Abstract

We can form a grating image with two gratings having different pitches with an extended light source. It is called generalized grating imaging or the Talbot–Lau effect. When we want to obtain high contrast image with pure absorption gratings or pure phase gratings, the separation between the two gratings is restricted. This corresponds to the Talbot condition. In this paper, we propose to use a combination of absorption grating and phase grating to relax the separation restriction. The theory of generalized grating imaging is applied to the system with this kind of grating. Simulations are performed for calculating contrast variation and show that the proposed system practically relaxes the Talbot condition. An experiment verifies the result of the simulation.

© 2012 Optical Society of America

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References

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  1. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (North Holland, 1989), pp. 3–108.
  2. P. Latimer, “Talbot plane patterns: grating images or interference effects?,” Appl. Opt. 32, 1078–1083 (1993).
    [CrossRef]
  3. S. Teng, L. Liu, J. Zu, Z. Luan, and D. Liu, “Uniform theory of the Talbot effect with partially coherent light illumination,” J. Opt. Soc. Am. A 20, 1747–1754 (2003).
    [CrossRef]
  4. R. Sudol and B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
    [CrossRef]
  5. L. García-Rodríguez, J. Alonso, and E. Bernabéu, “Grating pseudo-imaging with polychromatic and finite extension sources,” Opt. Express 12, 2529–2541 (2004).
    [CrossRef]
  6. G. J. Swanson and E. N. Leith, “Lau effect and generalized grating imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
    [CrossRef]
  7. G. J. Swanson and E. N. Leith, “Analysis of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. 2, 789–793 (1985).
    [CrossRef]
  8. D. Crespo, J. Alonso, and E. Bernabeu, “Generalized grating imaging using an extended monochromatic light source,” J. Opt. Soc. Am. A 17, 1231–1240 (2000).
    [CrossRef]
  9. D. Crespo, J. Alonso, and E. Bernabeu, “Experimental measurements of generalized grating images,” Appl. Opt. 41, 1223–1228 (2002).
    [CrossRef]
  10. K. Iwata, “Interpretation of generalized grating imaging,” J. Opt. Soc. Am. A 25, 2244–2250 (2008).
    [CrossRef]
  11. K. Iwata, “Interpretation of generalized grating imaging (further analysis and numerical calculation),” J. Opt. Soc. Am. A 25, 2939–2944 (2008).
    [CrossRef]
  12. K. Iwata, F. Kusunoki, K. Moriwaki, H. Fukuda, and T. Tomii, “Three-dimensional profiling using the Fourier transform method with a hexagonal grating projection,” Appl. Opt. 47, 2103–2107 (2008).
    [CrossRef]
  13. K. Iwata, Y. Sando, K. Satoh, and K. Moriwaki, “Application of generalized grating imaging to pattern projection in 3-D profilometry,” Appl. Opt. 50, 5115–5121 (2011).
    [CrossRef]
  14. E. Stoykova, G. Minchev, and V. Sainov, “Fringe projection with a sinusoidal phase grating,” Appl. Opt. 48, 4774–4784 (2009).
    [CrossRef]
  15. S. Yokozeki and T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt.157–158(1971).
  16. K. Patorski, “Incoherent superimposition of multiple self-imaging under plane wave-front illumination,” Appl. Opt. 25, 2396–2403 (1986).
    [CrossRef]
  17. J. H. Karp, T. K. Chan, and J. E. Ford, “Integrated diffractive shearing interferometry for adaptive wavefront sensing,” Appl. Opt. 47, 6666–6674 (2008).
    [CrossRef]
  18. K. Iwata, H. Fukuda, and K. Moriwaki, “Transmission-grating-based wave-front tilt sensor,” Appl. Opt. 48, 3961–3966 (2009).
    [CrossRef]
  19. S. P. Trivedi, S. Prakash, S. Rana, and O. Sasaki, “Real-time slope mapping and defect detection in bent plates using Talbot interferometry,” Appl. Opt. 49, 897–903 (2010).
    [CrossRef]
  20. A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef]
  21. H. Hamam, “Lau array illuminator,” Appl. Opt. 43, 2888–2894 (2004).
    [CrossRef]
  22. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304(2005).
    [CrossRef]
  23. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
    [CrossRef]
  24. K. Iwata, “X-ray shearing interferometer and generalized grating imaging,” Appl. Opt. 48, 886–892 (2009).
    [CrossRef]
  25. K. Hornberger, J. E. Sipe, and M. Arndt, “Theory of decoherence in a matter wave Talbot–Lau interferometer,” Phys. Rev. 70, 053608 (2004).
    [CrossRef]
  26. L. M. Sanchez-Breaa, J. Alonsoa, and E. Bernabeu, “Quasicontinuous pseudoimages in sinusoidal grating imaging using an extended light source,” Opt. Commun. 236, 53–58 (2004).
    [CrossRef]
  27. L. M. Sanchez-Brea, J. Saez-Landete, J. Alonso, and E. Bernabeu, “Invariant grating pseudoimaging using polychromatic light and a finite extension source,” Appl. Opt. 47, 1470–1477 (2008).
    [CrossRef]
  28. L. M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, “Continuous self-imaging regime with a double-grating mask,” Appl. Opt. 48, 5722–5727 (2009).
    [CrossRef]
  29. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), p. 325.

