Abstract

The application of Liouville’s theorem from statistical mechanics to a distribution of rays emanating from a light source provides a method of visualizing and calculating the maximum efficiency achievable when coupling the radiation to a receiver. It is shown that the maximum coupling efficiency depends only on measurable spatial and angular boundary conditions of the source and receiver together with knowledge of the source radiance distribution. Knowledge of the maximum coupling efficiency permits evaluation of any optical system which might be used to effect coupling. A specific example is given for coupling radiation from a light emitting diode to a cylindrical, multimode, optical waveguide.

© 1974 Optical Society of America

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References

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  1. R. C. Tolman, The Principles of Statistical Mechanics (Oxford U.P.Oxford,1938).
  2. D. Marcuse, Light Transmission Optics (Van Nostrand, Princeton, 1972), p. 92.
  3. H. Goldstein, Classical Mechanics (Addison Wesley, Reading, Mass.1960), pp. 30–33.
  4. Reference 3, p. 215.
  5. M. R. Holter, S. Nudelman, G. Suits, W. Wolfe, G. Zissis, Fundamentals of Infrared Technology (Macmillan, New York, 1962), p. 4.
  6. D. Marcuse, Bell Syst. Tech. J. 45, 743 (1966).
  7. D. Marcuse, Appl. Opt. 10, 494 (1971).
    [CrossRef] [PubMed]
  8. The meridional N.A. is really an underestimation of the light-gathering efficiency of a cylindrical fiber. Skew ray calculations show that an effective N.A. can be defined in terms of a fictitious angle γs, greater than γc. See N. S. Kapany, Fiber Optics, Principles and Applications (Academic Press, New York, 1967), pp. 30–34.

1971 (1)

1966 (1)

D. Marcuse, Bell Syst. Tech. J. 45, 743 (1966).

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison Wesley, Reading, Mass.1960), pp. 30–33.

Holter, M. R.

M. R. Holter, S. Nudelman, G. Suits, W. Wolfe, G. Zissis, Fundamentals of Infrared Technology (Macmillan, New York, 1962), p. 4.

Kapany, N. S.

The meridional N.A. is really an underestimation of the light-gathering efficiency of a cylindrical fiber. Skew ray calculations show that an effective N.A. can be defined in terms of a fictitious angle γs, greater than γc. See N. S. Kapany, Fiber Optics, Principles and Applications (Academic Press, New York, 1967), pp. 30–34.

Marcuse, D.

D. Marcuse, Appl. Opt. 10, 494 (1971).
[CrossRef] [PubMed]

D. Marcuse, Bell Syst. Tech. J. 45, 743 (1966).

D. Marcuse, Light Transmission Optics (Van Nostrand, Princeton, 1972), p. 92.

Nudelman, S.

M. R. Holter, S. Nudelman, G. Suits, W. Wolfe, G. Zissis, Fundamentals of Infrared Technology (Macmillan, New York, 1962), p. 4.

Suits, G.

M. R. Holter, S. Nudelman, G. Suits, W. Wolfe, G. Zissis, Fundamentals of Infrared Technology (Macmillan, New York, 1962), p. 4.

Tolman, R. C.

R. C. Tolman, The Principles of Statistical Mechanics (Oxford U.P.Oxford,1938).

Wolfe, W.

M. R. Holter, S. Nudelman, G. Suits, W. Wolfe, G. Zissis, Fundamentals of Infrared Technology (Macmillan, New York, 1962), p. 4.

Zissis, G.

M. R. Holter, S. Nudelman, G. Suits, W. Wolfe, G. Zissis, Fundamentals of Infrared Technology (Macmillan, New York, 1962), p. 4.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 45, 743 (1966).

Other (6)

The meridional N.A. is really an underestimation of the light-gathering efficiency of a cylindrical fiber. Skew ray calculations show that an effective N.A. can be defined in terms of a fictitious angle γs, greater than γc. See N. S. Kapany, Fiber Optics, Principles and Applications (Academic Press, New York, 1967), pp. 30–34.

R. C. Tolman, The Principles of Statistical Mechanics (Oxford U.P.Oxford,1938).

D. Marcuse, Light Transmission Optics (Van Nostrand, Princeton, 1972), p. 92.

H. Goldstein, Classical Mechanics (Addison Wesley, Reading, Mass.1960), pp. 30–33.

Reference 3, p. 215.

