Abstract

A novel method of determining phase from a modulated intensity pattern is described. A line integral of the gradient of the phase is used to reconstruct the phase, eliminating the necessity for complex methods of phase unwrapping. The new algorithm can be used with any technique that experimentally or theoretically yields the cosine and sine or the tangent of the phase. This phase-reconstruction process works effectively even in the regions of high-intensity gradients and is insensitive to the profile of the illuminating beams and to the shape of the domain boundaries.

© 1997 Optical Society of America

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References

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  1. J. E. Grievenkamp and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 501.
  2. C. Joenathan, Appl. Opt. 33, 4147 (1994).
    [CrossRef] [PubMed]
  3. O. Y. Kwon, D. Shough, and R. A. Williams, Opt. Lett. 12, 855 (1987).
    [CrossRef] [PubMed]
  4. M. Kujawinska, J. Opt. Soc. Am. A 5, 849 (1988).
    [CrossRef]
  5. M. Takeda, H. Ina, and S. Kobayashi, J. Opt. Soc. Am. 72, 156 (1982).
    [CrossRef]
  6. D. W. Robinson, in Interferogram Analysis, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 195.
  7. C. P. Brophy, J. Opt. Soc. Am. A 7, 537 (1990).
    [CrossRef]
  8. R. Onodera and Y. Ishii, J. Opt. Soc. Am. A 13, 139 (1996).
    [CrossRef]
  9. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 155.
  10. M. S. Scholl, J. Mod. Opt. 43, 1583 (1996).
    [CrossRef]

1996 (2)

1994 (1)

1990 (1)

1988 (1)

1987 (1)

1982 (1)

Brophy, C. P.

Bruning, J. H.

J. E. Grievenkamp and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 501.

Grievenkamp, J. E.

J. E. Grievenkamp and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 501.

Ina, H.

Ishii, Y.

Joenathan, C.

Kobayashi, S.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 155.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 155.

Kujawinska, M.

Kwon, O. Y.

Onodera, R.

Robinson, D. W.

D. W. Robinson, in Interferogram Analysis, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 195.

Scholl, M. S.

M. S. Scholl, J. Mod. Opt. 43, 1583 (1996).
[CrossRef]

Shough, D.

Takeda, M.

Williams, R. A.

Appl. Opt. (1)

J. Mod. Opt. (1)

M. S. Scholl, J. Mod. Opt. 43, 1583 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Lett. (1)

Other (3)

J. E. Grievenkamp and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 501.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 155.

D. W. Robinson, in Interferogram Analysis, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), p. 195.

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Figures (6)

Fig. 1
Fig. 1

Block diagram illustrating the principal steps of the wave-front reconstruction algorithm in phase-shifted interferometry (PSI).

Fig. 2
Fig. 2

Test wave front exhibits piston, tilt about the y axis, third-order spherical aberration, pentagonal fourth-order astigmatism, fourth-order coma along the x axis, and triangular fourth-order astigmatism with the base parallel to the y axis, given in terms of Zernike polynomials in Eqs.  (9).

Fig. 3
Fig. 3

Simulated interferometric pattern of the phase of Fig.  2. Note the presence of moiré patterns owing to inadequate sampling.

Fig. 4
Fig. 4

(a) Modulated intensity multiplied by the cosine of the unknown phase, Eq.  (1). (b) Modulated intensity multiplied by the sine of the unknown phase, Eq.  (2).

Fig. 5
Fig. 5

Phase reconstructed from the intensity-modulated cosine and sine functions.

Fig. 6
Fig. 6

Phase error, defined as the difference between the reconstructed and the original wave fronts, Δϕx, y=ϕrx, y-ϕx, y.

Equations (14)

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I1x, y; ξ=Imx, ycosϕx, y+ψx, y; ξ,
I2x, y; ξ=Imx, ysinϕx, y+ψx, y; ξ.
I1xx, y; ξ=Imxx, ycosϕx, y+ψx, y; ξ-ϕxx, y+ψxx, y; ξImx, y×sinϕx, y+ψx, y; ξ,
I2xx, y; ξ=Imxx, ysinϕx, y+ψx, y; ξ+ϕxx, y+ψxx, y; ξImx, y×cosϕx, y+ψx, y; ξ.
ϕxx, y=I2xx, yI1x, y-I2x, yI1xx, y/I1x, y2+I2x, y2-ψxx, y; ξ.
ϕyx, y=I2yx, yI1x, y-I2x, yI1yx, y/I1x, y2+I2x, y-ψyx, y; ξ.
ϕx, y-ϕx0, y0=y0yx0xϕx, y·ds,
ϕrx, y=y0yx0xϕx, y·ds.
ϕx, y=31
+-4.5x
+-31-6y2-6x2+6y4+12x2y2+6x4
+85xy4-10x3y2+x5
+13x-12xy2-12x3+10xy4+20x3y2+10x5
+6-4y3+12x2y+5y5-10x2y3-15x4y,

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