Abstract

A new method in interferometry based on on-off non-quadrature amplitude modulation for object phase retrieval is presented. Although the technique introduces inhomogeneous visibility and phase variations in the interferogram, it is shown that the phase retrieval of a given object is still possible. This method is implemented by using three beams and two Mach-Zehnder interferometers in series. One of the arms of the system is used as a probe beam and the other two are used as reference beams, yielding from their sum the conventional reference beam of a two-beam interferometer. We demonstrate that, if there is a phase difference within the range of (0,π) between these two beams, the effect of modulation in both amplitude and phase is generated for the case of on-off non-quadrature amplitude modulation. An analytical discussion is provided to sustain this method. Numerical and experimental results are also shown.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Schwider, “Advanced Evaluation Techniques in Interferometry,” in Progress in Optics, Vol. XXVIII, E. Wolf, ed., (Elsevier Science, 1990), pp. 274–276.
  2. D. Malacara, Optical Shop Testing (Wiley, New York, 2007), pp. 547–550.
  3. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A8(5), 822–827 (1991).
    [CrossRef]
  4. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett.29(2), 183–185 (2004).
    [CrossRef] [PubMed]
  5. X. Xu, L. Cai, H. Yuan, Q. Zhang, G. Lu, and C. Wang, “Phase shift selection for two-step generalized phase-shifting interferometry,” Appl. Opt.50(34), H171–H176 (2011).
    [CrossRef] [PubMed]
  6. A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng.43(3-5), 475–490 (2005).
    [CrossRef]
  7. C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol.5(6), 648–654 (1994).
    [CrossRef]
  8. G. S. Han and S. W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt.33(31), 7321–7325 (1994).
    [CrossRef] [PubMed]
  9. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt.13(11), 2693–2703 (1974).
    [CrossRef] [PubMed]
  10. Z. Zhigang, “Numerical analysis of optical bistability based on Fiber Bragg Grating cavity containing a high nonlinearity doped-fiber,” Opt. Commun.285, 521–526 (2011).
  11. Z. Zhi-Gang and Y. Wen-Xuan, “Theoretical an experimental investigation of all-optical switcing based on cascaded LPFGs separated by an erbium-doped fiber,” J. of Appl. Opt.109, 103106 (2011).
  12. Q. Yang, R. Zhou, and B. Zhao, “Principle of the moving-mirror-pair interferometer and the tilt tolerance of the double moving mirror,” Appl. Opt.47(13), 2486–2493 (2008).
    [CrossRef] [PubMed]
  13. J. C. Wyant and R. N. Shagam, “Use of Electronic Phase Measurement Techniques in Optical Testing,” Proc. ICO-11, Madrid, 659–662 (1978).
  14. D. Malacara, I. Rizo, and A. Morales, “Interferometry and the Doppler Effect,” Appl. Opt.8(8), 1746–1747 (1969).
    [CrossRef] [PubMed]
  15. T. Kiire, S. Nakadate, and M. Shibuya, “Phase-shifting interferometer based on changing the direction of linear polarization orthogonally,” Appl. Opt.47(21), 3784–3788 (2008).
    [CrossRef] [PubMed]
  16. T. Susuki and R. Hioki, “Translation of Light Frequency by a Moving Grating,” J. Opt. Soc. Am.57(12), 1551 (1967).
    [CrossRef]
  17. C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett.36(13), 2417–2419 (2011).
    [CrossRef] [PubMed]
  18. C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by the wave amplitude modulation: General case,” Opt. Lasers Eng.50(7), 905–909 (2012).
    [CrossRef]
  19. C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by amplitude modulation,” in: Interferometry research and applications in science and technology, Ivan Padron (Ed.), ISBN 978–953–51–0403–2, InTech, (2012).
  20. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt.21(14), 2470 (1982).
    [CrossRef] [PubMed]

2012 (1)

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by the wave amplitude modulation: General case,” Opt. Lasers Eng.50(7), 905–909 (2012).
[CrossRef]

2011 (4)

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett.36(13), 2417–2419 (2011).
[CrossRef] [PubMed]

X. Xu, L. Cai, H. Yuan, Q. Zhang, G. Lu, and C. Wang, “Phase shift selection for two-step generalized phase-shifting interferometry,” Appl. Opt.50(34), H171–H176 (2011).
[CrossRef] [PubMed]

Z. Zhigang, “Numerical analysis of optical bistability based on Fiber Bragg Grating cavity containing a high nonlinearity doped-fiber,” Opt. Commun.285, 521–526 (2011).