M. R. Holter, S. Nudelman, G. Suits, W. Wolfe, G. Zissis, Fundamentals of Infrared Technology (Macmillan, New York, 1962), p. 4.

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Figures (3)

Fig. 1
Fig. 1

Real space coordinate system. Vector S represents one point in phase space.

Fig. 2
Fig. 2

(a) One-dimensional source and receiver real space. (b) Source phase space. (c) Receiver phase space.

Fig. 3
Fig. 3

Planar source to be coupled to cylindrical optical waveguide.

Equations (43)

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P 1 P 2 n ( x , y , z ) d s = extremum .
d s = ( d x 2 + d y 2 + d z 2 ) 1 / 2 = ( 1 + x 2 + y 2 ) 1 / 2 d z
x = d x / d z , y = d y / d z ,
P 1 P 2 L ( x , y , x , y , z ) d z = extremum ,
L = n ( x , y , z , ) ( 1 + x 2 + y 2 ) 1 / 2 .
d d z ( L x ) - L x = 0 , d d z ( L x ) - L y = 0.
p x = L x ,             p y = L y .
H ( x , y , p x , p y ) = p x x + p y y - L ( x , y , x , y ) .
L x = d d z ( L x ) = d d z ( p x ) , L y = d d z ( L y ) = d d z ( p y ) .
H x = - L x = - d p x d z .
H p x = p x ( p x x + p y y - L ) = x + p x x p x + p y y p x - L x x p x - L y y p x .
H p x = x = d x d z .
H p y = y = d y d z .
H x = - d p x d z , H y = - d p y d z , H p x = d x d z , H p y = d y d z .
N z 0 = x y p x p y ρ ( x , y , p x , p y , z 0 ) d x d y d p x d p y .
d P = δ P ¯ · ρ ( x , y , p x , p y ) d x d y d p x d p y ,
d P = ρ ( x , y , p x , p y ) d x d y d p x d p y .
D ρ d z = 0
p x = L x = x [ n ( x , y , z ) ( 1 + x 2 + y 2 ) 1 / 2 ] = n ( x , y , z ) ( 1 + x 2 + y 2 ) - 1 / 2 d x d z p x = n ( x , y , z ) d x d z · d z d s = n d x d s .
p y = n d y d s .
p x = n sin α ,             p y = n sin β = n cos α sin β .
d p x = n cos α d α ,             d p y = n cos β d β = n cos α cos β d β .
d V = d x d y d p x d p y = d A n · n 2 cos 2 α cos β d α d β ,
d A = d A n · cos α cos β .
d a = S d α · S cos α d β .
d a S 2 = d Ω = cos α d α d β .
d V = n 2 d A d Ω .
d P d V = 1 n 2 lim Δ A 0 Δ Ω 0 Δ P Δ A Δ Ω = ρ
D d z ( N ) = 0.
E c ( max ) = { v = v r ρ s d V } max all v s ρ s d V .
{ v = v r ρ s d V } max
N ( x , α ) = N 0 .
E c ( max ) = ( X 2 - X 1 ) ( X 2 - X 1 ) sin α max .
N . A . = n 0 sin γ c = ( n 1 2 - n 2 2 ) 1 / 2 ,
ρ ( x , y , α , β ) = N 0 / n 0 2 .
E c ( max ) = ρ s v r d V ρ s v s d V = V r V s .
V r = 4 n 0 2 X Y 0 α max 0 β max = f ( α ) cos 2 α cos β d β d α d x d y .
V r = 4 A f n 0 2 0 γ c cos 2 α 0 f ( α ) cos β d β d α ,
V r = 4 n 0 2 A f 0 γ c cos α ( cos 2 α - cos 2 γ c ) 1 / 2 d α = π n 0 2 A f sin 2 γ c .
V r = π A f ( N . A . ) 2 .
V s = 4 n 0 2 X Y 0 π / 2 0 π / 2 cos 2 α cos β d α d β d x d y = π n 0 2 A s .
E c ( max ) = A f sin 2 γ c / A s = A f ( N . A . ) 2 / ( A s n 0 2 ) .
E c ( max ) = η A b ( N . A . ) 2 / ( A s n 0 2 ) = N A f ( N . A . ) 2 / ( A s n 0 2 ) .

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