Abstract

A generalized vector–matrix method is employed to determine the intensity of diffusely scattered light from a hyperboloid of one sheet. Solutions for both the elliptic and circular cross-section cases are obtained in integral form. For certain limiting conditions the latter results are shown to reduce to a form obtained earlier for a cylinder of circular cross section.

© 1983 Optical Society of America

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References

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  1. M. Sussman, J. Opt. Soc. Am. 48, 275 (1958).
    [CrossRef]
  2. W. R. Rambauske, R. G. Gruenzel, J. Opt. Soc. Am. 55, 315 (1965).
    [CrossRef]
  3. D. C. Look, J. Opt. Soc. Am. 55, 462 (1965).
  4. E. R. Lanczi, J. Opt. Soc. Am. 56, 873 (1966).
    [CrossRef]
  5. K. W. Brand, F. A. Spagnolo, J. Opt. Soc. Am. 57, 452 (1967).
    [CrossRef] [PubMed]
  6. F. A. Spagnolo, K. W. Brand, Appl. Opt. 7, 189 (1968).
    [CrossRef] [PubMed]
  7. F. A. Spagnolo, Appl. Opt. 11, 2890 (1972).
    [CrossRef] [PubMed]
  8. L. Brand, Advanced Calculus (Wiley, New York, 1955).

1972 (1)

1968 (1)

1967 (1)

1966 (1)

1965 (2)

1958 (1)

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Figures (2)

Fig. 1
Fig. 1

Hyperboloid of one sheet.

Fig. 2
Fig. 2

Section of hyperboloid in x,z plane.

Equations (53)

