Abstract

The coefficients for the expansion of a plane wave in terms of the eigenstates (modes) of a Selfoc® fiber are derived. This result is used as the basis for a suggestion for a new optical fiber whose spectral response could be any one of the functions used extensively in colorimetric instrumentation.

© 1975 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Messiah, Quantum Mechanics (North–Holland, Amsterdam, 1961), p. 359.
  2. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Ch. 16.
  3. Reference 1, Ch. 16.
  4. H. Kogelnik, in Proceedings of Symposium on Quasi-Optics (Polytechnic Press, New York, 1964).
  5. Reference 1, Ch. 5.
  6. W. Streifer and C. N. Kurtz, J. Opt. Soc. Am. 57, 779 (1967).
    [CrossRef]
  7. Reference 1, Appendix A.
  8. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, London, 1922), p. 20.
  9. W. Magnus and F. Oberhettinger, Formulas and Theory for Special Functions of Mathematical Physics (Springer, New York, 1966), Ch. 5.
  10. Reference 8, p. 395.
  11. Reference 8, p. 394.
  12. Reference 9, Ch. 6.
  13. Reference 9, Ch. 5.
  14. R. Eisberg, Fundamentals of Modern Physics (Wiley, New York, 1961), p. 57.
  15. A. W. Snyder and C. Pask, J. Opt. Soc. Am. 63, 806 (1973).
    [CrossRef]
  16. D. Marcuse, Light Transmission Optics (Van Nostrand–Reinhold, New York, 1972).
  17. P. M. Tannenbaum, J. Opt. Soc. Am. 64, 89 (1974).
    [CrossRef]
  18. Millimeter and Submillimeter Techniques, edited by R. A. Benson (ILI FFE Books, London, 1969), Ch. 18.
  19. Reference 16, p. 250.

1974 (1)

1973 (1)

1967 (1)

Eisberg, R.

R. Eisberg, Fundamentals of Modern Physics (Wiley, New York, 1961), p. 57.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Ch. 16.

Kogelnik, H.

H. Kogelnik, in Proceedings of Symposium on Quasi-Optics (Polytechnic Press, New York, 1964).

Kurtz, C. N.

Magnus, W.

W. Magnus and F. Oberhettinger, Formulas and Theory for Special Functions of Mathematical Physics (Springer, New York, 1966), Ch. 5.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand–Reinhold, New York, 1972).

Messiah, A.

A. Messiah, Quantum Mechanics (North–Holland, Amsterdam, 1961), p. 359.

Oberhettinger, F.

W. Magnus and F. Oberhettinger, Formulas and Theory for Special Functions of Mathematical Physics (Springer, New York, 1966), Ch. 5.

Pask, C.

Snyder, A. W.

Streifer, W.

Tannenbaum, P. M.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, London, 1922), p. 20.

J. Opt. Soc. Am. (3)

Other (16)

Millimeter and Submillimeter Techniques, edited by R. A. Benson (ILI FFE Books, London, 1969), Ch. 18.

Reference 16, p. 250.

D. Marcuse, Light Transmission Optics (Van Nostrand–Reinhold, New York, 1972).

A. Messiah, Quantum Mechanics (North–Holland, Amsterdam, 1961), p. 359.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Ch. 16.

Reference 1, Ch. 16.

H. Kogelnik, in Proceedings of Symposium on Quasi-Optics (Polytechnic Press, New York, 1964).

Reference 1, Ch. 5.

Reference 1, Appendix A.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, London, 1922), p. 20.

W. Magnus and F. Oberhettinger, Formulas and Theory for Special Functions of Mathematical Physics (Springer, New York, 1966), Ch. 5.

Reference 8, p. 395.

Reference 8, p. 394.

Reference 9, Ch. 6.

Reference 9, Ch. 5.

R. Eisberg, Fundamentals of Modern Physics (Wiley, New York, 1961), p. 57.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

FIG. 1
FIG. 1

Upper, the CIE system. Lower, the hydrogen-atom radial probability distribution function for several of the quantum number n, m. (From E. U. Condon and G. Shortley, The Theory of Atomic Spectra (Cambridge U. P., Cambridge, 1953).)

