Abstract

A theory of Fresnel images is presented. Only the Fresnel images of plane periodic objects viewed in monochromatic light are considered. The theory is in agreement with the experimental and computer research available in the literature. Photographs of Fresnel images of gratings are shown to verify certain aspects of the theory.

© 1965 Optical Society of America

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References

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  1. H. Talbot, Phil. Mag. 9, 401 (1836).
  2. Rayleigh, Phil. Mag. 11, 196 (1881).
  3. J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London) 70, 486, 497, 505 (1957).
    [Crossref]
  4. E. A. Hiedemann and M. A. Breazeale, J. Opt. Soc. Am. 49, 372 (1959).
    [Crossref]
  5. E. T. Burtt and W. T. Catton, Proc. Roy. Soc. (London) B157, 53 (1962).
  6. G. L. Rogers, Proc. Roy. Soc. (London) B157, 83 (1962).
  7. J. L. Oncley, M. Ludwig, and T. E. Thompson, Rev. Sci. Instr. 29, 985 (1958).
    [Crossref]
  8. J. M. Burch, Progr. Opt. 2, 75 (1963).
  9. J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London) 71, 533 (1958).
    [Crossref]
  10. G. L. Rogers, Brit. J. Appl. Phys. 14, 657 (1963).
    [Crossref]
  11. G. L. Rogers, Brit. J. Appl. Phys. 15, 594 (1964).
    [Crossref]
  12. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
  13. R. W. James, The Crystalline State (G. Bell and Sons, London, 1954), Vol. II.
  14. H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, Phil. Trans. Roy. Soc. (London) A242, 579 (1950).

1964 (1)

G. L. Rogers, Brit. J. Appl. Phys. 15, 594 (1964).
[Crossref]

1963 (2)

G. L. Rogers, Brit. J. Appl. Phys. 14, 657 (1963).
[Crossref]

J. M. Burch, Progr. Opt. 2, 75 (1963).

1962 (2)

E. T. Burtt and W. T. Catton, Proc. Roy. Soc. (London) B157, 53 (1962).

G. L. Rogers, Proc. Roy. Soc. (London) B157, 83 (1962).

1959 (1)

1958 (2)

J. L. Oncley, M. Ludwig, and T. E. Thompson, Rev. Sci. Instr. 29, 985 (1958).
[Crossref]

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London) 71, 533 (1958).
[Crossref]

1957 (1)

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London) 70, 486, 497, 505 (1957).
[Crossref]

1950 (1)

H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, Phil. Trans. Roy. Soc. (London) A242, 579 (1950).

1949 (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

1881 (1)

Rayleigh, Phil. Mag. 11, 196 (1881).

1836 (1)

H. Talbot, Phil. Mag. 9, 401 (1836).

Booker, H. G.

H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, Phil. Trans. Roy. Soc. (London) A242, 579 (1950).

Breazeale, M. A.

Burch, J. M.

J. M. Burch, Progr. Opt. 2, 75 (1963).

Burtt, E. T.

E. T. Burtt and W. T. Catton, Proc. Roy. Soc. (London) B157, 53 (1962).

Catton, W. T.

E. T. Burtt and W. T. Catton, Proc. Roy. Soc. (London) B157, 53 (1962).

Cowley, J. M.

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London) 71, 533 (1958).
[Crossref]

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London) 70, 486, 497, 505 (1957).
[Crossref]

Gabor, D.

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Hiedemann, E. A.

James, R. W.

R. W. James, The Crystalline State (G. Bell and Sons, London, 1954), Vol. II.

Ludwig, M.

J. L. Oncley, M. Ludwig, and T. E. Thompson, Rev. Sci. Instr. 29, 985 (1958).
[Crossref]

Moodie, A. F.

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London) 71, 533 (1958).
[Crossref]

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London) 70, 486, 497, 505 (1957).
[Crossref]

Oncley, J. L.

J. L. Oncley, M. Ludwig, and T. E. Thompson, Rev. Sci. Instr. 29, 985 (1958).
[Crossref]

Ratcliffe, J. A.

