776 OPTICS LETTERS / Vol. 33, No. 8 / April 15, 2008

Simple direct extraction of unknown phase shiftand wavefront reconstruction in generalized

phase-shifting interferometry: algorithmand experiments

X. F. Xu, L. Z. Cai,* Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, andX. X. Shen

Department of Optics, Shandong University, Jinan 250100, China*Corresponding author: lzcai@sdu.edu.cn

Received October 29, 2007; accepted February 28, 2008;posted March 12, 2008 (Doc. ID 89150); published April 8, 2008

An algorithm to extract the arbitrary unknown phase shift and then reconstruct the complex object wave ingeneralized phase-shifting interferometry (GPSI) without the iteration process and measurement of objectwave intensity is proposed. This method can be used for GPSI of any frame number �2. Both computersimulations with smooth and diffusing object surfaces and optical experiments have verified the effective-ness of this method over a wide range of phase shifts with very satisfactory results. © 2008 Optical Societyof America

OCIS codes: 090.2880, 100.3010, 100.2000, 120.5050.

Recently, phase-shifting interferometry (PSI) hasdrawn much attention because of its high precisionand wide applications in many areas [1–3]. The stan-dard PSI requires a special constant phase shift,2� /M (M is an integer �3), of the reference wave be-tween two adjacent steps. However, this requirementis often difficult to exactly meet in practice. To elimi-nate this inconvenience, a generalized method deal-ing with arbitrary phase shifts was developed byGreivenkamp [4]. In this method the amounts ofphase shifts still must be precisely known but are notnecessarily special values. On the other hand, severalphase retrieval algorithms with unknown phaseshifts have also been developed [5–8]. Usually thesemethods have substantial computation loads or addi-tional measurements. Some approaches to calculatethe unknown phase shifts and object wavefront bystatistical approaches [9,10] or iterative methods[11,12] have also been suggested. All of the methodsherein need three or more frames.

To further simplify the measurement, some re-searchers have investigated the possibility of usingonly two frames in PSI and obtained some useful re-sults [13–17]. However, in these methods the knowl-edge of either the phase shift or object wave intensityis needed [13–16] or the iteration process must beemployed [17].

Here we propose a novel approach to directly ex-tract the unknown phase shift from only two adjacentinterferograms and the constant reference wave in-tensity and then reconstruct the object wavefront. Wewill first explain its principles and then give its veri-fication by both simulations and experiments.

Taking two-step generalized phase-shifting inter-ferometry (GPSI) as an example, the intensities ofthe two interferograms can be expressed as

I1 = A2 + A2 + 2AoAr cos �, �1�

o r

0146-9592/08/080776-3/$15.00 ©

I2 = Ao2 + Ar

2 + 2AoAr cos�� − ��, �2�

where Ao and � are the real amplitude and phase dis-tributions of the object wave, respectively, in record-ing plane PH, Ar is the constant amplitude of an on-axis plane reference wave, and � is the arbitraryphase shift of the reference wave between two adja-cent steps, which is presumed in the range of 0���� to avoid possible ambiguity. Clearly, all I1, I2, Ao,and � are functions of coordinates �x ,y�, which areomitted here for brevity. Equations (1) and (2) yield

I2 − I1 = 4AoAr sin�� − �/2�sin��/2�, �3�

I2 + I1 = 2Ao2 + 2Ar

2 + 4AoAr cos�� − �/2�cos��/2�. �4�

By taking the average of the square of Eq. (3) and theaverage of Eq. (4) we have

��I2 − I1�2� = 16Ar2�Ao

2 sin2�� − �/2��sin2��/2�, �5�

�I2� + �I1� = 2�Ao2� + 2Ar

2 + 4Ar�Ao cos�� − �/2��cos��/2�.

�6�

Here and in the following the sign � � means averag-ing for all the pixels over the whole frame.

In the case where the object is at a certain dis-tance, z, away from the recording plane PH, the objectwave in plane PH is actually a Fresnel diffractionfield. As a result, as previously discussed [9,10,18,19],Ao and � can be considered mutually independentand the phase distribution of the diffraction field asnearly random. The first consideration allows one toseparate the amplitude and phase terms in bracketsin Eqs. (5) and (6), and the second leads to the ap-proximation �sin2��−� /2��=1/2 and �cos��−� /2��=0.

Then we have

2008 Optical Society of America

April 15, 2008 / Vol. 33, No. 8 / OPTICS LETTERS 777

��I2 − I1�2� = 4Ar2�Ao

2��1 − cos ��, �7�

�I1� + �I2� = 2��Ao2� + Ar

2�. �8�

To some extent, Eq. (8) reflects the energy conserva-tion in GPSI. From I1 and I2 and the measured con-stant intensity of reference wave Ir=Ar

2, we can cal-culate the unknown phase shift

� = cos−1�1 −��I2 − I1�2�

4Ir��I2�/2 + �I1�/2 − Ir�� . �9�

To reconstruct the object field, we can first calcu-late the object wave intensity by using the relation inthe two-step GPSI [17]

Io = �b − �b2 − 4c�1/2�/2, �10�

where b=I1+I2+2Ir cos � and c= ��I1+I2−2Ir�2+ �I1−I2�2 / tan2�� /2�� /4. Here we assume that Ar isgreater than the maximum of Ao as is the case inpractice to guarantee correct recording. Once Io isfound, the complex object field in plane PH can be re-trieved as [16]

O�x,y� =I1 − Io − Ir

2�Ir�1/2 + iI2 − I1 cos � − �1 − cos ���Io + Ir�

2�Ir�1/2 sin �,

�11�

and the complex object field in the original objectplane PO can be further calculated as the inverseFresnel transform of O�x ,y� by distance z.

