Abstract

In this paper we discuss the diffraction pattern resulting from the propagation of light past an opaque obstacle with a circular cross section. A mathematical description of the diffraction pattern is obtained in the Fresnel region using scalar diffraction theory and is presented in terms of the Lommel functions. This description is shown experimentally to be quite accurate, not only for near axis points within the shadow region but also well past the shadow’s edge into the directly illuminated region. The mathematical description is derived for spherical wave illumination and an isomorphic relation is developed relating it to plane wave illumination. The size of the central bright spot (as well as the subsequent diffraction rings), the axial intensity, and the intensity along the geometric shadow are characterized in terms of point source location and the distance of propagation past the circular obstacle.

© 1990 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. iii.
  2. S. G. Lipson, H. Lipson, Optical Physics (Cambridge U.P., London, 1969), p. 4.
  3. G. Mie, “Beitrage zur Optik Truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
  4. Ref. 1, Sec. 13.5.
  5. C. J. Bouwkamp, Dissertation, Groningen (1941).
  6. H. Osterberg, L. W. Smith, “Closed Solutions of Rayleigh’s Diffraction Integral for Axial Points,” J. Opt. Soc. Am. 51, 1050–1054 (1961).
    [CrossRef]
  7. V. N. Mahajan, “Axial Irradiance and Optimum Focusing of Laser Beams,” Appl. Opt. 22, 3042–3053 (1983).
    [CrossRef] [PubMed]
  8. R. E. English, N. George, “Diffraction Patterns in the Shadow of Disks and Obstacles,” Appl. Opt. 27, 1581–1587 (1988).
    [CrossRef] [PubMed]
  9. Ref. 1, p. 438. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 438.
  10. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1922), pp. 540–542.
  11. H. J. Weaver, Applications of Discrete and Continuous Fourier Analysis (Wiley, New York, 1983), pp. 217–223.
  12. J. Spanier, K. B. Oldham, An Atlas of Functions (Hemisphere Publishing, Washington, DC, 1987), p. 509.

1988 (1)

1983 (1)

1961 (1)

1908 (1)

G. Mie, “Beitrage zur Optik Truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. iii.

Ref. 1, p. 438. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 438.

Bouwkamp, C. J.

C. J. Bouwkamp, Dissertation, Groningen (1941).

English, R. E.

George, N.

Lipson, H.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U.P., London, 1969), p. 4.

Lipson, S. G.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U.P., London, 1969), p. 4.

Mahajan, V. N.

Mie, G.

G. Mie, “Beitrage zur Optik Truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Oldham, K. B.

J. Spanier, K. B. Oldham, An Atlas of Functions (Hemisphere Publishing, Washington, DC, 1987), p. 509.

Osterberg, H.

Smith, L. W.

Spanier, J.

J. Spanier, K. B. Oldham, An Atlas of Functions (Hemisphere Publishing, Washington, DC, 1987), p. 509.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1922), pp. 540–542.

Weaver, H. J.

H. J. Weaver, Applications of Discrete and Continuous Fourier Analysis (Wiley, New York, 1983), pp. 217–223.

Wolf, E.

Ref. 1, p. 438. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 438.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. iii.

Ann. Phys. (Leipzig) (1)

G. Mie, “Beitrage zur Optik Truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Other (8)

Ref. 1, p. 438. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. 438.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1922), pp. 540–542.

H. J. Weaver, Applications of Discrete and Continuous Fourier Analysis (Wiley, New York, 1983), pp. 217–223.

J. Spanier, K. B. Oldham, An Atlas of Functions (Hemisphere Publishing, Washington, DC, 1987), p. 509.

Ref. 1, Sec. 13.5.

C. J. Bouwkamp, Dissertation, Groningen (1941).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), p. iii.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U.P., London, 1969), p. 4.

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Figures (15)

Fig. 1
Fig. 1

Schematic representation of diffraction study geometry.

Fig. 2
Fig. 2

Geometric representation of conditions v/u > 1 and v/u < 1.

Fig. 3
Fig. 3

Analytic–experimental comparison (relative intensity vs radial distance s) for propagation distance z = 20 m.

Fig. 4
Fig. 4

Analytic–experimental comparison (relative intensity vs radial distance s) for propagation distance z = 10 m.

Fig. 5
Fig. 5

Analytic–experimental comparison (relative intensity vs radial distance s) for propagation distance z = 5 m.

Fig. 6
Fig. 6

Analytic–experimental comparison (relative intensity vs radial distance s) for propagation distance z = 2 m.

