Abstract

We apply the method of the line integration of the phase gradient to determine unambiguously the phase from several phase-shifted interferograms (intensity fringe patterns) without phase unwrapping. The ambiguities introduced owing to the multiple values of the arctangent function and to the necessity to invoke a priori knowledge in the regions of high-intensity gradients are avoided. A decentered wave front with circular boundaries is reconstructed from high-fringe-density interferograms with an error of less than 0.1 percent, thus demonstrating the feasibility of testing the off-axis optical elements with approximate reference components.

© 1999 Optical Society of America

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References

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  8. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
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  9. R. Onodera, Y. Ishii, “Phase-extraction analysis of laser-diode phase-shifting interferometry that is insensitive to changes in laser power,” J. Opt. Soc. Am. A 13, 139–146 (1996).
    [CrossRef]
  10. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 195–229.
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  12. G. Páez, M. Strojnik, “Fringe analysis and phase reconstruction from modulated intensity patterns,” Opt. Lett. 22, 1669–1971 (1997).
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  13. G. Páez, M. Strojnik, “Convergent, recursive phase reconstruction from noisy, modulated intensity patterns by use of synthetic interferograms,” Opt. Lett. 23, 406–408 (1998).
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  14. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 155.
  15. 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]
  16. M. S. Scholl, G. Páez Padilla, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infr. Phys. Technol. 38, 25–30 (1997).
    [CrossRef]
  17. M. S. Scholl, G. Páez Padilla, “Image-plane incidence for a baffled infrared telescope,” Infr. Phys. Technol. 38, 87–92 (1997).
    [CrossRef]

1998 (1)

1997 (3)

M. S. Scholl, G. Páez Padilla, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infr. Phys. Technol. 38, 25–30 (1997).
[CrossRef]

M. S. Scholl, G. Páez Padilla, “Image-plane incidence for a baffled infrared telescope,” Infr. Phys. Technol. 38, 87–92 (1997).
[CrossRef]

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

1996 (2)

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]

R. Onodera, Y. Ishii, “Phase-extraction analysis of laser-diode phase-shifting interferometry that is insensitive to changes in laser power,” J. Opt. Soc. Am. A 13, 139–146 (1996).
[CrossRef]

1995 (1)

1994 (1)

1990 (1)

1987 (3)

1985 (1)

1974 (1)

Brangaccio, D. J.

Brophy, C. P.

Bruning, J. H.

Cheng, Y.-Y.

Eiju, T.

Farrant, D. I.

Gallaghar, J. D.

Gallagher, J. D.

J. D. Gallagher, D. R. Herriott, “Wavefront measurement,” U.S. Patent3,694,088 (1972).

Greivenkamp, J. E.

Hariharan, P.

Herriott, D. R.

Hibino, K.

Ishii, Y.

Joenathan, C.

Korn, G. A.

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

Korn, T. M.

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

Kwon, O. Y.

Onodera, R.

Oreb, B. F.

Páez, G.

Páez Padilla, G.

M. S. Scholl, G. Páez Padilla, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infr. Phys. Technol. 38, 25–30 (1997).
[CrossRef]

M. S. Scholl, G. Páez Padilla, “Image-plane incidence for a baffled infrared telescope,” Infr. Phys. Technol. 38, 87–92 (1997).
[CrossRef]

Robinson, D. W.

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

Rosenfeld, D. P.

Scholl, M. S.

M. S. Scholl, G. Páez Padilla, “Image-plane incidence for a baffled infrared telescope,” Infr. Phys. Technol. 38, 87–92 (1997).
[CrossRef]

M. S. Scholl, G. Páez Padilla, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infr. Phys. Technol. 38, 25–30 (1997).
[CrossRef]

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]

Shough, D.

Strojnik, M.

White, A. D.

Williams, R. A.

Wyant, J. C.

Appl. Opt. (5)

Infr. Phys. Technol. (2)

M. S. Scholl, G. Páez Padilla, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infr. Phys. Technol. 38, 25–30 (1997).
[CrossRef]

M. S. Scholl, G. Páez Padilla, “Image-plane incidence for a baffled infrared telescope,” Infr. Phys. Technol. 38, 87–92 (1997).
[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. A (3)

Opt. Lett. (3)

Other (3)

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

J. D. Gallagher, D. R. Herriott, “Wavefront measurement,” U.S. Patent3,694,088 (1972).

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

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Figures (5)

Fig. 1
Fig. 1

Block diagram of the principal steps in the phase-reconstruction process by use of the method of line integration starting from four phase-shifted interferograms.

Fig. 2
Fig. 2

Phase, generated by an off-axis segment, exhibits piston, higher-order spherical aberration, and higher-order coma, given in terms of Zernike polynomials in Eq. (31). The surface shading was obtained assuming a Lambertian point-source illumination from the right-hand side.

Fig. 3
Fig. 3

The simulated interferometric patterns of a phase, shown in Fig. 2, with high-intensity gradients, decreased contrast, and moiré patterns. (a) Cosine of the phase, (b) sine of the phase, (c) negative cosine of the phase, (d) negative sine of the phase.

Fig. 4
Fig. 4

Phase reconstructed by use of the method of direct integration starting from four phase-shifted interferograms, shown in Fig. 3. It appears identical to the phase shown in Fig. 2.

