Abstract

The generalized grating imaging based on the equations derived in a previous paper [J. Opt. Soc. Am. A 25, 2244 (2008) ] is described. There the fundamental order imaging was interpreted for gratings with real diffraction coefficients. Here the equations for higher-order imaging are derived and discussed for more general gratings with complex diffraction coefficients. Based on the derived equations the contrast of the grating image is calculated for two examples. A discussion of the effect of the spectrum width of the light source is made and applied to the examples.

© 2008 Optical Society of America

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References

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  1. R. Sudol and B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107-1116 (1981).
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    [CrossRef]
  3. K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moire fringe explanation,” Opt. Acta 30, 745-758 (1983).
    [CrossRef]
  4. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E.Wolf, ed. (North Holland, 1989), Vol. 27, pp. 3-108.
    [CrossRef]
  5. 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]
  6. D. Crespo, J. Alonso, and E. Bernabeu, “Experimental measurements of generalized grating images,” Appl. Opt. 41, 1223-1228 (2002).
    [CrossRef] [PubMed]
  7. 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]
  8. K. Iwata, “Interpretation of generalized grating imaging,” J. Opt. Soc. Am. A 25, 2244-2250 (2008).
    [CrossRef]

2008

2004

2002

2000

1985

1983

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moire fringe explanation,” Opt. Acta 30, 745-758 (1983).
[CrossRef]

1981

Alonso, J.

Bernabeu, E.

Bernabéu, E.

Crespo, D.

García-Rodríguez, L.

Iwata, K.

Leith, E. N.

Patorski, K.

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moire fringe explanation,” Opt. Acta 30, 745-758 (1983).
[CrossRef]

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E.Wolf, ed. (North Holland, 1989), Vol. 27, pp. 3-108.
[CrossRef]

Sudol, R.

Swanson, G. J.

Thompson, B. J.

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Figures (4)

Fig. 1
Fig. 1

Schematic for explaining the generalized grating imaging.

Fig. 2
Fig. 2

Typical interfering rays in the grating imaging system.

Fig. 3
Fig. 3

Contrast of the generalized grating image for two Ronchi gratings. ( j = 1 , N I = 1 , M I = 1 ). (a) Overall contrast C ( 1 , 1 , 1 ) , (b) W, (c) C A ( 1 , 1 ) , (d) C B ( 1 , 1 ) . The unit of L 1 and L 2 is millimeter.

Fig. 4
Fig. 4

Contrast C ( 1 , 1 , 2 ) of the generalized grating image for a first Ronchi grating and a second phase grating. ( j = 1 , N I = 1 , M I = 2 ).

Equations (69)

