Abstract

Three theories of electromagnetic diffraction at an aperture in a screen, associated with the names of Kottler, Luneburg, Severin, and Vasseur, are reviewed and on the basis of these theories the coherence matrices relating to the far field are derived. These matrices will be used in Part II of this investigation to analyze systematically the structure of the far field as predicted by these theories.

© 1966 Optical Society of America

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References

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  1. F. Kottler, Ann. Physik 71, 457 (1923).
    [Crossref]
  2. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, 1964), p. 319.
  3. H. Severin, Z. Physik 129, 426 (1951).
    [Crossref]
  4. J. P. Vasseur, Ann. Phys. (Paris) 7, 506 (1952).
  5. B. Karczewski, Acta Phys. Polonica 5-6, 403 (1961).
  6. B. Karczewski, J. Opt. Soc. Am. 51, 1055 (1961).
    [Crossref]
  7. B. Karczewski and E. Wolf, J. Opt. Soc. Am. 56, 1214 (1966).
    [Crossref]
  8. For a reference to the original literature and for a discussion of the historical background see A. Rubinowicz, Acta Phys. Polonica 27, 435 (1965).
  9. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., Chap. 8.
  10. A. Sommerfeld, Optics (Academic Press Inc., New York, 1954), p. 197.
  11. G. Toraldo di Francia, Electromagnetic Waves (Interscience Publishers, Inc., New York, 1955), p. 221.
  12. J. A. Stratton and L. J. Chu, Phys. Rev. 56, 99, 316 (1939).
    [Crossref]
  13. F. Kottler, in Progress in Optics, Vol. VI, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam; John Wiley & Sons, Inc., New York, in preparation).
  14. The coherence matrix is normally defined by taking the time average of each element in the matrix (5.1) (cf. Ref. 15 or 9). However, since we are now dealing with the idealized case of strictly monochromatic wave, the time average may be omitted.
  15. E. Wolf, Nuovo Cimento 13, 1165 (1959).
    [Crossref]

1966 (1)

1965 (1)

For a reference to the original literature and for a discussion of the historical background see A. Rubinowicz, Acta Phys. Polonica 27, 435 (1965).

1961 (2)

B. Karczewski, Acta Phys. Polonica 5-6, 403 (1961).

B. Karczewski, J. Opt. Soc. Am. 51, 1055 (1961).
[Crossref]

1959 (1)

E. Wolf, Nuovo Cimento 13, 1165 (1959).
[Crossref]

1952 (1)

J. P. Vasseur, Ann. Phys. (Paris) 7, 506 (1952).

1951 (1)

H. Severin, Z. Physik 129, 426 (1951).
[Crossref]

1939 (1)

J. A. Stratton and L. J. Chu, Phys. Rev. 56, 99, 316 (1939).
[Crossref]

1923 (1)

F. Kottler, Ann. Physik 71, 457 (1923).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., Chap. 8.

Chu, L. J.

J. A. Stratton and L. J. Chu, Phys. Rev. 56, 99, 316 (1939).
[Crossref]

Karczewski, B.

Kottler, F.

F. Kottler, Ann. Physik 71, 457 (1923).
[Crossref]

F. Kottler, in Progress in Optics, Vol. VI, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam; John Wiley & Sons, Inc., New York, in preparation).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, 1964), p. 319.

Rubinowicz, A.

For a reference to the original literature and for a discussion of the historical background see A. Rubinowicz, Acta Phys. Polonica 27, 435 (1965).

Severin, H.

H. Severin, Z. Physik 129, 426 (1951).
[Crossref]

Sommerfeld, A.

A. Sommerfeld, Optics (Academic Press Inc., New York, 1954), p. 197.

Stratton, J. A.

J. A. Stratton and L. J. Chu, Phys. Rev. 56, 99, 316 (1939).
[Crossref]

Toraldo di Francia, G.

G. Toraldo di Francia, Electromagnetic Waves (Interscience Publishers, Inc., New York, 1955), p. 221.

Vasseur, J. P.

J. P. Vasseur, Ann. Phys. (Paris) 7, 506 (1952).

Wolf, E.

B. Karczewski and E. Wolf, J. Opt. Soc. Am. 56, 1214 (1966).
[Crossref]

E. Wolf, Nuovo Cimento 13, 1165 (1959).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., Chap. 8.

Acta Phys. Polonica (2)

For a reference to the original literature and for a discussion of the historical background see A. Rubinowicz, Acta Phys. Polonica 27, 435 (1965).

