Abstract

A technique is presented for determining the modal propagation properties of a homogeneous cylindrical dielectric waveguide of arbitrary cross sectional shape and index n1 embedded in a medium of index n2. Both the weakly guiding case in which n1n2 and the general case of arbitrary index difference are discussed theoretically. In both cases the approach is to derive integral representations for appropriate components of E and B. These satisfy the appropriate Helmholtz equations inside and outside the guide and also guarantee that the boundary conditions are satisfied. On expansion of the components in certain sets of basis functions, the representations become a set of linear equations. The vanishing of the determinant of this set yields the propagation constants of the various modes. Numerical results are given for weakly guiding fibers of various shapes. Among these are rectangles and ellipses, which make comparisons with previous work possible.

© 1979 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref] [PubMed]
  2. J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Sys. Tech. J. 48, 2133–2160 (1969).
    [Crossref]
  3. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Sys. Tech. J. 48, 2071–2102 (1969).
    [Crossref]
  4. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  5. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).
  6. L. Eyges, “Solution of Schrodinger and related equations for irregular and composite regions,” Ann. Phys. 81, 567–590 (1973).
    [Crossref]
  7. J. D. Love and A. W. Snyder, “Ray analysis of multimode optical fibres,” Ann. Telecommun. 32, 109–114 (1977);P. di Vita, “Theory of propagation in optical fibres: Ray Approach,” Ann. Telecommun. 32, 115–134 (1977).
  8. L. Eyges and P. D. Gianino, “Polarizabilities of rectangular dielectric cylinders and of a cube,” IEEE Trans. Ant. & Prop., July (1979).
    [Crossref]
  9. A. E. Nelson and L. Eyges, “Electromagnetic scattering from dielectric rods of arbitrary cross section,” J. Opt. Soc. Am. 66, 254–259 (1976).
    [Crossref]
  10. P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969)and, “Symmetry unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [Crossref]
  11. L. Eyges, “Fiber optic guides of noncircular cross section,” Appl. Opt. 17, 1673–1674 (1978).
    [Crossref] [PubMed]
  12. J. Allard, “Notes on squares and cubes,” Math. Mag. 37, 210–214 (1964).
    [Crossref]
  13. M. Gardner, “Mathematical games,” Sci. Am. 213, No. 3, 222–232 (1965).
    [Crossref]
  14. F. B. Hildebrand, Introduction to Numerical Analysis, (McGraw Hill, New York, 1956)p. 330.
  15. R. W. Hornbeck, Numerical Methods, (Quantum, New York, 1975)p. 65.
  16. C. Yeh, “Modes in weakly guiding elliptical optical fibres,” Opt. and Quant. Elect. 8, 43–47 (1976).
    [Crossref]

1979 (1)

L. Eyges and P. D. Gianino, “Polarizabilities of rectangular dielectric cylinders and of a cube,” IEEE Trans. Ant. & Prop., July (1979).
[Crossref]

1978 (1)

1977 (1)

J. D. Love and A. W. Snyder, “Ray analysis of multimode optical fibres,” Ann. Telecommun. 32, 109–114 (1977);P. di Vita, “Theory of propagation in optical fibres: Ray Approach,” Ann. Telecommun. 32, 115–134 (1977).

1976 (2)

1973 (1)

L. Eyges, “Solution of Schrodinger and related equations for irregular and composite regions,” Ann. Phys. 81, 567–590 (1973).
[Crossref]

1971 (1)

1969 (3)

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Sys. Tech. J. 48, 2133–2160 (1969).
[Crossref]

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Sys. Tech. J. 48, 2071–2102 (1969).
[Crossref]

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969)and, “Symmetry unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[Crossref]

1965 (1)

M. Gardner, “Mathematical games,” Sci. Am. 213, No. 3, 222–232 (1965).
[Crossref]

1964 (1)

J. Allard, “Notes on squares and cubes,” Math. Mag. 37, 210–214 (1964).
[Crossref]

Allard, J.

J. Allard, “Notes on squares and cubes,” Math. Mag. 37, 210–214 (1964).
[Crossref]

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Eyges, L.