2011 (1)

2010 (1)

2009 (4)

2008 (5)

2006 (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

2005 (1)

2004 (4)

H. Hamam, “Lau array illuminator,” Appl. Opt. 43, 2888–2894 (2004).
[CrossRef]

K. Hornberger, J. E. Sipe, and M. Arndt, “Theory of decoherence in a matter wave Talbot–Lau interferometer,” Phys. Rev. 70, 053608 (2004).
[CrossRef]

L. M. Sanchez-Breaa, J. Alonsoa, and E. Bernabeu, “Quasicontinuous pseudoimages in sinusoidal grating imaging using an extended light source,” Opt. Commun. 236, 53–58 (2004).
[CrossRef]

L. García-Rodríguez, J. Alonso, and E. Bernabéu, “Grating pseudo-imaging with polychromatic and finite extension sources,” Opt. Express 12, 2529–2541 (2004).
[CrossRef]

2003 (1)

2002 (1)

2000 (1)

1993 (1)

1990 (1)

1986 (1)

1985 (1)

G. J. Swanson and E. N. Leith, “Analysis of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. 2, 789–793 (1985).
[CrossRef]

1982 (1)

1981 (1)

1971 (1)

S. Yokozeki and T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt.157–158(1971).

Alonso, J.

Alonsoa, J.

L. M. Sanchez-Breaa, J. Alonsoa, and E. Bernabeu, “Quasicontinuous pseudoimages in sinusoidal grating imaging using an extended light source,” Opt. Commun. 236, 53–58 (2004).
[CrossRef]

Arndt, M.

K. Hornberger, J. E. Sipe, and M. Arndt, “Theory of decoherence in a matter wave Talbot–Lau interferometer,” Phys. Rev. 70, 053608 (2004).
[CrossRef]

Bernabeu, E.

Bernabéu, E.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), p. 325.

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Chan, T. K.

Cloetens, P.

Crespo, D.

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304(2005).
[CrossRef]

Diaz, A.

Ford, J. E.

Fukuda, H.

García-Rodríguez, L.

Hamam, H.

Hornberger, K.

K. Hornberger, J. E. Sipe, and M. Arndt, “Theory of decoherence in a matter wave Talbot–Lau interferometer,” Phys. Rev. 70, 053608 (2004).
[CrossRef]

Iwata, K.

Karp, J. H.

Kusunoki, F.

Latimer, P.

Leith, E. N.

G. J. Swanson and E. N. Leith, “Analysis of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. 2, 789–793 (1985).
[CrossRef]

G. J. Swanson and E. N. Leith, “Lau effect and generalized grating imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
[CrossRef]

Liu, D.

Liu, L.

Lohmann, A. W.

Luan, Z.

Minchev, G.

Moriwaki, K.

Patorski, K.

K. Patorski, “Incoherent superimposition of multiple self-imaging under plane wave-front illumination,” Appl. Opt. 25, 2396–2403 (1986).
[CrossRef]

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (North Holland, 1989), pp. 3–108.

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304(2005).
[CrossRef]

Prakash, S.

Rana, S.

Saez-Landete, J.

Sainov, V.

Sanchez-Brea, L. M.

Sanchez-Breaa, L. M.

