Abstract

The power of the trapped modes on a semi-infinite optical fiber illuminated by an incoherent source is determined. All possible modes are excited, each with approximately the same power when V → ∞, V=2πρ{n12n22}1/2/λ, where ρ is the fiber radius, λ the wavelength of light in vacuum, and n1, n2 are the refractive indices of the fiber and its surround, respectively. A ray-optical interpretation is given for the summed power of the modes. For V = ∞, the power corresponds to that found from classical geometric optics, treating all rays as if they are meridional. This result is independent of the degree of coherence of the source. The per cent error of meridional ray optics is 100/V when V is large. The total power within the fiber is the combined power of the trapped modes and the radiation field. In the limit V = ∞, the total power within the fiber at any position z along its axis is that given by classical geometric optics, i.e., that found by tracing all rays, skew and meridional. At the point z = ∞ for arbitrary V, the total power is that due to the trapped modes only.

© 1973 Optical Society of America

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References

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  1. A. W. Snyder, C. Pask, and D. J. Mitchell, J. Opt. Soc. Am. 63, 59 (1973).
    [Crossref] [PubMed]
  2. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 119 and 491.
  3. M. Beran and G. B. Parrent, Theory of Coherence (Prentice–Hall, Englewood Cliffs, N. J., 1964), pp. 53 and 57.
  4. H. H. Hopkins, in Advanced Optical Techniques, edited by A. C. S. van Heel (North–Holland, Amsterdam, 1967), p. 189.
  5. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138 (1969).
    [Crossref]
  6. Although the field is quasimonochromatic, it is still assumed to be incoherent (Refs. 2 and 3).
  7. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
    [Crossref]
  8. These are the surface or discrete modes that propagate along the fiber and not the continuous modes that account for radiation.
  9. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, New York, 1965), pp. 4–10.
  10. E. Snitzer, J. Opt. Soc. Am. 51, 1122 (1961).
    [Crossref]
  11. The approximation η≃ 1 (U/V)2{V2− U2}(−1/2)used in Ref. 13 leads to the incorrect conclusion that the total power <P> is less than the power within the fiber <PF>. Thus, as is often the case, the first term of an asymptotic expansion is more uniformly valid than the series with several terms.
  12. This result can be derived by considering the resolution of an aperture of the fiber diameter and the acceptance angle of the fiber based on meridional rays (Refs. 13 and 14).
  13. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [Crossref] [PubMed]
  14. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
    [Crossref] [PubMed]
  15. G. N. Watson, Theory of Bessel Functions (Cambridge, U.P., Cambridge, England, 1922), p. 477.
  16. R. J. Potter, J. Opt. Soc. Am. 51, 1079 (1961).
    [Crossref]
  17. A. W. S. Snyder and C. Pask, J. Opt. Soc. Am. 62, 998 (1972).
    [Crossref]
  18. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw–Hill, New York, 1961).
  19. N. S. Kapany, Fiber Optics (Academic, New York, 1967).

1973 (1)

1972 (1)

1971 (1)

1969 (3)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138 (1969).
[Crossref]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
[Crossref]

G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
[Crossref] [PubMed]

1961 (2)

Beran, M.

M. Beran and G. B. Parrent, Theory of Coherence (Prentice–Hall, Englewood Cliffs, N. J., 1964), pp. 53 and 57.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 119 and 491.

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, New York, 1965), pp. 4–10.

Gloge, D.

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw–Hill, New York, 1961).

Hopkins, H. H.

H. H. Hopkins, in Advanced Optical Techniques, edited by A. C. S. van Heel (North–Holland, Amsterdam, 1967), p. 189.

Kapany, N. S.

N. S. Kapany, Fiber Optics (Academic, New York, 1967).

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, New York, 1965), pp. 4–10.

Mitchell, D. J.

Parrent, G. B.

M. Beran and G. B. Parrent, Theory of Coherence (Prentice–Hall, Englewood Cliffs, N. J., 1964), pp. 53 and 57.

Pask, C.

Potter, R. J.

