Abstract

An expression for the transmitted fraction of the uniform and parallel radiation incident upon the coupler between two rectangular light pipes of different physical dimensions is derived. It is a function of the pipe dimension ratio and the angle of the incident radiation. It is shown that for radiation nearly parallel to the pipe axis, it is possible to design a coupler for any pipe dimension ratio so that all the radiation incident upon it is transmitted by it. Design curves for various incidence angles and the pipe dimensions are presented.

© 1976 Optical Society of America

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References

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  1. T. O. Poehler, R. Turner, Appl. Opt. 9, 971 (1970).
  2. W. R. Powell, Appl. Opt. 13, 952 (1974).
  3. M. D. Wagh, B. V. Rao, Appl. Opt. 15, 1331 (1976).
  4. C. I. Palmer, C. W. Leigh, S. H. Kimball, Plane and Spherical Trignometry (McGraw-Hill, New York, 1950), p. 129.

1976 (1)

1974 (1)

1970 (1)

Kimball, S. H.

C. I. Palmer, C. W. Leigh, S. H. Kimball, Plane and Spherical Trignometry (McGraw-Hill, New York, 1950), p. 129.

Leigh, C. W.

C. I. Palmer, C. W. Leigh, S. H. Kimball, Plane and Spherical Trignometry (McGraw-Hill, New York, 1950), p. 129.

Palmer, C. I.

C. I. Palmer, C. W. Leigh, S. H. Kimball, Plane and Spherical Trignometry (McGraw-Hill, New York, 1950), p. 129.

Poehler, T. O.

Powell, W. R.

Rao, B. V.

Turner, R.

Wagh, M. D.

Appl. Opt. (3)

Other (1)

C. I. Palmer, C. W. Leigh, S. H. Kimball, Plane and Spherical Trignometry (McGraw-Hill, New York, 1950), p. 129.

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Figures (4)

Fig. 1
Fig. 1

Coupling between two light pipes of dimensions 2a × c and 2b × c and the angles θ and ϕ defining the direction of incident radiation.

Fig. 2
Fig. 2

Coupling projection along the dimension c, the coordinate frame and the progress of a ray along the coupling.

Fig. 3
Fig. 3

Efficiency of a coupling as a function of the semicoupling angle ψ for various values of incident radiation angles ϕ and a fixed a/b ratio of 4.

Fig. 4
Fig. 4

Maximum semicoupling angle ψmax for perfect coupling with given a/b ratio and the incident radiation angle ψ.

Equations (33)

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( from ϕ n max + 1 > π / 2 , ϕ n max π / 2 )
n max = [ ( π - 2 ϕ ) / 4 ψ ] ,
( y n + 1 - y n ) = tan ϕ n ( z n + 1 - z n ) , ( a - y n + 1 ) = tan ψ z n + 1 ,
( y n + a ) = tan ψ z n .
z n + 1 = 2 a + ( tan ϕ n - tan ψ ) z n ( tan ϕ n + tan ψ ) = 2 a cos ( ϕ + 2 n ψ ) · cos ψ + z n · sin ( ϕ + 2 n - 1 ¯ ψ ) sin ( ϕ + 2 n + 1 ¯ ψ ) .
z n + 1 = 2 a cos ψ sin ( ϕ + 2 n + 1 ¯ ψ )             · i = 1 n cos ( ϕ + 2 i ψ ) + z 1 sin ( ϕ + ψ ) sin ( ϕ + 2 n + 1 ¯ ψ ) .
z n + 1 = 2 a cos ψ sin ( ϕ + 2 n + 1 ¯ ψ ) · cos ( ϕ + n + 1 ¯ ψ ) · sin ( n ψ ) sin ψ + z 1 · sin ( ϕ + ψ ) sin ( ϕ + 2 n + 1 ¯ ψ ) .
z n max + 1 ,
sin ( ϕ + 2 n max + 1 ¯ ψ ) sin ( ϕ + ψ ) by T .
2 cos ( ϕ + n + 1 ¯ ψ ) · sin ( n ψ ) + sin ( ϕ + ψ ) = sin ( ϕ + 2 n + 1 ¯ ψ ) ,
z n max + 1 = a tan ψ ( 1 - 1 / T ) + z 1 / T ,
z n max + 1 the length of the coupling
y 0 > b - ( a - b ) tan ϕ / tan ψ .
z 1 = ( a - y 0 ) / ( tan ϕ + tan ψ ) .
z n max + 1 length of the coupling = ( a - b ) / tan ψ
b ( 1 + R ) - a R < y 0 b T ( 1 + R ) - a R ,
z n max + 1 ( a - b ) / tan ϕ
b ( 1 - R ) + a R < - y 0 b T ( 1 - R ) + a R .
- b ( 1 - R ) - a R y 0 b ( 1 + R ) - a R .
η ϕ ψ = 1 2 a [ max { 0 , min { b T ( 1 + R ) - a R , a } - max { - a , b ( 1 + R ) - a R } } + max { 0 , min { b T ( 1 - R ) + a R , a } - max { - a , b ( 1 - R ) + a R } } + 2 b ] .
η ϕ ψ = min { 1 , ( b / a ) T } .
η ϕ > ψ = 1 2 a ( max { 0 , min { b T ( 1 + R ) - a R , a } - max { - a , b ( 1 + R ) - a R } } + max { 0 , min { a , b ( 1 + R ) - a R } + a } ) .
η ϕ > ψ = max { 0 , 1 + R 2 min { 1 , b a T } - R - 1 2 } ,
b T a .
sin ( ϕ + 2 n max + 1 ¯ ψ ) > sin ( ϕ + ψ ) .
T 1 for all ψ .
d T / d ψ = [ 2 n max + 1 ¯ cos ( ϕ + 2 n max + 1 ¯ ψ ) sin ( ϕ + ψ ) - cos ( ϕ + ψ ) · sin ( ϕ + 2 n max + 1 ¯ ψ ) ] / sin 2 ( ϕ + ψ ) .
lim ψ 0 Q 0 and d Q / d ψ < 0.
T max = lim ψ 0 sin ( ϕ + 2 n max + 1 ¯ ψ ) sin ( ϕ + ψ ) .
lim ψ 0 x ψ = lim ψ 0 n max ψ .
T max = 1 / sin ϕ .
a sin ϕ b ,
η e = η · η r ,

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