L. M. Sanchez-Breaa, J. Alonsoa, and E. Bernabeu, “Quasicontinuous pseudoimages in sinusoidal grating imaging using an extended light source,” Opt. Commun. 236, 53–58 (2004).
[CrossRef]

Sando, Y.

Sasaki, O.

Satoh, K.

Sipe, J. E.

K. Hornberger, J. E. Sipe, and M. Arndt, “Theory of decoherence in a matter wave Talbot–Lau interferometer,” Phys. Rev. 70, 053608 (2004).
[CrossRef]

Stampanoni, M.

Stoykova, E.

Sudol, R.

Suzuki, T.

S. Yokozeki and T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt.157–158(1971).

Swanson, G. J.

G. J. Swanson and E. N. Leith, “Analysis of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. 2, 789–793 (1985).
[CrossRef]

G. J. Swanson and E. N. Leith, “Lau effect and generalized grating imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
[CrossRef]

Teng, S.

Thomas, J. A.

Thompson, B. J.

Tomii, T.

Torcal-Milla, F. J.

Trivedi, S. P.

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304(2005).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), p. 325.

Yokozeki, S.

S. Yokozeki and T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt.157–158(1971).

Ziegler, E.

Zu, J.

Appl. Opt. (16)

P. Latimer, “Talbot plane patterns: grating images or interference effects?,” Appl. Opt. 32, 1078–1083 (1993).
[CrossRef]

R. Sudol and B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
[CrossRef]

D. Crespo, J. Alonso, and E. Bernabeu, “Experimental measurements of generalized grating images,” Appl. Opt. 41, 1223–1228 (2002).
[CrossRef]

K. Iwata, F. Kusunoki, K. Moriwaki, H. Fukuda, and T. Tomii, “Three-dimensional profiling using the Fourier transform method with a hexagonal grating projection,” Appl. Opt. 47, 2103–2107 (2008).
[CrossRef]

K. Iwata, Y. Sando, K. Satoh, and K. Moriwaki, “Application of generalized grating imaging to pattern projection in 3-D profilometry,” Appl. Opt. 50, 5115–5121 (2011).
[CrossRef]

E. Stoykova, G. Minchev, and V. Sainov, “Fringe projection with a sinusoidal phase grating,” Appl. Opt. 48, 4774–4784 (2009).
[CrossRef]

S. Yokozeki and T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt.157–158(1971).

K. Patorski, “Incoherent superimposition of multiple self-imaging under plane wave-front illumination,” Appl. Opt. 25, 2396–2403 (1986).
[CrossRef]

J. H. Karp, T. K. Chan, and J. E. Ford, “Integrated diffractive shearing interferometry for adaptive wavefront sensing,” Appl. Opt. 47, 6666–6674 (2008).
[CrossRef]

K. Iwata, H. Fukuda, and K. Moriwaki, “Transmission-grating-based wave-front tilt sensor,” Appl. Opt. 48, 3961–3966 (2009).
[CrossRef]

S. P. Trivedi, S. Prakash, S. Rana, and O. Sasaki, “Real-time slope mapping and defect detection in bent plates using Talbot interferometry,” Appl. Opt. 49, 897–903 (2010).
[CrossRef]

A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
[CrossRef]

H. Hamam, “Lau array illuminator,” Appl. Opt. 43, 2888–2894 (2004).
[CrossRef]

K. Iwata, “X-ray shearing interferometer and generalized grating imaging,” Appl. Opt. 48, 886–892 (2009).
[CrossRef]

L. M. Sanchez-Brea, J. Saez-Landete, J. Alonso, and E. Bernabeu, “Invariant grating pseudoimaging using polychromatic light and a finite extension source,” Appl. Opt. 47, 1470–1477 (2008).
[CrossRef]

L. M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, “Continuous self-imaging regime with a double-grating mask,” Appl. Opt. 48, 5722–5727 (2009).
[CrossRef]

J. Opt. Soc. Am. (2)

G. J. Swanson and E. N. Leith, “Lau effect and generalized grating imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
[CrossRef]

G. J. Swanson and E. N. Leith, “Analysis of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. 2, 789–793 (1985).
[CrossRef]

J. Opt. Soc. Am. A (4)

Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Opt. Commun. (1)

L. M. Sanchez-Breaa, J. Alonsoa, and E. Bernabeu, “Quasicontinuous pseudoimages in sinusoidal grating imaging using an extended light source,” Opt. Commun. 236, 53–58 (2004).
[CrossRef]

Opt. Express (2)

Phys. Rev. (1)

K. Hornberger, J. E. Sipe, and M. Arndt, “Theory of decoherence in a matter wave Talbot–Lau interferometer,” Phys. Rev. 70, 053608 (2004).
[CrossRef]

Other (2)

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (North Holland, 1989), pp. 3–108.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), p. 325.