B. Karczewski, Acta Phys. Polonica 5-6, 403 (1961).

Ann. Phys. (Paris) (1)

J. P. Vasseur, Ann. Phys. (Paris) 7, 506 (1952).

Ann. Physik (1)

F. Kottler, Ann. Physik 71, 457 (1923).
[Crossref]

J. Opt. Soc. Am. (2)

Nuovo Cimento (1)

E. Wolf, Nuovo Cimento 13, 1165 (1959).
[Crossref]

Phys. Rev. (1)

J. A. Stratton and L. J. Chu, Phys. Rev. 56, 99, 316 (1939).
[Crossref]

Z. Physik (1)

H. Severin, Z. Physik 129, 426 (1951).
[Crossref]

Other (6)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, 1964), p. 319.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., Chap. 8.

A. Sommerfeld, Optics (Academic Press Inc., New York, 1954), p. 197.

G. Toraldo di Francia, Electromagnetic Waves (Interscience Publishers, Inc., New York, 1955), p. 221.

F. Kottler, in Progress in Optics, Vol. VI, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam; John Wiley & Sons, Inc., New York, in preparation).

The coherence matrix is normally defined by taking the time average of each element in the matrix (5.1) (cf. Ref. 15 or 9). However, since we are now dealing with the idealized case of strictly monochromatic wave, the time average may be omitted.

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Figures (4)

F. 1
F. 1

Notation.

F. 2
F. 2

Notation.

F. 3
F. 3

Notation.

F. 4
F. 4

Notation: (K,L,M), (k,l,m) are triads of mutually orthogonal unit vectors. K is in the direction of incidence and k is in the direction of diffraction.

Equations (67)

