Abstract

We demonstrate the uniqueness and convergence of phase recovery from high-spatial-frequency and undersampled intensity data. Furthermore, this is accomplished without the ambiguities that arise in phase unwrapping and without the need to employ a priori information. The method incorporates the technique of line integration of the phase gradient to find the first approximation to the phase and the algorithm of synthetic interferograms to find the unknown phase with high accuracy. The method may be used with any experimental method that at a certain data processing step obtains generalized sine and cosine intensity functions.

© 2000 Optical Society of America

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References

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  1. G. Páez, M. Strojnik Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–116 (1999).
    [CrossRef]
  2. G. Paez, M. Strojnik Scholl, “Thermal contrast detected with a quantum detector,” Infrared Phys. Technol. 40, 261–265 (1999).
    [CrossRef]
  3. G. Paez, M. Strojnik, “Fringe analysis and phase reconstruction from modulated intensity patterns,” Opt. Lett. 22, 1669–1971 (1997).
    [CrossRef]
  4. G. Paez, M. Strojnik, “Convergent, recursive phase reconstruction from noisy, modulated intensity patterns using synthetic interferograms,” Opt. Lett. 23, 406–408 (1998).
    [CrossRef]
  5. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, 1993), pp. 195–229.
  6. M. Kujawinska, “Fresnel-field analysis of double-grating systems and their application in phase-stepping grating interferometers,” J. Opt. Soc. Am. A 5, 849–857 (1988).
    [CrossRef]
  7. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based tomography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  8. T. Kreis, “Digital holographic interference-phase measurement using Fourier transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
    [CrossRef]
  9. J. H. Bruning, D. R. Herriott, J. D. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  10. G. Paez, M. Strojnik, “Phase-shifted interferometry without phase unwrapping: reconstruction of a decentered wave front,” J. Opt. Soc. Am. A 16, 475–480 (1999).
    [CrossRef]
  11. M. Strojnik, G. Paez, “Testing the aspherical surfaces with the differential rotational-shearing interferometer,” in Fabrication and Testing of Aspherics, Vol. 24 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 119–123.
  12. M. S. Scholl, “Recursive exact ray trace equations through the foci of the tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
    [CrossRef]
  13. G. Páez Padilla, M. Strojnik School, “Recursive relations for ray-tracing through three-dimensional reflective confocal prolate spheroids,” Rev. Mex. Fis. 43, 875–886 (1997).
  14. J. L. Flores, G. Paez, M. Strojnik, “Design of a diluted aperture by use of the practical cutoff frequency,” Appl. Opt. 38, 6010–6018 (1999).
    [CrossRef]

1999 (4)

G. Páez, M. Strojnik Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–116 (1999).
[CrossRef]

G. Paez, M. Strojnik Scholl, “Thermal contrast detected with a quantum detector,” Infrared Phys. Technol. 40, 261–265 (1999).
[CrossRef]

G. Paez, M. Strojnik, “Phase-shifted interferometry without phase unwrapping: reconstruction of a decentered wave front,” J. Opt. Soc. Am. A 16, 475–480 (1999).
[CrossRef]

J. L. Flores, G. Paez, M. Strojnik, “Design of a diluted aperture by use of the practical cutoff frequency,” Appl. Opt. 38, 6010–6018 (1999).
[CrossRef]

1998 (1)

1997 (2)

G. Paez, M. Strojnik, “Fringe analysis and phase reconstruction from modulated intensity patterns,” Opt. Lett. 22, 1669–1971 (1997).
[CrossRef]

G. Páez Padilla, M. Strojnik School, “Recursive relations for ray-tracing through three-dimensional reflective confocal prolate spheroids,” Rev. Mex. Fis. 43, 875–886 (1997).

1996 (1)

M. S. Scholl, “Recursive exact ray trace equations through the foci of the tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
[CrossRef]

1988 (1)

1986 (1)

1982 (1)

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Flores, J. L.

Gallagher, J. D.

Herriott, D. R.

Ina, H.

Kobayashi, S.

Kreis, T.

Kujawinska, M.

Paez, G.

Páez, G.

G. Páez, M. Strojnik Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–116 (1999).
[CrossRef]

Páez Padilla, G.

G. Páez Padilla, M. Strojnik School, “Recursive relations for ray-tracing through three-dimensional reflective confocal prolate spheroids,” Rev. Mex. Fis. 43, 875–886 (1997).

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, 1993), pp. 195–229.

Rosenfeld, D. P.