L. Eyges and P. D. Gianino, “Polarizabilities of rectangular dielectric cylinders and of a cube,” IEEE Trans. Ant. & Prop., July (1979).
[Crossref]

L. Eyges, “Fiber optic guides of noncircular cross section,” Appl. Opt. 17, 1673–1674 (1978).
[Crossref] [PubMed]

A. E. Nelson and L. Eyges, “Electromagnetic scattering from dielectric rods of arbitrary cross section,” J. Opt. Soc. Am. 66, 254–259 (1976).
[Crossref]

L. Eyges, “Solution of Schrodinger and related equations for irregular and composite regions,” Ann. Phys. 81, 567–590 (1973).
[Crossref]

Gardner, M.

M. Gardner, “Mathematical games,” Sci. Am. 213, No. 3, 222–232 (1965).
[Crossref]

Gianino, P. D.

L. Eyges and P. D. Gianino, “Polarizabilities of rectangular dielectric cylinders and of a cube,” IEEE Trans. Ant. & Prop., July (1979).
[Crossref]

Gloge, D.

Goell, J. E.

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Sys. Tech. J. 48, 2133–2160 (1969).
[Crossref]

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis, (McGraw Hill, New York, 1956)p. 330.

Hornbeck, R. W.

R. W. Hornbeck, Numerical Methods, (Quantum, New York, 1975)p. 65.

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Love, J. D.

J. D. Love and A. W. Snyder, “Ray analysis of multimode optical fibres,” Ann. Telecommun. 32, 109–114 (1977);P. di Vita, “Theory of propagation in optical fibres: Ray Approach,” Ann. Telecommun. 32, 115–134 (1977).

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Sys. Tech. J. 48, 2071–2102 (1969).
[Crossref]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Nelson, A. E.

Snyder, A. W.

J. D. Love and A. W. Snyder, “Ray analysis of multimode optical fibres,” Ann. Telecommun. 32, 109–114 (1977);P. di Vita, “Theory of propagation in optical fibres: Ray Approach,” Ann. Telecommun. 32, 115–134 (1977).

Waterman, P. C.

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969)and, “Symmetry unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[Crossref]

Yeh, C.

C. Yeh, “Modes in weakly guiding elliptical optical fibres,” Opt. and Quant. Elect. 8, 43–47 (1976).
[Crossref]

Ann. Phys. (1)

L. Eyges, “Solution of Schrodinger and related equations for irregular and composite regions,” Ann. Phys. 81, 567–590 (1973).
[Crossref]

Ann. Telecommun. (1)

J. D. Love and A. W. Snyder, “Ray analysis of multimode optical fibres,” Ann. Telecommun. 32, 109–114 (1977);P. di Vita, “Theory of propagation in optical fibres: Ray Approach,” Ann. Telecommun. 32, 115–134 (1977).

Appl. Opt. (2)

Bell Sys. Tech. J. (2)

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Sys. Tech. J. 48, 2133–2160 (1969).
[Crossref]

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Sys. Tech. J. 48, 2071–2102 (1969).
[Crossref]

IEEE Trans. Ant. & Prop. (1)

L. Eyges and P. D. Gianino, “Polarizabilities of rectangular dielectric cylinders and of a cube,” IEEE Trans. Ant. & Prop., July (1979).
[Crossref]

J. Acoust. Soc. Am. (1)

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969)and, “Symmetry unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[Crossref]

J. Opt. Soc. Am. (1)

Math. Mag. (1)

J. Allard, “Notes on squares and cubes,” Math. Mag. 37, 210–214 (1964).
[Crossref]

Opt. and Quant. Elect. (1)

C. Yeh, “Modes in weakly guiding elliptical optical fibres,” Opt. and Quant. Elect. 8, 43–47 (1976).
[Crossref]

Sci. Am. (1)

M. Gardner, “Mathematical games,” Sci. Am. 213, No. 3, 222–232 (1965).
[Crossref]

Other (4)

F. B. Hildebrand, Introduction to Numerical Analysis, (McGraw Hill, New York, 1956)p. 330.

R. W. Hornbeck, Numerical Methods, (Quantum, New York, 1975)p. 65.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

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 (6)

FIG 1
FIG 1

Cross section of a cylindrical dielectric guide parallel to the z axis. The cross-sectional area is denoted by A, its bounding curve by L, the outward normal to the cylinder is n ̂ and the unit vector tangent to L is t ̂. The positive z axis is out of the plane of the paper.

