Abstract

Double grating interferometer is usually used to achieve phase information from distorted wave front by its temporal phase-shifting characteristic. In this paper, the spatial phase-shifting characteristic of double grating interferometer is presented. The explicit intensity distributions of interferograms produced by double gratings are derived with the scalar diffraction theory, and the stable phase shift is found between plus-first, zero and minus-first order interferograms. Results indicate that the phase shift only depends on the grating period and the distance between two gratings if no phase object exists. If phase object exists, it varies on the interferograms. But the phase shifts are equal at any special point of interferograms. In particular, the triple grating interferometer is presented to generate at least four phase shift interferograms simultaneously with the similar method.

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  1. . H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measureing interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974).
    [CrossRef] [PubMed]
  2. H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36(22), 5321–5324 (1997).
    [CrossRef] [PubMed]
  3. Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt. 47(19), D183–D189 (2008).
    [CrossRef] [PubMed]
  4. S. Wolfling, E. Lanzmann, N. Ben-Yosef, and Y. Arieli, “Wavefront reconstruction by spatial-phase-shift imaging interferometry,” Appl. Opt. 45(12), 2586–2596 (2006).
    [CrossRef] [PubMed]
  5. S. Wolfling, E. Lanzmann, M. Israeli, N. Ben-Yosef, and Y. Arieli, “Spatial phase-shift interferometry—a wavefront analysis technique for three-dimensional topometry,” J. Opt. Soc. Am. A 22(11), 2498–2509 (2005).
    [CrossRef]
  6. B. K. A. Ngoi, K. Venkatakrishnan, and N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40(19), 3211–3214 (2001).
    [CrossRef]
  7. H. Kihm and S.-W. Kim, “Fiber-diffraction interferometer for vibration desensitization,” Opt. Lett. 30(16), 2059–2061 (2005).
    [CrossRef] [PubMed]
  8. C. Han and B. Han, “Phase-shifting in achromatic moiré interferometry system,” Opt. Express 15(16), 9970–9976 (2007).
    [CrossRef] [PubMed]
  9. A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000).
    [CrossRef]
  10. G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express 16(11), 7806 (2008).
    [CrossRef] [PubMed]
  11. M. Kujawinska, L. Salbut, and K. Patorski, “Three-channel phase stepped system for moire interferometry,” Appl. Opt. 30(13), 1633–1635 (1991).
    [CrossRef] [PubMed]
  12. O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984).
    [CrossRef] [PubMed]
  13. A. Lohmann and O. Bryngdahl, “A Lateral Wavefront Shearing Interferometer with Variable Shear,” Appl. Opt. 6(11), 1934–1937 (1967).
    [CrossRef] [PubMed]
  14. K. Patorski and L. Salbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry Further considerations,” Opt. Acta (Lond.) 32, 1323–1331 (1985).
    [CrossRef]
  15. K. Patorski, “Talbot interferometry with increased shear,” Appl. Opt. 24(24), 4448–4453 (1985).
    [CrossRef] [PubMed]
  16. I. Amidror and R. D. Hersch, “Fourier-based analysis of phase shifts in the superposition of periodic layers and their moiré effects,” J. Opt. Soc. Am. A 13(5), 974–987 (1996).
    [CrossRef]
  17. M. Kujawinska, “Fresnel-field analysis of double-grating systems and their application in phase-stepping grating interferometers,” J. Opt. Soc. Am. A 5(6), 849–857 (1988).
    [CrossRef]
  18. G. W. R. Leibbrandt, G. Harbers, and P. J. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt. 35(31), 6151–6161 (1996).
    [CrossRef] [PubMed]
  19. H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36(22), 5321–5324 (1997).
    [CrossRef] [PubMed]
  20. B. Bon, J. Pierre, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 15 (1992).
  21. Y. Song, Y. Y. Chen, A. He, and Z. Zhao, “Theoretical analysis for moiré deflectometry from diffraction theory,” J. Opt. Soc. Am. A 26(4), 882–889 (2009).
    [CrossRef]
  22. M. Born, and E. Wolf, Principles of Optics (Cambridge, United Kingdom, 1999).
  23. G. Stoilov and T. Dragostinov, “Phase-stepping interferometry: Five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28(1), 61–69 (1997).
    [CrossRef]
  24. E. Keren, E. Bar-Ziv, I. Glatt, and O. Kafri, “Measurements of temperature distribution of flames by moire deflectometry,” Appl. Opt. 20(24), 4263–4266 (1981).
    [CrossRef] [PubMed]
  25. 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(31), 8092–8101 (2006).
    [CrossRef] [PubMed]

2009

2008

2007

2006

2005

2001

2000

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000).
[CrossRef]

1997

1996

1992

B. Bon, J. Pierre, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 15 (1992).

1991

1988

1985

K. Patorski and L. Salbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry Further considerations,” Opt. Acta (Lond.) 32, 1323–1331 (1985).
[CrossRef]

K. Patorski, “Talbot interferometry with increased shear,” Appl. Opt. 24(24), 4448–4453 (1985).
[CrossRef] [PubMed]

1984

1981

1974

1967

Amidror, I.

