Abstract

Obtaining the radial derivatives of wavefronts in projections is a critical step for volume optical computerized tomography. In this paper, the moiré effect of two identical circular gratings in acquiring the first-order radial derivative of a wavefront is analyzed. Based on scalar diffraction theory, the formation mechanism of circular gratings’ moiré fringes is derived. A more explicit analytical relation between moiré fringes of different diffraction orders and tested wavefront is obtained. The involved results will be useful for extracting the projection information that is used in three-dimensional reconstruction by volume OCT.

© 2012 Optical Society of America

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References

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  1. J. D. Posner and D. Dunn-Rankin, “Temperature field measurements of small, nonpremixed flames with use of an Abel inversion of holographic interferograms,” Appl. Opt. 42, 952–959 (2003).
    [CrossRef]
  2. D. A. Feikema, “Quantitative rainbow schlieren deflectometry as a temperature diagnostic for nonsooting spherical flames,” Appl. Opt. 45, 4826–4832 (2006).
    [CrossRef]
  3. Y. Song, B. Zhang, and A. He, “Algebraic iterative algorithm for deflection tomography and its application to density flow fields in a hypersonic wind tunnel,” Appl. Opt. 45, 8092–8101(2006).
    [CrossRef]
  4. H. Thayyullathil, R. M. Vasu, and R. Kanhirodan, “Quantitative flow visualization in supersonic jets through tomographic inversion of wavefronts estimated through shadow casting,” Appl. Opt. 45, 5010–5019 (2006).
    [CrossRef]
  5. X. Wan, S. Yu, G. Cai, Y. Gao, and J. Yi, “Three-dimensional plasma field reconstruction with multiobjective optimization emission spectral tomography,” J. Opt. Soc. Am. A 21, 1161–1171 (2004).
    [CrossRef]
  6. Y.-y. Chen, S. Yang, Z.-h. Li, and A.-z. He, “A model for arc plasma’s optical diagnosis by the measurement of the refractive index,” Opt. Commun. 284, 2648–2652 (2011).
    [CrossRef]
  7. T. M. Buzug, “Three-dimensional fourier-based reconstruction methods,” in Computed Tomography (Springer, 2008), pp. 303–401.
  8. O. Bryngdahl, “Reversed-radial-shearing interferometry,” J. Opt. Soc. Am. 60, 915–917 (1970).
    [CrossRef]
  9. A. W. Lohmann and D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
    [CrossRef]
  10. D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613–2624 (1972).
    [CrossRef]
  11. Q. Ru, N. Ohyama, and T. Honda, “Fringe scanning radial shearing interferometry with circular gratings,” Opt. Commun. 69, 189–192 (1989).
    [CrossRef]
  12. Q. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, “Constant radial shearing interferometry with circular gratings,” Appl. Opt. 28, 3350–3353 (1989).
    [CrossRef]
  13. C. Colautti, L. M. Zerbino, E. E. Sicre, and M. Garavaglia, “Lau effect using circular gratings,” Appl. Opt. 26, 2061–2062 (1987).
    [CrossRef]
  14. I. Glatt and O. Kafri, “Beam direction determination by moire deflectometry using circular gratings,” Appl. Opt. 26, 4051–4053 (1987).
    [CrossRef]
  15. C. Shakher and A. J. Pramila Daniel, “Talbot interferometer with circular gratings for the measurement of temperature in axisymmetric gaseous flames,” Appl. Opt. 33, 6068–6072 (1994).
    [CrossRef]
  16. M. Thakur, A. L. Vyas, and C. Shakher, “Measurement of temperature profile of a gaseous flame with a Lau phase interferometer that has circular gratings,” Appl. Opt. 41, 654–657 (2002).
    [CrossRef]
  17. Y. Park and S. Kim, “Determination of two-dimensional planar displacement by moiré fringes of concentric circle,” Appl. Opt. 33, 5171–5176 (1994).
    [CrossRef]
  18. Y. L. Lay and W. Y. Chen, “Rotation measurement using a circular moiré grating,” Opt. Laser Technol. 30, 539–544(1998).
    [CrossRef]
  19. Joseph W. Goodman, Introduction to Fourier Optics(Publishing House of Electronics Industry, Beijing, 2006).

