Abstract

Output coupling of light from a conically tapered optical fiber is investigated. Such a technique may be used to provide “wide-angle” illumination from a single fiber of low numerical aperture. Experimental results for a series of fibers of varying taper angles are presented. It is found experimentally that light which is coupled out of a taper is widely dispersed when the taper angle is large. Based upon modified geometrical optics we present a theory which qualitatively predicts the experimentally measured radiation intensity patterns.

© 1978 Optical Society of America

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References

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  1. P. K. Tien, G. Smolinsky, and R. J. Martin, “Radiation fields of a tapered film and novel film-to-fiber coupler,” IEEE Transactions on Microwave Theory and Technology,  MTT-23, p. 79 (1975).
    [Crossref]
  2. M. P. Lisitsa, L. I. Berezhinskii, and M. Y. Valakh, Fiber Optics, (Israel Program for Scientific Translations, New York, 1972), p. 36.
  3. D. Gloge, P. W. Smith, D. L. Bisbee, and E. L. Chinnock, “Optical fiber end preparation for low-loss splices,” Bell Sys. Tech. J.,  52, p. 1579 (1973).
    [Crossref]
  4. C. T. Chang and J. L. Bjorkstam, “Effect of nonuniform irradiance, and irradiance fluctuations, upon the response of photographic film,” J. Opt. Soc. Am., p. 1495 (1975).
    [Crossref]
  5. W. A. Gambling, J. P. Dakin, D. N. Payne, and H. R. D. Sunk, “Propagation model for multimode optical-fibre waveguide,” Electron. Lett.,  8, p. 260 (May1972).
    [Crossref]
  6. J. P. Dakin, W. A. Gambling, H. Matsumura, D. N. Payne, and H. R. D. Sunk, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun.,  7, p. 1 (1973).
    [Crossref]

1975 (2)

P. K. Tien, G. Smolinsky, and R. J. Martin, “Radiation fields of a tapered film and novel film-to-fiber coupler,” IEEE Transactions on Microwave Theory and Technology,  MTT-23, p. 79 (1975).
[Crossref]

C. T. Chang and J. L. Bjorkstam, “Effect of nonuniform irradiance, and irradiance fluctuations, upon the response of photographic film,” J. Opt. Soc. Am., p. 1495 (1975).
[Crossref]

1973 (2)

D. Gloge, P. W. Smith, D. L. Bisbee, and E. L. Chinnock, “Optical fiber end preparation for low-loss splices,” Bell Sys. Tech. J.,  52, p. 1579 (1973).
[Crossref]

J. P. Dakin, W. A. Gambling, H. Matsumura, D. N. Payne, and H. R. D. Sunk, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun.,  7, p. 1 (1973).
[Crossref]

1972 (1)

W. A. Gambling, J. P. Dakin, D. N. Payne, and H. R. D. Sunk, “Propagation model for multimode optical-fibre waveguide,” Electron. Lett.,  8, p. 260 (May1972).
[Crossref]

Berezhinskii, L. I.

M. P. Lisitsa, L. I. Berezhinskii, and M. Y. Valakh, Fiber Optics, (Israel Program for Scientific Translations, New York, 1972), p. 36.

Bisbee, D. L.

D. Gloge, P. W. Smith, D. L. Bisbee, and E. L. Chinnock, “Optical fiber end preparation for low-loss splices,” Bell Sys. Tech. J.,  52, p. 1579 (1973).
[Crossref]

Bjorkstam, J. L.

C. T. Chang and J. L. Bjorkstam, “Effect of nonuniform irradiance, and irradiance fluctuations, upon the response of photographic film,” J. Opt. Soc. Am., p. 1495 (1975).
[Crossref]

Chang, C. T.

C. T. Chang and J. L. Bjorkstam, “Effect of nonuniform irradiance, and irradiance fluctuations, upon the response of photographic film,” J. Opt. Soc. Am., p. 1495 (1975).
[Crossref]

Chinnock, E. L.

D. Gloge, P. W. Smith, D. L. Bisbee, and E. L. Chinnock, “Optical fiber end preparation for low-loss splices,” Bell Sys. Tech. J.,  52, p. 1579 (1973).
[Crossref]

Dakin, J. P.

J. P. Dakin, W. A. Gambling, H. Matsumura, D. N. Payne, and H. R. D. Sunk, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun.,  7, p. 1 (1973).
[Crossref]

W. A. Gambling, J. P. Dakin, D. N. Payne, and H. R. D. Sunk, “Propagation model for multimode optical-fibre waveguide,” Electron. Lett.,  8, p. 260 (May1972).
[Crossref]

Gambling, W. A.

J. P. Dakin, W. A. Gambling, H. Matsumura, D. N. Payne, and H. R. D. Sunk, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun.,  7, p. 1 (1973).
[Crossref]

W. A. Gambling, J. P. Dakin, D. N. Payne, and H. R. D. Sunk, “Propagation model for multimode optical-fibre waveguide,” Electron. Lett.,  8, p. 260 (May1972).
[Crossref]

Gloge, D.

