Abstract

Two gratings consist of an imaging system that forms a no-dispersion image of an object that is illuminated by white light. The process of double diffraction of the actual system is analyzed using the Fresnel diffraction approximation. One grating generates a number of spectral images of the diffraction orders of an object. The other grating combines one-order dispersive images into one and reconstructs a clear image. A bi-grating system in this case shows the same imaging capability as the lens, but its principle is completely different. An expression of the imaging rule is deduced that agrees well with experiments.

© 2009 Optical Society of America

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References

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  1. E. Lau, “Interference phenomenon on double gratings,” Ann. Phys. (Paris) 6, 417-423 (1948).
  2. J. Jahns and A. W. Lohmann, “The Lau effect: a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263-267 (1979).
    [CrossRef]
  3. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4-8 (1979).
    [CrossRef]
  4. G. J. Swanson and E. N. Leith, “Lau effect and grating imaging,” J. Opt. Soc. Am. 72, 552-555 (1982).
    [CrossRef]
  5. A. Olszak and L. Wronkowski, “Analysis of Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. (Bellingham) 36, 2149-2157 (1997).
    [CrossRef]
  6. D. Crespo, J. Alonso, and E. Bernabéu, “Generalized imaging using an extended monochromatic light source,” J. Opt. Soc. Am. A 17, 1231-1240 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
  8. L. García-Rodríguez, J. Alonso, and E. Bernabéu, “Grating pseudo-imaging with polychromatic and finite extension sources,” Opt. Express 12, 2529-2541 (2004).
    [CrossRef] [PubMed]
  9. J. J. Lunazzi and N. I. Rivera, “Pseudoscopic imaging in a double diffraction process with a slit,” Opt. Express 10, 1368-1373 (2002).
    [PubMed]
  10. J. J. Lunazzi and N. I. Rivera, “Pseudoscopic imaging in a double diffraction process with a slit: critical point properties,” J. Opt. Soc. Am. A 23, 1021-1026 (2006).
    [CrossRef]
  11. W. P. Zhang and W. L. Wei, “Method for spectrum imaging,” Proc. SPIE 4548, 99-102 (2001).
    [CrossRef]
  12. W. P. Zhang and X. R. He, “The spectral combination characteristic of grating and the bi-grating diffraction imaging effect,” Science in China Series G: Physics Mechanics and Astronomy 50(1), 1-5 (2007).
    [CrossRef]
  13. J. J. Lunazzi and N. I. Rivera, “White-light imaging in a two gratings diffraction process,” J. Optics-Opt. Soc. of India 37(2), 56-62 (2008). http://arxiv.org/abs/physics/0506148v1 (2005-2009 versions).
  14. X. R. He and W. P. Zhang, “Analysis of the bi-grating diffraction imaging effect by Fresnel diffraction theory,” Acta Opt. Sin. 27, 1371-1374 (2007) (in Chinese).

2008 (1)

J. J. Lunazzi and N. I. Rivera, “White-light imaging in a two gratings diffraction process,” J. Optics-Opt. Soc. of India 37(2), 56-62 (2008). http://arxiv.org/abs/physics/0506148v1 (2005-2009 versions).

2007 (2)

X. R. He and W. P. Zhang, “Analysis of the bi-grating diffraction imaging effect by Fresnel diffraction theory,” Acta Opt. Sin. 27, 1371-1374 (2007) (in Chinese).

W. P. Zhang and X. R. He, “The spectral combination characteristic of grating and the bi-grating diffraction imaging effect,” Science in China Series G: Physics Mechanics and Astronomy 50(1), 1-5 (2007).
[CrossRef]

2006 (1)

2004 (1)

2002 (2)

2001 (1)

W. P. Zhang and W. L. Wei, “Method for spectrum imaging,” Proc. SPIE 4548, 99-102 (2001).
[CrossRef]

2000 (1)

1997 (1)

A. Olszak and L. Wronkowski, “Analysis of Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. (Bellingham) 36, 2149-2157 (1997).
[CrossRef]

1982 (1)

1979 (2)

J. Jahns and A. W. Lohmann, “The Lau effect: a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263-267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4-8 (1979).
[CrossRef]

1948 (1)

E. Lau, “Interference phenomenon on double gratings,” Ann. Phys. (Paris) 6, 417-423 (1948).

Alonso, J.

