Abstract

There are many applications in biology and metrology where it is important to be able to measure both the amplitude and phase of an optical wave field. There are several different techniques for making this type of measurement, including digital holography and phase retrieval methods. In this paper we propose an analytical generalization of this two-step phase-shifting algorithm. We investigate how to reconstruct the object signal if both reference waves are different in phase and amplitude. The resulting equations produce two different solutions and hence an ambiguity remains as to the correct solution. Because of the complexity of the generalized analytical expressions we propose a graphical-vectorial method for solution of this ambiguity problem. Combining our graphical method with a constraint on the amplitude of the object field we can unambiguously determine the correct result. The results of the simulation are presented and discussed.

© 2012 Optical Society of America

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References

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2011 (6)

2010 (2)

2009 (5)

2008 (1)

2007 (1)

2006 (2)

2004 (1)

2002 (1)

Andrés, P.

Araiza-Esquivel, M. A.

Awatsuji, Y.

Belenguer, T.

Cai, L. Z.

Chang, C. C.

Chen, G. L.

Chen, P.

Devaney, A. J.

Dong, G. Y.

Estrada, J. C.

Fujii, A.

Gao, P.

Guo, C.-S.

Guo, P.

Guo, R.

Harder, I.

Imbe, M.

Javidi, B.

Jhou, Gu.

Kaneko, A.

Koyama, T.

Kubota, T.

Kuo, M. K.

Lancis, J.

Liao, J.

Lin, C. Y.

Liu, J.

Liu, J. P.

Lu, W.

Mantel, K.

Martinez-León, L.

Matoba, O.

Meneses-Fabian, C.

Meng, X. F.

Min, J.

Nercissian, V.

Nishio, K.

Nomura, T.

Pan, W.

Poon, T.

Poon, T. C.

Quiroga, J. A.

Rinehart, M. T.

Rivera-Ortega, U.

Servin, M.

Servín, M.

Shaked, N. T.

Shen, X. X.

Tahara, T.

Tajahuerce, E.

Ura, S.

Vargas, J.

Wang, H.-T.

Wang, Ji.

Wang, Y. V.

Wax, A.

Xu, X. F.

Yang, X. L.

Yao, B.

Yao, H. F.

Ye, T.

Zhang, L.

Zheng, J.

Zhu, Y.

Zhu, Y. Y.

Appl. Opt. (5)

Chin. Opt. Lett. (1)

J. Opt. Soc. Am. A (1)

Opt. Express (5)

Opt. Lett. (7)

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Figures (5)

Fig. 1.
Fig. 1.

Experimental setup illustrated schematically. D1 and D1, matrix detectors; AO, unknown complex amplitude of object wave; AR1 and AR2, complex amplitude distributions of two arbitrary reference waves; BS, beam splitter.

Fig. 2.
Fig. 2.

Graphical vectorial illustration of Eq. (4) at the complex plane for some possible values of the object value. (a) Illustration of some possible values which are an answer to Eqs. (4a) and (4b). (b) Illustration of some possible values which are an answer to Eqs. (4c) and (4d). A¯R1 is a known reference complex value, A¯Oobject complex value, c1 is a circle with radius |A¯1|=[I1]1/2 and center at the coordinate origin. c2 is a circle with radius |A¯1|=[I1]1/2 and center at the coordinate origin.

Fig. 3.
Fig. 3.

Graphical solution of Eq. (4). Example.

Fig. 4.
Fig. 4.

Mach–Zehnder interferometric setup for two-step PSI. BS1, BS2, beam splitters; λ/4, quarter-wave plate; P, polarizer; M1, M2, mirrors; GG, ground glass; Ob, object; D1, D2, detectors; MO, micro-objective; L1–L4 , lenses; WP, Wollaston prism; IP1, IP2, imaging planes.

Fig. 5.
Fig. 5.

Simulation results. (a) Amplitude object; (b), (c) interferograms I1, I2 accordingly; (d) reconstructed amplitude of object wave; (e) error regions by recovering object amplitude; (f) error regions by recovering object phase.