2011 (1)

Y.-y. Chen, S. Yang, Z.-h. Li, and A.-z. He, “A model for arc plasma’s optical diagnosis by the measurement of the refractive index,” Opt. Commun. 284, 2648–2652 (2011).
[CrossRef]

2006 (3)

2004 (1)

2003 (1)

2002 (1)

1998 (1)

Y. L. Lay and W. Y. Chen, “Rotation measurement using a circular moiré grating,” Opt. Laser Technol. 30, 539–544(1998).
[CrossRef]

1994 (2)

1989 (2)

Q. Ru, N. Ohyama, and T. Honda, “Fringe scanning radial shearing interferometry with circular gratings,” Opt. Commun. 69, 189–192 (1989).
[CrossRef]

Q. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, “Constant radial shearing interferometry with circular gratings,” Appl. Opt. 28, 3350–3353 (1989).
[CrossRef]

1987 (2)

1972 (2)

A. W. Lohmann and D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613–2624 (1972).
[CrossRef]

1970 (1)

Bryngdahl, O.

Buzug, T. M.

T. M. Buzug, “Three-dimensional fourier-based reconstruction methods,” in Computed Tomography (Springer, 2008), pp. 303–401.

Cai, G.

Chen, W. Y.

Y. L. Lay and W. Y. Chen, “Rotation measurement using a circular moiré grating,” Opt. Laser Technol. 30, 539–544(1998).
[CrossRef]

Chen, Y.-y.

Y.-y. Chen, S. Yang, Z.-h. Li, and A.-z. He, “A model for arc plasma’s optical diagnosis by the measurement of the refractive index,” Opt. Commun. 284, 2648–2652 (2011).
[CrossRef]

Colautti, C.

Dunn-Rankin, D.

Feikema, D. A.

Gao, Y.

Garavaglia, M.

Glatt, I.

Goodman, Joseph W.

Joseph W. Goodman, Introduction to Fourier Optics(Publishing House of Electronics Industry, Beijing, 2006).

He, A.

He, A.-z.

Y.-y. Chen, S. Yang, Z.-h. Li, and A.-z. He, “A model for arc plasma’s optical diagnosis by the measurement of the refractive index,” Opt. Commun. 284, 2648–2652 (2011).
[CrossRef]

Honda, T.

Q. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, “Constant radial shearing interferometry with circular gratings,” Appl. Opt. 28, 3350–3353 (1989).
[CrossRef]

Q. Ru, N. Ohyama, and T. Honda, “Fringe scanning radial shearing interferometry with circular gratings,” Opt. Commun. 69, 189–192 (1989).
[CrossRef]

Kafri, O.

Kanhirodan, R.

Kim, S.

Lay, Y. L.

Y. L. Lay and W. Y. Chen, “Rotation measurement using a circular moiré grating,” Opt. Laser Technol. 30, 539–544(1998).
[CrossRef]

Li, Z.-h.

Y.-y. Chen, S. Yang, Z.-h. Li, and A.-z. He, “A model for arc plasma’s optical diagnosis by the measurement of the refractive index,” Opt. Commun. 284, 2648–2652 (2011).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann and D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

Ohyama, N.

Q. Ru, N. Ohyama, and T. Honda, “Fringe scanning radial shearing interferometry with circular gratings,” Opt. Commun. 69, 189–192 (1989).
[CrossRef]

Q. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, “Constant radial shearing interferometry with circular gratings,” Appl. Opt. 28, 3350–3353 (1989).
[CrossRef]

Park, Y.

Posner, J. D.

Pramila Daniel, A. J.

Ru, Q.

Q. Ru, N. Ohyama, and T. Honda, “Fringe scanning radial shearing interferometry with circular gratings,” Opt. Commun. 69, 189–192 (1989).
[CrossRef]

Q. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, “Constant radial shearing interferometry with circular gratings,” Appl. Opt. 28, 3350–3353 (1989).
[CrossRef]

Shakher, C.