Scholl, M. S.

M. S. Scholl, “Recursive exact ray trace equations through the foci of the tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
[CrossRef]

Strojnik, M.

Strojnik Scholl, M.

G. Páez, M. Strojnik Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–116 (1999).
[CrossRef]

G. Paez, M. Strojnik Scholl, “Thermal contrast detected with a quantum detector,” Infrared Phys. Technol. 40, 261–265 (1999).
[CrossRef]

Strojnik School, M.

G. Páez Padilla, M. Strojnik School, “Recursive relations for ray-tracing through three-dimensional reflective confocal prolate spheroids,” Rev. Mex. Fis. 43, 875–886 (1997).

Takeda, M.

White, A. D.

Appl. Opt. (2)

Infrared Phys. Technol. (2)

G. Páez, M. Strojnik Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–116 (1999).
[CrossRef]

G. Paez, M. Strojnik Scholl, “Thermal contrast detected with a quantum detector,” Infrared Phys. Technol. 40, 261–265 (1999).
[CrossRef]

J. Mod. Opt. (1)

M. S. Scholl, “Recursive exact ray trace equations through the foci of the tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (2)

Rev. Mex. Fis. (1)

G. Páez Padilla, M. Strojnik School, “Recursive relations for ray-tracing through three-dimensional reflective confocal prolate spheroids,” Rev. Mex. Fis. 43, 875–886 (1997).

Other (2)

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, 1993), pp. 195–229.

M. Strojnik, G. Paez, “Testing the aspherical surfaces with the differential rotational-shearing interferometer,” in Fabrication and Testing of Aspherics, Vol. 24 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 119–123.

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Figures (4)

Fig. 1
Fig. 1

Measured (a) cosine and (b) sine of the unknown phase, with high-intensity gradients. The cosine may be interpreted as an interferogram. The regions of underdetection at the interferogram boundary are characterized by poor fringe visibility and moiré patterns.

Fig. 2
Fig. 2

Two synthetic interferograms are generated during one corrective iteration: (a) from the initially recovered phase, (b) from the converged phase. The regions of underdetection at the boundary of the interferogram in (a) are characterized by poor fringe visibility and the formation of moiré patterns. The intensity nonuniformity of (b) is due to the Gaussian illuminating beam.

Fig. 3
Fig. 3

Phase reconstructed from generalized cosine and sine intensity distributions after one corrective iteration. The phase is bounded within the interval [-0.025, to 76.081 rad]. The extend of the phase is in fact much larger than its range, as it increases from its minimum to its maximum value and then decreases back to the local minimum. Its slope undergoes a sign change.

Fig. 4
Fig. 4

Constant difference between the reconstructed phase and the original phase, Δϕ(1)(x, y)=ϕr(2)(x, y)-ϕ(x, y). The error range is [-0.027 to 0.030 rad]; its rms value is 0.0057 rad.

Equations (42)