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

e i k · r = 4 π l = 0 m = l l ( i ) l j l ( k r ) Y l , m * ( Ω k ) Y l , m ( Ω r )
e i k · r = n m η C n , m , η ( k ) ψ n , m , η ( r ) ,
C n , m , η ( k ) = ψ n , m , η * ( r ) e i k · r d 3 r .
ψ m , n , η ( ρ , ϕ , z ) = N ¯ ( n , m ) 1 / 2 α ( n + 1 ) / 2 ρ n exp ( α ρ 2 2 ) L m n ( α ρ 2 ) e i n ϕ × exp ( i 0 1 / 2 η z ) ,
α ( 0 δ ) 1 / 2 | k | / a .
k 2 η 2 k = 2 δ 1 / 2 0 1 / 2 a ( 2 m + n + 1 ) .
( ρ ) = 0 [ 1 δ ( ρ / a ) 2 ] .
C n , m , η ( k ) = M m , n 0 { ρ d ρ [ 0 2 π d ϕ exp ( i [ ρ { k x cos ϕ + k y sin ϕ } n ϕ ] ) ] [ exp ( i { 0 1 / 2 η k z } z ) d z ] × ρ n exp [ α ρ 2 / 2 ] L m n ( α ρ 2 ) } ,
M m , n { N ¯ ( n , m ) α n + 1 } 1 / 2 .
C n , m , η ( k ) = T m , n , η 0 ρ n + 1 J n ( k ρ ) L m n ( α ρ 2 ) exp [ α ρ 2 / 2 ] d ρ ,
T m , n , η 4 π 2 M m , n e i n ( θ k + 3 π / 2 ) δ ( 0 1 / 2 η k z )
k 2 = k x 2 + k y 2 , k z = | k | cos θ k ,
δ ( 0 1 / 2 η k z ) = ( 1 / 2 π ) exp ( i [ 0 1 / 2 η k z ] z ) d z
J n ( z ) = ( 1 / 2 π ) α 2 π + α e i ( n ϕ z sin ϕ ) d ϕ .
L m n ( x ) = x n / 2 e x m ! 0 e z z m + n / 2 J n ( 2 { z x } ) d z ,
C m , n , η ( k ) = W m , n , η 0 { ( 0 ρ exp [ α ρ 2 / 2 ] J n ( k ρ ) · J n ( 2 ρ { z α } ) d p ) × e z z m + n / 2 } d z ,
W m , n , η = T m , n , η α n / 2 / m ! .
0 t e p 2 t 2 J r ( a t ) J r ( b t ) d t = ( 1 / 2 p 2 ) exp ( a 2 + b 2 4 p 2 ) I r ( a b 2 p 2 ) ,
I r ( x ) = e i π r / 2 J r ( exp { i π / 2 } x ) .
C m , n , η ( k ) = P m , n , η exp ( k 2 / 2 α ) e i π n / 2 × 0 J n ( 2 i k t ) t 2 m + n + 1 exp ( α t 2 ) d t ,
P m , n , η 2 W m , n , η α m + n / 2 .
0 exp ( p 2 t 2 ) J r ( a t ) t μ 1 d t = { Γ ( 1 2 r + 1 2 μ ) ( 1 2 a / p ) r / 2 p μ Γ ( r + 1 ) } × exp ( a 2 / 4 p 2 ) F 1 ( 1 2 r 1 2 μ + 1 ; a 2 / 4 p 2 ) ,
C m , n , η ( k ) = 1 2 P m , n , η e i π ( m + 1 ) Γ ( m + n + 1 ) Γ ( n + 1 ) k n α m + n + 1 · exp ( k 2 / 2 α ) F 1 ( m ; n + 1 , k 2 / α ) .
L m n ( z ) = Γ ( m + n + 1 ) Γ ( n + 1 ) Γ ( m + 1 ) F 1 ( m , n + 1 , z ) ,
C m , n , η ( k ) = ( 1 ) n + m ( 2 π ) 2 N ¯ ( n , m ) 1 / 2 e i n ( θ k + π / 2 ) δ ( 0 1 / 2 η k z ) × k n α ( n + 1 ) / 2 exp ( k 2 / 2 α ) L m n ( k 2 / α ) .
P m , n ( k ) = | C m , n ( k ) | 2 × ( density of states ) = | C m , n ( k ) | 2 × ( 2 π k 2 sin θ k ) .
V = k a [ 0 ( a ) ] 1 / 2
n sin θ k [ 0 ( a ) ] 1 / 2 ,
k 2 / α V .
P m , n ( V ) R ( n , m ) V n + 1 exp ( V ) | L m n ( V ) | 2 ,