FIG. 2
FIG. 2

Plot of the function ρ(ϕ) given in Eq. (31) for 0 ≤ ϕπ/2 only, R = 1, and various values of N.

FIG. 3
FIG. 3

Modes of the weakly guiding rectangular guide with R = 2. Square points were taken from Goell’s Fig. 17.2 X’s represent results of a perturbation calculation for R 1 I the mode.

FIG. 4
FIG. 4

R 1 II mode of the weakly guiding rectangular guide as R changes from 1 to 10.

FIG. 5
FIG. 5

Modes of the weakly guiding elliptical guide with R = 2.

FIG. 6
FIG. 6

Modes of the cusp-shaped guide with N = 0.3 and R = 1.

Tables (3)

Tables Icon

TABLE I Dominant (noncutoff) R 1 I mode of the weakly guiding 2:1 rectangular guide. Listed values of P2 were obtained after a one-, two-, three-, or four-term truncation of the field expansion. The field intensity at two points on the perimeter is also given. The field intensity at ρ = 0 is 1

Tables Icon

TABLE II Results for the first 13 modes of the 2:1 rectangular guide, ordered according to the value of β at cutoff. Values of P2 were obtained for β = 2, from the roots of determinants 1 × 1 to 7 × 7 in dimension. Our designation and Goell’s are listed. For certain modes two-term truncation failed to locate the roots at this β value

Tables Icon

TABLE III Values of β at cutoff for elliptical guides with different aspect ratios and for a rectangular guide with R = 2. Asterisks indicate where the designations E 2 and E 3 should be Interchanged

Equations (113)