Bernabéu, E.

Crespo, D.

García-Rodríguez, L.

Gori, F.

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4-8 (1979).
[CrossRef]

He, X. R.

X. R. He and W. P. Zhang, “Analysis of the bi-grating diffraction imaging effect by Fresnel diffraction theory,” Acta Opt. Sin. 27, 1371-1374 (2007) (in Chinese).

W. P. Zhang and X. R. He, “The spectral combination characteristic of grating and the bi-grating diffraction imaging effect,” Science in China Series G: Physics Mechanics and Astronomy 50(1), 1-5 (2007).
[CrossRef]

Jahns, J.

J. Jahns and A. W. Lohmann, “The Lau effect: a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263-267 (1979).
[CrossRef]

Lau, E.

E. Lau, “Interference phenomenon on double gratings,” Ann. Phys. (Paris) 6, 417-423 (1948).

Leith, E. N.

Lohmann, A. W.

J. Jahns and A. W. Lohmann, “The Lau effect: a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263-267 (1979).
[CrossRef]

Lunazzi, J. J.

Olszak, A.

A. Olszak and L. Wronkowski, “Analysis of Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. (Bellingham) 36, 2149-2157 (1997).
[CrossRef]

Rivera, N. I.

Swanson, G. J.

Wei, W. L.

W. P. Zhang and W. L. Wei, “Method for spectrum imaging,” Proc. SPIE 4548, 99-102 (2001).
[CrossRef]

Wronkowski, L.

A. Olszak and L. Wronkowski, “Analysis of Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. (Bellingham) 36, 2149-2157 (1997).
[CrossRef]

Zhang, W. P.

W. P. Zhang and X. R. He, “The spectral combination characteristic of grating and the bi-grating diffraction imaging effect,” Science in China Series G: Physics Mechanics and Astronomy 50(1), 1-5 (2007).
[CrossRef]

X. R. He and W. P. Zhang, “Analysis of the bi-grating diffraction imaging effect by Fresnel diffraction theory,” Acta Opt. Sin. 27, 1371-1374 (2007) (in Chinese).

W. P. Zhang and W. L. Wei, “Method for spectrum imaging,” Proc. SPIE 4548, 99-102 (2001).
[CrossRef]

Acta Opt. Sin. (1)

X. R. He and W. P. Zhang, “Analysis of the bi-grating diffraction imaging effect by Fresnel diffraction theory,” Acta Opt. Sin. 27, 1371-1374 (2007) (in Chinese).

Ann. Phys. (Paris) (1)

E. Lau, “Interference phenomenon on double gratings,” Ann. Phys. (Paris) 6, 417-423 (1948).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Optics-Opt. Soc. of India (1)

J. J. Lunazzi and N. I. Rivera, “White-light imaging in a two gratings diffraction process,” J. Optics-Opt. Soc. of India 37(2), 56-62 (2008). http://arxiv.org/abs/physics/0506148v1 (2005-2009 versions).

Opt. Commun. (2)

J. Jahns and A. W. Lohmann, “The Lau effect: a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263-267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4-8 (1979).
[CrossRef]

Opt. Eng. (Bellingham) (1)

A. Olszak and L. Wronkowski, “Analysis of Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. (Bellingham) 36, 2149-2157 (1997).
[CrossRef]

Opt. Express (2)

Proc. SPIE (1)

W. P. Zhang and W. L. Wei, “Method for spectrum imaging,” Proc. SPIE 4548, 99-102 (2001).
[CrossRef]

Science in China Series G: Physics Mechanics and Astronomy (1)

W. P. Zhang and X. R. He, “The spectral combination characteristic of grating and the bi-grating diffraction imaging effect,” Science in China Series G: Physics Mechanics and Astronomy 50(1), 1-5 (2007).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

(a) Bi-grating imaging system, (b) spectral combination image.