H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, Phil. Trans. Roy. Soc. (London) A242, 579 (1950).

Rayleigh,

Rayleigh, Phil. Mag. 11, 196 (1881).

Rogers, G. L.

G. L. Rogers, Brit. J. Appl. Phys. 15, 594 (1964).
[Crossref]

G. L. Rogers, Brit. J. Appl. Phys. 14, 657 (1963).
[Crossref]

G. L. Rogers, Proc. Roy. Soc. (London) B157, 83 (1962).

Shinn, D. H.

H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, Phil. Trans. Roy. Soc. (London) A242, 579 (1950).

Talbot, H.

H. Talbot, Phil. Mag. 9, 401 (1836).

Thompson, T. E.

J. L. Oncley, M. Ludwig, and T. E. Thompson, Rev. Sci. Instr. 29, 985 (1958).
[Crossref]

Brit. J. Appl. Phys. (2)

G. L. Rogers, Brit. J. Appl. Phys. 14, 657 (1963).
[Crossref]

G. L. Rogers, Brit. J. Appl. Phys. 15, 594 (1964).
[Crossref]

J. Opt. Soc. Am. (1)

Phil. Mag. (2)

H. Talbot, Phil. Mag. 9, 401 (1836).

Rayleigh, Phil. Mag. 11, 196 (1881).

Phil. Trans. Roy. Soc. (London) (1)

H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, Phil. Trans. Roy. Soc. (London) A242, 579 (1950).

Proc. Phys. Soc. (London) (2)

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London) 70, 486, 497, 505 (1957).
[Crossref]

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London) 71, 533 (1958).
[Crossref]

Proc. Roy. Soc. (London) (3)

E. T. Burtt and W. T. Catton, Proc. Roy. Soc. (London) B157, 53 (1962).

G. L. Rogers, Proc. Roy. Soc. (London) B157, 83 (1962).

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Progr. Opt. (1)

J. M. Burch, Progr. Opt. 2, 75 (1963).

Rev. Sci. Instr. (1)

J. L. Oncley, M. Ludwig, and T. E. Thompson, Rev. Sci. Instr. 29, 985 (1958).
[Crossref]

Other (1)

R. W. James, The Crystalline State (G. Bell and Sons, London, 1954), Vol. II.

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Figures (9)

Fig. 1
Fig. 1

The source is in the plane x0′, y0′, the object is in the plane x, y, and the Fresnel images are observed in the plane x1, y1.

Fig. 2
Fig. 2

The hexagonal unit cell with the origin at γ=120° is shown in heavy lines; ● are the threefold positions. The unit-cell lattice (intensity) for n=3 is shown in dotted lines. 1 is rotated +30° with respect to the original, whereas 2 is rotated −90° with respect to the original.

Fig. 3
Fig. 3

Photograph of original grating of width ratio 0.44; β=0, q=0, and M=1.

Fig. 4
Fig. 4

Photograph of Fresnel image, in plane β = 1 4, of grating with width ratio 0.44; q=3.8 mm, M=1.03, m=2, and τ=0.88.

Fig. 5
Fig. 5

Photograph of original grating of width ratio 0.20; β=0, q=0, and M=1.

Fig. 6
Fig. 6

Photograph of Fresnel image, in plane β = 1 8, of grating with width ratio 0.20; q=1.6 mm, M=1.01, m=4, and τ=0.80.

Fig. 7
Fig. 7

Photograph of Fresnel image, in plane β = 1 4, of grating with width ratio 0.20; q=3.7 mm, M=1.03, m=2, and τ=0.40.

Fig. 8
Fig. 8

Photograph of Fresnel image, in plane β = 1 2, of grating with width ratio 0.20; q=7.7 mm, M=1.06, m=1 with a shift of origin of one-half period and τ=0.20.

Fig. 9
Fig. 9

Photograph of Fresnel image, in plane β=1, of grating with width ratio 0.20; q=17 mm and M=1.15. This is the first Fourier image.

