Abstract

It is well known that as the diameter of a dielectric waveguide decreases, more energy of the supported modes is conducted outside the guide. Thus the absorption coefficient of the waveguide material becomes only partially effective and the “effective absorption coefficient” is a function of the surrounding medium also. The effective absorption coefficient can be determined by calculating the fractions of energy conducted inside and outside the guide and applying the absorption coefficients of the two media. The results of such calculations for several low-order modes in fibers of small diameter and n. a. are presented. The effective absorption coefficient is plotted as a function of the fiber characteristic term R = πd/λ (the fiber n. a.). Experimental measurements have been made in low-n. a. fibers of absorbing core and transmitting coating of different R values, and photometric results in support of the theory are included.

© 1963 Optical Society of America

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References

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  1. D. G. Kiely, Dielectric Aerials (Methuen and Company, Ltd., London, 1953).
  2. C. M. McKinney, dissertation for Ph.D. degree, University of Texas, 1950.
  3. H. J. Wegener, thesis for degree of Dr. Ing., Technische Hoch schule, Berlin, 1944.
  4. W. M. Elsasser, J. Appl. Phys. 20, 1188 (1948).
  5. N. S. Kapany and J. J. Burke, J. Opt. Soc. Am. 51, 1067 (1961).
    [CrossRef]
  6. N. S. Kapany and J. J. Burke, Solid State Design 3, 35 (1962).
  7. N. S. Kapany, J. Opt. Soc. Am. 49, 770 (1959).
    [CrossRef]
  8. S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand, Inc., New York, 1943), pp. 77–81.
  9. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 131–137.
  10. N. S. Kapany, J. Opt. Soc. Am. 47, 413 (1957).
    [CrossRef]

1962 (1)

N. S. Kapany and J. J. Burke, Solid State Design 3, 35 (1962).

1961 (1)

1959 (1)

1957 (1)

1948 (1)

W. M. Elsasser, J. Appl. Phys. 20, 1188 (1948).

Burke, J. J.

N. S. Kapany and J. J. Burke, Solid State Design 3, 35 (1962).

N. S. Kapany and J. J. Burke, J. Opt. Soc. Am. 51, 1067 (1961).
[CrossRef]

Elsasser, W. M.

W. M. Elsasser, J. Appl. Phys. 20, 1188 (1948).

Kapany, N. S.

Kiely, D. G.

D. G. Kiely, Dielectric Aerials (Methuen and Company, Ltd., London, 1953).

McKinney, C. M.

C. M. McKinney, dissertation for Ph.D. degree, University of Texas, 1950.

Schelkunoff, S. A.

S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand, Inc., New York, 1943), pp. 77–81.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 131–137.

Wegener, H. J.

H. J. Wegener, thesis for degree of Dr. Ing., Technische Hoch schule, Berlin, 1944.

J. Appl. Phys. (1)

W. M. Elsasser, J. Appl. Phys. 20, 1188 (1948).

J. Opt. Soc. Am. (3)

Solid State Design (1)

N. S. Kapany and J. J. Burke, Solid State Design 3, 35 (1962).

Other (5)

D. G. Kiely, Dielectric Aerials (Methuen and Company, Ltd., London, 1953).

C. M. McKinney, dissertation for Ph.D. degree, University of Texas, 1950.

H. J. Wegener, thesis for degree of Dr. Ing., Technische Hoch schule, Berlin, 1944.

S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand, Inc., New York, 1943), pp. 77–81.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 131–137.

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Figures (5)

Fig. 1
Fig. 1

Ratio of the power transmitted inside the fiber core to total power transmitted in fiber of characteristic term R for HE1,m mode, where m = 1, 2, 3, 4, and 5.

Fig. 2
Fig. 2

Ratio of power transmitted inside the fiber core to the total power transmitted in HE1,1 and TE0,1 modes (nonlinear curve plotted to scale at left), and optical diameter of core (linear curves plotted to scale at right), as a function of R.

Fig. 3
Fig. 3

Ratio of effective absorption coefficient of fiber to that of bulk core material for seven values of the ratio of absorption coefficients of coating and core materials (nonlinear curves plotted to scale at left), and optical diameter of core for three values of δ = 1 − (n22/n11) (linear curves plotted to scale at right), as a function of R.

Fig. 4
Fig. 4

Photomicrographs of exit ends of fibers with filter glass cores coated in nonabsorbing glass, demonstrating the decrease in effective absorption coefficient with decreasing diameter: (a) Diam, 50 μ; length, 4.5 in. (b) Diam, 50 μ; length, 1 in. (c) Diam, 5-μ; length, 4.5 in. (d) Diam, 5 μ; length, 1 in. Core and coating of all fibers were uniformly illuminated by sodium source.

Fig. 5
Fig. 5

Optical arrangement used for measurement of effective absorption coefficient of fibers.

Tables (1)

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Table I Theoretical and experimental effective absorption coefficient for two fibers.a

Equations (10)

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α = ( α 1 P 1 + α 2 P 2 ) / P T ,
S ¯ z = 1 2 Re ( E × H * ) z .
E + = b f 0 ; E = 1 A 1 + A b f 2 ; E z = i β h b 1 + A f 1 ; H + = + i h ω μ ( K 1 2 / h 2 ) + A 1 + A E + ; H = i h ω μ ( K 1 2 / h 2 ) A 1 A E ; H z = i h ω μ A E z ;
f p = J p ( β r ) exp ( i ω t + i p ϕ i h z ) , β 2 + h 2 = K 1 2 = ω 2 1 μ , A = ( 1 u 2 + 1 q 2 ) / ( J 1 ( u ) u J 1 ( u ) + K 1 ( q ) q K 1 ( q ) ) , u = β d / 2 ; q = β d / 2 ; h 2 β = K 2 2 = ω 2 2 μ .
S ¯ z = 1 4 Re [ i ( E + H + * E H * ) ] .
0 2 π 0 d / 2 S ¯ z r d r d ϕ = P 1 ; 0 2 π d / 2 S ¯ r d r d ϕ = P 2 ,
P 2 P 1 = u 2 q 2 [ { ( k 2 + ξ 2 A ) [ K 0 2 ( q ) K 1 2 ( q ) 1 ] + γ ( k 2 ξ 2 A ) [ K 0 2 ( q ) K 1 2 ( q ) 1 4 q 2 ] } { ( 1 + ξ 2 A ) [ J 0 2 ( u ) J 1 2 ( u ) + 1 ] + γ ( 1 ξ 2 A ) [ J 0 2 ( u ) J 1 2 ( u ) + 1 4 q 2 ] } ] .
h = [ k 1 2 ( 2 u d ) 2 ] 1 2 ; θ = cos 1 ( u π n 1 d / λ ) .
J 0 ( u ) J 1 ( u ) = 1 u + ( 1 δ 2 ) ( u K 0 ( q ) q K 1 ( q ) + u q 2 ) [ R 2 q 4 ( R 2 u 2 δ ) + δ 2 4 ( u K 0 ( q ) q K 1 ( q ) + u q 2 ) 2 ] 1 / 2 ,
δ = 1 k 2 2 / k 1 2 , R = π d / λ ( n 1 2 n 2 2 ) 1 2 , q 2 = R 2 u 2 .