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I ( X ) = n = m = n = m = a n a n * b m b m * W ( μ 2 ) exp [ i Φ ( μ 2 ) ] exp [ i Ψ 1 ] exp [ i Ψ 2 ] exp [ i 2 π μ 1 ] ,
μ 1 ( n n , m m ) = α 1 ( n n ) p 1 + α 2 ( m m ) p 2 ,
W ( μ 2 ) = w ( x ) exp [ i 2 π μ 2 x ] d x ,
μ 2 ( n n , m m ) = β 1 ( n n ) p 1 + β 2 ( m m ) p 2 ,
Ψ 1 ( n n , m m ) = 2 π { ( n n ) ε 1 p 1 + ( m m ) ε 2 p 2 } ,
Ψ 2 ( n , m , n , m ) = ξ ( n 2 n 2 ) p 1 2 + η ( m 2 m 2 ) p 2 2 + ζ { ( n m n m ) } p 1 p 2 ,
ξ = π λ L 0 ( L 1 + L 2 ) L T ,
η = π λ L 2 ( L 0 + L 1 ) L T ,
ς = 2 π λ L 0 L 2 L T ,
α 1 = L 0 L T , α 2 = ( L 0 + L 1 ) L T ,
β 1 = 1 α 1 = ( L 1 + L 2 ) L T ,
β 2 = 1 α 2 = L 2 L T ,
L T = L 0 + L 1 + L 1 .
a n exp ( i ϕ n ) = a n exp ( i ϕ n ) ,
b m exp ( i ψ m ) = b m exp ( i ψ m ) ,
I ( X ) = n = m = n = m = a n a n b m b m W ( μ 2 ) cos [ Ψ 2 + { ( ϕ n ϕ n ) } { ( ψ m ψ m ) } ] cos [ 2 π μ 1 X + Ψ 1 ] .
n n = N , m m = M .
N = j N I , M = j M I ( j = an integer ) ,
I ( X ) = j = n = m = a n a n + j N I b m b m J M I × cos [ Ψ 2 ( n , m , n + j N I , m j M I ) + ( ϕ n ϕ n + j N I ) { ( ψ m ψ m j M I ) } ] W ( μ 2 ( j N I , j M I ) ) cos [ 2 π X μ 1 ( j N I , j M I ) + Ψ 1 ( j N I , j M I ) ] + n = m = N j N I M j M I a n a n + j N I b m b m J M I × cos [ Ψ 2 ( n , m , n + N , m M ) + ( ϕ n ϕ n + j N I ) { ( ψ m ψ m j M I ) } ] W ( μ 2 ( N , M ) ) cos [ 2 π X μ 1 ( N , M ) + Ψ 1 ( N , M ) ] .
W ( μ 2 ( N , M ) ) = sin ( 2 π S μ 2 ( N , M ) ) ( 2 π S μ 2 ( N , M ) ) .
S μ 2 ( N , M ) 1 .
S μ 2 ( N , M ) = S [ ( L 1 + L 2 ) N p 1 L 2 M p 2 ] L T = j S μ 2 ( N I , M I ) S [ ( L 1 + L 2 ) ( N j N I ) p 1 L 2 ( M j M I ) p 2 ] L T .
S ( L 1 + L 2 ) ( L T p 1 ) 1 , S L 2 ( L T p 2 ) 1 .
I ( X ) = j = amp ( j , N I , M I ) cos [ 2 π X j μ 1 ( N I , M I ) + j Ψ 1 ( N I , M I ) ] ,
amp ( j , N I , M I ) = W ( j μ 2 ( N I , M I ) ) n = m = a n b m a n + j N I b m j M I × cos [ Ψ 2 ( n , m , n + j N I , m j M I ) + ( ϕ n ϕ n + j N I ) { ( ψ m ψ m j M I ) } ] .
μ 1 ( N I , M I ) = [ L 0 N I p 1 ( L 0 + L 1 ) M I p 2 ] L T ,
μ 2 ( N I , M I ) = [ ( L 1 + L 2 ) N I p 1 L 2 M I p 2 ] L T ,
Ψ 1 ( N I , M I ) = 2 π { N I ε 1 p 1 M I ε 2 p 2 } ,
Ψ 2 ( n , m , n + j N I , m j M I ) = ( 2 n + j N I ) j E + ( 2 m j M I ) j F ,
E = ( π λ L 0 p 1 ) [ ( L 1 + L 2 ) N I p 1 L 2 M I p 2 ] L T = ( π λ L 0 p 1 ) μ 2 ( N I , M I ) ,