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E ( P ) = × Π m ( 1 / i k 0 ) × × Π e , H ( P ) = × Π e + ( 1 / i k 0 ) × × Π m , }
Π e = 1 4 π S n × H ( Q ) exp ( i k 0 r ) r d S , Π m = 1 4 π S n × E ( Q ) exp ( i k 0 r ) r d S , }
( I ) : E ( P ) = 2 × Π m , H ( P ) = ( 2 / i k 0 ) × × Π m , }
( II ) : E ( P ) = ( 2 / i k 0 ) × × Π e , H ( P ) = 2 × Π e , }
E ( i ) ( r ) = A exp [ i k 0 ( K r ) ] , H ( i ) ( r ) = B exp [ i k 0 ( K r ) ] ,
B = K × A .
n × E = n × E ( i ) , n × H = n × H ( i ) in a n × E = 0 , n × H = 0 on B
E ( e , m ) ( P ) = × Π m ( i ) ( 1 / i k 0 ) × × Π e ( i ) ,
H ( e , m ) ( P ) = × Π e ( i ) + ( 1 / i k 0 ) × × Π m ( i ) ,
Π e ( i ) = 1 4 π a n × H ( i ) ( Q ) exp ( i k 0 r ) r d S ,
Π m ( i ) = 1 4 π a n × E ( i ) ( Q ) exp ( i k 0 r ) r d S .
n × E = n × E ( i ) in a n × E = 0 on B . }
E ( m ) ( P ) = 2 × Π m ( i )
H ( m ) ( P ) = 2 / i k 0 × × Π m ( i ) .
n × H = n × H ( i ) in a n × H = 0 on B . }
E ( e ) ( P ) = ( 2 / i k 0 ) × × Π e ( i )
H ( e ) ( P ) = 2 × Π e ( i ) .
E ( e , m ) = 1 2 ( E ( m ) + E ( e ) ) , H ( e , m ) = 1 2 ( H ( m ) + H ( e ) ) . }
× Π m ( i ) k × ( F × A ) ,
× × Π m ( i ) = i k 0 { F × A k [ k ( F × A ) ] } ,
F = C n a exp [ i k 0 ( K k ) R ] d S ,
C = ( i k 0 / 4 π ) [ exp ( i k 0 r 0 ) r 0 ] .
E ( e , m ) ( P ) = k × ( F × A ) + ( F × B ) k ( F × B ) k H ( e , m ) ( P ) = k × ( F × B ) ( F × A ) + k ( F × A ) k ; }
E ( m ) ( P ) = 2 k × ( F × A ) H ( m ) ( P ) = 2 { F × A [ k ( F × A ) ] k } ; }
E ( e ) ( P ) = 2 { F × B [ k ( F × B ) ] k } H ( e ) ( P ) = 2 k × ( F × B ) . }
B = K × A .
k E ( e ) = k H ( e ) = 0 , k × E ( e ) = H ( e ) , }
J ( k ) = [ E l ( k ) E l * ( k ) E l ( k ) E m * ( k ) E m ( k ) E l * ( k ) E m ( k ) E m * ( k ) ] .
k 2 = l 2 = m 2 = 1 , k × l = m ,
I : k z / ( 1 k y 2 ) 1 2 , 0 , k x / ( 1 k y 2 ) 1 2 m : k x k y / ( 1 k y 2 ) 1 2 , ( 1 k y 2 ) 1 2 , k y k z / ( 1 k y 2 ) 1 2 . }
A x = A K K x + A L L x + A M M x = K z ( 1 K y 2 ) 1 2 A L K x K y ( 1 K y 2 ) 1 2 A M .
A y = ( 1 K y 2 ) 1 2 A M
A z = K x ( 1 K y 2 ) 1 2 A L K y K z ( 1 K y 2 ) 1 2 A M .
B L = A M , B M = A L ,
B x = K z ( 1 K y 2 ) 1 2 A M K x K y ( 1 K y 2 ) 1 2 A L B y = ( 1 K y 2 ) 1 2 A L , B z = K x ( 1 K y 2 ) 1 2 A M K y K z ( 1 K y 2 ) 1 2 A L .
E l ( m ) = 2 F { A L [ K z ( 1 k y 2 1 K y 2 ) 1 2 ] + A M [ k x k y ( 1 K y 2 1 k y 2 ) 1 2 + K x K y ( 1 k y 2 1 K y 2 ) 1 2 ] } , E m ( m ) = 2 F { A M [ k z ( 1 K y 2 1 k y 2 ) 1 2 ] } . }
E l ( e ) = 2 F { A L [ k z ( 1 K y 2 1 k y 2 ) 1 2 ] } , E m ( e ) = 2 F { A L [ k x k y ( 1 K y 2 1 k y 2 ) 1 2 K x K y ( 1 k y 2 1 K y 2 ) 1 2 + A M [ K z ( 1 k y 2 1 K y 2 ) 1 2 ] } . }
E k ( m ) = E k ( e ) = 0 .
a 1 ( m ) = K z ( 1 k y 2 1 K y 2 ) 1 2 , b 1 ( m ) = k x k y ( 1 K y 2 1 k y 2 ) 1 2 + K x K y ( 1 k y 2 1 K y 2 ) 1 2 , b 2 ( m ) = k z ( 1 K y 2 1 k y 2 ) 1 2 . }
a 1 ( e ) = k z ( 1 K y 2 1 k y 2 ) 1 2 , a 2 ( e ) = k x k y ( 1 K y 2 1 k y 2 ) 1 2 K x K y ( 1 k y 2 1 K y 2 ) 1 2 , b 2 ( e ) = K z ( 1 k y 2 1 K y 2 ) 1 2 . }
E l ( m ) = 2 F ( a 1 ( m ) A L + b 1 ( m ) A M ) , E m ( m ) = 2 F b 2 ( m ) A M . }
