Abstract

Closed-form solutions have been derived for the diffraction patterns at the focal plane of (1) a convergent wave of unit amplitude illuminating a segment of a circular aperture and (2) a convergent wave of Gaussian amplitude diffracted by an infinite edge. Photographs showing the main features of these edge transform patterns are presented together with computer-generated graphs.

© 1975 Optical Society of America

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References

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  1. A. Sommerfeld, Optics (Academic Press, New York, 1967).
  2. J. B. Keller, J. Opt. Soc. Am. 52, 116 (1962).
    [CrossRef]
  3. W. Braunbek, G. Laukien, Optik 9, 174 (1952).
  4. F. E. Borgnis, C. H. Papas, Randwertprobleme Der Mikrowellenphysik (Springer-Verlag, Berlin, Göttingen, 1955), pp. 101–157.
    [CrossRef]
  5. J. E. Pearson, T. C. McGill, S. Kurtin, A. Yariv, J. Opt. Soc. Am. 59, 1440 (1969).
    [CrossRef]
  6. H. Lipson, Optical Transforms (Academic Press, London, 1972).
  7. E. Lommel, Abh. Bayer Akad. 15, Abth, 2, 233, (1885).
  8. H. Struve, Mem. Acad. St. Petersbourgh 34, 1 (1886).
  9. H. Linfoot, E. Wolf, Proc. Phys. Soc. B 69, 823 (1956).
    [CrossRef]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  11. M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (Natl. Bureau of Standards, Washington, D.C., 1970).

1969 (1)

1962 (1)

1956 (1)

H. Linfoot, E. Wolf, Proc. Phys. Soc. B 69, 823 (1956).
[CrossRef]

1952 (1)

W. Braunbek, G. Laukien, Optik 9, 174 (1952).

1886 (1)

H. Struve, Mem. Acad. St. Petersbourgh 34, 1 (1886).

1885 (1)

E. Lommel, Abh. Bayer Akad. 15, Abth, 2, 233, (1885).

Borgnis, F. E.

F. E. Borgnis, C. H. Papas, Randwertprobleme Der Mikrowellenphysik (Springer-Verlag, Berlin, Göttingen, 1955), pp. 101–157.
[CrossRef]

Braunbek, W.

W. Braunbek, G. Laukien, Optik 9, 174 (1952).

Goodman, J. W.

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

Keller, J. B.

Kurtin, S.

Laukien, G.

W. Braunbek, G. Laukien, Optik 9, 174 (1952).

Linfoot, H.

H. Linfoot, E. Wolf, Proc. Phys. Soc. B 69, 823 (1956).
[CrossRef]

Lipson, H.

H. Lipson, Optical Transforms (Academic Press, London, 1972).

Lommel, E.

E. Lommel, Abh. Bayer Akad. 15, Abth, 2, 233, (1885).

McGill, T. C.

Papas, C. H.

F. E. Borgnis, C. H. Papas, Randwertprobleme Der Mikrowellenphysik (Springer-Verlag, Berlin, Göttingen, 1955), pp. 101–157.
[CrossRef]

Pearson, J. E.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic Press, New York, 1967).

Struve, H.

H. Struve, Mem. Acad. St. Petersbourgh 34, 1 (1886).

Wolf, E.

H. Linfoot, E. Wolf, Proc. Phys. Soc. B 69, 823 (1956).
[CrossRef]

Yariv, A.

Abh. Bayer Akad. (1)

E. Lommel, Abh. Bayer Akad. 15, Abth, 2, 233, (1885).

J. Opt. Soc. Am. (2)

Mem. Acad. St. Petersbourgh (1)

H. Struve, Mem. Acad. St. Petersbourgh 34, 1 (1886).

Optik (1)

W. Braunbek, G. Laukien, Optik 9, 174 (1952).

Proc. Phys. Soc. B (1)

H. Linfoot, E. Wolf, Proc. Phys. Soc. B 69, 823 (1956).
[CrossRef]

Other (5)

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

M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (Natl. Bureau of Standards, Washington, D.C., 1970).

F. E. Borgnis, C. H. Papas, Randwertprobleme Der Mikrowellenphysik (Springer-Verlag, Berlin, Göttingen, 1955), pp. 101–157.
[CrossRef]

H. Lipson, Optical Transforms (Academic Press, London, 1972).

A. Sommerfeld, Optics (Academic Press, New York, 1967).

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Figures (8)

Fig. 1
Fig. 1

(a) Radiation pattern at the back focal plane of a lens illuminated by a wave of uniform amplitude and truncated by the circular aperture shown in the inset. (b) Recorded intensity for the same illuminations as for (a) with the addition of an edge at the lens so as to reduce the opening to a semicircular aperture, shown in the inset.

Fig. 2
Fig. 2

Coordinate geometry for the aperture, consisting of a segment of a circle, and the back focal plane.

Fig. 3
Fig. 3

Normalized transmitted irradiance In is plotted logarithmically vs αx with y = 0 for a semicircular aperture curve (A), illuminated with a convergent wave of unit amplitude, and for a circular aperture, curve (B), illuminated with a convergent wave with amplitude equal to 0.5.

