Abstract

A rotated coordinate system is used to evaluate the irradiance distribution of a Fresnel diffraction pattern. Arbitrary diffracting objects are defined in a bounded input plane. Fresnel patterns for linear chirp functions are simply related to the single-slit pattern. Features of the single-slit pattern are developed for use as optical system design criteria.

© 1981 Optical Society of America

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References

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  1. B. J. Thompson, M. E. Krisl, Photogr. Sci. Eng. 21, No. 3, 109 (May/June1977).
  2. N. Takai, Opt. Commun. 14, 24 (1975).
    [CrossRef]
  3. e.g., M. V. Klein, Optics (Wiley, New York, 1970), Chap. 8.
  4. e.g., E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  5. T. Asakura, H. Mishina, Opt. Commun. 7, 38 (1973).
    [CrossRef]
  6. M. B. N. Butler, IEE Proc. 127, 118 (1980).
  7. B. J. Pernick, J. Kennedy, Appl. Opt. 19, 3224 (1980).
    [CrossRef] [PubMed]
  8. G. N. Lawrence, J. Opt. Soc. Am. 70, 1606A (1980).
  9. A. Sommerfeld, Optics, Lectures on Theoretical Physics, Vol. 4 (Academic, New York, 1964), Chap 5, Sec. 37.

1980 (3)

M. B. N. Butler, IEE Proc. 127, 118 (1980).

G. N. Lawrence, J. Opt. Soc. Am. 70, 1606A (1980).

B. J. Pernick, J. Kennedy, Appl. Opt. 19, 3224 (1980).
[CrossRef] [PubMed]

1977 (1)

B. J. Thompson, M. E. Krisl, Photogr. Sci. Eng. 21, No. 3, 109 (May/June1977).

1975 (1)

N. Takai, Opt. Commun. 14, 24 (1975).
[CrossRef]

1973 (1)

T. Asakura, H. Mishina, Opt. Commun. 7, 38 (1973).
[CrossRef]

Asakura, T.

T. Asakura, H. Mishina, Opt. Commun. 7, 38 (1973).
[CrossRef]

Brigham, E. O.

e.g., E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Butler, M. B. N.

M. B. N. Butler, IEE Proc. 127, 118 (1980).

Kennedy, J.

Klein, M. V.

e.g., M. V. Klein, Optics (Wiley, New York, 1970), Chap. 8.

Krisl, M. E.

B. J. Thompson, M. E. Krisl, Photogr. Sci. Eng. 21, No. 3, 109 (May/June1977).

Lawrence, G. N.

G. N. Lawrence, J. Opt. Soc. Am. 70, 1606A (1980).

Mishina, H.

T. Asakura, H. Mishina, Opt. Commun. 7, 38 (1973).
[CrossRef]

Pernick, B. J.

Sommerfeld, A.

A. Sommerfeld, Optics, Lectures on Theoretical Physics, Vol. 4 (Academic, New York, 1964), Chap 5, Sec. 37.

Takai, N.

N. Takai, Opt. Commun. 14, 24 (1975).
[CrossRef]

Thompson, B. J.

B. J. Thompson, M. E. Krisl, Photogr. Sci. Eng. 21, No. 3, 109 (May/June1977).

Appl. Opt. (1)

IEE Proc. (1)

M. B. N. Butler, IEE Proc. 127, 118 (1980).

J. Opt. Soc. Am. (1)

G. N. Lawrence, J. Opt. Soc. Am. 70, 1606A (1980).

Opt. Commun. (2)

N. Takai, Opt. Commun. 14, 24 (1975).
[CrossRef]

T. Asakura, H. Mishina, Opt. Commun. 7, 38 (1973).
[CrossRef]

Photogr. Sci. Eng. (1)

B. J. Thompson, M. E. Krisl, Photogr. Sci. Eng. 21, No. 3, 109 (May/June1977).

Other (3)

e.g., M. V. Klein, Optics (Wiley, New York, 1970), Chap. 8.

e.g., E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

A. Sommerfeld, Optics, Lectures on Theoretical Physics, Vol. 4 (Academic, New York, 1964), Chap 5, Sec. 37.

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Figures (4)

Fig. 1
Fig. 1

Region of integration for Eq. (6).

