Abstract

The Kirchhoff-Huygens equation is used to investigate wave propagation in optical systems, with large propagation Fresnel numbers NF and aperture-to-length ratios (a/L) which are not small. The limit of applicability of the Fresnel approximation is analytically established for a thin rectangular aperture. It is shown that the error introduced by the Fresnel approximation to the Kirchhoff integral is comparable to the effects of diffraction, computed by the approximation, times the dimensionless parameter πNF(a/2L)2.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. G. VanKampen, “An Asymptotic Treatment of Diffraction Problems,” Physica XIV, 575–589 (1949).
    [Crossref]
  2. Paul Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63, 1528–1543 (1973).
    [Crossref]
  3. Joseph W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 59.
  4. J. Boersma, “Computation of Fresnel Integrals,” Math of Computation 14, 380 (1960).
    [Crossref]
  5. A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956).
  6. F. D. Feiock, “Finite-size effects in Rayleigh scattering,” Phys. Rev. 169, 165–171 (1968).
    [Crossref]

1973 (1)

1968 (1)

F. D. Feiock, “Finite-size effects in Rayleigh scattering,” Phys. Rev. 169, 165–171 (1968).
[Crossref]

1960 (1)

J. Boersma, “Computation of Fresnel Integrals,” Math of Computation 14, 380 (1960).
[Crossref]

1949 (1)

N. G. VanKampen, “An Asymptotic Treatment of Diffraction Problems,” Physica XIV, 575–589 (1949).
[Crossref]

Boersma, J.

J. Boersma, “Computation of Fresnel Integrals,” Math of Computation 14, 380 (1960).
[Crossref]

Erdelyi, A.

A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956).

Feiock, F. D.

F. D. Feiock, “Finite-size effects in Rayleigh scattering,” Phys. Rev. 169, 165–171 (1968).
[Crossref]

Goodman, Joseph W.

Joseph W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 59.

Horwitz, Paul

VanKampen, N. G.

N. G. VanKampen, “An Asymptotic Treatment of Diffraction Problems,” Physica XIV, 575–589 (1949).
[Crossref]

J. Opt. Soc. Am. (1)

Math of Computation (1)

J. Boersma, “Computation of Fresnel Integrals,” Math of Computation 14, 380 (1960).
[Crossref]

Phys. Rev. (1)

F. D. Feiock, “Finite-size effects in Rayleigh scattering,” Phys. Rev. 169, 165–171 (1968).
[Crossref]

Physica (1)

N. G. VanKampen, “An Asymptotic Treatment of Diffraction Problems,” Physica XIV, 575–589 (1949).
[Crossref]

Other (2)

A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956).

Joseph W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 59.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

FIG. 1
FIG. 1

Contour for evaluation F(β, w).

Tables (3)

Tables Icon

TABLE I Propagation parameters for comparison of solutions.

Tables Icon

TABLE II Amplitude and phase of fields.

Tables Icon

TABLE III Summary of segments for contour integration.