Sicre, E. E.

Silva, D. E.

A. W. Lohmann and D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613–2624 (1972).
[CrossRef]

Song, Y.

Thakur, M.

Thayyullathil, H.

Tsujiuchi, J.

Vasu, R. M.

Vyas, A. L.

Wan, X.

Yang, S.

Y.-y. Chen, S. Yang, Z.-h. Li, and A.-z. He, “A model for arc plasma’s optical diagnosis by the measurement of the refractive index,” Opt. Commun. 284, 2648–2652 (2011).
[CrossRef]

Yi, J.

Yu, S.

Zerbino, L. M.

Zhang, B.

Appl. Opt. (11)

J. D. Posner and D. Dunn-Rankin, “Temperature field measurements of small, nonpremixed flames with use of an Abel inversion of holographic interferograms,” Appl. Opt. 42, 952–959 (2003).
[CrossRef]

D. A. Feikema, “Quantitative rainbow schlieren deflectometry as a temperature diagnostic for nonsooting spherical flames,” Appl. Opt. 45, 4826–4832 (2006).
[CrossRef]

Y. Song, B. Zhang, and A. He, “Algebraic iterative algorithm for deflection tomography and its application to density flow fields in a hypersonic wind tunnel,” Appl. Opt. 45, 8092–8101(2006).
[CrossRef]

H. Thayyullathil, R. M. Vasu, and R. Kanhirodan, “Quantitative flow visualization in supersonic jets through tomographic inversion of wavefronts estimated through shadow casting,” Appl. Opt. 45, 5010–5019 (2006).
[CrossRef]

D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613–2624 (1972).
[CrossRef]

Q. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, “Constant radial shearing interferometry with circular gratings,” Appl. Opt. 28, 3350–3353 (1989).
[CrossRef]

C. Colautti, L. M. Zerbino, E. E. Sicre, and M. Garavaglia, “Lau effect using circular gratings,” Appl. Opt. 26, 2061–2062 (1987).
[CrossRef]

I. Glatt and O. Kafri, “Beam direction determination by moire deflectometry using circular gratings,” Appl. Opt. 26, 4051–4053 (1987).
[CrossRef]

C. Shakher and A. J. Pramila Daniel, “Talbot interferometer with circular gratings for the measurement of temperature in axisymmetric gaseous flames,” Appl. Opt. 33, 6068–6072 (1994).
[CrossRef]

M. Thakur, A. L. Vyas, and C. Shakher, “Measurement of temperature profile of a gaseous flame with a Lau phase interferometer that has circular gratings,” Appl. Opt. 41, 654–657 (2002).
[CrossRef]

Y. Park and S. Kim, “Determination of two-dimensional planar displacement by moiré fringes of concentric circle,” Appl. Opt. 33, 5171–5176 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (3)

Y.-y. Chen, S. Yang, Z.-h. Li, and A.-z. He, “A model for arc plasma’s optical diagnosis by the measurement of the refractive index,” Opt. Commun. 284, 2648–2652 (2011).
[CrossRef]

A. W. Lohmann and D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

Q. Ru, N. Ohyama, and T. Honda, “Fringe scanning radial shearing interferometry with circular gratings,” Opt. Commun. 69, 189–192 (1989).
[CrossRef]

Opt. Laser Technol. (1)

Y. L. Lay and W. Y. Chen, “Rotation measurement using a circular moiré grating,” Opt. Laser Technol. 30, 539–544(1998).
[CrossRef]

Other (2)

Joseph W. Goodman, Introduction to Fourier Optics(Publishing House of Electronics Industry, Beijing, 2006).

T. M. Buzug, “Three-dimensional fourier-based reconstruction methods,” in Computed Tomography (Springer, 2008), pp. 303–401.

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Figures (11)

Fig. 1.
Fig. 1.

Optical configuration of Talbot interferometer.

Fig. 2.
Fig. 2.

Geometrical relationship of two displaced circular gratings.