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I = κ E π [ n · r s ] [ n · r o ] d S ,
I = κ E π ( r o Q r r Q r s + Q r c r s + r o c r r s + r o c c r s r Q 2 r + c Q r + r Q c + c c ) d S ,
Q = ( 2 q 1 q 6 q 5 q 6 2 q 2 q 4 q 5 q 4 2 q 3 ) , c = { q 7 q 8 q 9 } ,
q ( x , y , z ) = q 1 x 2 + q 2 y 2 + q 3 z 2 + q 4 y z + q 5 x z + q 6 x y + q 7 x + q 8 y + q 9 z 1 = 0.
r s = { sin ϕ s cos θ s sin ϕ s sin θ s cos ϕ s } , r o = { sin ϕ o cos θ o sin ϕ o sin θ o cos ϕ o } .
q ( x , y , z ) = x 2 a 2 + y 2 b 2 z 2 c 2 1 = 0.
Q = ( 2 / a 2 0 0 0 2 / b 2 0 0 0 2 / c 2 ) , c = { 0 0 0 } .
I = κ E π ( r o Q r r r s r Q 2 r ) d S .
r o Q r r Q r s = 4 [ sin ϕ o cos θ o a 2 ( x 2 a 2 sin ϕ s cos θ s + x y b 2 sin ϕ s sin θ s x z c 2 cos ϕ s ) + sin ϕ o sin θ o b 2 ( x y a 2 sin ϕ s cos θ s + y 2 b 2 sin ϕ s sin θ s y z c 2 cos ϕ s ) cos ϕ o c 2 ( x z a 2 sin ϕ s cos θ s + y z b 2 sin ϕ s sin θ s z 2 c 2 cos ϕ s ) ] ,
r Q 2 r = 4 [ x 2 a 4 + y 2 b 4 + z 2 c 4 ] .
x = a b ( c 2 + z 2 ) 1 / 2 cos θ c ( a 2 sin 2 θ + b 2 cos 2 θ ) 1 / 2 , y = a b ( c 2 + z 2 ) 1 / 2 sin θ c ( a 2 sin 2 θ + b 2 cos 2 θ ) 1 / 2 ,
d S = ( a b c 2 ) { a 4 [ c 4 + ( b 2 + c 2 ) z 2 ] sin 2 θ + b 4 [ c 4 + ( a 2 + c 2 ) z 2 ] cos 2 θ } 1 / 2 [ a 2 sin 2 θ + b 2 cos 2 θ ] 3 / 2 d θ d z .
r Q 2 r = 4 [ c 2 ( c 2 + z 2 ) ( a 4 sin 2 θ + b 4 cos 2 θ ) + a 2 b 2 ( a 2 sin 2 θ + b 2 cos 2 θ ) z 2 a 2 b 2 c 4 ( a 2 sin 2 θ + b 2 cos 2 θ ) ] .
I = κ E π { ( a b 5 ) sin ϕ o cos θ o sin ϕ s cos θ s K ( θ , z ) cos 2 θ d θ d z + ( a 3 b 3 ) sin ϕ o cos θ o sin ϕ s sin θ s K ( θ , z ) sin θ cos θ d θ d z ( a 2 b 4 c ) sin ϕ o cos θ o cos ϕ s K ( θ , z ) z cos θ d θ d z + ( a 3 b 3 ) sin ϕ o sin θ o sin ϕ s cos θ s K ( θ , z ) sin θ cos θ d θ d z + ( a 5 b ) sin ϕ o sin θ o sin ϕ s sin θ s K ( θ , z ) sin 2 θ d θ d z ( a 4 b 2 c ) sin ϕ o sin θ o cos ϕ s K ( θ , z ) z sin θ d θ d z ( a 2 b 4 c ) cos ϕ o sin ϕ s cos θ s K ( θ , z ) z cos θ d θ d z ( a 4 b 2 c ) cos ϕ o sin ϕ s sin θ s K ( θ , z ) z sin θ d θ d z + ( a 3 b 3 c 2 ) cos ϕ o cos ϕ s K ( θ , z ) z 2 d θ d z } .
K ( θ , z ) = ( c 2 + z 2 ) { a 4 [ c 4 + ( b 2 + c 2 ) z 2 ] sin 2 θ + b 4 [ c 4 + ( a 2 + c 2 ) z 2 ] cos 2 θ } 1 / 2 [ c 2 ( c 2 + z 2 ) ( a 4 sin 2 θ + b 4 cos 2 θ ) + a 2 b 2 ( a 2 sin 2 θ + b 2 cos 2 θ ) z 2 ] [ a 2 sin 2 θ + b 2 cos 2 θ ] 2 ,
K ( θ , z ) = ( c 2 + z 2 ) 1 / 2 { a 4 [ c 4 + ( b 2 + c 2 ) z 2 ] sin 2 θ + b 4 [ c 4 + ( a 2 + c 2 ) z 2 ] cos 2 θ } 1 / 2 [ c 2 ( c 2 + z 2 ) ( a 4 sin 2 θ + b 4 cos 2 θ ) + a 2 b 2 ( a 2 sin 2 θ + b 2 cos 2 θ ) z 2 ] [ a 2 sin 2 θ + b 2 cos 2 θ ] 2 ,
K ( θ , z ) = { a 4 [ c 4 + ( b 2 + c 2 ) z 2 ] sin 2 θ + b 4 [ c 4 + ( a 2 + c 2 ) z 2 ] cos 2 θ } 1 / 2 [ c 2 ( c 2 + z 2 ) ( a 4 sin 2 θ + b 4 cos 2 θ ) + a 2 b 2 ( a 2 sin 2 θ + b 2 cos 2 θ ) z 2 ] [ a 2 sin 2 θ + b 2 cos 2 θ ] 1 / 2 .
K ( θ , z ) a b ( c 2 + z 2 ) a 5 [ c 2 ( c 2 + z 2 ) + a 2 z 2 ] 1 / 2 = I ( z ) ,
K ( θ , z ) a b ( c 2 + z 2 ) 1 / 2 a 4 [ c 2 ( c 2 + z 2 ) + a 2 z 2 ] 1 / 2 = I ( z ) ,