Fig. 4
Fig. 4

Normalized transmitted irradiance In is plotted logarithmically vs αx and αy with d = 0, for a semicircular aperture illuminated with a convergent unit amplitude wave. P0 corresponds to a normalized irradiance of 0.25, and the range in αy is from −10 to +10, and in αx from −20 to 20.

Fig. 5
Fig. 5

Normalized transmitted irradiance In plotted logarithmically vs αx and αy with d = 0, for a semicircular aperture illuminated with a convergent unit amplitude wave. The interval along the αy axis is from −10 ≤ αy ≤ 10, and αx spans the relative maxima from P9 through P15; the point P9 corresponds to a normalized irradiance of 1.4 × 10−4.

Fig. 6
Fig. 6

Normalized transmitted irradiance In is plotted logarithmically vs 2πw0y/(λs) and 2πw0x/(λs) with d = 0, for an infinite edge illuminated with a convergent Gaussian wave. The maximum of In corresponds to a normalized irradiance of 1.0 while the horizontally ruled plane corresponds to an irradiance level of 10−5. The interval along the normalized x axis is from −80 ≤ 2πw0x/s ≤ 80 and along y from −10 ≤ 2πw0ys ≤ 10.

Fig. 7
Fig. 7

Recorded intensity for a convergent wave of Gaussian amplitude truncated by an infinite edge.

Fig. 8
Fig. 8

Transmitted irradiance I normalized to its d/w0 = 0 value is plotted logarithmically vs 2πw0x/(λs) for y = 0. The curves whose normalized intensity at 2πw0x/(λs) = 0 is greater than one correspond to negative values of d/w0.

Tables (1)

Tables Icon

Table I Nulls of Radiation Pattern for a Convergent Gaussian Wave Diffracted by an Offset Edge, Eq. (28)

Equations (31)