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

E ω ( r ) B ω ( r ) } = E ( ρ ) B ( ρ ) } e i k g z ,
( 2 + γ 1 2 ) ( E ( ρ ) B ( ρ ) ) = 0 inside the guide
( 2 γ 2 2 ) ( E ( ρ ) B ( ρ ) ) = 0 outside the guide
γ 1 2 = k 1 2 k g 2 , γ 2 2 = k g 2 k 2 2 .
Φ ( ρ ) = ( γ 1 2 + γ 2 2 ) A Φ ( ρ ) g γ 2 ( ρ , ρ ) d A ,
( 2 γ 2 2 ) g γ 2 ( ρ , ρ ) = δ ( ρ ρ ) .
Φ ( ρ ) = A [ δ ( ρ ρ ) 2 g γ 2 ] Φ d A + A g γ 2 2 Φ d A
A ( g γ 2 2 Φ Φ 2 g γ 2 ) d A = 0 , ( ρ inside A ) .
L ( g γ 2 Φ n Φ g γ 2 n ) d L = 0 , ( ρ inside A ) .
Φ ( ρ ) = L ( Φ + g γ 2 n Φ n | + g γ 2 ) d L , ρ outside .
0 = L ( Φ + g γ 2 n Φ n | + g γ 2 ) d L , ρ inside .
Φ ( ρ ) = s = 0 J s ( γ 1 ρ ) [ C s cos ( s ϕ ) + D s sin ( s ϕ ) ] .
R I : Φ ( ρ ) = s = 0 , 2 , 4 C s J s ( γ 1 ρ ) cos ( s ϕ ) ,
R II : Φ ( ρ ) = s = 1 , 3 , 5 C s J s ( γ 1 ρ ) cos ( s ϕ ) ,
R III : Φ ( ρ ) = s = 2 , 4 , 6 D s J s ( γ 1 ρ ) sin ( s ϕ ) ,
R IV : Φ ( ρ ) = s = 1 , 3 , 5 , D s J s ( γ 1 ρ ) sin ( s ϕ ) ,
Φ ( ρ ) = s = 0 , 4 , 8 C s J s ( γ 1 ρ ) cos ( s ϕ ) ,
Φ ( ρ ) = s = 2 , 6 , 10 C s J s ( γ 1 ρ ) cos ( s ϕ ) ,
Φ ( ρ ) = s = 2 , 6 , 10 D s J s ( γ 1 ρ ) sin ( s ϕ ) ,
Φ ( ρ ) = s = 4 , 8 , 12 D s J s ( γ 1 ρ ) sin ( s ϕ ) ,
Φ ( ρ ) = s = 1 , 3 , 5 C s J s ( γ 1 ρ ) cos ( s ϕ ) ,
Φ ( ρ ) = s = 1 , 3 , 5 D s J s ( γ 1 ρ ) sin ( s ϕ ) .
g γ 2 ( ρ , ρ ) = i 4 l = 0 l H l ( i γ 2 ρ ) J l ( γ 2 ρ ) × ( cos l ϕ cos l ϕ + sin l ϕ sin l ϕ ) , ρ < ρ
s C s G l s = 0 ,
s D s T l s = 0 ,
G l s = 0 2 π d ϕ [ F l s ( ρ ) cos l ϕ cos s ϕ + L l s ( ρ , ϕ ) ( s cos l ϕ sin s ϕ l sin l ϕ cos s ϕ ) ] ,
T l s = 0 2 π d ϕ [ F l s ( ρ ) sin l ϕ sin s ϕ L l s ( ρ , ϕ ) ( s sin l ϕ cos s ϕ l cos l ϕ cos s ϕ ) ] ,
F l s ( ρ ) = l [ γ 1 ρ H l ( i γ 2 ρ ) J s ( γ 1 ρ ) γ 2 ρ H l ( i γ 2 ρ ) J s ( γ 1 ρ ) ] ,
L l s ( ρ , ϕ ) = l H l ( i γ 2 ρ ) J s ( γ 1 ρ ) ρ ϕ / ρ .
det ( G l s ) = 0 ,
det ( T l s ) = 0 .
F l l ( γ 1 a ) H l ( i γ 2 a ) J l ( γ 1 a ) ( i γ 2 a ) H l ( i γ 2 a ) J l ( γ 1 a ) = 0
C l ( x ) = C l 1 ( x ) ( l / x ) C l ( x ) .
H l ( i x ) = ( 2 / π ) i ( l + 1 ) K l ( x ) .
P 2 = k g 2 k 2 2 k 1 2 k 2 2 = γ 2 2 γ 1 2 + γ 2 2 ,
β = b π ( k 1 2 k 2 2 ) 1 / 2 = b π ( γ 1 2 + γ 2 2 ) 1 / 2 = b k 0 π ( n 1 2 n 2 2 ) 1 / 2 ,
γ 1 ρ = π β ( 1 P 2 ) 1 / 2 ( ρ / b ) , γ 2 ρ = π P β ρ / b .
F l s ( ρ ) = 2 l β ρ b i ( l + 1 ) [ K l ( π P β ρ b ) J s 1 ( π β ρ b ( 1 P 2 ) 1 / 2 ) × ( 1 P 2 ) 1 / 2 + P J s ( π β ρ b ( 1 P 2 ) 1 / 2 ) K l 1 ( π β P ρ b ) ] + l ( l s ) 2 π i ( l + 1 ) K l ( π β P ρ b ) × J s ( π β ρ b ( 1 P 2 ) 1 / 2 ) ,
L l s ( ρ , ϕ ) = 2 π l i ( l + 1 ) K l ( π P β ρ b ) × J s ( π β ρ b ( 1 P 2 ) 1 / 2 ) ρ ϕ / ρ .
