Abstract

If one employs a diffraction-integral approach to wave propagation and diffraction, the connection between waves and conventional geometrical and diffracted rays is provided by the Method of Stationary Phase (MSP). However, conventional ray methods break down in focal regions because of the coalescense of stationary points. Then one may use the MSP to express the focused field in terms of aperture-plane Point-Spread-Function (PSF) rays. A tutorial review of these two ray techniques is given, and a number of applications are discussed with emphasis on the physical interpretation. Examples include focusing in free space or through a plane interface, plane-wave diffraction by a circular aperture, and diffraction of a Gaussian beam by a circular aperture followed by transmission into a biaxial crystal.

© 2002 Optical Society of America

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References

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  1. A. Rubinowicz, �??Zur Kirchho.schen Beugungstheorie,�?? Ann. Phys. Lpz. 73, 339-364 (1924 ).
    [CrossRef]
  2. J.B. Keller, �??The geometrical theory of diffraction,�?? Proc. Symp. on Microwave Optics, McGill University Press, Montreal, 1953.
  3. J.B. Keller, �??Diffraction by an aperture,�?? J. Appl. Phys. 28, 426-444 (1957).
    [CrossRef]
  4. J.B. Keller, �??Geometrical theory of diffraction,�?? J. Opt. Soc. Am. 52, 116-130 (1962).
    [CrossRef] [PubMed]
  5. J.J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol and Boston, 1986).
  6. H. Weyl, �??Ausbreitung elektromagnetischer Wellen uber einem ebenen Leiter,�?? Ann. Phys. Lpz. 60, 481-500 (1919).
    [CrossRef]
  7. H. Ling and S.W. Lee, �??Focusing of electromagnetic waves through a dielectric interface,�?? J. Opt. Soc. Am. A 1, 559-567 (1984).
    [CrossRef]
  8. J.J. Stamnes and D. Jiang, �??Focusing of two-dimensional electromagnetic waves through a plane interface,�?? Pure Appl. Opt. 7, 603-625 (1998).
    [CrossRef]
  9. D. Jiang and J.J. Stamnes, �??Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,�?? Pure Appl. Opt. 7, 627-641 (1998).
    [CrossRef]
  10. V. Dhayalan and J.J. Stamnes, �??Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,�?? Pure Appl. Opt. 7, 33-52 (1998).
    [CrossRef]
  11. J.J. Stamnes and D. Jiang, �??Focusing of electromagnetic waves into a uniaxial crystal,�?? Opt. Commun. 150, 251-262 (1998).
    [CrossRef]
  12. D. Jiang and J.J. Stamnes, �??Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,�?? Opt. Commun. 163, 55-71 (1999).
    [CrossRef]
  13. V. Dhayalan and J.J. Stamnes, �??Comparison of exact and asymptotic results for the focusing of electromagnetic waves through a plane interface,�?? Appl. Opt. 39, 6332-6340 (2001).
    [CrossRef]
  14. J.J. Stamnes and G. Sithambaranathan, �??Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium,�?? J. Opt. Soc. Am. A 18, 3119-3129 (2001).
    [CrossRef]
  15. G. Sithambaranathan and J.J. Stamnes, �??Transmission of a Gaussian beam into a biaxial crystal,�?? J. Opt. Soc. Am. A 18, 1662-1669 (2001).
    [CrossRef]
  16. J.J. Stamnes and V. Dhayalan, �??Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,�?? J. Opt. Soc. Am. A 18, 1670-1677 (2001).
    [CrossRef]
  17. G. Sithambaranathan and J.J. Stamnes, �??Analytical approach to the transmission of a Gaussian beam into a biaxial crystal,�?? accepted by Opt. Commun. (2002).
  18. P. Wolfe, �??A new approach to edge diffraction,�?? SIAM J. Appl. Math. 15, 1434-1469 (1967).
    [CrossRef]
  19. N. Sergienko, J.J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, and A. Friberg, �??Asymptotic methods for evaluation of di.ractive lenses,�?? J. Opt. A: Pure Appl. Opt. 1, 552-559 (1999).
    [CrossRef]
  20. J.J. Stamnes and N. Sergienko, �??Asymptotic analysis of imaging in the presence of a sinusoidal phase modulation,�?? J. Opt. A: Pure Appl. Opt. 2, 365-371 (2000).
    [CrossRef]
  21. M.V. Berry and C. Upstill, �??Catastrophe optics: morphologies of caustics and their diffraction patterns,�?? Progress in Optics, vol. XVIII, E. Wolf (ed.) (North-Holland, Amsterdam, 1980).
    [CrossRef]
  22. J.J. Stamnes, �??Diffraction, asymptotics, and catastrophes,�?? Opt. Acta 29, 823-842 (1982).
    [CrossRef]

Ann. Phys. Lpz. (2)

A. Rubinowicz, �??Zur Kirchho.schen Beugungstheorie,�?? Ann. Phys. Lpz. 73, 339-364 (1924 ).
[CrossRef]

H. Weyl, �??Ausbreitung elektromagnetischer Wellen uber einem ebenen Leiter,�?? Ann. Phys. Lpz. 60, 481-500 (1919).
[CrossRef]

