Abstract

Phase-shifting methods using interferogram normalization are often applied to smooth objects, for which the requirements for the normalization approach, including zero-order term elimination and the norm approximation condition, are easily achieved. Here we propose a three-step generalized phase-shifting method using the normalization approach for diffuse objects. In the proposed method, the zero-order terms are sufficiently suppressed by mutual subtraction of the phase-shifted holograms. The norm approximation condition is satisfied, and the complex field of the object wave can be estimated by the normalization approach when the hologram satisfies the phase randomness condition. We present an object wave retrieval algorithm using three phase-shifted holograms, in which estimation of phase-shift values is unnecessary. The proposed method is verified through simulations and optical experiments.

© 2019 Optical Society of America

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References

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2019 (1)

N. Yoshikawa, S. Namiki, and A. Uoya, “Generalized phase-shifting digital holography using normalized phase-shifted holograms,” Opt. Commun. 430, 391–399 (2019).
[Crossref]

2017 (2)

2016 (1)

2015 (3)

2014 (2)

2013 (2)

2012 (1)

2011 (1)

2010 (1)

2009 (2)

2008 (1)

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[Crossref]

2007 (2)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wavefront reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90, 121124 (2007).
[Crossref]

R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express 15, 5631–5640 (2007).
[Crossref]

2006 (1)

2004 (2)

2003 (1)

1999 (1)

1997 (1)

1991 (2)

Belenguer, T.

Cai, L. Z.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wavefront reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90, 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29, 183–185 (2004).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wavefront reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003).
[Crossref]

Carazo, J. M.

Chen, W.

Cheng, X. C.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[Crossref]

Deng, J.

Dong, G. Y.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wavefront reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90, 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
[Crossref]

Du, H.

Estrada, J. C.

Gao, P.

Geist, E.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Guo, H.

Han, B.

Hao, J.

Hao, Q.

Harder, I.

Hu, Y.

Imbe, M.

Kadono, H.

Kajihara, K.

Kasai, K.

Kawai, H.

Lai, G.

Li, C.

Li, R.

Lindlein, N.

Liu, Q.

Lu, R.

Lu, X.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119, 387–391 (2015).
[Crossref]

Lu, X. X.

Lu, Y.

Luo, C.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119, 387–391 (2015).
[Crossref]

Luo, C. S.

Lv, X.

Ma, S. Z.

Mantel, K.

Meng, X. F.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wavefront reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90, 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
[Crossref]

Muffoletto, R. P.

Namiki, S.

N. Yoshikawa, S. Namiki, and A. Uoya, “Generalized phase-shifting digital holography using normalized phase-shifted holograms,” Opt. Commun. 430, 391–399 (2019).
[Crossref]

Nomura, T.

Ohzu, H.

Qin, J.

Quiroga, J. A.

Shen, X. X.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wavefront reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90, 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
[Crossref]

Shiratori, T.

Sorzano, C. O. S.

Sun, P.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119, 387–391 (2015).
[Crossref]

Takaki, Y.

Tian, J.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119, 387–391 (2015).
[Crossref]

Tohline, J. E.

Toyooka, S.

Tyler, J. M.

Uoya, A.

N. Yoshikawa, S. Namiki, and A. Uoya, “Generalized phase-shifting digital holography using normalized phase-shifted holograms,” Opt. Commun. 430, 391–399 (2019).
[Crossref]

Vargas, J.

Wang, H.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119, 387–391 (2015).
[Crossref]

Wang, H. L.

Wang, J.

Wang, K.

Wang, Y. R.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wavefront reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90, 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
[Crossref]

Wang, Z.

Wu, D.

Xu, X. F.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wavefront reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90, 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
[Crossref]

Yamaguchi, I.

Yan, J.

Yang, X. L.

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wavefront reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90, 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29, 183–185 (2004).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wavefront reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003).
[Crossref]

Yao, B.

Yatagai, T.

Yoshikawa, N.

Zhang, H.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wavefront reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90, 121124 (2007).
[Crossref]

Zhang, T.

Zhang, Z.

Zhong, L.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119, 387–391 (2015).
[Crossref]

Zhong, L. Y.

Zhu, Q.

Appl. Opt. (6)

Appl. Phys. B (1)

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119, 387–391 (2015).
[Crossref]

Appl. Phys. Lett. (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wavefront reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90, 121124 (2007).
[Crossref]

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

Opt. Commun. (2)

X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, and X. C. Cheng, “Wavefront reconstruction by two-step generalized phase-shifting interferometry,” Opt. Commun. 281, 5701–5705 (2008).
[Crossref]

N. Yoshikawa, S. Namiki, and A. Uoya, “Generalized phase-shifting digital holography using normalized phase-shifted holograms,” Opt. Commun. 430, 391–399 (2019).
[Crossref]

Opt. Express (4)

Opt. Lett. (12)

N. Yoshikawa and K. Kajihara, “Statistical generalized phase-shifting digital holography with a continuous fringe-scanning scheme,” Opt. Lett. 40, 3149–3152 (2015).
[Crossref]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36, 1326–1328 (2011).
[Crossref]

J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step demodulation based on the Gram-Schmidt orthonormalization method,” Opt. Lett. 37, 443–445 (2012).
[Crossref]

