Abstract

A novel method to set the proper phase steps, as used in phase-stepped interferometry, is presented. It is indicated how and when this method can be used. With only two images one can deduce the relative phase step between them by calculating the correlation between the two images. The error of the proposed method is shown to be smaller than 0.1%.

© 1999 Optical Society of America

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References

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  1. Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef]
  2. D. Malacara, ed., Optical Shop Testing, in Pure and Applied Optics, 2nd ed. (Wiley, New York, 1992).
  3. J. Schwider, R. Burow, K.-E. Elssner, J. Grazanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  4. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  5. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.
  6. H. van Brug, “Zernike polynomials as basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36, 2788–2790 (1997).
    [CrossRef] [PubMed]
  7. H. van Brug, “Efficient Cartesian representation of Zernike polynomials in computer memory,” in Fifth International Topical Meeting on Education and Training in Optics, C. H. F. Velzel, ed., Proc. SPIE3190, 382–392 (1997).
    [CrossRef]

1997 (1)

1987 (1)

1985 (1)

1983 (1)

Burow, R.

Cheng, Y.-Y.

Eiju, T.

Elssner, K.-E.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.

Grazanna, J.

Hariharan, P.

Merkel, K.

Oreb, B. F.

Schwider, J.

Spolaczyk, R.

van Brug, H.

H. van Brug, “Zernike polynomials as basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36, 2788–2790 (1997).
[CrossRef] [PubMed]

H. van Brug, “Efficient Cartesian representation of Zernike polynomials in computer memory,” in Fifth International Topical Meeting on Education and Training in Optics, C. H. F. Velzel, ed., Proc. SPIE3190, 382–392 (1997).
[CrossRef]

Wyant, J. C.

Appl. Opt. (4)

Other (3)

H. van Brug, “Efficient Cartesian representation of Zernike polynomials in computer memory,” in Fifth International Topical Meeting on Education and Training in Optics, C. H. F. Velzel, ed., Proc. SPIE3190, 382–392 (1997).
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2.

D. Malacara, ed., Optical Shop Testing, in Pure and Applied Optics, 2nd ed. (Wiley, New York, 1992).

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Figures (6)

Fig. 1
Fig. 1

Intensity distribution of (a) the second, (b) the fifth, (c) the thirteenth Zernike polynomial, all shown with an amplitude of four wavelengths.

Fig. 2
Fig. 2

Error in (a) the π/4, (b) the π/3, (c) the 2π/3 phase-step retrieval as a function of the fringe density, shown on a logarithmic scale.

Fig. 3
Fig. 3

Error in the π/2 phase-step retrieval as a function of the fringe density, shown on (a) a linear and (b) a logarithmic scale.

Fig. 4
Fig. 4

Error in the π/4 phase-step retrieval as a function of the amplitude of (a) the fifth and (b) the thirteenth Zernike polynomial.

Fig. 5
Fig. 5

Intensity distribution of the second Zernike polynomial, having an amplitude of four wavelengths added to it with an amplitude of one wavelength (a) the fifth and (b) the thirteenth Zernike polynomial.

Fig. 6
Fig. 6

Error in the π/4 phase-step retrieval as a function of the fringe density. The fringe pattern contains the thirteenth Zernike polynomial with (a) amplitude 0.25 or (b) amplitude 1.0, and the second Zernike polynomial with a variable scaling.

Tables (1)

Tables Icon

Table 1 Error (%) Results of the Simulation for Several Often-Used Phase Step Sizesa

Equations (21)

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Ii=Ib1+M cosφ+i-1.5δi=0,, 3.
δ/2=arctan3I1-I2-I0-I3I0-I3+I1-I21/2.
Ii=Ib1+M cosφ+i-2δ i=0,, 4.
δ=arccos12I4-I0I3-I1.
I1=Ib11+M1 cosφ,
I2=Ib21+M2 cosφ+δ,
c12=I1I2-I1I2I12-I121/2I22-I221/2,
I1=12π02π I1dx=Ib1,
I2=12π02π I2dx=Ib2,
I1I2=12π02π I1I2dx=Ib1Ib2+Ib1Ib2M1M22cosδ,
I12=12π02π I12dx=Ib12+Ib12M122,
I22=12π02π I22dx=Ib22+Ib22M222,
c12=cosδ,
I1=1+cosi-12π/NPTS,
I2=1+cosi-12π/NPTS+δ,
Error%=|δcalc-δset|δset×100%.
ux, y=A expjWx, y2πλ.
Ix, y=IB1+M cosWx, y2πλ+δ,
Wx, y=ax,
Wx, y=a-1+2y2+2x2,
Wx, y=a1-6y2+6y4-6x2+12x2y2+6x4.

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