Fig. 2
Fig. 2

Geometry of diffraction of a spherical wave by a bi-grating imaging system.

Equations (24)

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n z 1 d 1 = w m z 2 d 2 ,
u 1 ( x 1 , y 1 ) = A R exp ( j k R 1 ) ,
R 1 = [ r 1 2 + ( x 1 x 11 ) 2 + ( y 1 y 11 ) 2 ] 1 2 .
R 1 r 1 + x 1 2 + y 1 2 2 x 1 x 11 2 y 1 y 11 2 r 1 .
u 1 ( x 1 , y 1 ) = A r exp ( j k r 1 ) exp ( j k x 1 2 + y 1 2 2 x 1 x 11 2 y 1 y 11 2 r 1 ) .
t 1 ( x 1 ) = n = + c n exp ( j 2 π n d 1 x 1 ) ,
t 2 ( x 2 ) = m = + c m exp ( j 2 π m d 2 x 2 ) .
C n = 1 d 1 d 1 2 d 1 2 t 1 ( x 1 ) exp ( j 2 π n d 1 x 1 ) d x 1 ,
C m = 1 d 2 d 2 2 d 2 2 t 2 ( x 2 ) exp ( j 2 π m d 2 x 2 ) d x 2 .
u 2 ( x 2 , y 2 ) = exp ( j k r 2 ) exp ( j k r 2 ) j λ i r 2 u 1 ( x 1 , y 1 ) t 1 ( x 1 ) exp [ j k ( x 1 2 + y 1 2 ) 2 r 2 ] exp [ j k ( x 2 x 1 + y 1 y 2 ) r 2 ] d x 1 d y 1 .
u 3 ( x 3 , y 3 ) = exp ( j k r 3 ) j λ i r 3 u 2 ( x 2 , y 2 ) t 2 ( x 2 ) exp [ j k x 2 2 + y 2 2 2 r 3 ] exp [ j k r 3 ( x 2 x 3 + y 2 y 3 ) ] d x 2 d y 2 .
u 2 ( x 2 , y 2 ) = A 1 n = + c n exp { j π 1 a 2 [ x 2 λ i r 2 ( n d 1 x 11 λ i r 1 ) ] 2 } exp [ j π 1 a 2 ( y 2 λ i r 2 + y 11 λ i r 1 ) 2 ] ,
( n λ i r 2 d 1 x 11 r 2 r 1 , y 11 r 2 r 1 , r 2 ( r 1 + r 2 ) r 1 ) .
u 3 ( x 3 , y 3 ) = A 2 n = + m = + c n c m exp [ j k ( x 3 2 + y 3 2 2 s x 0 x 3 s y 0 y 3 s ) ] ,
A 2 = A r 2 r 3 r 2 ( r 1 + r 2 ) r 1 r 3 ,
s = r 1 r 3 2 r 2 r 3 ( r 1 + r 2 ) r 2 ( r 1 + r 2 ) ,
x 0 = r 3 r 1 r 2 + r 1 n d 1 λ i r 3 x 11 r 2 + r 1 + r 3 m d 2 λ i ,
y 0 = r 3 y 11 r 2 + r 1 .
n r 1 d 1 = m ( r 1 + r 2 ) d 2 .
n z 1 d 1 = m z 2 d 2 ( cos θ cos φ z 1 z 2 cos θ cos φ + z 1 z 2 ) .
w = cos θ cos φ z 1 z 2 cos θ cos φ + z 1 z 2 .
cos θ cos φ = 1 ,
cos θ cos φ 1 ,
n d 1 = m d 2 ( 1 + r 2 r 1 ) ,

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