Z. Zhi-Gang and Y. Wen-Xuan, “Theoretical an experimental investigation of all-optical switcing based on cascaded LPFGs separated by an erbium-doped fiber,” J. of Appl. Opt.109, 103106 (2011).

2008 (2)

2005 (1)

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng.43(3-5), 475–490 (2005).
[CrossRef]

2004 (1)

1994 (2)

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol.5(6), 648–654 (1994).
[CrossRef]

G. S. Han and S. W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt.33(31), 7321–7325 (1994).
[CrossRef] [PubMed]

1991 (1)

1982 (1)

1974 (1)

1969 (1)

1967 (1)

Brangaccio, D. J.

Bruning, J. H.

Cai, L.

Cai, L. Z.

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol.5(6), 648–654 (1994).
[CrossRef]

Gallagher, J. E.

Han, G. S.

Herriott, D. R.

Hioki, R.

Itoh, K.

Kiire, T.

Kim, S. W.

Lai, G.

Liu, Q.

Lu, G.

Malacara, D.

Meneses-Fabian, C.

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by the wave amplitude modulation: General case,” Opt. Lasers Eng.50(7), 905–909 (2012).
[CrossRef]

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett.36(13), 2417–2419 (2011).
[CrossRef] [PubMed]

Morales, A.

Nakadate, S.

Patil, A.

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng.43(3-5), 475–490 (2005).
[CrossRef]

Player, M. A.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol.5(6), 648–654 (1994).
[CrossRef]

Rastogi, P.

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng.43(3-5), 475–490 (2005).
[CrossRef]

Rivera-Ortega, U.

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by the wave amplitude modulation: General case,” Opt. Lasers Eng.50(7), 905–909 (2012).
[CrossRef]

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett.36(13), 2417–2419 (2011).
[CrossRef] [PubMed]

Rizo, I.

Rosenfeld, D. P.

Shibuya, M.

Susuki, T.

Wang, C.

Wen-Xuan, Y.

Z. Zhi-Gang and Y. Wen-Xuan, “Theoretical an experimental investigation of all-optical switcing based on cascaded LPFGs separated by an erbium-doped fiber,” J. of Appl. Opt.109, 103106 (2011).

White, A. D.

Xu, X.

Yang, Q.

Yang, X. L.

Yatagai, T.

Yuan, H.

Zhang, Q.

Zhao, B.

Zhigang, Z.

Z. Zhigang, “Numerical analysis of optical bistability based on Fiber Bragg Grating cavity containing a high nonlinearity doped-fiber,” Opt. Commun.285, 521–526 (2011).

Zhi-Gang, Z.

Z. Zhi-Gang and Y. Wen-Xuan, “Theoretical an experimental investigation of all-optical switcing based on cascaded LPFGs separated by an erbium-doped fiber,” J. of Appl. Opt.109, 103106 (2011).

Zhou, R.

Appl. Opt. (7)

J. of Appl. Opt. (1)

Z. Zhi-Gang and Y. Wen-Xuan, “Theoretical an experimental investigation of all-optical switcing based on cascaded LPFGs separated by an erbium-doped fiber,” J. of Appl. Opt.109, 103106 (2011).

J. Opt. Soc. Am. (1)

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

Meas. Sci. Technol. (1)

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol.5(6), 648–654 (1994).
[CrossRef]

Opt. Commun. (1)

Z. Zhigang, “Numerical analysis of optical bistability based on Fiber Bragg Grating cavity containing a high nonlinearity doped-fiber,” Opt. Commun.285, 521–526 (2011).

Opt. Lasers Eng. (2)

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng.43(3-5), 475–490 (2005).
[CrossRef]

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by the wave amplitude modulation: General case,” Opt. Lasers Eng.50(7), 905–909 (2012).
[CrossRef]

Opt. Lett. (2)

Other (4)

J. C. Wyant and R. N. Shagam, “Use of Electronic Phase Measurement Techniques in Optical Testing,” Proc. ICO-11, Madrid, 659–662 (1978).