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, New York, 1965), pp. 4–10.

Snitzer, E.

Snyder, A. W.

A. W. Snyder, C. Pask, and D. J. Mitchell, J. Opt. Soc. Am. 63, 59 (1973).
[Crossref] [PubMed]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138 (1969).
[Crossref]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
[Crossref]

Snyder, A. W. S.

Toraldo di Francia, G.

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge, U.P., Cambridge, England, 1922), p. 477.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 119 and 491.

Appl. Opt. (1)

IEEE Trans. Microwave Theory Tech. (2)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138 (1969).
[Crossref]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
[Crossref]

J. Opt. Soc. Am. (5)

Other (11)

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw–Hill, New York, 1961).

N. S. Kapany, Fiber Optics (Academic, New York, 1967).

The approximation η≃ 1 (U/V)2{V2− U2}(−1/2)used in Ref. 13 leads to the incorrect conclusion that the total power <P> is less than the power within the fiber <PF>. Thus, as is often the case, the first term of an asymptotic expansion is more uniformly valid than the series with several terms.

This result can be derived by considering the resolution of an aperture of the fiber diameter and the acceptance angle of the fiber based on meridional rays (Refs. 13 and 14).

G. N. Watson, Theory of Bessel Functions (Cambridge, U.P., Cambridge, England, 1922), p. 477.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 119 and 491.

M. Beran and G. B. Parrent, Theory of Coherence (Prentice–Hall, Englewood Cliffs, N. J., 1964), pp. 53 and 57.

H. H. Hopkins, in Advanced Optical Techniques, edited by A. C. S. van Heel (North–Holland, Amsterdam, 1967), p. 189.

These are the surface or discrete modes that propagate along the fiber and not the continuous modes that account for radiation.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, New York, 1965), pp. 4–10.

Although the field is quasimonochromatic, it is still assumed to be incoherent (Refs. 2 and 3).

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Figures (2)

Fig. 1
Fig. 1

A semi-infinite fiber of arbitrary cross section illuminated by an incoherent source (IS). The n’s are refractive indices and μ is the magnetic permeability of vacuum.

Fig. 2
Fig. 2

The percent error of meridional-ray optics, for use in determining the summed time-averaged power of the trapped modes transmitted within a circular optical fiber illuminated as in Fig. 1. The solid curve is determined numerically from Eq. (40a), the dashed curve is the asymptotic analysis Eq. (40b). ρ is the fiber radius, λ the wavelength in vacuum, and n1, n2 the refractive indices of the fiber and its surround, respectively. The maxima of the solid curve occur at V equal to the cutoff values of the modes, i.e., V = 0, 2.405, 3.832, 5.135, etc. At V = 2.405, the error is approximately 110% and as V approaches zero the percent error approaches infinity.

Tables (1)

Tables Icon

Table I Comparison of numerical and asymptotic expressions for number of propagating modes N, power of the trapped modes transmitted within the fiber 〈PF〉, and percent error of meridionalray optics for various values of the dimensionless parameter V denned by Eq. (1). We assume the excitation condition of Fig. 1.

Equations (59)