Appl. Opt. (1)

J. Appl. Phys. (1)

J.B. Keller, �??Diffraction by an aperture,�?? J. Appl. Phys. 28, 426-444 (1957).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (2)

N. Sergienko, J.J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, and A. Friberg, �??Asymptotic methods for evaluation of di.ractive lenses,�?? J. Opt. A: Pure Appl. Opt. 1, 552-559 (1999).
[CrossRef]

J.J. Stamnes and N. Sergienko, �??Asymptotic analysis of imaging in the presence of a sinusoidal phase modulation,�?? J. Opt. A: Pure Appl. Opt. 2, 365-371 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

J.J. Stamnes, �??Diffraction, asymptotics, and catastrophes,�?? Opt. Acta 29, 823-842 (1982).
[CrossRef]

Opt. Commun. (3)

G. Sithambaranathan and J.J. Stamnes, �??Analytical approach to the transmission of a Gaussian beam into a biaxial crystal,�?? accepted by Opt. Commun. (2002).

J.J. Stamnes and D. Jiang, �??Focusing of electromagnetic waves into a uniaxial crystal,�?? Opt. Commun. 150, 251-262 (1998).
[CrossRef]

D. Jiang and J.J. Stamnes, �??Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,�?? Opt. Commun. 163, 55-71 (1999).
[CrossRef]

Proc. Symp. on Microwave Optics (1)

J.B. Keller, �??The geometrical theory of diffraction,�?? Proc. Symp. on Microwave Optics, McGill University Press, Montreal, 1953.

Progress in Optics (1)

M.V. Berry and C. Upstill, �??Catastrophe optics: morphologies of caustics and their diffraction patterns,�?? Progress in Optics, vol. XVIII, E. Wolf (ed.) (North-Holland, Amsterdam, 1980).
[CrossRef]

Pure Appl. Opt. (3)

J.J. Stamnes and D. Jiang, �??Focusing of two-dimensional electromagnetic waves through a plane interface,�?? Pure Appl. Opt. 7, 603-625 (1998).
[CrossRef]

D. Jiang and J.J. Stamnes, �??Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,�?? Pure Appl. Opt. 7, 627-641 (1998).
[CrossRef]

V. Dhayalan and J.J. Stamnes, �??Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,�?? Pure Appl. Opt. 7, 33-52 (1998).
[CrossRef]

SIAM J. Appl. Math. (1)

P. Wolfe, �??A new approach to edge diffraction,�?? SIAM J. Appl. Math. 15, 1434-1469 (1967).
[CrossRef]

Other (1)

J.J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol and Boston, 1986).

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Figures (3)

Fig. 1.
Fig. 1.

Geometry for focusing through a circular aperture. The Geometrical Shadow Boundary (GSB) is described by rays passing through the focus and the aperture boundary.

Fig. 2.
Fig. 2.

Diffraction of a plane wave that is normaly incident upon a circular aperture.

Fig. 3.
Fig. 3.

Axial intensities of the transmitted beam. (a) Computed from Eq. (34) with an aperture radius of a = 5 mm (solid curve) and a = 2 mm (dashed curve). (b) Computed from Eq. (34) (solid curve) and from Eq. (39) (dashed curve) for a = 2 mm.

Equations (47)