J. Schwider, “Advanced Evaluation Techniques in Interferometry,” in Progress in Optics, Vol. XXVIII, E. Wolf, ed., (Elsevier Science, 1990), pp. 274–276.

D. Malacara, Optical Shop Testing (Wiley, New York, 2007), pp. 547–550.

C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by amplitude modulation,” in: Interferometry research and applications in science and technology, Ivan Padron (Ed.), ISBN 978–953–51–0403–2, InTech, (2012).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Three-beam Mach-Zehnder interferometer. MO Microscope objective, PH Pinhole, L Convergence lens, BS Beam splitter, AF on-off amplitude filter, A Bean amplitude, and CCD camera.

Fig. 2
Fig. 2

Phase, Amplitude, and Phase and amplitude modulation cases for Δ ϕ 31 = 3π /4 , where A r =0.4,0.7,1.0,1.3,1.6 for N=16 phase-steps with a fundamental step of π/8 .

Fig. 3
Fig. 3

Non-quadrature amplitude modulation and its effects in changes of phase and visibility in an interferogram for the cases: PM, AM, and PAM.

Fig. 4
Fig. 4

Numerical simulation of optical fields at the output of three-beam Mach-Zehnder interferometer: (a)-(c) Amplitudes and phases, (d1) Phase difference ϕ= ϕ 2 ϕ 1 considered as the phase object, and (d2) Phase-difference Δ ϕ 31 = ϕ 3 ϕ 1 within the range ( 0,π ) .

Fig. 5
Fig. 5

Numerical simulation of phase variations because of binary NQAM: (a) Δ ϕ r1 =0 when A F 1 =1 and A F 3 =0 , (b) Δ ϕ r1 Δ ϕ 31 /2 when A F 1 =A F 3 =1 , and (c) Δ ϕ r1 =Δ ϕ 31 when A F 1 =0 and A F 3 =1 .

Fig. 6
Fig. 6

Intensities obtained at interferometer’s output for the all cases of on-off amplitude modulation, except the trivial case, A F 1 =A F 2 =A F 3 =0 .

Fig. 7
Fig. 7

Normalized interferograms and theirs phase spatial variations indicated above of each interferogram as result of on-off NQAM.

Fig. 8
Fig. 8

(a) wrapped phase, (b) unwrapped phase, and (c) comparison by subtraction of proposed phase and unwrapped phase.

Fig. 9
Fig. 9

Intensities obtained experimentally at output of three-beam Mach-Zehnder interferometer for the all cases of binary amplitude modulation, except the trivial case, A F 1 =A F 2 =A F 3 =0 .

Fig. 10
Fig. 10

Experimental results obtained from the intensities in Fig. (9): (a)-(c) Normalized interferograms, (d)-(f) Phase difference variations introduced between each couple of interferograms.

Fig. 11
Fig. 11

Phase retrieval from the experimental normalized interferograms and theirs phase difference variations: (a) wrapped phase and (b) unwrapped phase.

Fig. 12
Fig. 12

Qualitative estimation of phase variations between each couple of interferograms by using the adjustments of ellipses: (a)

Equations (28)

