Abstract

The power transmittance and directivity of light beams propagated along straight circular symmetrical dielectric tubes and emitted from the end surface of the bore are derived from geometrical optics. The radial power distribution in the beam is included as a distribution function F(ρ,θ). The calculations apply to the case of Fresnel reflection losses. Scattering and absorption phenomena and the contribution of rays refracted into the tube wall are not taken into consideration. Numerical results are compared with experimental values obtained with pulsed light beams characterized by an unsymmetric radial power distribution.

© 1966 Optical Society of America

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References

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  1. D. G. Kiely, Dielectric Aerials (Methuen Co., London, 1953).
  2. E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961).
    [Crossref]
  3. R. J. Potter, J. Opt. Soc. Am. 51, 1079 (1961).
    [Crossref]
  4. R. J. Potter, E. Donath, and R. Tynan, J. Opt. Soc. Am. 53, 256 (1963).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959), Chap.1.5.
  6. K. Vogel, Nature 207, 281 (1965).
    [Crossref]
  7. H. J. Caulfield, Nature 208, 773 (1965).
    [Crossref]

1965 (2)

K. Vogel, Nature 207, 281 (1965).
[Crossref]

H. J. Caulfield, Nature 208, 773 (1965).
[Crossref]

1963 (1)

1961 (2)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959), Chap.1.5.

Caulfield, H. J.

H. J. Caulfield, Nature 208, 773 (1965).
[Crossref]

Donath, E.

Kiely, D. G.

D. G. Kiely, Dielectric Aerials (Methuen Co., London, 1953).

Potter, R. J.

Snitzer, E.

Tynan, R.

Vogel, K.

K. Vogel, Nature 207, 281 (1965).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959), Chap.1.5.

J. Opt. Soc. Am. (3)

Nature (2)

K. Vogel, Nature 207, 281 (1965).
[Crossref]

H. J. Caulfield, Nature 208, 773 (1965).
[Crossref]

Other (2)

D. G. Kiely, Dielectric Aerials (Methuen Co., London, 1953).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959), Chap.1.5.

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Figures (8)

F. 1
F. 1

Cross sectional view of a dielectric tube indicating the path of rays emitted from the source S, propagated along the tube, and accepted by the beam-scanning aperture in the reference plane A2. Rays refracted into medium n2 are not shown. In the reference plane A1 the power distribution in the beam is described by the distribution function F(ρ,θ). Δϕ(0) and Δϕ(1) indicate the space cone for rays with a constant number of reflections N. B0 and B1 define the intersection of A1 with the limiting rays for N = 0 and N = 1.

F. 2
F. 2

Geometry in the reference plane A1. The power distribution in the beam is described by a function F(ρ,θ). Broken circles refer to experiment and indicate the position of the modes in the light beam, centered around F1(0) and F2(0) at the distance c1 and c2 from the optical axis. Each mode contributes to the total power radiated through the surface element ρdρdθ. The arrow indicates the experimentally determined direction of the electric vector. For c1 = c2 = 0.6 mm, F1(r) = 0.77 exp(−r2) and F2(r) = 0.32 exp(−0.4r2).

F. 3
F. 3

Power transmittance calculated for circular polarization, F(ρ,θ)=1, b=0, and n12=1.50 and plotted as function of a=di/(b+l). Solid lines refer to ϕmax=15.5°, s=1.0, and to scale I. Broken lines refer to ϕmax=5°, s=1.0, and to scale II.

F. 4
F. 4

Partial power transmittance plotted as function of ϕin for circular polarization and for different values of N. Solid lines refer to n12=1.60, broken lines to n12=1.40.

F. 5
F. 5

Experimental setup used for measuring the radiation characteristics of light beams emitted from the bore of dielectric tubes.

F. 6
F. 6

Energy transmittance plotted as function of a=di/ (b+l). Symbols (cf. Table I) refer to experimental values measured with a pulsed light beam. Lines refer to numerical results calculated for the corresponding parameter values and for circular (broken line) and linear (solid line) polarization.

F. 7
F. 7

Directivity plotted as function of the scaling factor m. Symbols (cf. Table I) refer to experimental values measured with a pulsed light beam. Solid lines refer to numerical results calculated for the corresponding parameter values.

F. 8
F. 8

Directivity plotted as function of the scaling factor m. Symbols (cf. Table I) refer to experimental values measured with a pulsed light beam. Solid lines refer to numerical results calculated for the corresponding parameter values.

Tables (1)

Tables Icon

Table I Tube measurements and parameter values.

Equations (13)

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T = P Ω / P in , D = P Ω / P out .
P Ω = P 0 N = 0 N = N max ρ 1 ρ 2 0 2 π R ( N , ρ , θ ) F ( ρ , θ ) ρ d ρ d θ .
R = tan 2 ( α β ) / tan 2 ( α + β ) R = sin 2 ( α β ) / sin 2 ( α + β )
tan γ 1 = [ cos ( α β ) / cos ( α + β ) ] tan γ 0 ,
R ( N , ρ , θ ) = N = 1 N = N max { R ( ρ ) sin 2 [ γ N 1 ( θ ) ] + R ( ρ ) cos 2 [ γ N 1 ( θ ) ] } , N 1 = 1 , N = 0 }
tan γ N 1 ( θ ) = [ cos ( α β ) / cos ( α + β ) ] N 1 tan γ 0 ( θ ) .
R ( N , ρ , θ ) = { 1 2 [ R 2 ( ρ ) + R 2 ( ρ ) ] } N 1 .
ρ 1 ( N ) = 1 2 ( 2 N 1 ) a z N > 0 = 0 N = 0 }
ρ 2 ( N ) = 1 2 ( 2 N + 1 ) a z N 0 .
N 1 2 [ ( m + s + 1 ) / m ]
ρ 2 ( N ) = 1 2 [ ( 2 N + m ) / ( 1 + m ) ] a z .
F 1 ( r ) = 0.77 exp ( 1.0 r 2 )
F 2 ( r ) = 0.32 exp ( 0.4 r 2 )