Abstract

The Rayleigh–Sommerfeld solutions and the Luneburg–Severin–Vasseur and Severin–Vasseur solutions of diffraction problems are shown to satisfy certain reciprocity theorems in the case of a spherical incident wave. These new reciprocity theorems have very simple relations to those known in Kirchhoff’s theory of diffraction.

© 1966 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 381.
  2. A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Polska Akademia Nauk, Warszawa, 1957), p. 69; see also Acta Phys. Polon. 20, 9 (1961).
  3. F. Kottler, Ann. Physik. 71, 457 (1923).
    [Crossref]
  4. A. Kujawski, Acta Phys. Polon. 21, 597 (1962).
  5. A. Kujawski, Acta Phys. Polon. 25, 7 (1964).
  6. O. Laporte and J. Meixner, Z. Physik 153, 129 (1958).
    [Crossref]
  7. See, for example, Rayleigh (J. W. Strutt), Phil. Mag. 43, 259 (1897); see also his Scientific Papers (Cambridge University Press, Cambridge, England, 1903), Vol. 4, p. 283.
  8. A. Sommerfeld, Optics (Academic Press Inc., New York, 1954), p. 199; see also Ref. 2, pp. 77–83.
  9. C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
    [Crossref]
  10. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., New York, 1962), p. 271.
  11. See Ref. 2, p. 242.
  12. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1964), p. 319.
  13. H. Severin, Z. Physik. 129, 426 (1951).
    [Crossref]
  14. J. P. Vasseur, Ann. Phys. (Paris) 7, 506 (1952).
  15. R. Barakat, J. Opt. Soc. Am. 55, 992 (1965).

1965 (1)

1964 (1)

A. Kujawski, Acta Phys. Polon. 25, 7 (1964).

1962 (1)

A. Kujawski, Acta Phys. Polon. 21, 597 (1962).

1958 (1)

O. Laporte and J. Meixner, Z. Physik 153, 129 (1958).
[Crossref]

1954 (1)

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[Crossref]

1952 (1)

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

1951 (1)

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

1923 (1)

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

1897 (1)

See, for example, Rayleigh (J. W. Strutt), Phil. Mag. 43, 259 (1897); see also his Scientific Papers (Cambridge University Press, Cambridge, England, 1903), Vol. 4, p. 283.

See, for example, Rayleigh (J. W. Strutt), Phil. Mag. 43, 259 (1897); see also his Scientific Papers (Cambridge University Press, Cambridge, England, 1903), Vol. 4, p. 283.

Barakat, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 381.

Bouwkamp, C. J.

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., New York, 1962), p. 271.

Kottler, F.

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

Kujawski, A.

A. Kujawski, Acta Phys. Polon. 25, 7 (1964).

A. Kujawski, Acta Phys. Polon. 21, 597 (1962).

Laporte, O.

O. Laporte and J. Meixner, Z. Physik 153, 129 (1958).
[Crossref]

Luneburg, R. K.

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

Meixner, J.

O. Laporte and J. Meixner, Z. Physik 153, 129 (1958).
[Crossref]

Rayleigh,

See, for example, Rayleigh (J. W. Strutt), Phil. Mag. 43, 259 (1897); see also his Scientific Papers (Cambridge University Press, Cambridge, England, 1903), Vol. 4, p. 283.

Rubinowicz, A.

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Polska Akademia Nauk, Warszawa, 1957), p. 69; see also Acta Phys. Polon. 20, 9 (1961).

Severin, H.

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

Sommerfeld, A.

A. Sommerfeld, Optics (Academic Press Inc., New York, 1954), p. 199; see also Ref. 2, pp. 77–83.

Strutt, J. W.

See, for example, Rayleigh (J. W. Strutt), Phil. Mag. 43, 259 (1897); see also his Scientific Papers (Cambridge University Press, Cambridge, England, 1903), Vol. 4, p. 283.

Vasseur, J. P.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 381.

Acta Phys. Polon. (2)

A. Kujawski, Acta Phys. Polon. 21, 597 (1962).

A. Kujawski, Acta Phys. Polon. 25, 7 (1964).

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. (1)

Phil. Mag. (1)

See, for example, Rayleigh (J. W. Strutt), Phil. Mag. 43, 259 (1897); see also his Scientific Papers (Cambridge University Press, Cambridge, England, 1903), Vol. 4, p. 283.

Rept. Progr. Phys. (1)

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[Crossref]

Z. Physik (1)

O. Laporte and J. Meixner, Z. Physik 153, 129 (1958).
[Crossref]

Z. Physik. (1)

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

Other (6)

A. Sommerfeld, Optics (Academic Press Inc., New York, 1954), p. 199; see also Ref. 2, pp. 77–83.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 381.

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Polska Akademia Nauk, Warszawa, 1957), p. 69; see also Acta Phys. Polon. 20, 9 (1961).

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., New York, 1962), p. 271.

See Ref. 2, p. 242.

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

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Figures (1)

Fig. 1
Fig. 1

Illustrating the notation.