Fig. 5
Fig. 5

Phase error is defined as the difference between the reconstructed and the original wave front, Δϕ(x, y)=ϕr(x, y)-ϕ(x, y). Some error is noted in the regions, characterized by high fringe density, low contrast, and the appearance of the moiré pattern. We note a 0.0184-rad rms error for a phase with 30.4781 rms rad.

Equations (32)

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Ei(x, y)=[Ii(x, y)]1/2 exp[iϕi(x, y)].
I(x, y)=Ib(x, y)+Im(x, y)cos[ϕ1(x, y)-ϕ2(x, y)].
ϕ1(x, y)-ϕ2(x, y)=ϕ(x, y)+Δ.
I(x, y)=Ib(x, y)+Im(x, y)cos[ϕ(x, y)+Δ].
Ii(x, y)=Ib(x, y)+Im(x, y)cos[ϕ(x, y)+Δi].
I1(x, y)=Ib(x, y)+Im(x, y)cos[ϕ(x, y)],
I2(x, y)=Ib(x, y)-Im(x, y)sin[ϕ(x, y)],
I3(x, y)=Ib(x, y)-Im(x, y)cos[ϕ(x, y)],
I4(x, y)=Ib(x, y)+Im(x, y)sin[ϕ(x, y)].
Im(x, y)cos[ϕ(x, y)]=I1(x, y)-I3(x, y).
Im(x, y)sin[ϕ(x, y)]=I4(x, y)-I2(x, y).
bI(x, y)/bx=Ix(x, y)=Dx{I(x, y)},
bI(x, y)/by=Iy(x, y)=Dy{I(x, y)}.
Dx{Im(x, y)cos[ϕ(x, y)]}=-Im(x, y)sin[ϕ(x, y)]ϕx(x, y)+Imx(x, y)cos[ϕ(x, y)].
Dx{Im(x, y)sin[ϕ(x, y)]}=Im(x, y)cos[ϕ(x, y)]ϕx(x, y)+Imx(x, y)sin[ϕ(x, y)].
Dy{Im(x, y)cos[ϕ(x, y)]}=-Im(x, y)sin[ϕ(x, y)]ϕy(x, y)+Imy(x, y)cos[ϕ(x, y)].
Dy{Im(x, y)sin[ϕ(x, y)]}=Im(x, y)cos[ϕ(x, y)]ϕy(x, y)+Imy(x, y)sin[ϕ(x, y)].
ϕx(x, y)=(Dx{Im(x, y)sin[ϕ(x, y)]}×Im(x, y)cos[ϕ(x, y)]-Im(x, y)sin[ϕ(x, y)]×Dx{Im(x, y)cos[ϕ(x, y)]})/({Im(x, y)cos[ϕ(x, y)]}2+{Im(x, y)sin[ϕ(x, y)]}2).
ϕy(x, y)=(Dy{Im(x, y)sin[ϕ(x, y)]}×Im(x, y)cos[ϕ(x, y)]-Im(x, y)sin[ϕ(x, y)]×Dy{Im(x, y)cos[ϕ(x, y)]})/({Im(x, y)cos[ϕ(x, y)]}2+{Im(x, y)sin[ϕ(x, y)]}2).
ϕ(x, y)=iϕx(x, y)+jϕy(x, y).
f(x, y)=(Dx{Im(x, y)sin[ϕ(x, y)]}×Im(x, y)cos[ϕ(x, y)]-Im(x, y)sin[ϕ(x, y)]×Dx{Im(x, y)cos[ϕ(x, y)]})/({Im(x, y)cos[ϕ(x, y)]}2+{Im(x, y)sin[ϕ(x, y)]}2),
g(x, y)=(Dy{Im(x, y)sin[ϕ(x, y)]}×Im(x, y)cos[ϕ(x, y)]-Im(x, y)sin[ϕ(x, y)]×Dy{Im(x, y)cos[ϕ(x, y)]})/({Im(x, y)cos[ϕ(x, y)}]2+{Im(x, y)sin[ϕ(x, y)]}2).
ϕ(x, y)=h(x, y)+u(x)+v(y).
ϕx(x, y)=f(x, y)=hx(x, y)+ux(x),
ϕy(x, y)=g(x, y)=hy(x, y)+vy(y).
ϕ(x, y)-ϕ(x0, y)=x0xf(x, y)dx=h(x, y)+u(x)-h(x0, y)-u(x0),
ϕ(x, y)-ϕ(x, y0)=y0yg(x, y)dy=h(x, y)+v(y)-h(x, y0)-v(y0).
ϕ(x, y0)-ϕ(x0, y0)=x0xf(x, y0)dx=h(x, y0)+u(x)-h(x0, y0)-u(x0),
ϕ(x0, y)-ϕ(x0, y0)=y0yg(x0, y)dy=h(x0, y)+v(y)-h(x0, y0)-v(y0).
ϕr(x, y)=ϕ(x, y)-ϕ(x0, y0)=h(x, y)+u(x)+v(y)-h(x0, y0)-u(x0)-v(y0).
W(x, y)=(5)1piston+(-5)(1-6y2-6x2+6y4+12x2y2+6x4)higher-orderspherical+(1)(3x-12xy2-12x3+10xy4+20x3y2+10x5)higher-ordercoma
Δϕ(x, y)=ϕr(x, y)-ϕ(x, y).

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