Fig. 7
Fig. 7

Microdensitometer intensity scan (dots) and intensity profile predicted by code (solid line) for a propagation distance z = 10 m. Inset shows photograph of a diffraction pattern.

Fig. 8
Fig. 8

Analytical results showing invariance of a central bright spot with respect to the location of the point source (a = 5 mm, λ = 0.5145 μm, z = 10 m).

Fig. 9
Fig. 9

Experimental results showing invariance of a central bright spot with respect to the location of the point source (a = 5 mm, λ = 0.5145 μm, z = 10 m).

Fig. 10
Fig. 10

Isomnorphic relationship: s is constant; z is variable.

Fig. 11
Fig. 11

Isomorphic relationship: z is constant; s is variable.

Fig. 12
Fig. 12

Experimental demonstration of isomorphic propagation for a scale factor d/(z + d) = 1/3: (a) diffraction pattern for spherical wave illumination with d = 5 m and z = 10 m; (b) diffraction pattern for plane wave illumination with z ^ = 3.33 m.

Fig. 13
Fig. 13

Zero-order Lommel functions U0(u,v) and V0(u,v).

Fig. 14
Fig. 14

First-order Lommel functions U1(u,v) and V1(u,v).

Fig. 15
Fig. 15

Second-order Lomnmel functions U2(u,v) and V2(u,v).

Tables (1)

Tables Icon

Table I Number of Terms Required to Evaluate the V0(u,v) Lommel Functions with Remainder < 0.0001

Equations (71)

