into two perpendicularly linearly polar-ized beams by the polarizing beam splitter, PBS. Thereflection beam (s-polarization) is expanded by thebeam expander BE1 and illuminates the specimen.The transmitted wave is magnified by themicroscopeobjective, MO, and is collimated by the lens L1 into aplane wave as the object wave O. The transmissionbeam (p-polarization), serving as the reference waveR, is reflected by mirror M2 and expanded by thebeam expander BE2. The relative intensity of theO and R beams is adjusted by the polarizer, P. TheO and R beams are combined by the nonpolarizingbeam splitter NPBS1, where there is no angular off-set between the two beams. The perpendicularly lin-early polarized object and reference waves passthrough the parallel phase-shifting unit that weproposed above, and form the two parallel phase-shifting interferograms with the phase shift π=2. Arectangular aperture is used to ensure that only halfof the digital camera plane is illuminated by eachinterferogram.

3. Two-Step Phase-Shifting Algorithm

The processing algorithm for reconstruction of thephase-shifting interferograms is described below.The two π=2 phase-shifted interferograms, whichare acquired in a single camera exposure, can bemathematically expressed as follows:

�I1 ¼ IR þ IS þ 2

ffiffiffiffiffiffiffiffiffiffiIRIS

pcosφ

I2 ¼ IR þ IS − 2ffiffiffiffiffiffiffiffiffiffiIRIS

psinφ

; ð3Þ

where IR and IS are the reference and object inten-sity distributions, respectively, and φ is the total

phase difference including the specimen φS andthe background phase φB. From Eqs. (3) we have

�2

ffiffiffiffiffiffiffiffiffiffiIRIS

pcosφ ¼ I1 − IR − IS

−2ffiffiffiffiffiffiffiffiffiffiIRIS

psinφ ¼ I2 − IR − IS

: ð4Þ

By using the relation of sin2φþ cos2φ ¼ 1, the follow-ing expression can be obtained:

4IRIS ¼ ðI1 − IR − ISÞ2 þ ðI2 − IR − ISÞ2: ð5Þ

To guarantee correct reconstruction with this meth-od, the reference intensity IR must be large enoughcompared to the object intensity IS [20]. Thus, the in-tensity of the object wave and the overall wrappedphase is calculated as follows:

8<:

ISðx; yÞ ¼ ðI1þI2Þ−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðI1þI2Þ2−2ðI1−IRÞ2−2ðI2−IRÞ2

p2

~φðx; yÞ ¼ arctan�−

I2−IS−IRI1−IS−IR

�modð2πÞ

: ð6Þ

After removal of the 2π ambiguity by a phase un-wrapping process [21], the total phase informationφðx; yÞ can be obtained.

To remove the background phase, the phase retrie-val procedure requires measuring the backgroundphase of the setup without the specimen [22]. Threemeasurements are taken without the presence of thespecimen: the object arm intensity I0S, the referencearm intensity I0R, and one shot of parallel phase-shifting interferogram ðI1; I2Þ. Using these data,the wrapped background phase ~φB can be calculatedas follows:

~φB ¼ arctan�−

I02 − I0S − I0RI01 − I0S − I0R

�modð2πÞ: ð7Þ

Fig. 2. (Color online) Experimental setup for the parallel two-step phase-shifting interferomety. NF, variable neutral filter; P, linearpolarizer; PBS, polarizing beam splitter; NPBS1, NPBS2, nonpolarizing beam splitters; M1–M4, mirrors; BE1, BE2, beam expanders;MO, microscope objective; L1–L3, lenses; QW, quarter-wave plate.

6614 APPLIED OPTICS / Vol. 49, No. 34 / 1 December 2010

Then, an unwrapping algorithm is applied to solve2π ambiguities in ~φB, which yields φB. Finally, thespecimen phase is calculated by φS ¼ φ − φB.

This method is not limited to the rate at which thedynamic process changes. The only limiting factorhere is the frame rate of the CCD, because the infor-mation for each observation of the specimen is ac-quired in a single camera exposure, as well as thea priori knowledge of the reference intensity IR,which is assumed as constant during the observationperiod. Thus, the dynamic process measurementscan be performed by obtaining parallel recordingsof two π=2 phase-shifted on-axis interferogramsðI1; I2Þ, using the system illustrated in Fig. 2.

4. Experiment Results

An experiment was conducted to verify the validity ofthe proposed parallel phase-shifting interferometry.In our setup, the laser beams are expanded to adiameter of 10 mm with beam expanders BE1 andBE2. The microscope objective used has a magnifica-tion M ¼ 40 and numerical aperture NA ¼ 0:6. Thefocal length of each lens L2 and L3 is 75 mm, andthe total magnification of the system is 22:5×. Thetest phase object is a quartz refractive microlens ar-ray and the interferograms were acquired by a CCDcamera with 1024ðHÞ × 768ðVÞ pixels of pixel size4:65 μmðHÞ × 4:65 μmðVÞ. The reference beam is ad-justed to be enough stronger than the object beam toguarantee correct reconstruction with the proposedmethod [20].

Figure 3(a) shows the two parallel phase-shiftinginterferograms of the microlens array acquired ata single camera exposure. I1 and I2 are side by sidein the plane transversal to the light propagation di-rection. From Fig. 3(a), it is clear that I1 and I2 haverelative phase shift of π=2, which is introduced by thepolarization unit. The relatively broad region be-tween I1 and I2 is determined by the focal lengthof lens L3 and the angle between mirrors M3 andM4 with respect to the vertical and horizontal direc-tions, respectively. The final reconstructed phase dis-

tribution from the interferograms according to thealgorithm described in Section 3 is shown in Fig. 3(b).Phase unwrapping is performed numerically using anoniterative least-squares algorithm [21]. The re-peatability of the setup, which is defined as the rootmean square (RMS) of the phase difference for twomeasurements with the same specimen-free setup,is measured to be around λ=50.

5. Conclusion

We have presented a configuration for parallel two-step phase shifting at the same time by usingMichelson-like architecture and a polarization unit.The experimental results show that the proposedmethod can reconstruct instantaneously the ampli-tude and phase of the object with a one-shotmeasure-ment; thus, it can be used for measuring movingobjects and dynamic processes. For further applica-tion, this configuration can be developed to performparallel four-step phase-shifting interferometry.

We thank Dr. Guanghua Cheng from Xi’an Insti-tute of Optics and Precision Mechanics, ChineseAcademy of Sciences for providing with the micro-scopic specimen for the experimental test. Thisresearch is supported by the National NaturalScience Foundation of China (NSFC) (61077005,10874240).

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Fig. 3. (Color online) Measurement results of the quartz refractive microlens array: (a) two parallel phase-shifting interferogramsacquired at a single camera exposure, (b) reconstructed pseudo-three-dimensional profile of the microlens array after removal of the back-ground phase.

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