Abstract

Calculations have been made of mode properties of optical fibers with ring index profiles. Three idealized profiles with regions of depressed index on axis were chosen for study. The quantities of interest are the propagation constant for various modes, the group delay, and the near and far field mode patterns. Near and far field patterns are compared with Gaussian TEM laser modes. The results are particularly relevant to studies of nonlinear interactions in fiber waveguides.

© 1975 Optical Society of America

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  1. C. A. Burrus, E. L. Chinnock, D. Gloge, W. S. Holden, Tingye Li, R. D. Standley, D. B. Keck, Proc. IEEE 61, 1498 (1973).
    [CrossRef]
  2. D. Marcuse, W. L. Mammel, Bell Syst. Tech. J. 52, 423 (1973);M. M. Z. Kharadly, J. E. Lewis, Proc. Inst. Electr. Eng. 116, 214 (1969).
    [CrossRef]
  3. R. H. Stolen, J. E. Bjorkholm, A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
    [CrossRef]
  4. A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-17, 1130 (1969).
    [CrossRef]
  5. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  6. N. W. McLachlan, Bessel Functions for Engineers (Oxford U. P., New York, 1955).
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  8. G. Biernson, D. J. Kinsley, IEEE Trans. Microwave Theory Tech. MTT-13, 345 (1965).
    [CrossRef]
  9. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970);Chan Kei Biu, Ph.D. Thesis, University of London (1972).
    [CrossRef]
  10. S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), MIT Rad. Lab. Series Vol. 12, p. 194.
  11. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  12. J. A. Giordmaine, S. L. Shapiro, U.S. Patent3,571,607 (1971).
  13. R. H. Stolen, E. P. Ippen, Appl. Phys. Lett. 22, 276 (1973).
    [CrossRef]
  14. M. A. Duguay, J. W. Hansen, Phys. Abstr. 73, 4090 (1970);R. H. Stolen, A. Ashkin, Appl. Phys. Lett. 22, 294 (1973).
    [CrossRef]

1974

R. H. Stolen, J. E. Bjorkholm, A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
[CrossRef]

1973

C. A. Burrus, E. L. Chinnock, D. Gloge, W. S. Holden, Tingye Li, R. D. Standley, D. B. Keck, Proc. IEEE 61, 1498 (1973).
[CrossRef]

D. Marcuse, W. L. Mammel, Bell Syst. Tech. J. 52, 423 (1973);M. M. Z. Kharadly, J. E. Lewis, Proc. Inst. Electr. Eng. 116, 214 (1969).
[CrossRef]

R. H. Stolen, E. P. Ippen, Appl. Phys. Lett. 22, 276 (1973).
[CrossRef]

1971

1970

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970);Chan Kei Biu, Ph.D. Thesis, University of London (1972).
[CrossRef]

M. A. Duguay, J. W. Hansen, Phys. Abstr. 73, 4090 (1970);R. H. Stolen, A. Ashkin, Appl. Phys. Lett. 22, 294 (1973).
[CrossRef]

1969

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

1966

1965

G. Biernson, D. J. Kinsley, IEEE Trans. Microwave Theory Tech. MTT-13, 345 (1965).
[CrossRef]

Ashkin, A.

R. H. Stolen, J. E. Bjorkholm, A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
[CrossRef]

Biernson, G.

G. Biernson, D. J. Kinsley, IEEE Trans. Microwave Theory Tech. MTT-13, 345 (1965).
[CrossRef]

Bjorkholm, J. E.

R. H. Stolen, J. E. Bjorkholm, A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
[CrossRef]

Burrus, C. A.

C. A. Burrus, E. L. Chinnock, D. Gloge, W. S. Holden, Tingye Li, R. D. Standley, D. B. Keck, Proc. IEEE 61, 1498 (1973).
[CrossRef]

Chinnock, E. L.

C. A. Burrus, E. L. Chinnock, D. Gloge, W. S. Holden, Tingye Li, R. D. Standley, D. B. Keck, Proc. IEEE 61, 1498 (1973).
[CrossRef]

Duguay, M. A.