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T ( ξ , η ) = circ [ ( ξ 2 + η 2 ) 1 / 2 / a ] { 1 2 + [ sgn ( ξ d ) ] / 2 } ,
circ [ ( ξ 2 + η 2 ) 1 / 2 / a ] { = 1 when ( ξ 2 + η 2 ) / a 2 1 , = 0 otherwise ,
sgn ( ξ d ) { = 1 when ξ d > 0 , = 1 when ξ d < 0.
U ( ξ , η ) = exp [ ( i π / λ s ) ( ξ 2 + η 2 ) ] .
V ( x , y , z = s ) = i exp [ i 2 π s / λ ] λ s × U exp { i π λ s [ ( ξ x ) 2 + ( η y ) 2 ] } d ξ d η ,
U ( ξ , η ) = T ( ξ , η ) U ( ξ , η ) .
V ( x , y , z = s ) = { i exp [ i 2 π s / λ i π ( x 2 + y 2 ) / ( λ s ) ] / λ s } ( I 1 + I 2 ) .
I 1 = 1 2 circ [ ( ξ 2 + η 2 ) 1 / 2 a ] × exp [ i 2 π ( x ξ + y η ) / ( λ s ) ] d ξ d η ,
I 1 = π a 2 { J 1 [ α ( x 2 + y 2 ) 1 / 2 ] / α ( x 2 + y 2 ) 1 / 2 } ,
I 2 = 1 2 circ [ ( ξ 2 + η 2 ) 1 / 2 a ] sgn ( ξ d ) × exp [ i 2 π ( x ξ + y η ) / ( λ s ) ] d ξ d η .
I 2 = 1 2 λ s π y a + a sgn ( ξ d ) sin [ 2 π y λ s ( a 2 ξ 2 ) 1 / 2 ] × exp [ i 2 π x ξ / ( λ s ) ] d ξ .
I 2 = a 2 π 2 π α y { i d / a 1 sin [ α y ( 1 u 2 ) 1 / 2 ] sin ( α x u ) d u 0 d / a sin [ α y ( 1 u 2 ) 1 / 2 ] cos ( α x u ) d u } .
I 2 = a 2 π 2 π α y { n = 0 ( 1 ) n J 2 n + 1 ( α y ) [ i m = 0 J 2 m + 1 ( α x ) g 1 ( m , n ) m = 0 ( 1 1 2 δ 0 m ) ( 1 ) m J 2 m ( α x ) g 2 ( m , n ) ] } ,
g 1 ( m , n ) = cos [ ( 2 m + 2 n + 3 ) sin 1 ( d / a ) ] 2 m + 2 n + 3 + cos [ ( 2 m + 2 n + 1 ) sin 1 ( d / a ) ] 2 m + 2 n + 1 + cos [ ( 2 n 2 m + 1 ) sin 1 ( d / a ) ] 2 n 2 m + 1 + cos [ ( 2 m 2 n + 1 ) sin 1 ( d / a ) ] 2 m 2 n + 1 ,
g 2 ( m , n ) = sin [ ( 2 m + 2 n + 2 ) sin 1 ( d / a ) ] 2 m + 2 n + 2 + sin [ ( 2 m + 2 n ) sin 1 ( d / a ) ] 2 m + 2 n + sin [ ( 2 n 2 m + 2 ) sin 1 ( d / a ) ] 2 n 2 m + 2 + sin [ ( 2 m 2 n ) sin 1 ( d / a ) ] 2 m + 2 n .
V ( x , y , z = s ) = i λ s exp ( i k s ) exp [ i π ( x 2 + y 2 ) / ( λ s ) ] π a 2 × ( J 1 [ α ( x 2 + y 2 ) 1 / 2 ] / [ α ( x 2 + y 2 ) 1 / 2 ] + 2 π α y { n = 0 ( 1 ) n J 2 n + 1 ( α y ) [ i m = 0 J 2 m + 1 ( α x ) g 1 ( m , n ) m = 0 ( 1 1 2 δ 0 m ) ( 1 ) m J 2 m ( α x ) g 2 ( m , n ) ] } ) ,
V ( x , y , z = s ) = i λ s exp ( i k s ) exp [ i π ( x 2 + y 2 ) / ( λ s ) ] π a 2 × { J 1 [ α ( x 2 + y 2 ) 1 / 2 ] / [ α ( x 2 + y 2 ) 1 / 2 ] + 2 i π α y n = 0 m = 0 ( 1 ) n J 2 n + 1 ( α y ) J 2 m + 1 ( α x ) g 0 ( m , n ) } .
g 0 ( m , n ) = { 1 / [ 2 ( m + n ) + 3 ] } + { 1 / [ 2 ( m + n ) + 1 ] } { 1 / [ 2 ( n m ) + 1 ] } + { 1 / [ 2 ( m n ) + 1 ] } .
V 0 V 0 * = ( ( π a 2 / λ s ) { J 1 [ α ( x 2 + y 2 ) 1 / 2 ] / α ( x 2 + y 2 ) 1 / 2 } ) 2 ,
V ( 0 , y , z = s ) = ( i / λ s ) exp ( i k s ) × exp [ i π y 2 / ( λ s ) ] π a 2 [ J 1 ( α | y | ) / α | y | ] ,
V ( x , 0 , z = s ) = i λ s exp ( i k s ) exp [ i π x 2 / ( λ s ) ] π a 2 × [ J 1 ( α x ) α x + 4 i π α x m = 1 ( 4 m 2 4 m 2 1 ) J 2 m ( α x ) ] .
4 i π α x m = 1 [ 4 m 2 4 m 2 1 ] J 2 m ( α x ) = i α x H 1 ( α x ) ,
4 i π α x m = 1 [ 4 m 2 4 m 2 1 ] J 2 m ( α x ) = J 1 ( α x ) α x + H 1 ( 1 ) ( | α x | ) α x + 2 i π α x [ 1 + 1 ( α x ) 2 3 ( α x ) 4 + ] ,
I 2 = π a 2 i α x m = 0 ( α y 2 4 x ) m m ! H m + 1 ( α x )
V ( x , y , z = s ) = i λ s exp ( i k s ) exp [ i π ( x 2 + y 2 ) / ( λ s ) ] π a 2 { J 1 [ α ( x 2 + y 2 ) 1 / 2 ] α ( x 2 + y 2 ) 1 / 2 + i α x m = 0 ( α y 2 4 x ) m m ! H m + 1 ( α x ) } .
U g ( ξ , η ) = exp [ i π ( ξ 2 + η 2 ) / ( λ s ) ( ξ 2 + η 2 ) / w 0 2 ] ,
T ( ξ , η ) = 1 2 [ 1 + sgn ( ξ d ) ] .
V g ( x , y , z = s ) = [ i / ( 2 λ s ) ] exp ( i k s ) exp [ i π ( x 2 + y 2 ) / ( λ s ) ] π w 0 2 exp [ ( π w 0 / λ s ) 2 ( x 2 + y 2 ) ] { 1 erf [ ( d / w 0 ) ( i π x w 0 / λ s ) ] } ,
V g ( x , y , z = s ) = [ i / ( 2 λ s ) ] exp ( i k s ) exp [ i π ( x 2 + y 2 ) / ( λ s ) ] exp [ i 2 π x d / ( λ s ) ] π w 0 2 exp { [ π w 0 y / ( λ s ) ] 2 ( d / w 0 ) 2 } w [ π w 0 x / ( λ s ) + i d / w 0 ] .
V g ( x , y , z = s ) = [ 1 / ( λ s ) ] exp ( i k s ) exp [ i π ( x 2 + y 2 ) / ( λ s ) ] exp [ i 2 π x d / ( λ s ) ] π w 0 2 exp { [ π w 0 y / ( λ s ) ] 2 } ( d / w 0 1 ) { 0.256 / [ π w 0 x / ( λ s ) ] } .
V s g ( x , y , z = s ) = [ i / ( 2 λ s ) ] exp ( i k s ) exp [ i π ( x 2 + y 2 ) / ( λ s ) ] π w 0 2 exp [ ( π w 0 / λ s ) 2 ( x 2 + y 2 ) ] { erf [ ( d 2 / w 0 ) ( i π x w 0 / λ s ) ] erf [ ( d 1 / w 0 ) ( i π x w 0 / λ s ) ] } .

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