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
[Crossref]

Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34, 1288–1290 (2009).
[Crossref]

P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34, 3553–3555 (2009).
[Crossref]

T. Nomura and M. Imbe, “Single-exposure phase-shifting digital holography using a random-phase reference wave,” Opt. Lett. 35, 2281–2283 (2010).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wavefront reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29, 183–185 (2004).
[Crossref]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671–1673 (2004).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
[Crossref]

H. Kadono and S. Toyooka, “Statistical interferometry based on the statistics of speckle phase,” Opt. Lett. 16, 883–885 (1991).
[Crossref]

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

D. Malacara, ed., Optical Shop Testing, 3rd ed. (Wiley, 2007).

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Figures (12)

Fig. 1.
Fig. 1. Experimental configuration: beam expander, BE; beam splitter, BS; moving mirror, MM; and object, OBJ. The picture shown in the upper middle part is the diffuse object used in the experiment.
Fig. 2.
Fig. 2. Procedure of the object wave retrieval algorithm using normalized holograms.
Fig. 3.
Fig. 3. (a) Amplitude, (b) constant phase, and (c) random phase of the target object.
Fig. 4.
Fig. 4. Phase-shifted holograms of the constant-phase object in (a)–(c) inline configuration ( α = 0 ° ) and (d)–(f) off-axis configuration ( α = 0.18 ° ).
Fig. 5.
Fig. 5. (a) Intensity and (b) phase of reconstructed object wave for constant-phase object and (c) cross-sectional view of the red line in (a) inline configuration ( α = 0 ° ) and (d)–(f) off-axis configuration ( α = 0.18 ° ).
Fig. 6.
Fig. 6. Average error between the original and estimated object waves as a function of the angle of incidence of the reference wave.
Fig. 7.
Fig. 7. Phase-shift estimation error as a function of the angle of incidence of the reference wave.
Fig. 8.
Fig. 8. (a) Intensity and (b) phase of the reconstructed object wave for random-phase object and (c) cross-sectional view of the red line in (a) inline configuration ( α = 0 ° ) and (d)–(f) off-axis configuration ( α = 0.18 ° ).
Fig. 9.
Fig. 9. Typical randomly phase-shifted holograms of a diffuse object.
Fig. 10.
Fig. 10. Object waves retrieved by (a), (b) normalization, (c) statistical, and (d) constant methods, where the reconstructed image has been calculated by the (a) shifted Fresnel transform and (b)–(d) angular spectrum method.
Fig. 11.
Fig. 11. Phase-shift values of Δ ϕ 01 and Δ ϕ 12 estimated by the normalization method.
Fig. 12.
Fig. 12. Correlation of the reconstructed intensity image between the first and following frames. The object wave was retrieved by the normalization method (NORM), statistical method (STAT), and constant method (CONST).

Tables (1)

Tables Icon

Table 1. Calculation Time for Object Wave Retrieval

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

I i ( x , y ) = | O | 2 + | R | 2 + 2 | O | | R | cos ( θ ϕ i ) ,
I p q = 4 | O | | R | sin ( θ ϕ p + ϕ q 2 ) sin ( Δ ϕ p q 2 ) ,
I p q = [ n = 1 N 4 2 | O n | 2 | R | 2 sin 2 ( θ n ϕ p + ϕ q 2 ) sin 2 ( Δ ϕ p q 2 ) ] 1 / 2 = 4 | R | | sin ( Δ ϕ p q 2 ) | [ 1 2 n = 1 N | O n | 2 { 1 cos ( 2 θ n ϕ p ϕ q ) } ] 1 / 2 4 c p q | R | sin ( Δ ϕ p q 2 ) 1 2 n = 1 N | O n | 2 ,
I p q I p q = c p q | Q n | sin ( θ n ϕ p + ϕ q 2 ) ,
I a = I 01 I 01 + I 20 I 20 = 2 | Q n | cos ( θ n ϕ 0 2 ϕ 1 + ϕ 2 4 ) sin ( Δ ϕ 12 4 ) ,
I s = I 01 I 01 I 20 I 20 = 2 | Q n | sin ( θ n ϕ 0 2 ϕ 1 + ϕ 2 4 ) cos ( Δ ϕ 12 4 ) .
I a 2 M sin ( Δ ϕ 12 4 ) ,
I s 2 M cos ( Δ ϕ 12 4 ) ,
I a I a = | Q n | M cos ( θ n ϕ 0 2 ϕ 1 + ϕ 2 4 ) ,
I s I s = | Q n | M sin ( θ n ϕ 0 2 ϕ 1 + ϕ 2 4 ) ,
θ n = tan 1 ( I s / I s I a / I a ) + Δ ϕ 01 2 + Δ ϕ 12 4 + ϕ 0 ,
Δ ϕ 12 = 4 tan 1 ( I a I s ) .
Δ ϕ 01 = 2 sin 1 { I 01 I 12 sin [ 2 tan 1 ( I a I s ) ] } .
| O n | ( I a I a ) 2 + ( I s I s ) 2 .
| O n | e j θ n ( I a I a ) 2 + ( I s I s ) 2 exp { j [ tan 1 ( I s / I s I a / I a ) + ϕ ] } .

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