M. A. Duguay, J. W. Hansen, Phys. Abstr. 73, 4090 (1970);R. H. Stolen, A. Ashkin, Appl. Phys. Lett. 22, 294 (1973).
[CrossRef]

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970);Chan Kei Biu, Ph.D. Thesis, University of London (1972).
[CrossRef]

Giordmaine, J. A.

J. A. Giordmaine, S. L. Shapiro, U.S. Patent3,571,607 (1971).

Gloge, D.

C. A. Burrus, E. L. Chinnock, D. Gloge, W. S. Holden, Tingye Li, R. D. Standley, D. B. Keck, Proc. IEEE 61, 1498 (1973).
[CrossRef]

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

Hansen, J. W.

M. A. Duguay, J. W. Hansen, Phys. Abstr. 73, 4090 (1970);R. H. Stolen, A. Ashkin, Appl. Phys. Lett. 22, 294 (1973).
[CrossRef]

Holden, W. S.

C. A. Burrus, E. L. Chinnock, D. Gloge, W. S. Holden, Tingye Li, R. D. Standley, D. B. Keck, Proc. IEEE 61, 1498 (1973).
[CrossRef]

Ippen, E. P.

R. H. Stolen, E. P. Ippen, Appl. Phys. Lett. 22, 276 (1973).
[CrossRef]

Keck, D. B.

C. A. Burrus, E. L. Chinnock, D. Gloge, W. S. Holden, Tingye Li, R. D. Standley, D. B. Keck, Proc. IEEE 61, 1498 (1973).
[CrossRef]

Kinsley, D. J.

G. Biernson, D. J. Kinsley, IEEE Trans. Microwave Theory Tech. MTT-13, 345 (1965).
[CrossRef]

Kitano, I.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970);Chan Kei Biu, Ph.D. Thesis, University of London (1972).
[CrossRef]

Kogelnik, H.

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970);Chan Kei Biu, Ph.D. Thesis, University of London (1972).
[CrossRef]

Li, T.

Li, Tingye

C. A. Burrus, E. L. Chinnock, D. Gloge, W. S. Holden, Tingye Li, R. D. Standley, D. B. Keck, Proc. IEEE 61, 1498 (1973).
[CrossRef]

Mammel, W. L.

D. Marcuse, W. L. Mammel, Bell Syst. Tech. J. 52, 423 (1973);M. M. Z. Kharadly, J. E. Lewis, Proc. Inst. Electr. Eng. 116, 214 (1969).
[CrossRef]

Marcuse, D.

D. Marcuse, W. L. Mammel, Bell Syst. Tech. J. 52, 423 (1973);M. M. Z. Kharadly, J. E. Lewis, Proc. Inst. Electr. Eng. 116, 214 (1969).
[CrossRef]

Matsumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970);Chan Kei Biu, Ph.D. Thesis, University of London (1972).
[CrossRef]

McLachlan, N. W.

N. W. McLachlan, Bessel Functions for Engineers (Oxford U. P., New York, 1955).

Shapiro, S. L.

J. A. Giordmaine, S. L. Shapiro, U.S. Patent3,571,607 (1971).

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), MIT Rad. Lab. Series Vol. 12, p. 194.

Snyder, A. W.

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

Standley, R. D.

C. A. Burrus, E. L. Chinnock, D. Gloge, W. S. Holden, Tingye Li, R. D. Standley, D. B. Keck, Proc. IEEE 61, 1498 (1973).
[CrossRef]

Stolen, R. H.

R. H. Stolen, J. E. Bjorkholm, A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
[CrossRef]

R. H. Stolen, E. P. Ippen, Appl. Phys. Lett. 22, 276 (1973).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970);Chan Kei Biu, Ph.D. Thesis, University of London (1972).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

R. H. Stolen, E. P. Ippen, Appl. Phys. Lett. 22, 276 (1973).
[CrossRef]

R. H. Stolen, J. E. Bjorkholm, A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
[CrossRef]

Bell Syst. Tech. J.

D. Marcuse, W. L. Mammel, Bell Syst. Tech. J. 52, 423 (1973);M. M. Z. Kharadly, J. E. Lewis, Proc. Inst. Electr. Eng. 116, 214 (1969).
[CrossRef]

IEEE J. Quantum Electron.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970);Chan Kei Biu, Ph.D. Thesis, University of London (1972).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

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

G. Biernson, D. J. Kinsley, IEEE Trans. Microwave Theory Tech. MTT-13, 345 (1965).
[CrossRef]

Phys. Abstr.