D. Gloge, P. W. Smith, D. L. Bisbee, and E. L. Chinnock, “Optical fiber end preparation for low-loss splices,” Bell Sys. Tech. J.,  52, p. 1579 (1973).
[Crossref]

Lisitsa, M. P.

M. P. Lisitsa, L. I. Berezhinskii, and M. Y. Valakh, Fiber Optics, (Israel Program for Scientific Translations, New York, 1972), p. 36.

Martin, R. J.

P. K. Tien, G. Smolinsky, and R. J. Martin, “Radiation fields of a tapered film and novel film-to-fiber coupler,” IEEE Transactions on Microwave Theory and Technology,  MTT-23, p. 79 (1975).
[Crossref]

Matsumura, H.

J. P. Dakin, W. A. Gambling, H. Matsumura, D. N. Payne, and H. R. D. Sunk, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun.,  7, p. 1 (1973).
[Crossref]

Payne, D. N.

J. P. Dakin, W. A. Gambling, H. Matsumura, D. N. Payne, and H. R. D. Sunk, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun.,  7, p. 1 (1973).
[Crossref]

W. A. Gambling, J. P. Dakin, D. N. Payne, and H. R. D. Sunk, “Propagation model for multimode optical-fibre waveguide,” Electron. Lett.,  8, p. 260 (May1972).
[Crossref]

Smith, P. W.

D. Gloge, P. W. Smith, D. L. Bisbee, and E. L. Chinnock, “Optical fiber end preparation for low-loss splices,” Bell Sys. Tech. J.,  52, p. 1579 (1973).
[Crossref]

Smolinsky, G.

P. K. Tien, G. Smolinsky, and R. J. Martin, “Radiation fields of a tapered film and novel film-to-fiber coupler,” IEEE Transactions on Microwave Theory and Technology,  MTT-23, p. 79 (1975).
[Crossref]

Sunk, H. R. D.

J. P. Dakin, W. A. Gambling, H. Matsumura, D. N. Payne, and H. R. D. Sunk, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun.,  7, p. 1 (1973).
[Crossref]

W. A. Gambling, J. P. Dakin, D. N. Payne, and H. R. D. Sunk, “Propagation model for multimode optical-fibre waveguide,” Electron. Lett.,  8, p. 260 (May1972).
[Crossref]

Tien, P. K.

P. K. Tien, G. Smolinsky, and R. J. Martin, “Radiation fields of a tapered film and novel film-to-fiber coupler,” IEEE Transactions on Microwave Theory and Technology,  MTT-23, p. 79 (1975).
[Crossref]

Valakh, M. Y.

M. P. Lisitsa, L. I. Berezhinskii, and M. Y. Valakh, Fiber Optics, (Israel Program for Scientific Translations, New York, 1972), p. 36.

Bell Sys. Tech. J. (1)

D. Gloge, P. W. Smith, D. L. Bisbee, and E. L. Chinnock, “Optical fiber end preparation for low-loss splices,” Bell Sys. Tech. J.,  52, p. 1579 (1973).
[Crossref]

Electron. Lett. (1)

W. A. Gambling, J. P. Dakin, D. N. Payne, and H. R. D. Sunk, “Propagation model for multimode optical-fibre waveguide,” Electron. Lett.,  8, p. 260 (May1972).
[Crossref]

IEEE Transactions on Microwave Theory and Technology (1)

P. K. Tien, G. Smolinsky, and R. J. Martin, “Radiation fields of a tapered film and novel film-to-fiber coupler,” IEEE Transactions on Microwave Theory and Technology,  MTT-23, p. 79 (1975).
[Crossref]

J. Opt. Soc. Am. (1)

C. T. Chang and J. L. Bjorkstam, “Effect of nonuniform irradiance, and irradiance fluctuations, upon the response of photographic film,” J. Opt. Soc. Am., p. 1495 (1975).
[Crossref]

Opt. Commun. (1)

J. P. Dakin, W. A. Gambling, H. Matsumura, D. N. Payne, and H. R. D. Sunk, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun.,  7, p. 1 (1973).
[Crossref]

Other (1)

M. P. Lisitsa, L. I. Berezhinskii, and M. Y. Valakh, Fiber Optics, (Israel Program for Scientific Translations, New York, 1972), p. 36.

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Figures (7)

FIG. 1
FIG. 1

Schematic diagram in cross section of a tapered cylindrical optical fiber with taper angle equal to α. Angles used in the analysis are shown. The angles Θi and Θr are incident and refracted angles, respectively, while Θ is the angle measured with respect to the z axis in the experiment. The values of Θi change gradually from Θc to Θc − 2α. Each one of these individual angles corresponds to a particular subzone in moving from point A to point B.