Fig. 3.
Fig. 3.

Geometrical diagram for diffraction wave vector.

Fig. 4.
Fig. 4.

Spectrum of double circular gratings.

Fig. 5.
Fig. 5.

Schematic diagram of double gratings’ diffraction effect.

Fig. 6.
Fig. 6.

The result of two different filters for first-order spectrum.

Fig. 7.
Fig. 7.

Moiré fringes on twice Talbot distance: (a) upper part of first-order spectrum filtering and (b) lower part of first-order spectrum filtering; (c) zeroth-order fciltering.

Fig. 8.
Fig. 8.

Moiré fringes on 2.5 times Talbot distance: (a) upper part of first-order spectrum filtering; (b) lower part of first-order spectrum filtering; (c) zeroth-order filtering.

Fig. 9.
Fig. 9.

Simulated intensity distributions with zeroth-order filtering.

Fig. 10.
Fig. 10.

Moiré fringes on twice Talbot distance with a propane flame: (a) upper part of first-order spectrum filtering; (b) lower part of first-order spectrum filtering, and (c) zeroth-order filtering.

Fig. 11.
Fig. 11.

Moiré fringes on 2.5 times Talbot distance with a propane flame: (a) upper part of first-order spectrum filtering; (b) lower part of first-order spectrum filtering, and (c) zeroth-order filtering.

Equations (43)

