Abstract

An integral-equation approach is developed to compute the propagation coefficients and field distributions of the modes of an open cylindrical waveguide surrounded by a vacuum. The general theory applies to arbitrary graded-index profiles and arbitrarily shaped cross sections. Lossy and uniaxially anisotropic media, with longitudinal extraordinary axis, are included. To investigate the accuracy of the method, we have implemented the integral-equation method for a lossless symmetrical slab waveguide with step- and graded-index profile. Results are given for isotropic and anisotropic waveguides of the indicated kind. The permittivity profiles considered are step-index and two different quadratic profiles.

© 1980 Optical Society of America

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References

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  1. A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
    [Crossref]
  2. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref] [PubMed]
  3. H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), Chap 5.4, pp. 487–500.
  4. A. G. Gronthoud and H. Blok, “The influence of bulk losses and bulk dispersion on the propagation properties of surface waves in a radially inhomogeneous optical waveguide,” Opt. Quantum Electron. 10, 95–106 (1978).
    [Crossref]
  5. C. Yeh, K. Ha, S. B. Dong, and W. P. Brown, “Single mode optical waveguides,” Appl. Opt. 18, 1490–1504 (1979).
    [Crossref] [PubMed]
  6. L. W. Kantorowitsch and W. I. Krylow, Näherungsmethoden der hoheren Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, 1956), Chap. II 4.5, pp. 144–148.
  7. S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Analysis and fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Proceedings of the Optical Communication Conference, Amsterdam, 1979, Chaps. 10.1-1–10.1-4 (unpublished).
  8. H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), Chap. 2.3, pp. 93–105.
  9. A. T. de Hoop, Modern Topics in Electromagnetics and Antennas, Stevenage, Peter Peregrinus, 1977, PPL Conference Publications13, Chap. 6, pp. 6.1–6.59.
  10. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New York, 1973), Chap. 5.3a,b, pp. 455–462 and Chap. 5.6, p. 549.

1979 (1)

1978 (1)

A. G. Gronthoud and H. Blok, “The influence of bulk losses and bulk dispersion on the propagation properties of surface waves in a radially inhomogeneous optical waveguide,” Opt. Quantum Electron. 10, 95–106 (1978).
[Crossref]

1971 (1)

1969 (1)

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[Crossref]

Adams, M. J.

S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Analysis and fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Proceedings of the Optical Communication Conference, Amsterdam, 1979, Chaps. 10.1-1–10.1-4 (unpublished).

Blok, H.

A. G. Gronthoud and H. Blok, “The influence of bulk losses and bulk dispersion on the propagation properties of surface waves in a radially inhomogeneous optical waveguide,” Opt. Quantum Electron. 10, 95–106 (1978).
[Crossref]

Brown, W. P.

de Hoop, A. T.

A. T. de Hoop, Modern Topics in Electromagnetics and Antennas, Stevenage, Peter Peregrinus, 1977, PPL Conference Publications13, Chap. 6, pp. 6.1–6.59.

Dong, S. B.

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New York, 1973), Chap. 5.3a,b, pp. 455–462 and Chap. 5.6, p. 549.

Gloge, D.

Gronthoud, A. G.

A. G. Gronthoud and H. Blok, “The influence of bulk losses and bulk dispersion on the propagation properties of surface waves in a radially inhomogeneous optical waveguide,” Opt. Quantum Electron. 10, 95–106 (1978).
[Crossref]

Ha, K.

Kantorowitsch, L. W.

L. W. Kantorowitsch and W. I. Krylow, Näherungsmethoden der hoheren Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, 1956), Chap. II 4.5, pp. 144–148.

Krylow, W. I.

L. W. Kantorowitsch and W. I. Krylow, Näherungsmethoden der hoheren Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, 1956), Chap. II 4.5, pp. 144–148.

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New York, 1973), Chap. 5.3a,b, pp. 455–462 and Chap. 5.6, p. 549.

Norman, S. R.

S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Analysis and fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Proceedings of the Optical Communication Conference, Amsterdam, 1979, Chaps. 10.1-1–10.1-4 (unpublished).

Payne, D. N.

S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Analysis and fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Proceedings of the Optical Communication Conference, Amsterdam, 1979, Chaps. 10.1-1–10.1-4 (unpublished).

Smith, A. M.

S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Analysis and fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Proceedings of the Optical Communication Conference, Amsterdam, 1979, Chaps. 10.1-1–10.1-4 (unpublished).

Snyder, A. W.

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[Crossref]

Unger, H.-G.

H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), Chap. 2.3, pp. 93–105.

H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), Chap 5.4, pp. 487–500.

Yeh, C.