K ( θ , z ) a b 1 a 3 [ c 2 ( c 2 + z 2 ) + a 2 z 2 ] 1 / 2 = I ( z ) .
I = κ E π { a 6 sin ϕ o cos θ o sin ϕ s cos θ s cos 2 θ d θ I ( z ) d z + a 6 sin ϕ o cos θ o sin ϕ s sin θ s sin θ cos θ d θ I ( z ) d z ( a 6 c ) sin ϕ o cos o cos ϕ s cos θ d θ I ( z ) z d z + a 6 sin ϕ o sin θ o sin ϕ s cos θ s sin θ cos θ d θ I ( z ) d z + a 6 sin ϕ o sin θ o sin ϕ s sin θ s sin 2 θ d θ I ( z ) d z ( a 6 c ) sin ϕ o sin θ o cos ϕ s sin θ d θ I ( z ) z d z ( a 6 c ) cos ϕ o sin ϕ s cos θ s cos θ d θ I ( z ) z d z ( a 6 c ) cos ϕ o sin ϕ s sin θ s sin θ d θ I ( z ) z d z + ( a 6 c 2 ) cos ϕ o cos ϕ s d θ I ( z ) z 2 d z } .
I ( z ) c ( 1 a 5 ) ,
I ( z ) c 0 ,
I ( z ) c 0.
I = κ E a π { sin ϕ o cos θ o sin ϕ s cos θ s cos 2 θ d θ d z + sin ϕ o cos θ o sin ϕ s sin θ s sin θ cos θ d θ d z + sin ϕ o sin θ o sin ϕ s cos θ s sin θ cos θ d θ d z + sin ϕ o sin θ o sin ϕ s sin θ s sin 2 θ d θ d z } .
I = κ E a h π { sin ϕ o sin ϕ s [ cos θ o cos θ s cos 2 θ d θ + sin ( θ o + θ s ) sin θ cos θ d θ + sin θ o sin θ s sin 2 θ d θ ] } .
x 2 a 2 + y 2 b 2 z 2 c 2 1 = 0.
x = ρ cos θ , y = ρ sin θ ,
y x = tan θ .
x = ( a b c ) [ c 2 + z 2 b 2 + a 2 tan 2 θ ] 1 / 2 ,
y = ( a b c ) [ c 2 + z 2 b 2 cot 2 θ + a 2 ] 1 / 2 ,
R = i ˆ x + j ˆ y + k ˆ z .
( R θ ) i ˆ ( x θ ) = ( a 3 b c ) ( c 2 + z 2 ) 1 / 2 tan θ sec 2 θ [ b 2 + a 2 tan 2 θ ] 3 / 2 ,
( R θ ) j ˆ ( y θ ) = ( a b 3 c ) ( c 2 + z 2 ) 1 / 2 cot θ c s c 2 θ [ b 2 cot 2 θ + a 2 ] 3 / 2 ,
( R θ ) k ˆ ( z θ ) = 0 ,
( R z ) i ˆ ( x z ) = ( a b c ) z ( c 2 + z 2 ) 1 / 2 [ b 2 + a 2 tan 2 θ ] 1 / 2 ,
( R z ) j ˆ ( y z ) = ( a b c ) z ( c 2 + z 2 ) 1 / 2 [ b 2 cot 2 θ + a 2 ] 1 / 2 ,
( R z ) k ˆ ( z z ) = 1.
d S = [ ( R z × R θ ) · ( R z × R θ ) ] 1 / 2 d θ d z = | R z × R θ | d θ d z ,
R θ i ˆ ( x θ ) + j ˆ ( y θ ) + k ˆ ( z θ ) ,
R z i ˆ ( x z ) + j ˆ ( y z ) + k ˆ ( z z ) ,
( R z × R θ ) = | i ˆ j ˆ k ˆ ( x z ) ( y z ) ( z z ) ( x θ ) ( y θ ) ( z θ ) |
( R z × R θ ) i ˆ = ( a b 3 c ) ( c 2 + z 2 ) 1 / 2 cot θ c s c 2 θ [ b 2 cot 2 θ + a 2 ] 3 / 2 ,
( R z × R θ ) j ˆ = ( a 3 b c ) ( c 2 + z 2 ) 1 / 2 tan θ sec 2 θ [ b 2 + a 2 tan 2 θ ] 3 / 2 ,
( R z × R θ ) k ˆ = ( a 2 b 2 c 2 ) z [ b 2 cot θ + a 2 tan θ ] [ b 2 + a 2 tan 2 θ ] 1 / 2 [ b 2 cot 2 θ + a 2 ] 1 / 2 [ b 2 cos 2 θ + a 2 sin 2 θ ] ,
| R z × R θ | 2 = ( R z × R θ ) i ˆ 2 + ( R z × R θ ) j ˆ 2 + ( R z × R θ ) k ˆ 2 ,
| R z × R θ | 2 = ( a 2 b 2 c 4 ) { a 4 [ c 4 + ( b 2 + c 2 ) z 2 ] sin 2 θ + b 4 [ c 4 + ( a 2 + c 2 ) z 2 ] cos 2 θ } [ a 2 + sin 2 θ + b 2 cos 2 θ ] 3 .
d S = ( a b c 2 ) { a 4 [ c 4 + ( b 2 + c 2 ) z 2 ] sin 2 θ + b 4 [ c 4 + ( a 2 + c 2 ) z 2 ] cos 2 θ } 1 / 2 [ a 2 sin 2 θ + b 2 cos 2 θ ] 3 / 2 d θ d z .
x 2 a 2 z 2 c 2 = 1.
z = c ( x 2 a 2 1 ) 1 / 2 ,
tan β = ( d z d x ) z = h / 2 = ( c a h ) [ 4 c 2 + h 2 ] 1 / 2 .
β = tan 1 { ( c a h ) [ 4 c 2 + h 2 ] 1 / 2 } .
[ π 2 tan 1 { ( c a h ) [ 4 c 2 + h 2 ] 1 / 2 } ] ϕ o , ϕ s [ π 2 + tan 1 { ( c a h ) [ 4 c 2 + h 2 ] 1 / 2 } ] .

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