Equations (28)

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{I1=IR1+IO+2IR1IOcos(φR1φO)I2=IR2+IO+2IR2IOcos(φR2φO),
[IR1+IR22cos(d)IR1IR2]IO2+[2cos(d)(J2+J1)IR1IR22J1IR22J2IR14sin2(d)IR1IR2]IO+J12IR22cos(d)J1J2IR1IR2+J22IR1=0,
{φO=φR1±arccos(I1IR1IO2IR1IO)φO=φR2±arccos(I2IR2IO2IR2IO).
{U¯1=U¯R1+U¯OI1=|U¯1|U¯2=U¯R2+U¯OI2=|U¯2|.
h=AO,max=2|SΔO1RO2||U¯R1U¯R2|=|U¯R1×U¯R2||U¯R1U¯R2|=|U¯R1||U¯R2||sin(δR)||U¯R1|2+|U¯R2|22|U¯R1||U¯R2|cos(δR).
{I1=IR1+IO+2IR1IOcos(φR1φO)I2=IR2+IO+2IR2IOcos(φR2φO).
φR1φO=x;φR2φR1=d;I1IR1=J1;I2IR2=J2.
{2IR1IOcos(x)=J1IO2IR2IOcos(x+d)=J2IO,
{cos(x)=J1IO2IR1IOcos(x+d)=J2IO2IR2IO.
{cos((x+d2)d2)=J1IO2IR1IOcos((x+d2)+d2)=J2IO2IR2IO,
{cos(d/2)cos(x+d/2)+sin(d/2)sin(x+d/2)=J1IO2IR1IOcos(d/2)cos(x+d/2)sin(d/2)sin(x+d/2)=J2IO2IR2IO.
{2cos(d/2)cos(x+d/2)=J1IO2IR1IO+J2IO2IR2IO2sin(d/2)sin(x+d/2)=J1IO2IR1IOJ2IO2IR2IO,
{cos(x+d/2)=J1IO4cos(d/2)IR1IO+J2IO4cos(d/2)IR2IOsin(x+d/2)=J1IO4sin(d/2)IR1IOJ2IO4sin(d/2)IR2IO,
d0;dπ.
1=[J1IO4cos(d/2)IR1IO+J2IO4cos(d/2)IR2IO]2+[J1IO4sin(d/2)IR1IOJ2A24sin(d/2)IR2IO]2,
16IO=[J1IOcos(d/2)IR1+J2IOcos(d/2)IR2]2+[J1IOsin(d/2)IR1J2IOsin(d/2)IR2]2,
16IO=(J1IO)2cos2(d/2)IR1+2(J1IO)(J2IO)cos2(d/2)IR1IR2+(J2IO)2cos2(d/2)IR2+(J1IO)2sin2(d/2)IR12(J1IO)(J2IO)sin2(d/2)IR1IR2+(J2IO)2sin2(d/2)IR2,
16IO=(J1IO)2sin2(d/2)+cos2(d/2)(J1IO)2cos2(d/2)sin2(h)IR1+2(J1IO)(J2IO)sin2(d/2)2(J1IO)(J2IO)cos2(d/2)cos2(d/2)sin2(d/2)IR1IR2+(J2IO)2sin2(d/2)+(J2IO)2cos2(d/2)cos2(d/2)sin2(d/2)IR2,
16IO=4(J1IO)2sin2(d)IR18cos(d)(J1IO)(J2IO)sin2(d)IR1IR2+4(J2IO)2sin2(d)IR2,
4IO=(J1IO)2sin2(d)IR12cos(d)(J1IO)(J2IO)sin2(d)IR1IR2+(J2IO)2sin2(d)IR2.
4sin2(d)IR1IR2IO=(J1IO)2IR22cos(d)(J1IO)(J2IO)IR1IR2+(J2IO)2IR1
4sin2(d)IR1IR2IO=(J122J1IO+IO2)IR22cos(d)(J1J2(J2+J1)IO+IO2)IR1IR2+(J222J2IO+IO2)IR1
4sin2(d)IR1IR2IO=J12IR22J1IR2IO+IR2IO22cos(d)J1J2IR1IR2+2cos(d)(J2+J1)IR1IR2IO2cos(d)IR1IR2IO2+J22IR12J2IR1IO+IR1IO2
[IR1+IR22cos(d)IR1IR2]IO2+[2cos(d)(J2+J1)IR1IR22J1IR22J2IR14sin2(d)IR1IR2]IO+J12IR22cos(d)J1J2IR1IR2+J22IR1=0
{I1IR1+IO=2IR1IOcos(φR1φO)I2IR2+IO=2IR2IOcos(φR2φO),
{cos(φR1φO)=I1IR1+IO2IR1IOcos(φR2φO)=I2IR2+IO2IR2IO.
{φR1φO=±arccos(I1IR1+IO2IR1IO)φR2φO=±arccos(I2IR2+IO2IR2IO),
{φO=φR1±arccos[I1IR1+IO2IR1IO]φO=φR2±arccos[I2IR2+IO2IR2IO].

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