Appl. Opt. (2)

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[Crossref]

Opt. Quantum Electron. (1)

A. G. Gronthoud and H. Blok, “The influence of bulk losses and bulk dispersion on the propagation properties of surface waves in a radially inhomogeneous optical waveguide,” Opt. Quantum Electron. 10, 95–106 (1978).
[Crossref]

Other (6)

H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), Chap 5.4, pp. 487–500.

L. W. Kantorowitsch and W. I. Krylow, Näherungsmethoden der hoheren Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, 1956), Chap. II 4.5, pp. 144–148.

S. R. Norman, D. N. Payne, M. J. Adams, and A. M. Smith, “Analysis and fabrication of single-mode fibers exhibiting extremely low polarization birefringence,” Proceedings of the Optical Communication Conference, Amsterdam, 1979, Chaps. 10.1-1–10.1-4 (unpublished).

H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), Chap. 2.3, pp. 93–105.

A. T. de Hoop, Modern Topics in Electromagnetics and Antennas, Stevenage, Peter Peregrinus, 1977, PPL Conference Publications13, Chap. 6, pp. 6.1–6.59.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New York, 1973), Chap. 5.3a,b, pp. 455–462 and Chap. 5.6, p. 549.

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Figures (7)

FIG. 1
FIG. 1

(a) Waveguiding structure and coordinate system employed. (b) Cross section of the waveguide. (c) Permittivity/permeability profile (longitudinal section).

FIG. 2
FIG. 2

Waveguiding structure having a plane of symmetry x = 0. The generating subdomain D e of D is shaded.

FIG. 3
FIG. 3

Symmetrical slab waveguide of thickness 2a.

FIG. 4
FIG. 4

Functions kz/k0 (phase-delay parameter), ∂kz/∂k0 (group-delay parameter) and ½ 2 k z / k 0 2 (pulse-broadening parameter) of the TM0 mode of an anisotropic step-index slab waveguide with r ( o ) = 2.25 and r ( e ) = 2.25 (full), r ( e ) = 1.25 (dashed), r ( e ) = 3.25 (dash-dot).

FIG. 5
FIG. 5

Permittivity profiles of the symmetrical slab waveguide: step-index profile, r = 1.01 (full); quadratic permittivity profile, r,max = 1.01, Δ = 2.475 × 10−3 (dashed); quadratic permittivity profile, r,max = 1.01, Δ = 4.950 × 10−3 (dash-dot).

FIG. 6
FIG. 6

Functions kz/k0 (phase-delay parameter), ∂kz/∂k0 (group-delay parameter) and ½ 2 k z / k 0 2 (pulse-broadening parameter) of the TE0 and TE1 modes of a slab waveguide with step-index profile (full); quadratic permittivity profile, r,max = 1.01, Δ = 2.475 × 10−3 (dashed); quadratic permittivity profile, r,max = 1.01, Δ = 4.950 × 10−3 (dash-dot).

FIG. 7
FIG. 7

Field component ey (x) of the (odd) TE1 mode in the slab waveguide (a) at k0a = 2.602 × 101 (near cut-off) and (b) at k0a = 1.101 × 102 (far from cut-off) for the step-index r = 1.01 profile (full); for quadratic permittivity profile, r,max = 1.01, Δ = 2.475 × 10−3 (dashed); for quadratic permittivity profile, r,max = 1.01, Δ = 4.950 × 10−3 (dash-dot).

Tables (8)

Tables Icon

TABLE I Electromagnetic properties of the different domains in a cross sectional plane.

Tables Icon

TABLE II Even and odd field distributions in a configuration with x = 0 as a plane of symmetry (dependences on irrelevant parameters are omitted).

Tables Icon

TABLE III Definitions of the components πE and πM for odd and even modes to be used in Eqs. (11) and (12); g(x,x′,kz) = g(x,x′,kz) −g(−x,x′,kz) and g+ (x,x′,kz) = g(x,x′,kz) + g(−x,x′,kz) (dependences on irrelevant parameters are omitted).

Tables Icon

TABLE IV Some values of kz/k0 for the step-index, r = 1.01 symmetrical slab waveguide, obtained by the integral-equation method with 8 and 16 matching points; values from the analytical expression for comparison. Average C.P.U. time was 10 s for N = 8 matching points, and 35 s for n = 16 matching points.

Tables Icon

TABLE V Some values of kz/k0 for the step-index r = 2.25 symmetrical slab waveguide, obtained by the integral-equation method with 8 and 16 matching points; values from the analytical expression for comparison. Average C.P.U. times as in Table IV.

Tables Icon

TABLE VI Some values of kz/k0 of the TM0 mode for the negatively and positively uniaxial symmetrical step-index slab waveguide with r ( o ) = 2.25. The extraordinary direction is along the z axis. Values are obtained by the integral-equation method with eight matching points, and from the analytical expression. Average C.P.U. time as in Table IV.