Arieli, Y.

Awatsuji, Y.

Bar-Ziv, E.

Ben-Yosef, N.

Bon, B.

B. Bon, J. Pierre, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 15 (1992).

Brangaccio, D. J.

Breidne, M.

B. Bon, J. Pierre, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 15 (1992).

Bruning, . H.

Bryngdahl, O.

Chen, Y. Y.

Dragostinov, T.

G. Stoilov and T. Dragostinov, “Phase-stepping interferometry: Five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28(1), 61–69 (1997).
[CrossRef]

Gallagher, J. E.

Glatt, I.

Han, B.

Han, C.

Harbers, G.

He, A.

Herriott, D. R.

Hersch, R. D.

Hettwer, A.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000).
[CrossRef]

Israeli, M.

Kafri, O.

Kaneko, A.

Keren, E.

Kihm, H.

Kim, S.-W.

Koyama, T.

Kranz, J.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000).
[CrossRef]

Kubota, T.

Kujawinska, M.

Kunst, P. J.

Kwon, O. Y.

Lanzmann, E.

Leibbrandt, G. W. R.

Lohmann, A.

Matoba, O.

Meneses-Fabian, C.

Ngoi, B. K. A.

Nishio, K.

Patorski, K.

Pierre, J.

B. Bon, J. Pierre, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 15 (1992).

Robledo-Sánchez, C.

Rodriguez-Zurita, G.

Rosenfeld, D. P.

Salbut, L.

M. Kujawinska, L. Salbut, and K. Patorski, “Three-channel phase stepped system for moire interferometry,” Appl. Opt. 30(13), 1633–1635 (1991).
[CrossRef] [PubMed]

K. Patorski and L. Salbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry Further considerations,” Opt. Acta (Lond.) 32, 1323–1331 (1985).
[CrossRef]

Schreiber, H.

Schwider, J.

Sivakumar, N. R.

Song, Y.

Stoilov, G.

G. Stoilov and T. Dragostinov, “Phase-stepping interferometry: Five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28(1), 61–69 (1997).
[CrossRef]

Tahara, T.

Toto-Arellano, N.-I.

Ura, S.

Vázquez-Castillo, J. F.

Venkatakrishnan, K.

White, A. D.

Wolfling, S.

Wosinski, L.

B. Bon, J. Pierre, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 15 (1992).

Zhang, B.

Zhao, Z.

Appl. Opt.

A. Lohmann and O. Bryngdahl, “A Lateral Wavefront Shearing Interferometer with Variable Shear,” Appl. Opt. 6(11), 1934–1937 (1967).
[CrossRef] [PubMed]

. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measureing interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974).
[CrossRef] [PubMed]

E. Keren, E. Bar-Ziv, I. Glatt, and O. Kafri, “Measurements of temperature distribution of flames by moire deflectometry,” Appl. Opt. 20(24), 4263–4266 (1981).
[CrossRef] [PubMed]

K. Patorski, “Talbot interferometry with increased shear,” Appl. Opt. 24(24), 4448–4453 (1985).
[CrossRef] [PubMed]

M. Kujawinska, L. Salbut, and K. Patorski, “Three-channel phase stepped system for moire interferometry,” Appl. Opt. 30(13), 1633–1635 (1991).
[CrossRef] [PubMed]

H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36(22), 5321–5324 (1997).
[CrossRef] [PubMed]

H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36(22), 5321–5324 (1997).
[CrossRef] [PubMed]

G. W. R. Leibbrandt, G. Harbers, and P. J. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt. 35(31), 6151–6161 (1996).
[CrossRef] [PubMed]

B. K. A. Ngoi, K. Venkatakrishnan, and N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40(19), 3211–3214 (2001).
[CrossRef]

S. Wolfling, E. Lanzmann, N. Ben-Yosef, and Y. Arieli, “Wavefront reconstruction by spatial-phase-shift imaging interferometry,” Appl. Opt. 45(12), 2586–2596 (2006).
[CrossRef] [PubMed]

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(31), 8092–8101 (2006).
[CrossRef] [PubMed]

Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt. 47(19), D183–D189 (2008).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Acta (Lond.)