ρ ( ϕ ) = b [ ( cos 2 ϕ / R 2 ) N + ( sin 2 ϕ ) N ] 1 / 2 N
R 1 I
E 11 x , y
R 1 II
R 21 x , y
R 1 IV
E 12 x , y
R 2 I
E 31 x , y
R 1 III
E 22 x , y
R 2 II
E 41 x , y
R 2 IV
R 3 I
R 4 I
R 2 III
R 3 II
R 3 IV
R 4 II
R = 1
R = 1.2
R = 1.5
R = 2.0
R = 2.0
E 1 I
R 1 I
E 1 II
R 1 II
E 1 IV
R 1 IV
E 1 III
R 1 III
E 2 I
R 2 I
E 3 I
R 3 I
E 2 IV
R 2 IV
E 2 II
R 3 II
E 3 II
R 2 II
E 3 IV
E 4 I
R 4 I
E 2 III
R 2 III
E = i 1 k 0 2 k g 2 [ k 0 ( × B z ) + k g ( E z ) ] ,
E t = i γ 1 2 ( k g E z t k 0 B z n ) ,
B t = i γ 1 2 ( k g B z t + 1 k 0 E z n ) .
E z ( ρ ) = ( γ 1 2 + γ 2 2 ) E z ( ρ ) g γ 2 ( ρ , ρ ) d A .
1 γ 1 2 ( k g E z t k 0 B z n ) = 1 γ 2 2 ( k g E z t + k 0 B z n ) + .
B z n | + B z n | = ( γ 2 2 γ 1 2 + 1 ) B z n | + k g k 0 ( γ 2 2 γ 1 2 + 1 ) E z t .
E z n | + E z n | = ( 1 γ 2 2 2 γ 1 2 + 1 ) E z n | k g 2 k 0 ( γ 2 2 γ 1 2 + 1 ) B z t .
ξ ( ρ ) = L g γ 2 ( ρ , ρ ) σ ( ρ ) d L ,
ξ n | + ξ n | = σ ( ρ ) ,
E z ( ρ ) = ( γ 1 2 + γ 2 2 ) A E z ( ρ ) g γ 2 ( ρ , ρ ) d A 1 2 γ 1 2 L [ ( 1 γ 2 2 + 2 γ 1 2 ) E z n | ] + k g k 0 ( γ 1 2 + γ 2 2 ) B z t ] g γ 2 ( ρ , ρ ) d L .
B z ( ρ ) = ( γ 1 2 + γ 2 2 ) A B z ( ρ ) g γ 2 ( ρ , ρ ) d A γ 1 2 + γ 2 2 γ 1 2 L ( B z n | k g k 0 E z t ) g γ 2 ( ρ , ρ ) d L
0 = L E z g γ 2 n d L 1 γ 2 2 2 γ 1 2 L g γ 2 E z n | d L 1 2 γ 1 2 k g k 0 ( γ 1 2 + γ 2 2 ) L B z t g γ 2 d L ,
0 = L B z g γ 2 n d L γ 2 2 γ 1 2 L B z n | g γ 2 d L + k g k o ( γ 1 2 + γ 2 2 γ 1 2 ) L E z t g γ 2 d L .
E z ( ρ ) = s = A s J s ( γ 1 ρ ) exp ( i s ϕ ) ,
B z ( ρ ) = s = B s J s ( γ 1 ρ ) exp ( i s ϕ ) .
g γ 2 ( ρ , ρ ) = i 4 l = J l ( γ 2 ρ ) H l ( i γ 2 ρ ) e i l ( ϕ ϕ ) .
n ̂ = ( ρ ̂ ρ ϕ ̂ d ρ / d ϕ ) / ( d L / d ϕ ) ,
s = [ A s ( M l s + 1 γ 2 2 2 γ 1 2 N l s ) + B s 2 Q l s ] = 0 ,
s = [ A s Q l s + B s ( M l s + γ 2 2 γ 1 2 N l s ) ] = 0 , l = 0 , ± 1 , 2 , ,
M l s = 0 2 π e i ( s l ) ϕ J s ( γ 1 ρ ) × [ i γ 2 ρ H l ( i γ 2 ρ ) i l ρ ( d ρ d ϕ ) H l ( i γ 2 ρ ) ] d ϕ ,
N l s = 0 2 π e i ( s l ) ϕ H l ( i γ 2 ρ ) × ( γ 1 ρ J s ( γ 1 ρ ) i s ρ d ρ d ϕ J s ( γ 1 ρ ) ) d ϕ ,
Q l s = k g k 0 ( 1 + γ 2 2 γ 1 2 ) 0 2 π e i ( s l ) ϕ H l ( i γ 2 ρ ) × ( γ 1 J s ( γ 1 ρ ) d ρ d ϕ + i s J s ( γ 1 ρ ) ) d ϕ .
M l s = 2 π i γ 2 a J l ( γ 1 a ) H l ( i γ 2 a ) δ l s ,
N l s = 2 π ( γ 2 2 a / γ 1 ) J l ( γ 1 a ) H l ( i γ 2 a ) δ l s ,
Q l s = 2 π ( k g / k o ) ( 1 + γ 2 2 / γ 1 2 ) i l J l ( γ 1 a ) H l ( i γ 2 a ) δ l s .
( 1 k 0 2 / y ) ( J l ( y ) / J l ( y ) ) ( 2 k 0 2 / x ) ( H l ( i x ) / H l ( i x ) ) ] × [ J l ( y ) / y J l ( y ) H l ( i x ) / x H l ( i x ) ] = l 2 k g 2 ( y 2 x 2 ) 2 / x 4 y 4 .