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U ( s , θ , z ) = - z 2 π 0 2 π a U ( ρ , φ , 0 ) 1 r r [ exp ( ikr ) r ] ρ d ρ d φ ,
k = 2 π λ , U ( ρ , φ , 0 ) = exp ( i k r ) r ( spherical wave from a point source ) , r = ρ 2 + d 2 , r = s 2 + ρ 2 + z 2 - 2 ρ s cos ( θ - φ ) ,
r [ exp ( ikr ) r ] = ( i k - 1 / r ) exp ( ikr ) r i k exp ( ikr ) r .
U ( s , θ , z ) = - i k 2 π z d 0 2 π 0 exp [ i k ( z + d ) ] × exp { i k [ ρ 2 2 ( 1 z + 1 d ) + s 2 2 z - 2 s ρ cos ( θ - φ ) 2 z ] } ρ d ρ d φ , = - i k 2 π z d exp [ i k ( z + d ) ] exp ( i k s 2 2 z ) 0 2 π a × exp [ i k ( ρ 2 2 ) ( z + d z d ) ] exp [ - iks ρ cos ( θ - φ ) z ] ρ d ρ d φ .
0 2 π exp [ i x cos ( θ - φ ) ] d φ = 2 π J 0 ( x ) ,
U ( s , z ) = - i k z d exp [ i k ( z + d ) ] exp ( i k s 2 / 2 z ) 0 J 0 ( k s ρ / z ) × exp [ i k ( ρ 2 2 ) ( z + d z d ) ] ρ d ρ .
ρ = a t ,             u = k a 2 ( z + d z d ) ,             and v = kas z ,
U ( s , z ) = - i exp [ i k ( z + d ) ] ( z + d ) u exp ( i k s 2 / 2 z ) 1 J 0 ( v t ) exp ( i u t 2 / 2 ) tdt .
v u = s d a ( z + d ) .
s = a ( z + d ) d .
u exp ( - i u / 2 ) 1 J 0 ( v t ) exp ( i u t 2 / 2 ) tdt = V 1 ( u , v ) + i V 0 ( u , v ) ,
U ( s , z ) = exp [ i k ( z + d ) ] z + d exp ( i k s 2 / 2 z ) exp ( i u / 2 ) [ V 0 ( u , v ) - i V 1 ( u , v ) ] .
I ( s , z ) = U ( s , z ) 2 = 1 ( z + d ) 2 [ V 0 ( u , v ) 2 + V 1 ( u , v ) 2 ] .
U ( s , z ) = - i exp [ i k ( z + d ) ] ( z + d ) u exp ( i k s 2 / 2 z ) [ 0 J 0 ( v t ) exp ( i u t 2 / 2 ) tdt - 0 1 J 0 ( v t ) exp ( i u t 2 / 2 ) tdt ] .
0 J 0 ( v t ) exp ( i u t 2 / 2 ) tdt = i u exp ( - i v 2 / 2 u ) .
u exp ( - i u / 2 ) 0 1 J 0 ( v t ) exp ( i u t 2 / 2 ) tdt = U 1 ( u , v ) - i U 2 ( u , v ) .
U ( s , z ) = exp [ i k ( z + d ) ] z + d exp ( i k s 2 / 2 z ) × [ exp ( - i v 2 / 2 u ) + U 2 ( u , v ) exp ( i u / 2 ) + i U 1 ( u , v ) exp ( i u / 2 ) ] .
I ( s , z ) = U ( s , z ) 2 = 1 ( z + d ) 2 [ 1 + U 1 ( u , v ) 2 + U 2 ( u , v ) 2 - 2 U 1 ( u , v ) sin ( u 2 + v 2 2 u ) + 2 U 2 ( u , v ) cos ( u 2 + v 2 2 u ) ] .
I ( 0 , z ) = 1 ( z + d ) 2 ( on - axis intensity ) ,
I ( s , z ) = 1 ( z + d ) 2 [ V 0 ( v , v ) 2 + V 1 ( v , v ) 2 ] .
V 0 ( v , v ) = J 0 ( v ) - J 2 ( v ) + J 4 ( v ) - J 6 ( v ) + , = ½ [ J 0 ( v ) + J 0 ( v ) - 2 J 2 ( v ) + 2 J 4 ( v ) - 2 J 6 ( v ) + ] = ½ [ J 0 ( v ) + cos ( v ) ] ,
V 1 ( v , v ) = ½ [ 2 J 1 ( v ) - 2 J 3 ( v ) + 2 J 5 ( v ) - 2 J 7 ( v ) + ] = ½ sin ( v ) .
I ( s , z ) = 1 4 ( z + d ) 2 [ 1 + J 0 ( v ) 2 + 2 J 0 ( v ) cos ( v ) ] .
d I ( s , z ) d s = 1 ( z + d ) 2 d d s [ V 0 ( u , v ) 2 + V 1 ( u , v ) 2 ] = 0 ,
d d s [ V 0 ( u , v ) 2 + V 1 ( u , v ) 2 ] = 0.
d d s ( V 0 ( u , v ) 2 + V 1 ( u , v ) 2 ) = d v d s d d v [ V 0 ( u , v ) 2 + V 1 ( u , v ) 2 ] = k a z d d v [ V 0 ( u , v ) 2 + V 1 ( u , v ) 2 ] = 0.
v V 0 ( u , v ) [ V 1 ( u , v ) + V - 1 ( u , v ) ] = 0.
V 0 ( u , v ) J 1 ( v ) = 0.
s d = 0.38 λ z a ,
s b = 0.61 λ z a .
u = k a 2 ( z + d z d ) ,
v = kas z .
u = k a 2 z ^ .
v = k a s ^ z ^ .
z ^ = z ( d z + d ) and s ^ = s ( d z + d ) .
U n ( u , v ) = m = 0 ( - 1 ) m ( u v ) n + 2 m J n + 2 m ( v ) .