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u I x 2 y 2 z 2 = A g ( x , y ) exp [ ik f ( x , y ) ] dxdy ,
g ( x , y ) = 1 i λ z 2 R 2 1 R 1 R 2 ; f ( x , y ) = R 2 R 1 ,
R j = ( x x j ) 2 + ( y y j ) 2 + z j 2 ( j = 1,2 ; x 1 = y 1 = 0 ) .
u IS x 2 y 2 z 2 = 2 π σ k H 1 2 exp [ ik f ( x s , y s ) ] g ( x s , y s ) ,
H = { 2 f x 2 2 f y 2 ( 2 f x y ) 2 } x s , y s ,
σ = { 1 if H < 0 i if H > 0 , { 2 f x 2 } x s , y s > 0 i if H > 0 , { 2 f x 2 } x s , y s < 0 .
u IS x 2 y 2 z 2 = exp [ ik ( R 2 s R 1 s ) ] R 2 s R 1 s ,
u IB 0,0 z 2 = z 2 R 2 exp [ ik ( R 2 R 1 ) ] R 2 R 1 ,
R j = a 2 + z j 2 ( j = 1,2 ) .
u I 0,0 z 2 = exp ( ik z ˜ ) z ˜ + z 2 R 2 exp [ ik ( R 2 R 1 ) ] R 2 R 1 ; z ˜ = z 2 z 1 .
u I 0,0 z 2 = 1 i λ a 2 z 1 z 2 exp [ ik ( z ˜ u 4 ) ] sinc ( u 4 ) ,
u = k a 2 z 1 z 2 z ˜ .
u I D R f oc ( r , z 2 ) =
= a z 2 2 π R 1 a λ ar { exp [ ik ( R 2 R 1 a ) + 4 ] R 2 R 2 [ a r R 2 a R 1 a ] + exp [ ik ( R 2 + R 1 a ) 4 ] R 2 + R 2 + [ a + r R 2 + a R 1 a ] } ,
R 1 a = z 1 2 + a 2 ; R 2 ± = z 2 2 + ( a ± r ) .
u I x 2 y 2 z 2 = A u i x y 0 h x 2 x y 2 y z 2 dxdy ,
u i x y 0 = exp ( ik R 1 ) R 1 ,
h x 2 x y 2 y z 2 = g PSF ( k x , k y ) exp [ i f PSF ( k x , k y ) ] d k x d k y .
g PSF = ( 1 2 π ) 2 ; f PSF = k x ( x 2 x ) + k y ( y 2 y ) + k z z 2 ,
k z = k 2 k x 2 k y 2 .
h S x 2 x y 2 y z 2 = 1 i λ z 2 R 2 exp ( ik R 2 ) R 2 .
h Exact x 2 x y 2 y z 2 = 1 2 π z 2 ( exp ( ik R 2 ) R 2 ) = h S x 2 x y 2 y z 2 ( 1 + i k R 2 ) ,
E t = ω μ 1 π c 2 k 2 2 A exp [ i k 1 R 1 ] R 1 h ( x x , y y ) d x d y ,
h ( x x , y y ) = g ( k x , k y ) exp [ if k x k y ; x x y y ] d k x d k y .
g ( k x , k y ) = k 2 2 k y k z 1 k t 2 T TE k t × e ̂ z + k 2 k 1 k x k t 2 T TM k t × ( k t × e ̂ z ) ,
f k x k y ; x x y y = k x ( x x ) + k y ( y y ) + k z 1 z 0 + k z 2 ( z z 0 ) ,
x 2 = r cos β , y 2 = r sin β ; x = ρ sin ϕ , y = ρ cos ϕ ,
u I ( r , z 2 ) = u I S ( r , z 2 ) + u ID ( r , z 2 ) ,
u I S ( r , z 2 ) = exp ( ikz ) ,
u I D ( r , z 2 ) = 0 2 π a g ( ρ , ϕ ) exp [ ikf ( ρ , ϕ ) ] dρdϕ ,
g ( ρ , ϕ ) = 1 i λ z 2 R 2 ρ R 2 ; f ( ρ , ϕ ) = R 2 = z 2 2 + r 2 + ρ 2 2 ρ r cos ( ϕ β ) .
u I D ( r , z 2 ) = 0 2 π g B ( ϕ ) exp [ ikf ( ϕ ) ] d ϕ ,
g B ( ϕ ) = 1 2 π z 2 R 2 a a r cos ( ϕ β ) ; f ( ϕ ) = R 2 = z 2 2 + r 2 + a 2 2 ar cos ( ϕ β ) .
u I D R ( r , z 2 ) = a z 2 2 π λ ar { exp ( ik R 2 + 4 ) R 2 ( a r ) + exp ( ik R 2 + 4 ) R 2 + ( a + r ) } ,
R 2 ± = z 2 2 + ( a ± r ) .
u I ( 0 , z 2 ) = exp ( ik z 2 ) ( 1 z 2 R 2 exp [ ik ( R 2 z 2 ) ] ) ; R 2 = z 2 2 + a 2 ,
u I ( 0 , z 2 ) = 2 i exp [ i ( k z 2 u 4 ) ] sin ( u 4 ) ; u = k a 2 z 2 .
u I disk ( 0 , z 2 ) = z 2 R 2 exp ( ik R 2 ) .
u x y z = exp ( i k ( 1 ) Z 1 ) i λ ( 1 ) ( Z 1 , x Z 1 , y ) 1 2 2 n ( 1 ) n ( 1 ) + n 1
A exp [ x 2 + y 2 2 σ 0 2 ] exp { i k ( 1 ) [ ( x x ) 2 2 Z 1 , x + ( y y ) 2 2 Z 1 , y ] } dx dy .
Z j = z 0 + n j n ( 1 ) ( z z 0 ) ,
Z 1 , x = z 0 + n ( 1 ) n 1 ( z z 0 ) ; Z 1 , y = z 0 + n 1 n ( 1 ) n 3 2 ( z z 0 ) ,
Z 2 , x = z 0 + n ( 1 ) n 2 ( z z 0 ) ; Z 2 , y = z 0 + n 2 n ( 1 ) n 3 2 ( z z 0 ) .
u x y z = exp ( i k ( 1 ) Z ) i λ ( 1 ) Z ¯ 2 1 + n r
A exp [ x 2 + y 2 2 σ 0 2 ] exp { i k ( 1 ) [ ( x x ) 2 + ( y y ) 2 Z ¯ ] } dx dy ,
Z = z 0 + n r ( z z 0 ) ; Z ¯ = z 0 + 1 n r ( z z 0 ) .
u 0,0 z = exp ( i k ( 1 ) Z ) 2 1 + n r σ 0 i σ 0 ( Z ¯ ) exp [ i arctan ( k ( 1 ) σ 0 2 Z ¯ ) ] [ 1 exp ( a 2 2 σ 0 2 ) exp ( i u ¯ 2 ) ] ,

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