K. Patorski and L. Salbut, “Optical differentiation of distorted gratings using Talbot and double diffraction interferometry Further considerations,” Opt. Acta (Lond.) 32, 1323–1331 (1985).
[CrossRef]

Opt. Eng.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39(4), 960–966 (2000).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

G. Stoilov and T. Dragostinov, “Phase-stepping interferometry: Five-frame algorithm with an arbitrary step,” Opt. Lasers Eng. 28(1), 61–69 (1997).
[CrossRef]

Opt. Lett.

Pure Appl. Opt.

B. Bon, J. Pierre, L. Wosinski, and M. Breidne, “Double grating phase stepping interferometry for testing aspherics,” Pure Appl. Opt. 1, 15 (1992).

Other

M. Born, and E. Wolf, Principles of Optics (Cambridge, United Kingdom, 1999).

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Figures (10)

Fig. 1
Fig. 1

Optical configuration of the double grating interferometer.

Fig. 2
Fig. 2

the shift b between Grating G1 and G2.

Fig. 3
Fig. 3

Interferograms with the Talbot distance.

Fig. 4
Fig. 4

Interferograms with the sub Talbot distance.

Fig. 5
Fig. 5

Optical configuration of the double grating interferometer with spatial filters

Fig. 6
Fig. 6

Interferograms with spatial filters (a) plus-first order (b) minus-first order

Fig. 7
Fig. 7

Interferograms with the propane flame.

Fig. 8
Fig. 8

(a) Relation between Py1 and number of fringes with different grating period. (b) Relation between Px2 and the distance between gratings with different grating period.

Fig. 9
Fig. 9

Approximate infinite width fringe interferograms with propane flame.

Fig. 10
Fig. 10

Shearing interferograms generated by the triple grating interferometer.

Tables (2)

Tables Icon

Table 1 Coefficients before the partial derivatives in Eq. (29) and Eq. (30) a

Tables Icon

Table 2 Coefficients before the partial derivatives in Eq. (27) and Eq. (28) a

Equations (35)