E l ( e ) = 2 F a 1 ( e ) A L E m ( e ) = 2 F ( a 2 ( e ) A L + b 2 ( e ) A M ) . }
E l ( e , m ) = 2 F ( a 1 ( e , m ) A L + b 1 ( e , m ) A M ) , E m ( e , m ) = 2 F ( a 2 ( e , m ) A L + b 2 ( e , m ) A M ) , }
a 1 ( e , m ) = 1 2 ( a 1 ( m ) + a 1 ( e ) ) , a 2 ( e , m ) = 1 2 a 2 ( e ) , b 1 ( e , m ) = 1 2 b 1 ( m ) , b 2 ( e , m ) = 1 2 ( b 2 ( m ) + b 2 ( e ) ) . }
J l l ( m ) = 4 | F | 2 [ ( a 1 ( m ) ) 2 | A L | 2 + ( b 1 ( m ) ) 2 | A M | 2 + 2 a 1 ( m ) b 1 ( m ) | A L | | A M | cos δ ] , J l m ( m ) = 4 | F | 2 [ b 1 ( m ) b 2 ( m ) | A M | 2 + a 1 ( m ) b 2 ( m ) | A L | | A M | e i δ ] , J m l ( m ) = J l m ( m ) * , J m m ( m ) = 4 | F | 2 ( b 2 ( m ) ) 2 | A M | 2 . }
δ = δ 1 δ 2 .
J l l ( e ) = 4 | F | 2 ( a 1 ( e ) ) 2 | A L | 2 , J l m ( e ) = 4 | F | 2 [ a 1 ( e ) a 2 ( e ) | A L | 2 + a 1 ( e ) b 2 ( e ) | A L | | A M | e i δ ] , J m l ( e ) = J l m ( e ) * , J m m ( e ) = 4 | F | 2 [ ( a 2 ( e ) ) 2 | A L | 2 + ( b 2 ( e ) ) 2 | A M | 2 + 2 a 2 ( e ) b 2 ( e ) | A L | | A M | cos δ ] , }
J l l ( e , m ) = 4 | F | 2 [ ( a 1 ( e , m ) ) 2 | A L | 2 + ( b 1 ( e , m ) ) 2 | A M | 2 + 2 a 1 ( e , m ) b 1 ( e , m ) | A L | | A M | cos δ ] , J l m ( e , m ) = 4 | F | 2 [ a 1 ( e , m ) a 2 ( e , m ) | A L | 2 + b 1 ( e , m ) b 2 ( e , m ) | A M | 2 + a 1 ( e , m ) b 2 ( e , m ) | A L | | A M | e i δ + a 2 ( e , m ) b 1 ( e , m ) | A L | | A M | e i δ ] , J m l ( e , m ) = J l m ( e , m ) * , J m m ( e , m ) = 4 | F | 2 [ ( a 2 ( e , m ) ) 2 | A L | 2 + ( b 2 ( e , m ) ) 2 | A M | 2 + 2 a 2 ( e , m ) b 2 ( e , m ) | A L | | A M | cos δ ] . }
× Π m ( i ) = 1 4 π a ( exp i k 0 r r ) × ( n × E ( i ) ) d S = 1 4 π a exp i k 0 r r ( i k 0 1 r ) ( r r ) × ( n × E ( i ) ) d S ,
× Π m ( i ) = i k 0 4 π a exp [ i k 0 ( r + K R ) ] r × ( r r ) × ( n × A ) d S .
r / r r 0 / r = k ,
r r 0 k R , 1 / r 1 / r 0 .
× Π m ( i ) k × ( F × A ) ,
F = C n a exp [ i k 0 ( K k ) R ] d a ,
C = ( i k 0 / 4 π ) exp ( i k 0 r ) / r 0 .
× ( u × v ) = u v v u + v ( u ) u ( v ) ,
× × Π m ( i ) = 1 4 π a [ k 0 2 × ( n × E ( i ) ) exp ( i k 0 r ) r + W ] d S ,
W = [ ( n × E ( i ) ) ] × { [ exp ( i k 0 r ) / r ] ( i k 0 1 / r ) ( r / r ) } .
ξ = x Q x P , η = y Q y P , ζ = z Q z P .
W x = ( a x x P + a y y P + a z z P ) × [ exp ( i k 0 r ) r ( i k 1 r ) ξ r ] ,
a = n × E ( i ) .
x P { exp ( i k 0 r ) r ( i k 1 r ) ξ r } = { ( i k 0 1 r ) 2 ξ 2 r 2 + ( i k 0 1 r ) ( r 2 ξ 2 r 3 ) + ξ 2 r 4 } exp ( i k 0 r ) r , y P { exp ( i k 0 r ) r ( i k 1 r ) ξ r } = { ( i k 0 1 r ) 2 ξ η r 2 ( i k 0 1 r ) ξ η r 3 + ξ η r 4 } exp ( i k 0 r ) r , }
x P [ exp ( i k 0 r ) r ( i k 1 r ) ξ r ] k 0 2 ξ 2 r 2 exp ( i k 0 r ) r , y P [ exp ( i k 0 r ) r ( i k 1 r ) ξ r ] k 0 2 ξ η r 2 exp ( i k 0 r ) r , }
W x k 0 2 [ a x ξ 2 r 2 + a y ξ η r 2 + a z ξ ζ r 2 ] exp ( i k 0 r ) r = k 0 2 k x ( k a ) exp ( i k 0 r ) r , }
W = k 0 2 k ( k a ) exp ( i k 0 r ) / r .
× × Π m ( i ) 1 4 π k 0 2 a { n × E ( i ) [ k ( n × E ( i ) ) ] k } ( exp i k 0 r / r ) d S .
× × Π m ( i ) ( i k 0 ) { F × A [ k ( F × A ) ] k } .