FIG. 2
FIG. 2

The calculated (dashed curve) and experimental (solid curve) radiation intensity / vs angle Θ plotted for a tapered fiber with α = 2°.

FIG. 3
FIG. 3

The calculated (dashed curve) and experimental (solid curve) radiation intensity i vs angle Θ plotted for a tapered fiber with α = 11.4°.

FIG. 4
FIG. 4

The calculated (dashed curve) and experimental (solid curve) radiation intensity i vs angle Θ plotted for a tapered fiber with α = 22.5°.

FIG. 5
FIG. 5

Microscopic photographs of a tapered optical fiber with (a) α = 2° and (b) α = 22.5°. Magnification is the same for both pictures. The diameter of the fiber in (b) is approximately 140 μm.

FIG. 6
FIG. 6

The input (dashed curve) and output (solid curve) radiation intensity i vs angle Θ plotted for the nylon-coated Corning step-index multimode fiber with both ends cleaved (≃ 1.5 m length). The input consisted of a focused argon laser beam (10× microscope objective) of diameter 1.6 mm.

FIG. 7
FIG. 7

Photographs of radiation patterns from tapered fibers with taper angles of (a) 2° and (b) 22.5°. Neither picture is perfectly circular because they were taken with a camera off-axis to avoid blocking the pattern.

Equations (17)

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Θ c = sin - 1 ( η 0 / η i ) = sin - 1 ( 1.0 / 1.46 ) = 43.23° ,
Θ i ( m ) Θ c - ( m - 1 ) 2 α / N .
Θ r ( m ) = sin - 1 [ η i sin Θ i ( m ) ] .
Θ ( m ) = π / 2 - α / 2 - sin - 1 [ η i sin Θ i ( m ) ] .
P i ( m ) = E i H i sin Θ i A ( m ) = ( i / μ i ) 1 / 2 E i 2 sin Θ i A ( m ) .
D 0 sin ( π / 2 - Θ c + α ) sin ( π 2 - α 2 ) = A E = l sin ( 2 Θ c - 2 α ) sin ( π 2 - Θ c + 2 α ) .
l D 0 = cos ( α / 2 ) sin ( 2 Θ c - 2 α ) cos ( Θ c - α ) cos ( Θ c - 2 α ) = 2 cos ( α / 2 ) sin ( Θ c - α ) cos ( Θ c - 2 α ) .
P r ( m ) = ( 0 / μ 0 ) 1 / 2 E r 2 cos Θ r Δ s ( m ) ,
Δ s ( m ) = D 0 / 2 - ( m / N ) l sin ( α / 2 D 0 / 2 - [ ( m - 1 ) / N ] l sin ( α / 2 ) 2 π y ( d y csc α 2 ) = π l N ( D 0 - 2 m - 1 N l sin α 2 ) .
A ( m ) = ( l sin ( α / 2 ) N ) - 1 D 0 / 2 - ( m / N ) l sin ( α / 2 ) D 0 / 2 - [ ( m - 1 ) / N ] l sin ( α / 2 ) π y 2 d y = π [ ( D 0 2 ) 2 + ( D 0 2 ) ( - 2 m + 1 N ) l sin α 2 + 3 m 2 - 3 m + 1 3 N 2 l 2 sin 2 α 2 ] .
Δ s ( m ) A ( m ) = l / N D 0 + ( - 2 m + 1 ) ( l / N D 0 ) 2 sin ( α / 2 ) [ ¼ + ½ ( - 2 m + 1 ) ( l / N D 0 ) sin ( α / 2 ) + ( m 2 - m + ) ( l / N D 0 ) 2 sin 2 ( α / 2 ) ] .
P r ( m ) P i ( m ) = ( 0 ) 1 / 2 E r 2 cos Θ r ( i ) 1 / 2 E i 2 sin Θ i Δ s ( m ) A ( m ) .
T ( 0 ) 1 / 2 E r 2 cos Θ r ( i ) 1 / 2 E i 2 cos Θ i = 1 2 ( sin ( 2 Θ i ) sin ( 2 Θ r ) sin 2 ( Θ i + Θ r ) cos 2 ( Θ i - Θ r ) + sin ( 2 Θ i ) sin ( 2 Θ r ) sin 2 ( Θ i + Θ r ) ) .
P r ( m ) = Δ s ( m ) 2 A ( m ) ( sin ( 2 Θ i ) sin ( 2 Θ r ) sin 2 ( Θ i + Θ r ) cos 2 ( Θ i - Θ r ) + sin ( 2 Θ i ) sin ( 2 Θ r ) sin 2 ( Θ i + Θ r ) ) cot Θ i P i ( m ) .
P i ( m + 1 ) = P i ( m ) - P r ( m ) .
I ( m ) = P r ( m ) / [ Θ ( m ) - Θ ( m - 1 ) ] .
Θ ( m ) = π 2 - α 2 - sin - 1 [ η i sin ( Θ c - ( m - 1 ) N ( 2 α ) ] .