Abstract

The mode structure of a W-fiber has been discussed in conjunction with that of an SC-fiber using the WKB approximation, and the leaky mode attenuation coefficient has been derived using Poynting’s vector theorem. Furthermore, a simplified expression for the attenuation coefficient of the W-fiber has been obtained and compared with that of the SC-fiber. Comparisons between the calculated and measured spatial transient responses have also been carried out and shown to agree well with each other. The results are useful in the study of W-fiber properties.

© 1977 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-10, 879 (1974).
    [CrossRef]
  2. S. Kawakami, S. Nishida, M. Sumi, Proc. IEEE 123, 586 (1976).
  3. S. Onoda, T. P. Tanaka, M. Sumi, Appl. Opt. 15, 1930 (1976).
    [CrossRef] [PubMed]
  4. See, for example, A. W. Snyder, Appl. Phys. 4, 273 (1975).
    [CrossRef]
  5. M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 389 (1975).
    [CrossRef]
  6. M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
    [CrossRef]
  7. F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
    [CrossRef]
  8. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  9. M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 240 (1975).
    [CrossRef]
  10. D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).
  11. C. Pask, A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 65, 356 (1975).
    [CrossRef]
  12. S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-11, 130 (1975).
    [CrossRef]
  13. A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 956 (1974).
    [CrossRef]
  14. A. W. Snyder, D. J. Mitchell, Opto-electronics 6, 287 (1974).
    [CrossRef]
  15. T. P. Tanaka, S. Onoda, M. Sumi, Appl. Opt. 15, 1121 (1976).
    [CrossRef] [PubMed]
  16. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions, NBS Appl. Math. Series 55 (Government Printing Office, Washington, D.C., 1970).
  17. R. Sammut, A. W. Snyder, Appl. Opt. 15, 477 (1976).
    [CrossRef] [PubMed]
  18. M. J. Adams, D. N. Payne, F. M. E. Sladen, Opt. Commun. 17, 204 (1976).
    [CrossRef]
  19. A. W. Snyder, C. Pask, Opt. Commun. 15, 314 (1975).
    [CrossRef]

1976

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
[CrossRef]

F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
[CrossRef]

S. Kawakami, S. Nishida, M. Sumi, Proc. IEEE 123, 586 (1976).

R. Sammut, A. W. Snyder, Appl. Opt. 15, 477 (1976).
[CrossRef] [PubMed]

S. Onoda, T. P. Tanaka, M. Sumi, Appl. Opt. 15, 1930 (1976).
[CrossRef] [PubMed]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Opt. Commun. 17, 204 (1976).
[CrossRef]

T. P. Tanaka, S. Onoda, M. Sumi, Appl. Opt. 15, 1121 (1976).
[CrossRef] [PubMed]

1975

C. Pask, A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 65, 356 (1975).
[CrossRef]

A. W. Snyder, C. Pask, Opt. Commun. 15, 314 (1975).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 240 (1975).
[CrossRef]

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-11, 130 (1975).
[CrossRef]

See, for example, A. W. Snyder, Appl. Phys. 4, 273 (1975).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 389 (1975).
[CrossRef]

1974

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-10, 879 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, Opto-electronics 6, 287 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 956 (1974).
[CrossRef]

1973

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

1971

Adams, M. J.

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Opt. Commun. 17, 204 (1976).
[CrossRef]

F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 240 (1975).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 389 (1975).
[CrossRef]

Gloge, D.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

D. Gloge, Appl. Opt. 10, 2252 (1971).
[CrossRef] [PubMed]

Kawakami, S.

S. Kawakami, S. Nishida, M. Sumi, Proc. IEEE 123, 586 (1976).

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-11, 130 (1975).
[CrossRef]

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-10, 879 (1974).
[CrossRef]

Marcatili, E. A. J.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Mitchell, D. J.

Nishida, S.

S. Kawakami, S. Nishida, M. Sumi, Proc. IEEE 123, 586 (1976).

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-11, 130 (1975).
[CrossRef]

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-10, 879 (1974).
[CrossRef]

Onoda, S.

Pask, C.

Payne, D. N.

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Opt. Commun. 17, 204 (1976).
[CrossRef]

F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 240 (1975).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 389 (1975).
[CrossRef]

Sammut, R.

Sladen, F. M. E.

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
[CrossRef]

F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Opt. Commun. 17, 204 (1976).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 240 (1975).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 389 (1975).
[CrossRef]

Snyder, A. W.

R. Sammut, A. W. Snyder, Appl. Opt. 15, 477 (1976).
[CrossRef] [PubMed]

A. W. Snyder, C. Pask, Opt. Commun. 15, 314 (1975).
[CrossRef]

C. Pask, A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 65, 356 (1975).
[CrossRef]

See, for example, A. W. Snyder, Appl. Phys. 4, 273 (1975).
[CrossRef]

A. W. Snyder, D. J. Mitchell, Opto-electronics 6, 287 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, J. Opt. Soc. Am. 64, 956 (1974).
[CrossRef]

Sumi, M.

Tanaka, T. P.

Appl. Opt.

Appl. Phys.

See, for example, A. W. Snyder, Appl. Phys. 4, 273 (1975).
[CrossRef]

Appl. Phys. Lett.

F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
[CrossRef]

Bell Syst. Tech. J.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Electron. Lett.

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 389 (1975).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 12, 281 (1976).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 240 (1975).
[CrossRef]

IEEE J. Quantum Electron.