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V = 2 π ρ λ [ n 1 2 n 2 2 ] 1 2 ,
E S = 2 ( μ / 1 ) 1 4 E 0 Re { x ˆ cos ψ ( x , y , t ) + y ˆ sin ψ ( x , y , t ) } e i { ϕ ( x , y , t ) ω t } ,
E = Re p a p e p ( x , y ) e i ( ω t + β p z ) ,
H = Re p a p h p ( x , y ) e i ( ω t + β p z ) ,
h p = ( 1 μ ) 1 2 z ˆ × e p ,
S e p × h q * · z ˆ d S = ( 1 μ ) 1 2 S e p · e q * d S = δ p q ,
a p = 2 ( 1 μ ) 1 2 S F e p * · { x ˆ cos ψ + y ˆ sin ψ } e i ϕ d S ,
P = S E × H · z ˆ d S .
P = Re lim T 1 2 T T T P d t .
P = 1 2 p | a p | 2 ,
P F = 1 2 p , q a p a q * e i ( β p β q ) z C p q ,
C p q = ( 1 μ ) 1 2 S F e p · e q * d s .
C p q = η p = Power of the p th mode within the fiber Total power of the p th mode .
a p a q * = 4 ( 1 μ ) 1 2 S F S F d S d S × e i ( ϕ ϕ ) e p · ( x ˆ cos ψ + y ˆ sin ψ ) × e q * · ( x ˆ cos ψ + y ˆ sin ψ ) ,
T T e i ( ϕ ϕ ) f ( t ) d t = δ ( x x ) δ ( y y ) T T f ( t ) d t ,
1 2 T T T cos 2 ψ d t = 1 2 T T T sin 2 ψ d t = 1 2 ,
1 2 T T T cos ψ sin ψ d t = 0 ,
a p a q * = 4 ( 1 μ ) 1 2 S F d S × { 1 2 ( e p · x ˆ ) ( e q · x ˆ ) + 1 2 ( e p · y ˆ ) ( e q · y ˆ ) }
= 2 C p q .
P = p η p ,
P F = p η p 2 + P cross ,
P cross = p , q p q C p q 2 e i ( β p β q ) z .
P F η p 2 .
P = P F + P 0 .
P P F p 1 = N ,
n = S F ( 2 π ) 2 d k x d y y × 2 .
N = S F ( 2 π ) 2 A ( k x , k y ) × 2 ,
| k | 2 = k x 2 + k y 2 = ( 2 π n 1 λ ) 2 k z 2 .
max | k | 2 = ( 2 π λ ) 2 ( n 1 2 n 2 2 ) ,
P F = ( n 1 2 n 2 2 λ 2 ) S F ,
η l m = ( U l m V ) 2 { ( W l m U l m ) 2 + K l 1 2 ( W l m ) K l ( W l m ) K l 2 ( W l m ) } ,
V 2 = U l m 2 + W l m 2 ,
U l m K l 1 ( W l m ) J l ( U l m ) = W l m J l 1 ( U l m ) K l ( W l m ) ,
p η p = 2 m = 1 M ( V ) { η 1 m + 2 l 2 L ( V ) η l m } ,
η l m 1 U l m 2 V 3 .
N V 2 2 ,
U l m ( 2 ξ ) 1 2 ,
P 0 V 3 / 2 η ( ξ ) d ξ V 2 2 { 1 1 2 V } ,
P F 0 V 2 / 2 η 2 ( ξ ) d ξ V 2 2 { 1 1 V } .
P 0 P 1 2 V .
P S = 2 3 ( 2 π ρ n 1 λ ) 2 = 2 3 V 2 δ
δ = 1 ( n 2 n 1 ) 2 .
P F P S 3 4 δ ( 1 1 V ) .
( P F P S ) GO = ( P F P S ) MRO + ( P F P S ) SRC
= 3 δ 2 ,
( P F P S ) MRO = ( P F P S ) SRC ,
( P F P S ) V = = 1 2 ( P F P S ) GO .
= [ ( P F P S ) MRO ( P F P S ) ] / ( P F P S )
1 V .
C p q = 2 I ( ξ p ξ q ) 1 2 U p U q V 2 ,
I = 1 ( U p 2 U q 2 ) ( U p J l ( U p ) J l 1 ( U p ) U p J l ( U q ) J l 1 ( U q ) ) .
C p q U p U q / V 3 ,
P cross = 2 p , q p q C p q 2 cos ( β p β q ) z ,
| P cross | 2 p , q p q C p q 2 .
E ( r ) = i e i k r 2 π r k × S F z ˆ × E S ( x ) e i k · x d S ,
P S = 1 2 Re S E ( r ) × H * ( r ) · z ˆ d S ,
H ( r ) = ( μ ) 1 2 k ˆ × E ( r ) .
P S = 2 3 π S F ( 2 n 1 λ ) 2 .
P S = 2 3 V 2 δ .