V n ( u , v ) = m = 0 ( - 1 ) m ( v u ) n + 2 m J n + 2 m ( v ) .
R | ( v u ) n + 2 m + 2 J n + 2 m + 2 ( v ) | ( v u ) n + 2 m + 2 .
R ( v u ) 2 m + 2 ,
m ln ( R ) 2 ln ( v / u ) - 1.
v = kas z = 2 π a 2 λ z ( u v ) = 60.45.
U 0 ( u , v ) = J 0 ( v ) - ( u v ) 2 J 2 ( v ) + ( u v ) 4 J 4 ( v ) + .
U 0 ( 0 , v ) = J 0 ( v ) ,
V 0 ( u , v ) = J 0 ( v ) - ( v u ) 2 J 2 ( v ) + ( v u ) 4 J 4 ( v ) - .
V 0 ( u , v ) J 0 ( v ) ,
U 1 ( u , v ) = ( u v ) J 1 ( v ) - ( u v ) 3 J 3 ( v ) + ( u v ) 5 J 5 ( v ) - ,
V 1 ( u , v ) = ( v x ) J 1 ( v ) - ( v u ) 3 J 3 ( v ) + ( v u ) 5 J 5 ( v ) - .
U 1 ( , v ) ( v ) J 1 ( v ) .
V 1 ( u , v ) ( v u ) J 1 ( v ) ,
U n ( u , v ) = m = 0 ( - 1 ) m ( u v ) n + 2 m J n + 2 m ( v ) = ( u v ) n J n ( v ) + m = 1 ( - 1 ) m ( u v ) n + 2 m J n + 2 m ( v ) .
U n ( u , v ) = ( u v ) n J n ( v ) + k = 0 ( - 1 ) k + 1 ( u v ) n + 2 k + 2 J n + 2 k + 2 ( v ) = ( u v ) n J n ( v ) - k = 0 ( - 1 ) k ( u v ) n + 2 + 2 k J n + 2 + 2 k ( v ) .
U n + 2 ( u , v ) + U n ( u , v ) = ( u v ) n J n ( v ) .
V n + 2 ( u , v ) + V n ( u , v ) = ( v u ) n J n ( v ) .
U - n ( u , v ) = m = 0 ( - 1 ) m ( u v ) 2 m - n J 2 m - n ( v ) = m = 0 n - 1 ( - 1 ) m ( u v ) 2 m - n J 2 m - n ( v ) + m = n ( - 1 ) m ( u v ) 2 m - n J 2 m - n ( v ) .
U - n ( u , v ) = m = 0 n - 1 ( - 1 ) m ( u v ) 2 m - n J 2 m - n ( v ) + k = 0 ( - 1 ) k + n ( u v ) 2 k + n J 2 k + n ( v ) = m = 0 n - 1 ( - 1 ) m ( u v ) 2 m - n J 2 m - n ( v ) + ( - 1 ) n k = 0 ( - 1 ) k ( u v ) 2 k + n J 2 k + n ( v ) , U - n ( u , v ) = ( - 1 ) n U n ( u , v ) + m = 0 n - 1 ( - 1 ) m ( u v ) 2 m - n J 2 m - n ( v ) .
V - n ( u , v ) = ( - 1 ) n V n ( u , v ) + m = 0 n - 1 ( - 1 ) m ( v u ) 2 m - n J 2 m - n ( v ) .
U - 1 ( u , v ) + U 1 ( u , v ) = ( v u ) J - 1 ( v ) ,
V - 1 ( u , v ) + V 1 ( u , v ) = ( u v ) J - 1 ( v ) .
U n ( u , v ) u = m = 0 ( n + 2 m ) 1 u ( u v ) n + 2 m J n + 2 m ( v ) .
2 k v J k ( v ) = J k - 1 ( v ) + J k + 1 ( v ) ,
U n ( u , v ) u = m = 0 ( - 1 ) m 1 2 ( u v ) n + 2 m - 1 [ J n + 2 m - 1 ( v ) + J n + 2 m + 1 ( v ) ] = 1 2 m = 0 ( - 1 ) m ( u v ) n + 2 m - 1 J n + 2 m - 1 ( v ) + 1 2 ( v u ) 2 m = 0 ( - 1 ) m ( u v ) n + 2 m + 1 J n + 2 m + 1 ( v ) .
U n ( u , v ) u = 1 2 [ U n - 1 ( u , v ) + ( v u ) 2 U n + 1 ( u , v ) ] .
V n ( u , v ) u = - 1 2 [ V n + 1 ( u , v ) + ( v u ) 2 V n - 1 ( u , v ) ] .
U n ( u , v ) v = m = 0 ( - 1 ) m 1 v ( u v ) n + 2 m [ - ( n + 2 m ) ] J n + 2 m ( v ) + m = 0 ( - 1 ) m ( u v ) n + 2 m J n + 2 m ( v ) v .
2 J k ( v ) v = J k - 1 ( v ) - J k + 1 ( v ) , 2 k v J k ( v ) = J k - 1 ( v ) + J k + 1 ( v ) ,
U n ( u , v ) v = - m = 0 ( - 1 ) m ( u v ) n + 2 m J n + 2 m + 1 ( v ) .
U n ( u , v ) v = - ( v u ) U n + 1 ( u , v ) .
V n ( u , v ) v = ( v u ) V n - 1 ( u , v ) .
V 0 ( u , v ) v = v u V - 1 ( u , v ) ,
V 1 ( u , v ) v = v u V 0 ( u , v ) .
U n ( u , v ) u = 1 2 [ U n - 1 ( u , v ) + ( v u ) 2 U n + 1 ( u , v ) ] , V n ( u , v ) u = - 1 2 [ V n - 1 ( u , v ) + ( v u ) 2 V n - 1 ( u , v ) ] , U n ( u , v ) v = - ( v u ) U n + 1 ( u , v ) , V n ( u , v ) v = ( v u ) V n - 1 ( u , v ) .

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