M. A. Duguay, J. W. Hansen, Phys. Abstr. 73, 4090 (1970);R. H. Stolen, A. Ashkin, Appl. Phys. Lett. 22, 294 (1973).
[CrossRef]

Proc. IEEE

C. A. Burrus, E. L. Chinnock, D. Gloge, W. S. Holden, Tingye Li, R. D. Standley, D. B. Keck, Proc. IEEE 61, 1498 (1973).
[CrossRef]

Other

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), MIT Rad. Lab. Series Vol. 12, p. 194.

J. A. Giordmaine, S. L. Shapiro, U.S. Patent3,571,607 (1971).

N. W. McLachlan, Bessel Functions for Engineers (Oxford U. P., New York, 1955).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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Figures (5)

Fig. 1
Fig. 1

Four index profiles: (a) uniform core; (b), (c) ring profiles; (d) tube waveguide.

Fig. 2
Fig. 2

Plots of normalized effective refractive index b vs normalized frequency V for the lowest ten modes of four different core index profiles. The parameters b and V are defined in Eqs. (10) and (7). The dotted lines in (b), (c), and (d) are the LP01 and LP02 modes from the uniform core case of Fig. 1(a). The intensities of the linearly polarized modes are illustrated schematically in 2(a).

Fig. 3
Fig. 3

Plots of the normalized group delay d(Vb)/dV vs normalized frequency V.

Fig. 4
Fig. 4

Normalized electric field for the LP01, LP02, LP11, and LP21 modes. These patterns are all calculated for fibers with V = 8 and have the same power. Also included are TEM laser cavity modes fit to the homogeneous core fiber modes.

Fig. 5
Fig. 5

Far field patterns for the fields in Fig. 4. The ordinate is proportional to electric field, and the abscissa is a normalized radius defined in Eq. (13).

Equations (15)

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E y = F f l ( r ) cos l θ
E y = F f l ( r ) sin l θ ,
0 2 π 0 f l 2 ( r ) cos 2 l θ rdrd θ = 1.0
f l ( r ) = A l J l ( ρ u 1 ) u 1 = r 2 ( k 2 n 1 2 β 2 ) 1 / 2
f l ( r ) = A l I l ( ρ w 1 ) ; w 1 = r 2 ( β 2 k 2 n 1 2 ) 1 / 2 .
f l ( r ) = B l J l ( ρ u 2 ) + C l Y l ( ρ u 2 ) , u 2 = r 2 ( k 2 n 2 2 β 2 ) 1 / 2 ,
f l ( r ) = D l K l ( ρ w 3 ) , w 3 = r 2 ( β 2 n 3 2 k 2 ) 1 / 2 .
V 2 π r 2 λ ( n 2 2 n 3 2 ) 1 / 2 = ( w 3 2 + u 2 2 ) 1 / 2 .
u 1 2 = ( n 1 n 3 n 2 n 3 ) V 2 w 3 2 .
b ( V ) = 1 ( u 2 2 / V 2 ) ,
b ( β / k n 3 ) / ( n 2 n 3 ) .
τ = L c { [ d ( n k / d k ) ] + ( n 2 n 3 ) [ d ( V b / d V ) ] } .
F l ( P ) = 0 f l ( ρ ) J l ( 2 ρ P ) ρ d ρ .
U ( P ) = 2 π cos l θ exp [ i ( 1 l ) ] F ( P ) / λ L , P = R / R c = R / ( λ L / π r 2 ) ,
TEM 00 : E = G 1 exp ( r 2 / w 1 2 ) ; TEM 01 : E = G 2 ( 1 2 r 2 w 2 2 ) exp ( r 2 / w 2 2 ) ; TEM 10 : E = G 3 ( r / w 3 ) cos θ exp ( r 2 / w 3 2 ) ; TEM 20 : E = G 4 ( r / w 4 ) 2 cos 2 θ exp ( r 2 / w 4 2 ) .

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