Obviously the phase-shift extraction method de-scribed herein for the two-step GPSI can be extendedto GPSI of any frame number �2, since any unknownphase shift can be calculated in this way by using itstwo adjacent interferograms. Once all of the phaseshifts are obtained, the complex object field in theplane PH can be reconstructed with the correspond-ing formula of GPSI [9,10].

A series of computer simulations have been madeto examine the performance of our algorithm. Thesimulation optical setup and the relevant parametersare the same as those used in [9]. The object surfacefor test here is expressed as

W�x,y� = 20� sin��x + x0�2 + �y + y0�2�/36

+ ��x − x1�2 + �y − y1�2�/�2R�, �12�

where R=100 cm, x0=y0=200 pixels, x1=y1=150pixels, and the image size is 512�512 with a15 m�15 m pixel pitch. The wavelength �=532 nm and z=216.5 mm. The three-dimensional(3D) shape and phase map of the object are shown inFigs. 1(a) and 1(b), respectively. Similar pictures inFigs. 1(d) and 1(e) are for the same surface but with aGaussian noise (zero mean and � /20 standard varia-tion for height fluctuation h) added to simulate adiffusing surface. The amplitude of object wave inplane PO decreases from 1 at the center to 0.03 at theedge to simulate the possible change of object lightintensity. For both the smooth and diffusing surfaces,

two interferograms are computer generated with an

assumed phase shift �=1.4000 rad, and the phaseshifts extracted by Eq. (9) are 1.3987 and 1.3992 rad,respectively. With these extracted phase shifts wehave calculated the object wave O�x ,y� in plane PH byEq. (11) for both surfaces and further retrieved thecorresponding object waves in plane PO whose phasemaps are shown in Figs. 1(c) and 1(f). We can see thatthe relative error between the preset and extractedphase shifts is �0.1% in both cases, and the differ-ence between Figs. 1(b) and 1(c) or 1(e) and 1(f) is dif-ficult to recognize.

To estimate the feasibility of this method for differ-ent unknown phase shifts, we have made a series ofsimulations as described herein for both the smoothand diffusing object surfaces with different � from0.1 to 3.1 rad with an interval of 0.1 or 0.2 rad. Fig-ure 2 shows the blind searching results of the phaseshifts, where the diagonal line marked by crossesrepresents the real values of �, and the two curvesmarked by circles and triangles stand for the ex-tracted values of � for the smooth and diffusing ob-ject surfaces, respectively. Figure 2 shows that thephase-shift extraction results are satisfactory overthe wide range of 0.1���2.8 rad. Specifically, thephase-shift extraction error is �0.005 rad for 0.1���2.2 and 0.1���2.8 rad for the smooth and diffus-ing surfaces, respectively, and the phase-shift rangesto assure relative errors �0.5% (0.2%) in these twocases are 0.1–2.4 �0.8–2.0� and 0.6–2.8�1.0–2.6� rad, respectively. From these data we canconclude that the usable phase shift for precise ex-

Fig. 1. Simulation results of phase shift extraction andwavefront reconstruction. (a) Original smooth object sur-face, (b) phase map of (a), (c) reconstructed phase map of(a), (d) original diffusing object surface, (e) phase map of(d), and (f) reconstructed phase map of (d).

traction may be chosen from 0.6 to 2.4 rad or, more

778 OPTICS LETTERS / Vol. 33, No. 8 / April 15, 2008

strictly, from 1 to 2 rad, including the most fre-quently used phase shift in standard PSI, 0.5�. Aninteresting phenomenon is that for a large � a diffus-ing object may yield smaller phase-shift extractionerrors than a similar smooth object; this fact may bedue to the stronger scattering of the diffusing objectand thus a phase distribution in plane PH better de-scribed by random approximation. Similar simula-tions for different Rs have also been made, leading tosimilar results.

Optical experiments of GPSI have also been car-ried out. The experimental setup is similar to thatshown in Fig. 1 in [13] but without the use of randomphase masks. The central part of 1024�1024 pixelson the recording CCD is chosen, �=632.8 nm and z=15.31 cm. The object in plane PO is a United StatesAir Force (USAF) resolution target. With a presetphase shift of 1.5708 rad we obtained two interfero-

Fig. 2. Results of phase-shift extraction for both thesmooth and diffusing object surfaces.

Fig. 3. Optical experimental results: (a) first interfero-gram I1, (b) second interferogram I2, (c) reconstructed im-age of a resolution target, and (d) enlarged central partof (c).

grams as shown in Figs. 3(a) and 3(b). The calculatedphase shift is 1.5657 by using Eq. (9) and the average�Ir� as Ir. The retrieved object image with this phaseshift value, Eq. (11), and inverse Fresnel transform isgiven in Fig. 3(c), and its central part is enlarged inFig. 3(d). Evidently the reconstructed image has highquality with little noise. Many other experimentshave also yielded similar results.

In summary, we have proposed a new method to ex-tract the unknown phase shift in a two-frame GPSIand then reconstruct the complex object field. A se-ries of computer simulations with both the smoothand diffusing objects have verified the effectivenessof this algorithm over a wide range of phase shifts,and optical experiments have further yielded verysatisfactory wavefront retrieval results. This algo-rithm is simple and convenient, since only one for-mula is needed to extract the unknown phase shift,and no iteration process or measurement of objectwave intensity is necessary. It can be used for GPSIof any frame number �2.

We acknowledge support by the National NaturalScience Foundation (grant 60777008) of China.

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