F = ( π λ L 2 p 2 ) [ ( L 0 + L 1 ) M I p 2 L 0 N I p 1 ] L T = ( π λ L 2 p 2 ) μ 1 ( N I , M I ) .
L 20 = L 1 p 2 M I ( p 1 N I p 2 M I ) .
μ 10 ( N I , M I ) = [ ( L T 0 L 1 L 20 ) N I p 1 ( L T 0 L 20 ) M I p 2 ] L T 0 = [ N I p 1 M I p 2 ] ,
L T 0 = L 0 + L 1 + L 20 .
1 P 0 = N I p 1 + M I p 2 ,
1 P = μ 1 ( N I , M I ) = L T 0 ( P 0 L T ) .
μ 2 ( N I , M I ) = [ ( L T L 0 ) N I p 1 ( L T L 0 L 1 ) M I p 2 ] L T = [ 1 P 0 1 P ] .
amp ( j , N I , M I ) = W ( j μ 2 ( N I , M I ) ) n = m = a n b m a n + j N I b m j M I × cos [ Ψ 2 ( n , m , n + j N I , m + j M I ) + ( ϕ n ϕ n + j N I ) + { ψ m ψ m j M I ) ] .
amp ( j , N I , M I ) = W ( j μ 2 ( N I , M I ) ) A ( j , N I ) B ( j , M I ) ,
A ( j , N I ) = n = a n a n + j N I cos [ ( 2 n + j N I ) j E + ( ϕ n ϕ n + j N I ) ] ,
B ( j , M I ) = m = b m b m j M I cos [ ( 2 m j M I ) j F + ( ψ m ψ m j M I ) ] ,
A 0 ( j , N I ) = n = a n a n + j N I cos [ ϕ n ϕ n + j N I ] ,
B 0 ( j , M I ) = m = b m b m j M I cos [ ( 2 m j M I ) j F 0 + ( ψ m ψ m j M I ) ] ,
F 0 = π λ L 1 N I ( p 1 p 2 ) .
δ θ = j μ 2 ( N I , M I ) λ .
Δ θ = j μ I ( N I , M I ) λ .
C ( j , N I , M I ) = [ amp ( j , N I , M I ) + amp ( j , N I , M I ) ] amp ( 0 , N I , M I ) .
A ( j , N I ) = A ( j , N I ) , B ( j , M I ) = B ( j , M I ) .
C ( j , N I , M I ) = 2 W ( j μ 2 ) C A ( j , N I ) C B ( j , M I ) ,
C A ( j , N I ) = A ( j , N I ) A ( 0 , N I ) ,
C B ( j , M I ) = B ( j , M I ) B ( 0 , M I ) .
a 0 = b 0 = 1 2 , a 2 n 1 = b 2 n 1 = ( 1 ) n + 1 [ ( 2 n 1 ) π ] ,
a 2 n = b 2 n = 0 ( n 0 ) .
B ( 1 , 1 ) = 2 b 0 b 1 cos [ F 0 ] .
F 0 = ν π , ν = 0 , ± 1 , ± 2 , .
L 1 = ν p 1 p 2 λ .
a 0 = 1 2 , b 0 = 0 , a 2 n 1 = b 2 n 1 = ( 1 ) n + 1 [ ( 2 n 1 ) π ] ,
a 2 n = b 2 n = 0 ( n 0 ) .
B ( 1 , 2 ) = [ 1 ( 2 3 ) cos ( 4 F 0 ) ( 2 15 ) cos ( 8 F 0 ) ] π 2 .
4 F 0 = ( 2 ν + 1 ) π ,
L 1 = ( 2 ν + 1 ) p 1 p 2 ( 4 λ ) .
amp s ( j , N I , M I ) = W ( j μ 2 ) g ( λ ) A ( j , N I , M I , λ ) B ( j , N I , M I , λ ) d λ .
amp s ( j , N I , M I ) = A 0 ( j , N I , M I ) B 0 s ( j , N I , M I ) ,
B 0 s ( j , N I , M I ) = m = b m b m j M I g ( λ ) cos [ G λ + ( ψ m ψ m j M I ) ] d λ ,
G = ( 2 m j M I ) j N I L 1 π ( p 1 p 2 ) .
g ( λ ) = { 1 δ for λ 0 δ < λ < λ 0 + δ 0 otherwise } ,
B 0 s ( j , N I , M I ) = m = 2 b m b m j M I cos [ G λ 0 + ψ m ψ m j M I ] sin ( G δ ) G δ .
G δ < π 2 ,
δ < p 1 p 2 [ 2 ( 2 m j M I ) j N I L 1 ] .

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