Abstract

Rays have been traced through hollow straight light pipes, condensing cones, mitered right angles, and toroidal right-angle bends. The results presented include transmittance and average number of reflections in representative elements. Condensing cones should be kept short to minimize reflection losses, and mitered corners are superior even to ideal bends in all cases of practical interest. A formula is derived for the average number of reflections for all rays—skew and meridional—in a straight pipe. The discussion is limited to elements of circular cross section, and the emphasis is on applications to the far infrared.

© 1969 Optical Society of America

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References

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  1. R. C. Ohlman, P. L. Richards, and M. Tinkham, J. Opt. Soc. Am. 48, 531 (1958).
    [Crossref]
  2. Robert J. Potter, J. Opt. Soc. Am. 51, 1079 (1961).
    [Crossref]
  3. N. S. Kapany, Fiber Optics (Academic Press Inc., New York. 1967).
  4. P. L. Richards, J. Opt. Soc. Am. 54, 1474 (1964).
    [Crossref]
  5. W. Benesch and J. Strong, J. Opt. Soc. Am. 41, 252 (1951).
    [Crossref]
  6. Donald E. Williamson, J. Opt. Soc. Am. 42, 712 (1952).
    [Crossref]
  7. K. D. Möller, V. P. Tomaselli, L. R. Skube, and B. K. McKenna, J. Opt. Soc. Am. 55, 1233 (1965).
    [Crossref]
  8. See D. H. Martin, Ed. Spectroscopic Techniques for Far Infrared, Submillimetre, and Millimetre Waves (North-Holland Publ. Co., Amsterdam, 1967), p. 375.

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Figures (15)

Fig. 1
Fig. 1

Polar coordinates of the entrance point of the ray on the end of the light pipe. The view is looking into the pipe.

Fig. 2
Fig. 2

Coordinates describing the cone of light illuminating any point P on the entrance end of the light pipe.

Fig. 3
Fig. 3

The projection of a skew ray on the entrance face of a straight light pipe.

Fig. 4
Fig. 4

Two ways of looking at the path of a meridional ray through a straight light pipe of length l, diameter d.

Fig. 5
Fig. 5

Geometrical parameters of a condensing cone.

Fig. 6
Fig. 6

Output distribution for annular input to condensing cone. Solid line: input angle = 6°, dotted line: input angle = 12°. The dashed vertical lines are the solutions of Eq. (14) for these two cases. Condensor parameters are s = 0.5 cm, c = 0.167 cm, x = 7.5 cm. The numbers across the top are the number of reflections (for meridional rays) corresponding to each peak.

Fig. 7
Fig. 7

Transmittance of condensers for annular input as a function of input angle. All condensors had s = 0.5 cm, c = 0.167 cm; the numbers on the curves are the length (in cm).

Fig. 8
Fig. 8

Transmittance of condensers for solid cones of radiation incident, illustrating the small difference in geometric transmittance for different lengths. The condenser had demagnification = 3, for which the cutoff angle is 19.5°.

Fig. 9
Fig. 9

Average number of reflections in traversing condensers, as a function of θmax. All condensors had s = 0.5 cm, c = 0.167 cm; the lengths in cm are indicated on the curves.

Fig. 10
Fig. 10

Construction for mitered right-angle corner.

Fig. 11
Fig. 11

Three possible fates for meridional rays in a right-angle corner. θ = input angle, θ′ = output angle, each measured from the respective cylinder axis.

Fig. 12
Fig. 12

Transfer functions or output distribution for a toroid of R/d = 20, for three different annular inputs.

Fig. 13
Fig. 13

Geometrical transmittance of toroids (solid lines) and a mitered right-angle corner (dashed lines). The upper dashed line represents the fraction of the total energy exiting from the corner (regardless of solid angle); the lower dashed curve represents the transmittance with solid angle preserved, which is the relevant quantity.

Fig. 14
Fig. 14

Average number of reflections for toroids of various R/d as a function of θmax.

Fig. 15
Fig. 15

Comparison of net transmittance of toroids and mitered corner, taking into account the reflection loss (using reflectance = 99%). Over-all transmittance is plotted against θmax.

Tables (1)

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Table I Calculated reflectance values.

Equations (17)

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F = tan - 1 ( 1 / 2 θ max ) .
i · n = r · n ,
( i × n ) · r = 0.
L = l / cos θ .
D merid = d / sin θ ,
n merid = ( l / d ) tan θ .
D skew = 2 ( r 2 - ρ 2 sin 2 γ ) 1 2 ,
n skew = l tan θ / [ 2 ( r 2 - ρ 2 sin 2 γ ) 1 2 ] .
log R = log T N ¯ .
R n avg R N ¯ .
N ¯ = 0 2 π 0 θ m 0 2 π 0 2 π 0 r n ( ρ d ρ d ϕ ) d γ ( sin θ d θ d ψ ) 0 2 π 0 θ m 0 2 π 0 2 π 0 r ( ρ d ρ d ϕ ) d γ ( sin θ d θ d ψ ) .
N ¯ skew = 4 π l d { log tan [ ( π / 4 ) + ( θ max / 2 ) ] - sin θ max } 1 - cos θ max .
N ¯ skew 8 3 π l d θ max = 0.85 ( l / d ) θ max ,
N ¯ merid 2 3 l d θ max = 0.67 ( l / d ) θ max .
x = s ( 1 - c / s ) cos θ max ( c / s ) sin α - sin θ max .
c / s = sin θ max / sin α .
sin α = ( s / c ) ( sin θ max + tan η cos θ max ) ,