Tables (1)

Tables Icon

Table I Values of γ and Ω.

Equations (57)

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u ( r 1 ) = ( i λ p q ) - 1 exp ( i ψ 0 ) source s ( r 0 ) object t ( r ) × exp [ ( i π / λ p ) ( r - r 0 ) 2 ] × exp [ ( i π / λ q ) ( r - r 1 ) 2 ] d σ d σ 0 ,
a ( r ) b ( r ) = a ( r ) b ( r - r ) d σ .
u ( r 1 ) = ( i λ p q ) - 1 exp ( i ψ 0 ) [ ( { s ( r 1 ) exp [ ( i π / λ p ) r 1 2 ] } t ( r 1 ) ) exp [ ( i π / λ q ) r 1 2 ] ] .
g ( r 1 ) = ( i λ p q ) - 1 exp [ i ψ 0 ] ( { exp [ ( i π / λ p ) ( r 1 - r 0 ) 2 ] t ( r 1 ) } exp [ ( i π / λ q ) r 1 2 ] ) .
g ( r 1 ) = ( M / i λ q ) t ( r ) exp [ ( i π M / λ q ) ( r - R ) 2 ] d σ ,
g ( r 1 ) = ( M / i λ q ) t ( R ) exp [ ( i π M / λ q ) R 2 ] ,
t ( r ) = [ - 1 ( M / i λ q ) g ( R ) ] exp [ - ( i π M / λ q ) R 2 ] .
u ( r 1 ) = g ( r 1 ) s ( r 0 ) d σ 0 .
I ( r 1 ) = g ( r 1 ) 2 S ( r 0 ) d σ 0 .
t ( r ) = t c ( r ) ϕ ( r ) ,
ϕ ( r ) = H , K = - δ ( x - H a ) δ ( y - K b ) .
g ( r 1 ) = ( M / i λ q ) { t c ( R ) ϕ ( R ) exp [ ( i π M / λ q ) R 2 ] } .
T ( r * ) = t ( r ) exp ( i 2 π r · r * ) d σ ,
t ( r ) = T ( r * ) exp ( - i 2 π r · r * ) d σ * .
Z ( r * ) = exp [ - ( i π λ q / M ) ( h a * + k b * ) 2 ] .
g ( x 1 , y 1 ) = A h , k = - T ( h , k ) Z ( h , k ) × exp { - i 2 π ( h X a + k Y b ) } .
t ( X , Y ) = 1 A h , k = - T ( h , k ) exp { - i 2 π ( h X a + k Y b ) } .
Z ( h , k ) = exp [ - i π λ q ( M R a R b a b sin 2 γ ) - 1 × ( R b 2 h 2 + R a 2 k 2 - 2 h k R a R b cos γ ) ] .
q = ν Ω a b M / λ ,
Ω = 2 R a R b sin 2 γ .
q = ν Ω a b M / λ + q β ,
q β = β Ω a b M / λ
g ( x 1 , y 1 ) = ( M / i λ q ) [ t c ( X , Y ) Δ ( X , Y ) ] ,
Δ ( x 1 , y 1 ) = ϕ ( X , Y ) exp [ ( i π M / λ q ) ( X 2 + Y 2 ) ] .
ϕ ( x , y ) = 1 A h , k = - exp [ - i 2 π ( h x a + k y b ) ] .
Δ ( x 1 , y 1 ) = ( i λ q M ) 1 A h , k = - Z ( h , k ) × exp [ - i 2 π ( h X a + k Y b ) ] .
h = n H + u , where 0 u n - 1 ; k = n K + v , where 0 v n - 1.
Δ ( x 1 , y 1 ) = ( i λ q M ) 1 A H , K = - exp [ - i 2 π ( n H X a + n K Y b ) ] × n - 1 n - 1 u , v = 0 Z ( u , v ) exp [ - i 2 π ( u X a + v Y b ) ] .