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

I n ( x,y )=a( x,y )+b( x,y )cos[ ϕ( x,y )+ α n ]
E k ( x,y,z,t )= A k ( x,y ) e i[ k z zωt+ ϕ k ( x,y ) ] =A F k A kM ( x,y ) e i[ k z zωt+ ϕ k ( x,y ) ]
E r ( x,y )= A r ( x,y )exp{ i[ k z zωt+ ϕ r ( x,y ) ] },
A r 2 ( x,y )= A 1 2 ( x,y )+ A 3 2 ( x,y )+2 A 1 ( x,y ) A 3 ( x,y )cosΔ ϕ 31 ( x,y ),
tan ϕ r ( x,y )= A 1 ( x,y )sin ϕ 1 ( x,y )+ A 3 ( x,y )sin ϕ 3 ( x,y ) A 1 ( x,y )cos ϕ 1 ( x,y )+ A 3 ( x,y )cos ϕ 3 ( x,y ) ,
tanΔ ϕ r1 ( x,y )= A 3 ( x,y )sinΔ ϕ 31 ( x,y ) A 1 ( x,y )+ A 3 ( x,y )cosΔ ϕ 31 ( x,y ) ,
A 1 ( x,y )= A r ( x,y ) sinΔ ϕ 31 ( x,y ) sin[ Δ ϕ 31 ( x,y )Δ ϕ r1 ( x,y ) ]; Δ ϕ 31 0,π
A 3 ( x,y )= A r ( x,y ) sinΔ ϕ 31 ( x,y ) sinΔ ϕ r1 ( x,y )
I( A 1 , A 2 , A 3 )= A 1 2 + A 2 2 + A 3 2 +2 A 1 A 2 cosϕ+2 A 1 A 3 cosΔ ϕ 31 +2 A 2 A 3 cos( ϕΔ ϕ 31 ),
I( A 1 , A 2 , A 3 )= A r 2 ( A 1 , A 3 )+ A 2 2 +2 A r ( A 1 , A 3 ) A 2 cos( ϕΔ ϕ r1 ),
V( A 1 , A 2 , A 3 )= 2 A r ( A 1 , A 3 ) A 2 A r 2 ( A 1 , A 3 )+ A 2 2 = 2 A 2 A 1 2 + A 3 2 +2 A 1 A 3 cosΔ ϕ 31 A 1 2 + A 2 2 + A 3 2 +2 A 1 A 3 cosΔ ϕ 31 ,
I( A 1 ,0,0 )= A 1M 2 = I 1 ,
I( 0, A 2 ,0 )= A 2M 2 = I 2 ,
I( 0,0, A 3 )= A 3M 2 = I 3 ,
I( A 1 , A 2 ,0 )= A 1M 2 + A 2M 2 +2 A 1M A 2M cosϕ= I 12 ,
I( A 1 ,0, A 3 )= A 1M 2 + A 3M 2 +2 A 1M A 3M cosΔ ϕ 31 = I r ,
I( 0, A 2 , A 3 )= A 2M 2 + A 3M 2 +2 A 2M A 3M cos( ϕΔ ϕ 31 )= I 23
I( A 1 , A 2 , A 3 )= A r 2 ( A 1M , A 3M )+ A 2M 2 +2 A r ( A 1M , A 3M ) A 2M cos( ϕΔ ϕ r1 )=I
cosΔ ϕ 31 = I r I 1 I 3 2 I 1 I 3 ; Δ ϕ 31 ( 0,π ),
tanΔ ϕ r1 = 4 I 1 I 3 ( I r I 1 I 3 ) 2 I r + I 1 I 3 ,
cosΔ ϕ r1 = I r + I 1 I 3 2 I r I 1 ; Δ ϕ r1 [ 0,Δ ϕ 31 ],
I ¯ 12 =cosϕ= I 12 I 1 I 2 2 I 1 I 2 ,
I ¯ =cos( ϕΔ ϕ r1 )= I I r I 2 2 I r I 2 ,
I ¯ 23 =cos( ϕΔ ϕ 31 )= I 23 I 2 I 3 2 I 2 I 3 ,
tanϕ= I ¯ 23 I ¯ 12 cosΔ ϕ 31 I ¯ 12 sinΔ ϕ 31 = 2 I 1 ( I 23 I 2 I 3 )( I 12 I 1 I 2 )( I r I 1 I 3 ) ( I 12 I 1 I 2 ) 4 I 1 I 3 ( I r I 1 I 3 ) 2 ,
tanϕ= I ¯ I ¯ 12 cosΔ ϕ r1 I ¯ 12 sinΔ ϕ r1 = 2 I 1 ( I I r I 2 )( I 12 I 1 I 2 )( I r + I 1 I 3 ) ( I 12 I 1 I 2 ) 4 I 1 I 3 ( I r I 1 I 3 ) 2 ,
tanϕ= I ¯ cosΔ ϕ 31 I ¯ 23 cosΔ ϕ r1 I ¯ sinΔ ϕ 31 I ¯ 23 sinΔ ϕ r1 = ( I I r I 2 )( I r I 1 I 3 )( I 23 I 2 I 3 )( I r + I 1 I 3 ) ( I I r I 23 I 3 ) 4 I 1 I 3 ( I r I 1 I 3 ) 2
A kM ( x,y )=exp( ( x x k ) 2 + ( y y k ) 2 σ k 2 ); ϕ k ( x,y )= ( x c k ) 2 ( y d k ) 2 h k 2 ,

Metrics