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f(x,y,z)=18π2ϑ=0πγ=02π2pγ,ϑ(ξ)ξ2sin(ϑ)dγdϑ,
u1(r,φ)exp[ikΦ(r,φ)],
g1(r,φ)=(m)Cmexp(i2πmra),
u1+(r,φ)=u1(r,φ)(m)Cmexp(2πmra).
u2(r,φ)=exp(ikΔ)iλΔ002πu1(r,φ)(m)Cmexp(2πmra)×exp[iπr2+r22rrcos(φφ)λΔ]rddφ,
u1(r,φ)=(p)up(r)exp(ipφ).
u2(r,φ)=exp(ikΔ)iλΔ0(m)Cmexp(i2πmra)exp[iπr2+r2λΔ](p)up(r)×02πexp(ipφ)exp[iπ2rrcos(φφ)λΔ]dφrdr.
u2(r,φ)=exp(ikΔ)iλΔ(m)(p)Cmexp[ip(φπ2)]×0up(r)(λΔrr)1/2{exp[iπ[(r+r)2λΔp214+2mra]]+exp[iπ[(rr)2λΔ+p2+14+2mra]]}dr.
u2(r,φ)=exp(ikΔ)N(m)(p)Cmexp[ip(φπ2)]×0up(r)(λΔrr)1/2exp[iπ(ma)2λΔ]{exp[iπ(p214)]exp[i2πmra]δ(r+r+mλΔa)+exp[iπ(p2+14)]exp[i2πmra]δ(rr+mλΔa)}dr.
u2(r,φ)=M(m)Cmu1(rmλΔa,φ)exp[iπ(ma)2λΔ]exp[i2πmra].
rs2=r2+ε2+2εrcosφ=(r+εcosφ)2+ε2sin2φ.
g2(r,φ)=(n)Cnexp[i2πn(r+εcosφ)a].
u2+(r,φ)=M(m)(n)CmCnu1(rmλΔa,φ)exp[i2π(m+n)ra]exp[i2πnεacosφ]exp[iπ(ma)2λΔ].
u2+(r,φ)=M(m)(n)Amn(r,φ)exp[i2π(m+n)ra],
k⃗0(m+n)(r,φ)=cosθcosφx⃗+cosθsinφy⃗+sinθz⃗=λ(m+n)acosφx⃗+λ(m+n)asinφy⃗+sinθz⃗.
[ξ,η](m+n)(r,φ)=[(m+n)acosφ,(m+n)asinφ].
[ξ,η](m+n)(r,φ+π)=[(m+n)acosφ,(m+n)asinφ].
[ρ,φ](m+n)(r,φ)=[(m+n)a,φ],[ρ,φ](m+n)(r,φ+π)=[(m+n)a,φ].
[ρ,φ](m+n)(r,φ+π)=[ρ,φ](m+n)(r,φ).
F1(U(ρ,φ))=F1(U(0,φ))=u2+(r,φ)|m+n=0,φ[0,2π).
F1(U(ρ,φ))=F1(U(1a,φ))=u2+(r,φ)|m+n=1+u2+(r,φ+π)|m+n=1,φ[θ0,θ0+π),
F1(U(ρ,φ))=F1(U(1a,φ))=u2+(r,φ)|m+n=1+u2+(r,φ+π)|m+n=1,φ[θ0+π,θ0+2π)=u2+(r,φ)|m+n=1+u2+(r,φ+π)|m+n=1,φ[θ0,θ0+π).
u2+(r,φ)=Mexp(ikC)(m)(n)CmCnexp[iπ(ma)2λΔ]exp(i2πnεacosφ)exp[i2π(m+n)ra].
u(r,φ)|m+n=0=Mexp(ikC)(m)CmCmexp[iπ(ma)2λΔ]exp(i2πmεacosφ).
u(r,φ)|m+n=0=M[C02exp(ikC)+2C12exp(iπλΔa2)exp(ikC)cos(2πεacosφ)].
I(r,φ)|m+n=0=u(r,φ)×u*(r,φ)=M{C04+4C02C12cos(πλΔa2)cos[2πεacosφ]+4C14cos2[2πεacosφ]}.
u(r,φ)|m+n=1=MC0C1exp(ikC)exp(i2πra)[exp(i2πεacosφ)+exp(iπλΔa2)],
u(r,φ)|m+n=1=MC0C1exp(ikC)exp(i2πra)[exp(i2πεacosφ)+exp(iπλΔa2)].
I(r,φ)|m+n=1=2MC02C12[1+cos(2πεacosφ+πλΔa2)],
I(r,φ)|m+n=1=2MC02C12[1+cos(2πεacosφπλΔa2)].
I(r,φ)|m+n=0=M[C02±2C12cos(2πεacosφ)]2,
I(r,φ)|m+n=1=2MC02C12[1±cos(2πεacosφ)],
I(r,φ)|m+n=1=2MC02C12[1±cos(2πεacosφ)].
I(r,φ)|m+n=0=M[C04+4C14cos2(2πεacosφ)],
I(r,φ)|m+n=1=2MC02C12[1±cos(2πεacosφ+π2)],
I(r,φ)|m+n=1=2MC02C12[1±cos(2πεacosφπ2)].
u2+(r,φ)=M(m)(n)CmCnexp[iπ(ma)2λΔ]exp[i2πnεacosφ]exp[i2π(m+n)ra]{exp[ikΦ(r,φ)]exp[ikΦ(r,φ)r(mλΔa)]}.
u(r,φ)|m+n=0=M{C02exp[ikϕ(r,φ)]+2C12exp[iπλΔa2]exp[ikϕ(r,φ)]cos[2πϕ(r,φ)rΔa+2πεacosφ]},
u(r,φ)|m+n=1=MC0C1exp(i2πra)exp[ikϕ(r,φ)]×{exp(i2πεacosφ)+exp[iπλΔa2]exp[ikϕ(r,φ)rλΔa]},
u(r,φ)|m+n=1=MC0C1exp(i2πra)exp[ikϕ(r,φ)]×{exp(i2πεacosφ)+exp[iπλΔa2]exp[ikϕ(r,φ)rλΔa]}.
I(r,φ)|m+n=0=M{C04+4C02C12cos(πλΔa2)cos[2πϕ(r,φ)rΔa+2πεacosφ]+4C14cos2[2πϕ(r,φ)rΔa+2πεacosφ]},
I(r,φ)|m+n=1=2MC02C12{1+cos[2πεacosφ+2πϕ(r,φ)rΔa+πλΔa2]},
I(r,φ)|m+n=1=2MC02C12{1+cos[2πεacosφ+2πϕ(r,φ)rΔaπλΔa2]}.

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