Δ ( x 1 , y 1 ) = ( i λ q M ) H , K = - C ( H , K , n , N ) × δ ( X - H a / n ) δ ( Y - K b / n ) ,
C ( H , K , n , N ) = n - 2 n - 1 n - 1 u , v = 0 Z ( u , v ) × exp { - i 2 π ( H u / n + K v / n ) } .
C ( H , n , N ) = n - 1 u = 0 n - 1 Z ( u ) exp ( - i 2 π H u / n ) .
Z ( u , v ) = exp [ - ( i 2 π N / n ) ( R b 2 u 2 + R a 2 v 2 - 2 u v R a R b cos γ ) ] .
Z ( u ) = exp [ - ( i 2 π N u 2 / n ) ] ,
Z ( u , v ) = exp [ - ( i 2 π N / n ) ( u 2 + v 2 ) ] ,
Z ( u , v ) = exp [ - ( i 2 π N / n ) ( u 2 + u v + v 2 ) ] .
ϕ ( x , y ) = ξ , η = - δ ( x - ξ a ) δ ( y - η a ) ,
Δ ( x 1 , y 1 ) = exp [ i π M λ q ( X 2 + Y 2 ) ] ξ , η = - exp [ i π 2 β ( ξ 2 + η 2 ) ] × exp [ - i π β ( ξ X a + η Y a ) ] ,
ξ = 2 N H + U , where 0 U 2 N - 1 ; η = 2 N K + V , where 0 V 2 N - 1.
Δ ( x 1 , y 1 ) = ( i λ q M ) H , K = - C ( H , K , n , N ) × δ ( X - H a / n ) δ ( Y - K a / n ) .
C ( H , K , n , N ) = ( 2 n N ) - 1 exp [ i π ( H 2 + K 2 ) 2 n N - i π 2 ] × 2 N - 1 2 N - 1 U , V = 0 exp [ i n π ( U 2 + V 2 ) 2 N ] × exp [ - i π N ( H U + K V ) ] .
C ( H , n , N ) = ( 2 n N ) - 1 2 exp ( i π H 2 2 n N - i π 4 ) × U = 0 2 N - 1 exp ( i n π U 2 2 N ) exp ( - i π H U N ) .
t ( x , y ) = H , K = - t c ( x - H a , y - K b ) .
g ( x 1 , y 1 ) = H , K = - C ( H , k , n , N ) × t c ( X - H a / n , Y - K b / n ) .
g ( x 1 ) = H = - C ( H , n , N ) t c ( X - H a / n ) .
w = a / m exp ( i α ) ,
z = b / m exp ( i α ) .
C ( W , Z , 3 L , N ) = ( 3 N L ) - 1 exp [ i 2 π 3 N L ( W 2 + W Z + Z 2 ) - i π 2 ] × N - 1 N - 1 U , V = 0 exp [ i 2 π L N ( U 2 + U V + V 2 ) ] × exp [ - i 2 π N ( W U + Z V ) ] .
C ( W , Z , 3 L , 1 ) = ( 3 L ) - 1 × exp [ ( i 2 π / 3 L ) ( W 2 + W Z + Z 2 ) - ( i π / 2 ) ] .
C ( 0 , 8 , 1 ) = 1 2 exp ( - i π / 4 ) , C ( 2 , 8 , 1 ) = C ( 6 , 8 , 1 ) = 1 2 , C ( 4 , 8 , 1 ) = 1 2 exp ( i 3 π / 4 ) .
θ ( H , K , 2 , 1 ) = π ( H 2 + H K + K 2 ) + π .
θ ( W , Z , 3 L , 1 ) = ( 2 π / 3 L ) ( W 2 + W Z + Z 2 ) - 1 2 π .
θ ( W , Z , 3 , 2 ) = ( 4 π / 3 ) ( W 2 + W Z + Z 2 ) + 1 2 π .
P ( r 1 ) = g ( r 1 ) g * ( r 1 + r 1 ) d σ 1 .
χ = λ ( 2 a β m ) - 1 .
θ min = τ χ .
θ min = w λ ( 2 a 2 β ) - 1 ,
θ max = λ ( 4 a β m ) - 1 .