Fig. 2
Fig. 2

Intensity distribution in the Fresnel diffraction pattern of a slit for K = 79π.

Fig. 3
Fig. 3

Smooth pattern averaged over the interval 2π/K to eliminate high spatial frequency ringing.

Fig. 4
Fig. 4

Ringing spatial frequency νF and effective aperture a′ vary with Fresnel number K.

Equations (22)

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U o ( x o ) = - a a U i ( x i ) exp ( j k x i 2 / 2 z ) exp ( - j k x o x i / z ) d x i ,
P ( x o ) = U o ( x o ) U o * ( x o ) = - a a - a a U i ( x i ) U i * ( x ¯ i ) exp [ j k ( x i 2 - x ¯ i 2 ) / 2 z ] · exp [ - j k x o ( x i - x ¯ i ) / z ] d x i d x ¯ i .
ξ = x i / a ,             ξ ¯ = x ¯ i / a ,             ν = x o / a ,             K = k a 2 / z ,
P ( ν , K ) = - 1 1 - 1 1 U i ( ξ ) U i * ( ξ ¯ ) exp [ j K ( ξ 2 - ξ ¯ 2 ) / 2 ] · exp [ - j K ν ( ξ - ξ ¯ ) ] d ξ d ξ ¯ .
r = ξ - ξ ¯ ,             s = ξ + ξ ¯ ,             ξ = ( s + r ) / 2 , ξ ¯ = ( s - r ) / 2 ,             d ξ d ξ ¯ = d r d s .
P ( ν , K ) = [ R ( r , s ) exp ( j K r s / 2 ) d s ] exp ( - j K ν r ) d r ,
U i ( ξ ) = - c n exp ( j n 2 π ξ )
R ( r , s ) = n m c n c m * R n m , R n m = exp [ j π ( n + m ) r ] exp [ j π ( n - m ) s ] .
I - ( r ) = 4 sin { ( 2 + r ) [ K r + 2 π ( n - m ) ] / 2 } K r + 2 π ( n - m )
I + ( r ) = 4 sin { ( 2 - r ) [ K r + 2 π ( n - m ) ] / 2 } K r + 2 π ( n - m )
ν ¯ = ν - π ( n + m ) / K ,             α = 2 π ( n - m ) / K ,
P n m = - 2 + α α ( 4 / K r ) sin [ K r ( 2 + r + α ) / 2 ] cos [ K ν ¯ ( r + α ) ] d r + α 2 + α ( 4 / K r ) sin [ K r ( 2 - r - α ) / 2 ] cos [ K ν ¯ ( r + α ) ] d r .
P ( ν , K ) = n m c n c m * P n m .
P S ( V , K ) = 0 2 ( 8 / K r ) sin [ K r ( 2 - r ) / 2 ] cos ( V r ) d r ,
U i ( ξ ) = exp [ j ( A + B ξ + C ξ 2 ) ] ,
R ( r , s ) = exp [ j ( B r + C r s ) ]
P C ( V , K ) = { exp [ j ( K + 2 C ) r s / 2 ] d s } exp [ - j ( K ν - B ) r ] d r .
I - ( r ) = [ 4 / ( K + 2 C ) r ] sin [ ( K + 2 C ) r ( 2 + r ) / 2 ]
I + ( r ) = [ 4 / ( K + 2 C ) r ] sin [ ( K + 2 C ) r ( 2 - r ) / 2 ]
P C ( V , K ) = 0 2 [ 8 ( K + 2 C ) r ] sin [ ( K + 2 C ) r ( 2 - r ) / 2 ] cos [ ( V - B ) r ] d r .
P C ( V , K ) = P S [ ( V - B ) , ( K + 2 C ) ] .
P C ( far field ) = P S [ ( ω a - B ) , 2 C ] .

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