Equations (31)

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u 1 ( L , P ) = 1 2 π A exp ( i k ρ ) ρ n ( exp ( i k s ) s ) d S ,
u 2 ( L , P ) = - 1 2 π A exp ( i k s ) s n ( exp ( i k ρ ) ρ ) d S ,
u 1 ( L , P ) = u ( i ) ( P ) when P is in A = 0 when P is on ( + ) } ,
u 2 ( L , P ) / n = u ( i ) ( P ) / n when P is in A = 0 when P is on ( + ) } .
u K ( L , P ) = 1 2 [ u 1 ( L , P ) + u 2 ( L , P ) ] .
u 1 ( L , P ) = u 2 ( P , L ) u 2 ( L , P ) = u 1 ( P , L ) } .
u K ( L , P ) = u K ( P , L ) .
u 1 ( L , P ) = u 2 ( P , L ) u 2 ( L , P ) = u 1 ( P , L ) } .
u 1 ( L , P ) + u 1 ( L , P ) = u 2 ( P , L ) + u 2 ( P , L ) u 2 ( L , P ) + u 2 ( L , P ) = u 1 ( P , L ) + u 1 ( P , L ) } .
E ( p L , P ) = ( · { p L [ exp ( i k r ) / r ] } ) + k 2 p L [ exp ( i k r ) / r ] ,
H ( p L , P ) = - i k × { p L [ exp ( i k r ) / r ] } ,
E ( m L , P ) = i k [ exp ( i k r ) / r ] × m L ,
H ( m L , P ) = ( · { m L [ exp ( i k r ) / r ] } ) + k 2 m L [ exp ( i k r ) / r ] .
p P · E ( p L , P ) = p L · E ( p P , L ) ,             m P · H ( p L , P ) = - p L · E ( m P , L ) ,
m P · H ( m L , P ) = m L · H ( m P , L ) ,             p P · E ( m L , P ) = - m L · H ( p P , L ) .
E 1 = 1 2 π A [ i k ( n × H ( i ) ) exp ( i k s ) s + ( n · E ( i ) ) exp ( i k s ) s ] d S + 1 2 π i k C ( d s · H ( i ) ) exp ( i k s ) s ,
H 1 = 1 2 π A ( n × H ( i ) ) × exp ( i k s ) s d S ,
E 2 = 1 2 π A ( n × E ( i ) ) × exp ( i k s ) s d S ,
H 2 = 1 2 π A [ - i k ( n × E ( i ) ) exp ( i k s ) s + ( n · H ( i ) ) exp ( i k s ) s ] d S - 1 2 π i k C ( ds · E ( i ) ) exp ( i k s ) s .
n × E 1 = n × E ( i ) in A n × E 1 = 0 on ( + ) } .
n × H 2 = n × H ( i ) in A n × H 2 = 0 on ( + ) } .
E K = 1 2 ( E 1 + E 2 ) H K = 1 2 ( H 1 + H 2 ) } .
p P · E 1 ( p L , P ) = p L · E 2 ( p P , L ) m P · H 1 ( m L , P ) = m L · H 2 ( m P , L ) } ,
p P · E 2 ( p L , P ) = p L · E 1 ( p P , L ) m P · H 2 ( m L , P ) = m L · H 1 ( m P , L ) } ,
m P · H 1 ( p L , P ) = - p L · E 2 ( m P , L ) p P · E 1 ( m L , P ) = - m L H 2 ( p P , L ) } ,
m P · H 2 ( p L , P ) = - p L · E 1 ( m P , L ) p P · E 2 ( m L , P ) = - m L · H 1 ( p P , L ) } .
m P · H 1 ( p L , P ) = - p L · E 2 ( m P , L ) .
m P · H 1 ( p L , P ) = m P 2 π A [ n × ( - i k exp ( i k ρ ) ρ × p L ) ] × exp ( i k s ) s d S = i k m P 2 π A [ p L ( n · exp ( i k ρ ) ρ ) - ( n · p L ) exp ( i k ρ ) ρ ] × exp ( i k s ) s d S = i k 2 π A [ ( n · exp ( i k ρ ) ρ ) ( m P × p L ) · exp ( i k s ) s - ( n · p L ) m P · ( exp ( i k ρ ) ρ × exp ( i k s ) s ) ] d S .
- p L · E 2 ( m P , L ) = - p L 2 π A [ - n × ( i k exp ( i k s ) s × m P ) ] × exp ( i k ρ ) ρ d S = i k p L 2 π A { ( exp ( i k s ) s × m P ) × ( n · exp ( i k ρ ) ρ ) - n × [ ( exp ( i k s ) s × m P ) · exp ( i k ρ ) ρ ] } d S = i k 2 π A [ ( n · exp ( i k ρ ) ρ ) ( m P × p L ) · exp ( i k s ) s - ( n · p L ) m · ( exp ( i k ρ ) ρ × exp ( i k s ) s ) ] d S .
p P · E K ( p L , P ) = p L · E K ( p P , L ) ,             m P · H K ( p L , P ) = - p L · E K ( m P , L ) ,
m P · H K ( m L , P ) = m L · H K ( m P , L ) ,             p P · E K ( m L , P ) = - m L · H K ( p P , L ) .