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Ic(x, y; ξ)=Im(x, y)cos[ϕ(x, y)+ψ(x, y; ξ)],
Is(x, y; ξ)=Im(x, y)sin[ϕ(x, y)+ψ(x, y; ξ)],
Icx(x, y; ξ)=Imx(x, y)cos[ϕ(x, y)+ψ(x, y; ξ)]-[ϕx(x, y)+ψx(x, y; ξ)]Im(x, y)×sin[ϕ(x, y)+ψ(x, y; ξ)],
Isx(x, y; τ)=Imx(x, y)sin[ϕ(x, y)+ψ(x, y; ξ)]+[ϕx(x, y)+ψx(x, y; ξ)]Im(x, y)×cos[ϕ(x, y)+ψ(x, y; ξ)].
ϕx(x, y)=[Isx(x, y)Ic(x, y)-Is(x, y)Icx(x, y)]/[Ic(x, y)2+Is(x, y)2]-ψx(x, y; ξ),
ϕy(x, y)=[Isy(x, y)Ic(x, y)-Is(x, y)Icy(x, y)]/[Ic(x, y)2+Is(x, y)2]-ψy(x, y; ξ).
ϕ(x, y)-ϕ(x0, y0)=y0yx0xϕ(x, y)·ds.
ϕr(x, y)=y0yx0xϕ(x, y)·ds+ϕ(x0, y0).
cos{[ϕr(n)(x, y)]-[ϕ(x, y)+ψ(x, y; ξ)]}=cos[ϕr(n)(x, y)]cos[ϕ(x, y)+ψ(x, y; ξ)]+sin[ϕr(n)(x, y)]sin[ϕ(x, y)+ψ(x, y; ξ)],
sin{[ϕr(n)(x, y)]-[ϕ(x, y)+ψ(x, y; ξ)]}=sin[ϕr(n)(x, y)]cos[ϕ(x, y)+ψ(x, y; ξ)]-cos[ϕr(n)(x, y)]sin[ϕ(x, y)+ψ(x, y; ξ)].
Im(x, y)cos{[ϕr(n)(x, y)]-[ϕ(x, y)+ψ(x, y; ξ)]}=Im(x, y)cos[ϕr(n)(x, y)]cos[ϕ(x, y)+ψ(x, y; ξ)]+Im(x, y)sin[ϕr(n)(x, y)]sin[ϕ(x, y)ψ(x, y; ξ)],
Im(x, y)sin{[ϕr(n)(x, y)]-[ϕ(x, y)+ψ(x, y; ξ)]}=Im(x, y)sin[ϕr(n)(x, y)]cos[ϕ(x, y+ψ(x, y; ξ)]-Im(x, y)cos[ϕr(n)(x, y)]sin[ϕ(x, y)+ψ(x, y; ξ)].
Ic(n)(x, y; ξ)=Im(x, y)cos{[ϕr(n)(x, y)]-[ϕ(x, y)+ψ(x, y; ξ)]},
Is(n)(x, y; ξ)=Im(x, y)sin{[ϕr(n)(x, y)]-[ϕ(x, y)+ψ(x, y; ξ)]}.
Δϕ(n)(x, y)=ϕr(n)(x, y)-ϕ(x, y).
Ic(n)(x, y; ξ)=Im(x, y)cos{Δϕ(n)(x, y)-ψ(x, y; ξ)},
Is(n)(x, y; ξ)=Im(x, y)sin{Δϕ(n)(x, y)-ψ(x, y; ξ)}.
ϕr(n+1)(x, y)=ϕr(n)(x, y)+Δϕr(n)(x, y).
ϕr(n+1)(x, y)=ϕr(1)(x, y)+1nΔϕr(n)(x, y).
(n+1)(x, y)=Is(n+1)(x, y)=Im(x, y)|sin{[ϕr(n+1)(x, y)]-[ϕ(x, y)+ψ(x, y; ξ)]}|0  forevery(x, y).
(n+1)(x, y)=Is(n+1)(x, y; ξ)=Im(x, y)|sin[Δϕ(n+1)(x, y)-ψ(x, y; ξ)]|0
forevery(x, y).
Ic(n+1)(x, y)=Im(x, y)cos{[ϕr(n+1)(x, y)]-[ϕ(x, y)+ψ(x, y; ξ)]}.
ϕr(f)(x, y)=ϕr(n+1)(x, y)=ϕr(1)(x, y)+1nΔϕr(n)(x, y).
ϕr(f)(x, y)=ϕ(x, y)+δ(f)(x, y).
|ϕ(x, y)-ϕr(n)(x, y)|<|ϕ(x, y)-ϕr(n+1)(x, y)|.
|ϕ(x, y)-ϕr(n)(x, y)|<|ϕ(x, y)-ϕr(n)(x, y)+Δϕr(n)(x, y)|.
|-Δϕ(n)(x, y)|<|+Δϕr(n)(x, y)-Δϕ(n)(x, y)|.
ϕr(n+1)(x, y)=ϕr(f)(x, y)=ϕ(x, y)+δ(n)(x, y).
ϕr(n+1)(x, y)=ϕc(x, y)(x, y).
ϕr(n+1)(x, y)=ϕc(x, y)=ϕ(x, y)+Δϕc(x, y).
(n+1)(x, y)=Im(x, y)|sin[ϕ(x, y)+Δϕc(x, y)-ϕ(x, y)]|0.
(n+1)(x, y)=Im(x, y)|sin[Δϕc(x, y)]|0.
|sin[Δϕc(x, y)]|00.
mπ-γ0Δϕc(x, y)mπ+γ0.
ϕm(x, y)=ϕ(x, y)+mπ.
|ϕ(x, y)-ϕr(1)(x, y)|<π/2.
ϕ(ρ, θ)=7.6piston,
+ρ sin θtiltabouttheyaxis,
+7.6(6ρ4-6ρ2+1)
third-ordersphericalaberration.
Δϕ(1)(x, y)=ϕr(2)(x, y)-ϕ(x, y).

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