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 <i>V</i>→∞, <i>V</i> = 2πρ (n<sup>2</sup><sub>1</sub> - n<sup>2</sup><sub>2</sub>)<sup>½</sup> /λ, where ρ is the fiber radius, λ the wavelength of light in vacuum, and n<sub>1</sub>, n<sub>2</sub> 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 <i>V</i>=∞, 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/<i>V</i> when <i>V</i> is large. The total power within the fiber is the combined power of the trapped modes and the radiation field. In the limit <i>V</i>=∞, 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 <i>V</i>, the total power is that due to the trapped modes only.

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  1. A. W. Snyder, C. Pask, and D. J. Mitchell, J. Opt. Soc. Am. 63, 59 (1973).
  2. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 119 and 491.
  3. M. Beran and G. B. Parrent, Jr., 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).
  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).
  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).
  11. The approximation η ≈ 1 - (U / V)2 {V2 - U2}(-½) used in Ref. 13 leads to the incorrect conclusion that the total power <P > is less than the power within the fiber <P F >. 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).
  14. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
  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).
  17. A. W. S. Snyder and C. Pask, J. Opt. Soc. Am. 62, 998 (1972).
  18. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw—Hill, New York, 1961).
  19. N. S. Kapany, Fiber Optics (Academic, New York, 1967).

Beran, M.

M. Beran and G. B. Parrent, Jr., 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.

di Francia, G. Toraldo

G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).

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.

D. Gloge, Appl. Opt. 10, 2252 (1971).

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.

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

Parrent, Jr., G. B.

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

Pask, C.

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

A. W. S. Snyder and C. Pask, J. Opt. Soc. Am. 62, 998 (1972).

Potter, R. J.

R. J. Potter, J. Opt. Soc. Am. 51, 1079 (1961).

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.

E. Snitzer, J. Opt. Soc. Am. 51, 1122 (1961).

Snyder, A. W.

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

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

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

Snyder, A. W. S.

A. W. S. Snyder and C. Pask, J. Opt. Soc. Am. 62, 998 (1972).

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.

Other (19)

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

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

M. Beran and G. B. Parrent, Jr., 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.

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

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

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

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.

E. Snitzer, J. Opt. Soc. Am. 51, 1122 (1961).

The approximation η ≈ 1 - (U / V)2 {V2 - U2}(-½) used in Ref. 13 leads to the incorrect conclusion that the total power <P > is less than the power within the fiber <P F >. 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).

D. Gloge, Appl. Opt. 10, 2252 (1971).

G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).

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

R. J. Potter, J. Opt. Soc. Am. 51, 1079 (1961).

A. W. S. Snyder and C. Pask, J. Opt. Soc. Am. 62, 998 (1972).

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

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

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