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

U p = ( i / λ ) U A   e - i k R   ( cos θ / R )   d A .
k = 2 π / λ ,
R = [ ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 ] 1 / 2
- a x 1 a , - b y 1 b .
L = z 2 - z 1 ,
x 2 = a x , x 1 = a x , y 2 = b y , y 1 = b y .
R = L   { 1 + 1 2   [ ( a L ) 2   ( x - x ) 2 + ( b L ) 2   ( y - y ) 2 ] - 1 8 [ ( a L ) 2   ( x - x ) 2 + ( b L ) 2   ( y - y ) 2 ] 2 } .
cos θ / R = L / R 2
U p ( x , y ) = i ( N x N y ) 1 / 2 - 1 1 U A ( x , y ) × exp - i π [ N x ( x - x ) 2 + N y ( y - y ) 2 ]   d x   d y ,
N x = a 2 / λ L ,             N y = b 2 / λ L
max   { N x   ( a 2 L ) 2 , N y   ( a 2 L ) 2 , N x   ( b 2 L ) 2 , N y   ( b 2 L ) 2 } 1.
( 2 x 1 2 + 2 y 1 2 + 2 z 1 2 + k 2 )   ψ = 0.
ψ ( x 1 , y 1 , z 1 ) = U ( x 1 , y 1 , z 1 )   e - i k z 1 .
2 ψ z 1 2 = e - i k z 1   ( - k 2 - 2 i k z 1 + 2 z 1 2 )   U ( x 1 , y 1 , z 1 ) .
( 2 x 1 2 + 2 y 1 2 - 2 i k z 1 )   U ( x 1 , y 1 , z 1 ) = 0.
2 U z 1 2 ( 2 x 1 2 + 2 y 1 2 - 2 i k z 1 )   U ,
b a < L .
R λ = L λ + 1 2 N x ( x - x ) 2 [ 1 - α ( x - x ) 2 ] + 1 2 N y ( y - y ) 2 ,
α = ( a / 2 L ) 2 .
cos θ / R = 1 / { L [ 1 + 4 α ( x - x ) 2 ] } .
U p ( x , y ) = i ( N x N y ) 1 / 2 [ U A ( x , y ) / 1 + 4 α ( x - x ) 2 ] × exp ( - i π { N x ( x - x ) 2 [ 1 - α ( x - x ) 2 ] + N y ( y - y ) 2 } ) d x d y .
U p ( x , y ) = U ( x ) V ( y ) ,
V ( y ) = ( i N y ) 1 / 2 - 1 1 e - i π N y ( y - y ) 2 d y .
U ( α , x ) = ( i N x ) 1 / 2 × - 1 1 exp { - i π N x ( x - x ) 2 [ 1 - α ( x - x ) 2 ] } 1 + 4 α ( x - x ) 2 d x .
U ( β , x ) = ( i / 2 ) 1 / 2 { F [ β , π N x ( 1 - x ) 2 ] + F [ β , π N x ( 1 + x ) 2 ] } ,
β = α / π N x = λ / 4 π L ,
F ( β , w ) = 0 w e - i t ( 1 - β t ) 1 + 4 β t d t ( 2 π t ) 1 / 2 .
lim β 0 F ( β , w ) = I F ( w ) = 0 w e - i t d t 2 π t
w = π N x ( 1 ± x ) 2
B = 1 - 2 β w 1 - 2 ( a / L ) 2
U 1 ( x ) = 1 + [ ( 1 + i ) / 2 ] { H [ π N x ( 1 - x ) 2 ] + H [ π N x ( 1 + x ) 2 ] } ,
H ( w ) = i e - i w / ( 2 π w ) 1 / 2 .
U 2 ( x ) = 1 + [ ( 1 + i ) / 2 ] { G [ π N x ( 1 - x ) 2 ] + G [ π N x ( 1 + x ) 2 ] } ,
G ( w ) = H ( w ) e i β w 2 .
U 2 ( 0 ) = 1 + [ ( 1 + i ) i 2 / ( 2 π w 0 ) 1 / 2 ] = 0.98944 e - 0.010 723 i ,
U 1 ( 0 ) = 1 + [ ( 1 + i ) i / ( 2 π w 0 ) 1 / 2 ] = 0.98944 e + 0.010 723 i .
D U 1 ( x ) = U 1 ( x ) - 1 = [ ( 1 + i ) / 2 ] { H [ π N x ( 1 - x ) 2 ] + H [ π N x ( 1 + x ) 2 ] } ,
Δ U = U 2 ( x ) - U 1 ( x ) = [ ( 1 + i ) / 2 ] { E [ π N x ( 1 - x ) 2 ] + E [ π N x ( 1 + x ) 2 ] } ,
E ( w ) = [ e i β w 2 - 1 ] H ( w ) i β w 2 H ( w ) .
β w o 2 = π ( a / 2 L ) 2 N x .
U ( x ) = { 1 x < 1 0 x > 0
I c = c e - i h ( t ) 1 + γ t d t ( 2 π t ) 1 / 2 ,
h ( t ) = t - β t 2 ,
( 1 - 2 β t ) 1 - 2 β w = 1 - α ( 1 ± x ) 2 1 - ( a / L ) 2 > 0.
F ( β w ) = ( - i / 2 π ) 1 / 2 { - I 1 + I 4 } ,
I 1 = 0 e - r - i β r 2 1 - i γ r d r r ,
I 4 = e - i w ( 1 - β w ) 0 e - B r - i β r 2 1 + γ ( w - i r ) d r ( r + i w ) 1 / 2 ,
B = 1 - 2 β w .
I 1 = 2 0 e - s 2   ( e - i β s 4 1 - i γ s 2 )   d s .
I 1 = π { 1 + ( i / 4 ) ( 2 γ - 3 β ) - ( 3 / 32 ) ( 8 γ 2 - 20 β γ + 35 β 2 ) } .
B = 1 - 2 β w 1 - ( a / L ) 2
w = π N x ( 1 ± x ) 2 .
e - B r / ( i w ) 1 / 2 ( 1 + γ w )
F ( β , w ) = 1 - i 2 ( 1 + i 4 ( 2 γ - 3 β ) - 3 32 ( 8 γ 2 - 20 β γ + 35 β 2 ) ) + i e - i w ( 1 - β w ) ( 2 π w ) 1 / 2 ( 1 + γ w ) [ 1 B + i B 2 1 + 3 γ w 2 w ( 1 + γ w ) - 1 B 3 ( 2 i β + 3 + 10 γ w + 15 γ 2 w 2 ( 2 w ) 2   ( 1 + γ w ) 2 ) + 3 B 4 ( β ( 1 + 3 γ w ) w ( 1 + γ w ) - i 5 + 21 γ w + 35 γ 2 w 2 + 35 γ 3 w 3 ( 2 w ) 3   ( 1 + γ w ) 3 ) ] .
I F ( w ) = 1 - i 2 + i e - i w ( 2 π w ) 1 / 2   [ 1 - 3 ( 2 w ) 2 + 3 · 5 · 7 ( 2 w ) 4 + i ( 1 2 w - 3 · 5 ( 2 w ) 3 + ) ] .