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Δ2Fdλ.
u1(x,y)exp[ikφ(x,y)].
g1(x,y)=(m)amexp[i2πmd(xcosα2ysinα2)].
u1+(x,y)=u1(x,y)(m)amexp[i2πmd(xcosα2ysinα2)]         =F1{U1(u,v)(m)amδ(umdcosα2,v+mdsinα2)}         =F1{(m)amU1(umdcosα2,v+mdsinα2)}.
U1+(u,v)=(m)amU1(umdcosα2,v+mdsinα2).
U2(u,v)=U1+(u,v)exp[ikΔ11λ2(u2+v2)]           =exp[ikΔ11λ2(u2+v2)](m)amU1(umdcosα2,v+mdsinα2).
g2(x,y)=(n)anexp[i2πnd(xcosα2+ysinα2b)].
U2+(u,v)=(m)(n)amanexp(i2πnbd)U1(um+ndcosα2,v+mndsinα2)             ×exp{ikΔ11λ2[(undcosα2)2+(vndsinα2)2]}.
U2+(u,v)=(m)(n)amanexp(i2πnbd)U1(um+ndcosα2,v+mndsinα2)       ×exp{ikΔ1[1λ22(u2+v2)]}exp(iΔ1λn2πd2)       ×exp[i2πΔ1λnd(ucosα2+vsinα2)].
U3(u,v)=(m)(n)amanexp(i2πnbd)U1(um+ndcosα2,v+mndsinα2)       ×exp{ik(Δ1+Δ2)[1λ22(u2+v2)]}exp(iΔ1λn2πd2)       ×exp[i2πΔ1λnd(ucosα2+vsinα2)].
u3(x,y)=(m)(n)amanexp(i2πnbd)exp[iπλΔ2(m2+n2+2mncosαd2)]       ×exp(iπλΔ1m2d2)exp{i2π(m+ndcosα2xmndsinα2y)}       ×{u1(xλΔmdcosα2λΔ2ndcosα2,y+λΔmdsinα2λΔ2ndsinα2)       exp(ikΔ)iλΔexp[iπλΔ(x2+y2)]}.
u3(x,y)=exp(ikΔ)(m)(n)amanexp(i2πnbd)exp[iπλΔ2(m2+n2+2mncosαd2)]       ×exp(iπλΔ1m2d2)exp{i2π(m+ndcosα2xmndsinα2y)}       ×u1(xλΔmdcosα2λΔ2ndcosα2,y+λΔmdsinα2λΔ2ndsinα2).       
[(λΔm/d)cos(α/2)+(λΔ2n/d)cos(α/2),(λΔm/d)sin(α/2)+(λΔ2n/d)sin(α/2)]
u3(x,y)=exp(ikΔ)exp(ikC)(m)(n)amanexp(i2πnbd)       ×exp[iπλΔ2(m2+n2+2mncosαd2)]       ×exp(iπλΔ1m2d2)exp{i2π(m+ndcosα2xmndsinα2y)}.
I0(x,y)=a04+4a14cos2(4πdysinα22πbd)+4a02a12cos(4πdysinα22πbd)cos[πλd2(4Δ2sin2α2+Δ1)].
I+1(x,y)=2a02a12{1+cos[4π(1dsinα2y)+πλΔ1d22πbd]}.
I1(x,y)=2a02a12{1+cos[4π(1dsinα2y)πλΔ1d22πbd]}.
I0t(x,y)={[a02+2a12cos(4πdysinα22πbd)]2     Δ1=2ld2λ   [a022a12cos(4πdysinα22πbd)]2     Δ1=(2l+1)d2λ.
I+1t(x,y)=2a02a12{1+cos[4π(1dsinα2y)+Kπ2πbd]}.
I1t(x,y)=2a02a12{1+cos[4π(1dsinα2y)Kπ2πbd]}.
I0st(x,y)=a04+4a14cos2(4πdysinα22πbd).
I+1st(x,y)=2a02a12{1+cos[4π(1dsinα2y)+π22πbd]}.
I1st(x,y)=2a02a12{1+cos[4π(1dsinα2y)π22πbd]}.
u3(x,y)=exp(ikΔ)(m)(n)amanexp(i2πnbd)exp[iπλΔ2(m2+n2+2mncosαd2)]       ×exp(iπλΔ1m2d2)exp{i2π(m+ndcosα2xmndsinα2y)}       ×exp[ikφ(xλΔmdcosα2λΔ2ndcosα2,y+λΔmdsinα2λΔ2ndsinα2)].
φ(xλΔmdcosα2λΔ2ndcosα2,y+λΔmdsinα2λΔ2ndsinα2)=φ(x,y)φ(x,y)x(λΔmdcosα2+λΔ2ndcosα2)φ(x,y)y(λΔmdsinα2+λΔ2ndsinα2)+122φ(x,y)x2(λΔmdcosα2+λΔ2ndcosα2)2+122φ(x,y)y2(λΔmdsinα2+λΔ2ndsinα2)2+2φ(x,y)xy(λΔmdcosα2+λΔ2ndcosα2)×(λΔmdsinα2+λΔ2ndsinα2)+.
I+1(x,y)=2a02a12{1+cos[4π(1dsinα2y)+P1+(πλΔ1d2P2)2πbd]}.
I1(x,y)=2a02a12{1+cos[4π(1dsinα2y)+P1(πλΔ1d2P2)2πbd]}.
P1=φ(x,y)x2πΔ1dcosα2φ(x,y)y2π(Δ+Δ2)dsinα2.
P2=2φ(x,y)x2πλ(Δ2Δ22)d2cos2α2+2φ(x,y)y2πλ(Δ2Δ22)d2sin2α2   2φ(x,y)xy2πλ(Δ2+Δ22)d2sinα2cosα2.
P1=φ(x,y)x2πΔ1dcosα2φ(x,y)y2πΔ1dsinα2.
P2=2φ(x,y)x2πλΔ12d2cos2α2+2φ(x,y)y2πλΔ12d2sin2α22φ(x,y)xy2πλΔ12d2sinα2cosα2.
I+1(x,y)=2a02a12{1+cos[4π(1dsinα2y)+φ(x,y)x2πΔ1dcosα2+πλΔ1d22πbd]}.
I1(x,y)=2a02a12{1+cos[4π(1dsinα2y)+φ(x,y)x2πΔ1dcosα2πλΔ1d22πbd]}.
Py1=2πΔ1dsinα22πNλd2λ2.
Px2πλΔ12d2cos2α2+2πFΔ1d2d2λ2.

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