Abstract

The purpose of this study is to let a plane periodic grating act as an imaging element for general objects. Using the Fresnel approximation and the Kirchhoff diffraction theory, we investigated an object wave diffracted by a grating. We demonstrated that an image of an original virtual object can be produced by the nth-order monochromatic diffracted wave of a grating. The impulse response of this imaging system and some imaging properties of this imaging system were obtained.

© 2009 Optical Society of America

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References

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    [CrossRef]
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2008

2007

W. Zhang and X. He, Sci. China, Ser. G 50, 1 (2007).
[CrossRef]

2002

2001

W. Zhang and W. Wei, Proc. SPIE 4548, 99 (2001).
[CrossRef]

1992

1981

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Express

Proc. SPIE

W. Zhang and W. Wei, Proc. SPIE 4548, 99 (2001).
[CrossRef]

Sci. China, Ser. G

W. Zhang and X. He, Sci. China, Ser. G 50, 1 (2007).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (2)

Fig. 1
Fig. 1

Plane periodic transmission grating illuminated by a monochromatic spherical wave.

Fig. 2
Fig. 2

Two pictures captured by a CCD camera: (a) the original virtual object produced by a positive lens for a transmission grating, (b) images formed by the first-order diffracted light waves on a surface of a flat frosted glass.

Equations (15)

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T ( x , y ) = P ( x , y ) n = + t n exp ( i 2 π n f x ) ,
U 0 ( x , y ) = 1 d 1 exp ( i k d 1 ) exp [ i k ( x ϵ ) 2 + ( y η ) 2 2 d 1 ] ,
U 0 ( x , y ) = U 0 ( x , y ) T ( x , y ) .
U 2 ( x 2 , y 2 ) = 1 i λ d 1 d 2 exp [ i k ( d 2 d 1 ϵ 2 + η 2 2 d 1 + x 2 2 + y 2 2 2 d 2 ) ] n = + t n + P ( x , y ) exp { i k [ ( 1 2 d 1 + 1 2 d 2 ) ( x 2 + y 2 ) + ( ϵ d 1 x 2 d 2 + λ n f ) x + ( η d 1 y 2 d 2 ) y ] } d x d y .
U 2 ( x 2 , y 2 ) = 1 i λ d 1 2 exp [ i k x 2 2 + y 2 2 ( ϵ 2 + η 2 ) 2 d 1 ] n = + t n + P ( x , y ) exp { i k d 1 [ ( ϵ + d 1 λ n f x 2 ) x + ( η y 2 ) y ] } d x d y .
U 2 , n ( x 2 , y 2 ) = t n i λ d 1 2 exp [ i k x 2 2 + y 2 2 ( ϵ 2 + η 2 ) 2 d 1 ] × + P ( x , y ) exp { i k d 1 [ ( ϵ + d 1 λ n f x 2 ) x + ( η y 2 ) y ] } d x d y ,
U 2 , n ( x 2 , y 2 ) = t n i λ d 1 2 exp [ i k x 2 2 + y 2 2 ( ϵ 2 + η 2 ) 2 d 1 ] + exp { i k d 1 [ ( ϵ + d 1 λ n f x 2 ) x + ( η y 2 ) y ] } d x d y = i λ t n exp [ i k ( λ n f ϵ + 1 2 d 1 f 2 λ 2 n 2 ) δ ( ϵ + d 1 f λ n x 2 , η y 2 ) ] .
U 2 , n to U 2 , n ( x 2 , y 2 ) = t n i λ d 1 2 + P ( x , y ) exp { i k d 1 [ ( ϵ + d 1 λ n f x 2 ) x + ( η y 2 ) y ] } d x d y .
h ( x 2 , y 2 ; ϵ , η ) = l { δ ( x 1 ϵ , y 1 η ) } = U 2 , n ( x 2 , y 2 ) .
h ( x 2 , y 2 ; ϵ , η ) = h ( x 2 ϵ , y 2 η ) .
U 1 ( x 1 , y 1 ) = + U 1 ( ϵ , η ) δ ( x 1 ϵ , y 1 η ) d ϵ d η .
U 1 ( x 2 , y 2 ) = l { U 1 ( x 1 , y 1 ) } = + U 1 ( ϵ , η ) h ( x 2 ϵ , y 2 η ) d ϵ d η .
| U 1 ( x 2 , y 2 ) | 2 = | + U 1 ( ϵ , η ) h ( x 2 ϵ , y 2 η ) d ϵ d η | 2 .
h I ( x 2 , y 2 ; ϵ , η ) = | h ( x 2 , y 2 ; ϵ , η ) | 2 .
I 1 ( x 2 , y 2 ) = l { I 1 ( x 1 , y 1 ) } = + I 1 ( ϵ , η ) l { δ I ( x 1 ϵ , y 1 η ) } d ϵ d η = + I 1 ( ϵ , η ) h I ( x 2 ϵ , y 2 η ) d ϵ d η ,

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