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-10, 879 (1974).
[CrossRef]

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-11, 130 (1975).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

M. J. Adams, D. N. Payne, F. M. E. Sladen, Opt. Commun. 17, 204 (1976).
[CrossRef]

A. W. Snyder, C. Pask, Opt. Commun. 15, 314 (1975).
[CrossRef]

Opto-electronics

A. W. Snyder, D. J. Mitchell, Opto-electronics 6, 287 (1974).
[CrossRef]

Proc. IEEE

S. Kawakami, S. Nishida, M. Sumi, Proc. IEEE 123, 586 (1976).

Other

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions, NBS Appl. Math. Series 55 (Government Printing Office, Washington, D.C., 1970).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Geometry and refractive index profile of a W-fiber in comparison with those of two types of SC-fibers.

Fig. 2
Fig. 2

Squared magnitude of radial wave vector components of a leaky mode with an azimuthal mode order ν.

Fig. 3
Fig. 3

Mode structure of a W-fiber on the assumption that ν and μ are continuous.

Fig. 4
Fig. 4

Mode structure of a W-fiber obtained by taking into account the mode discreteness.

Fig. 5
Fig. 5

Spatial transient of the transmitted power within the core of a W-fiber, which is excited by an incoherent light source.

Fig. 6
Fig. 6

Comparison of the calculated and the measured spatial transient of the transmitted power.

Fig. 7
Fig. 7

Calculated and measured NAs as a function of the distance from the light source.

Fig. 8
Fig. 8

Calculated and measured NAs as a function of the normalized intermediate layer thickness.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

u 2 = ( 2 π n 0 λ a ) 2 - ( β a ) 2 ,
w 2 = ( 2 π p n 0 λ a ) 2 - ( β a ) 2 ,
w ^ 2 = ( 2 π q n 0 λ a ) 2 - ( β a ) 2 ,
v 2 = u 2 - w 2 = ( 2 π n 0 λ a ) 2 ( 1 - p 2 ) ,
v ^ 2 = u 2 - w ^ 2 = ( 2 π n 0 λ a ) 2 ( 1 - q 2 ) ,
0 < w 2 ,
w ^ 2 < ν 2 < u 2 .
w 2 < 0 < ν 2 < u 2 ,
μ = 1 π r 1 a [ ( u a ) 2 - ( ν r ) 2 ] 1 / 2 d r ,
= ν π { [ ( u ν ) 2 - 1 ] 1 / 2 - tan - 1 [ ( u ν ) 2 - 1 ] 1 / 2 } .
μ t m = ν π { [ ( v ν ) 2 - 1 ] 1 / 2 - tan - 1 [ ( v ν ) 2 - 1 ] } .
μ l m = ν π ( v ^ ν - tan - 1 v ^ ν ) .
N t = ( v 2 / 2 ) ,
N l = v ^ 2 - ( v 2 / 2 ) ,
E z = { A J ν + 1 ( u r a )             r < a ; B H ν + 1 ( 1 ) ( w ^ r a ) + C H ν + 1 ( 2 ) ( w ^ r a )             a < r < s a ; D H ν + 1 ( 2 ) ( w r a )             s a < r ;
H z = - j β ω μ 0 E z ;
( E ϕ E r ) = ω μ 0 β ( - H r H ϕ ) ;
= { A β a u J ν ( u r a ) ; B β a w ^ H ν ( 1 ) ( w ^ r a ) + C β a w ^ H ν ( 2 ) ( w ^ r a ) ; D β a w H ν ( 2 ) ( w r a ) ;
H ν + 1 ( 2 ) ( s w ^ ) H ν + 1 ( 2 ) ( w ^ ) [ J ( u ) - H ( 1 ) w ^ ) ] [ H ( 2 ) ( s w ) - H ( 2 ) ( s w ^ ) ] - H ν + 1 ( 1 ) ( s w ^ ) H ν + 1 ( 1 ) ( w ^ ) [ J ( u ) - H ( 2 ) ( w ^ ) ] [ H ( 2 ) ( s w ) - H ( 1 ) ( s w ^ ) ] = 0 ;
J ( u ) = J ν ( u ) u J ν + 1 ( u ) ;
H ( i ) ( w ) = H ν + 1 ( i ) ( w ) w H ν + 1 ( i ) ( w ) ( i = 1 , 2 ) .
α w a = - 1 P ( z ) d P ( z ) d z ,
α w = a Re A s E × H * · k d A Re A c E × H * · z d A ,
α w 4 π β a | K w ( δ ) H ν ( 2 ) ( s w ) | 2 { 1 - J ν - 1 ( u ) J ν + 1 ( u ) [ J ν ( u ) ] 2 } - 1 ,
K w ( δ ) = J ( u ) - H ( 1 ) ( w ^ ) J ( u ) S H ( 2 ) ( s w ^ ) H ( 2 ) ( s w ) - H ( 1 ) ( s w ^ ) × H ( 1 ) ( s w ^ ) - H ( 2 ) ( s w ^ ) H ( 1 ) ( w ^ ) - H ( 2 ) ( w ^ ) H ν + 1 ( 2 ) ( s w ^ ) H ν + 1 ( 2 ) ( w ^ ) .
α w α s v 2 / v ^ 2 1 - ( 1 - v 2 v ^ 2 ) { 1 - tan h 2 [ δ ( ν 2 - w ^ 2 ) 1 / 2 ] } × { exp { - 2 δ [ ( ν 2 - w ^ 2 ) 1 / 2 - ( ν 2 - w 2 ) 1 / 2 ] } w < ν exp [ - 2 δ ( ν 2 - w ^ 2 ) 1 / 2 ] w > ν ,

Metrics