Tables Icon

TABLE VII Some values of kz/k0 for a graded-index symmetrical slab waveguide having a quadratic permittivity profile with r,max = 1.01, A = 2.475 × 10−3, obtained by the integral-equation method with N = 8 and N = 16 matching points. Average C.P.U. times as in Table IV.

Tables Icon

TABLE VIII Some values of kz/k0 for a graded-index symmetrical slab waveguide having a quadratic permittivity profile with r,max = 1.01, Δ = 4.950 × 10−3, obtained by the integral-equation method with N = 8 and N = 16 matching points. Average C.P.U. times as in Table IV.

Equations (46)

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electric field constituent = e ( x , y , k z , ω ) exp ( j k z z + j ω t ) .
V 2 = a 2 ( ω 2 max μ max ω 2 0 μ 0 ) ,
× h = j ω d ,
× e = j ω b ,
d = ( o ) e T + ( e ) e z î z = 0 e + χ E ( o ) e T + χ E ( e ) e z î z ,
b = μ ( o ) h T + μ ( e ) h z î z = μ 0 h + χ M ( o ) h T + χ ( e ) h z î z .
× h j ω 0 e = j E ,
× e + j ω μ 0 h = j M ,
j E = j ω 0 ( χ E ( o ) e T + χ E ( e ) e z î z ) ,
j M = j ω μ 0 ( χ M ( o ) h T + χ M ( e ) h z î z ) .
e = ( j ω 0 ) 1 ( · φ E ) j ω μ 0 π E × π M ,
h = ( j ω μ 0 ) 1 ( · π M ) j ω 0 π M + × π E ,
π E = D g ( r T , r T , k z ) j E ( r T ) d A ( r T ) ,
π M = D g ( r T , r T , k z ) j M ( r T ) d A ( r T ) ,
g ( r T , r T , k z ) = ( j / 4 ) H 0 ( 2 ) ( j k T | r T r T | ) ,
rect m ( r T ) = { 1 when r T D m 0 when r T D m .
[ A ] [ f ] = [ f ] ,
D D [ e ( r T ) × h ( r T ) ] · î z d A ( r T ) = ½ .
π E = D | y = 0 | j E ( x ) g 1 ( x , x , k z ) d x ,
π M = D | y = 0 | j M ( x ) g 1 ( x , x , k z ) d x ,
g 1 ( x , x , k z ) = ( 2 k T ) 1 exp ( k T | x x | ) .
e y = j ω μ 0 π E , y + j k z π M , x + x π M , z ,
h x = ( j ω μ 0 ) 1 ( x 2 π M , x j k z x π M , z ) j ω 0 π M , x + j k z π E , y ,
h z = ( j ω μ 0 ) 1 ( j k z x π M , x k z 2 π M , z ) j ω 0 π M , z + x π E , y ,
h y = j ω 0 π M , y j k z π E , x x π E , z ,
e x = ( j ω 0 ) 1 ( x 2 π E , x j k z x π E , z ) j ω μ 0 π E , x j k z π M , y ,
e z = ( j ω 0 ) 1 ( j k z x π E , x k z 2 π E , z ) j ω μ 0 π E , z x π M , y ,
r ( x ) = r , max ( 1 2 Δ | x / a | 2 ) when a x a r ( x ) = 1 when < x a or a > x > .
D
D
e x even ( x ) = [ e x ( x ) e x ( x ) ] / 2
e x odd ( x ) = [ e x ( x ) + e x ( x ) ] / 2
e y , z even ( x ) = [ e y , z ( x ) + e y , z ( x ) ] / 2
e y , z odd ( x ) = [ e y , z ( x ) e y , z ( x ) ] / 2
h x even ( x ) = [ h x ( x ) + h x ( x ) ] / 2
h x odd ( x ) = [ h x ( x ) h x ( x ) ] / 2
h y , z even ( x ) = [ h y , z ( x ) h y , z ( x ) ] / 2
h y , z odd ( x ) = [ h y , z ( x ) + h y , z ( x ) ] / 2
π E , x even = D e g ( x , x , k z ) j E , x even ( x ) d A ( x )
π E , x odd = D e g + ( x , x , k z ) j E , x odd ( x ) d A ( x )
π E , y , z even = D e g + ( x , x , k z ) j E , y , z even ( x ) d A ( x )
π E , y , z odd = D e g ( x , x , k z ) j E , y , z odd ( x ) d A ( x )
π M , x even = D e g + ( x , x , k z ) j M , x even ( x ) d A ( x )
π M , x odd = D e g ( x , x , k z ) j M , x odd ( x ) d A ( x )
π M , y , z even = D e g ( x , x , k z ) j M , y , z even ( x ) d A ( x )
π M , y , z odd = D e g + ( x , x , k z ) j M , y , z odd ( x ) d A ( x )