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Figures (9)

Fig. 1.
Fig. 1.

Optical system for generalized grating imaging.

Fig. 2.
Fig. 2.

Contrast diagrams for 3 systems. (a) System S1, (b) system S2, and (c) system S3.

Fig. 3.
Fig. 3.

Rays for the imaging system with sinusoidal gratings. (A) tp for both gratings G1 and G2 and (B) ta for G1 and tp for G2.

Fig. 4.
Fig. 4.

Absorption-phase grating.

Fig. 5.
Fig. 5.

Contrast diagrams for three systems. (a) System S4, (b) system S5, and (c) system S6.

Fig. 6.
Fig. 6.

Fabricated gratings used in the experiment. (a) Absorption grating (b) Absorption-phase grating.

Fig. 7.
Fig. 7.

An example of the projected fringe at L1=7mm and LT=663mm.

Fig. 8.
Fig. 8.

Contrast obtained by experiment for the system S5.

Fig. 9.
Fig. 9.

Contrast obtained by simulation for the systems S5 (solid lines) and S6 (broken lines).

Tables (1)

Tables Icon

Table 1. Diffraction Coefficients or Two Gratings Used in the Contrast Calculationa

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

an=an,bm=bm.
L20=L1p2/MIp1/NIp2/MI.
μ10(NI,MI)=p1/NIp2/MIp1/NIp2/MI
P0=1/μ10(NI,MI).
μ1(NI,MI)=μ10(NI,MI)LT0/LT
P=P0LT/LT0.
μ2(NI,MI)=μ10(NI,MI)μ1(NI,MI).
S(L1+L2)/(LTp1)1,SL2/(LTp2)1.
I(X)=j=amp(j,NI,MI)cos[2πXjμ1(NI,MI)+jΨ1(NI,MI)],
amp(j,NI,MI)=W(jμ2(NI,MI))A(j,NI)B(j,MI),A(j,NI)=n=anan+jNIcos[(2n+jNI)jE],B(j,MI)=m=bmbmjMIcos[(2mjMI)jF].
W(jμ2)=SSexp[i2πjμ2x]dx=sin(2πjSμ2)/(2πjSμ2),Ψ1(NI,MI)=2π{NIε1/p1MIε2/p2},
E=(πλL0/p1)μ2(NI,MI),F=(πλL2/p2)μ1(NI,MI).
C(j,NI,MI)=2amp(j,NI,MI)amp(0,NI,MI)=2W(jμ2)A(j,NI)B(j,MI)A(0,NI)B(0,MI).
amp0(j,NI,MI)=A0(j,NI,MI)B0(j,NI,MI),
A0(j,NI)=n=anan+jNI,B0(j,MI)=m=bmbmjMIcos[(2mjMI)jF0],
F0=πλNIL1/(p1p2).
C0(j,NI,MI)=2A0(j,NI)B0(j,MI)A0(0,NI)B0(0,MI).
ta(x)=1+cos(2πx/(pa/2)).
q0=1,q±1=0,q±2=1/2,andq±n=0elsewhere.
tb(x)=cos(2πx/pb).
q0=0,q±1=1/2,andq±n=0elsewhere.
A0(0,2)=3/2,A0(1,2)=1,B0(0,2)=3/2,B0(1,2)=cos[F0].
A0(0,2)=1/2,A0(1,2)=1/4B0(0,2)=1/2,B0(1,2)=1/4.
L2=100L1.
1L1+1L2=λ2h(p2/MI)2,
L1=2hp1p2λNIMI.
F0=2hπ.
q0=w{1+exp(iϕ)}/p,q±n=1nπ{[(1)n+exp(iϕ)]sin(nπw/p)}forn=1,2,3.
C=ImaxIminImax+Imin.

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