Abstract

The propagation of cylindrical dielectric waveguide modes near cutoff and far from cutoff are considered. The relative amounts of Ez and Hz, and the transverse components of the field are determined for both sets of hybrid modes. With the radial dependence of the z components of the field in the central dielectric given by Jn(ur/a), the transverse components far from cutoff are given by Jn±1(ur/a), where u is a parameter found from the boundary conditions and which fixes the scale of the Bessel function relative to the boundary r = a. The two values n + 1 and n − 1 correspond to the two sets of modes. The designations of the hybrid modes are discussed. Field plots for the lower order modes are given.

© 1961 Optical Society of America

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References

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  1. Rayleigh, Phil. Mag. 43, 125 (1897).
    [CrossRef]
  2. D. Hondros and P. Debye, Ann. Physik 32, 465 (1910).
    [CrossRef]
  3. O. Schriever, Ann. Physik 63, 645 (1920).
    [CrossRef]
  4. J. R. Corson, S. P. Mead, and S. A. Schelkunoff, Bell System Tech. J. 15, 310 (1936).
    [CrossRef]
  5. S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1943), p. 425.
  6. R. E. Beam, M. M. Astrahan, W. C. Jakes, H. M. Wachowski, and W. L. Firestone, Northwestern University Report ATI 94929, Chap. V (1949).
  7. M. Abele, Nuovo cimento 5, 274 (1948).
    [CrossRef]
  8. E. Snitzer and J. W. Hicks, J. Opt. Soc. Am. 49, 1128 (1959), Abstract TB36; H. Osterberg, E. Snitzer, M. Polanyi, R. Hilberg, and J. W. Hicks, ibid. 49, 1128 (1959), Abstract TB37.
  9. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), Chap. V.
  10. H. Wegner, Air Material Command Microfilm ZWV/FB/RE/2018, R8117F831.
  11. S. P. Schlesinger and D. D. King, I.R.E. Trans. M.T.T.-6, 291 (1958).
  12. J. F. Reintzes and G. T. Coate, Principles of Radar (McGraw-Hill Book Company, Inc., New York, 1952), 3rd ed., Chap. 8, p. 609.

1959 (1)

E. Snitzer and J. W. Hicks, J. Opt. Soc. Am. 49, 1128 (1959), Abstract TB36; H. Osterberg, E. Snitzer, M. Polanyi, R. Hilberg, and J. W. Hicks, ibid. 49, 1128 (1959), Abstract TB37.

1958 (1)

S. P. Schlesinger and D. D. King, I.R.E. Trans. M.T.T.-6, 291 (1958).

1948 (1)

M. Abele, Nuovo cimento 5, 274 (1948).
[CrossRef]

1936 (1)

J. R. Corson, S. P. Mead, and S. A. Schelkunoff, Bell System Tech. J. 15, 310 (1936).
[CrossRef]

1920 (1)

O. Schriever, Ann. Physik 63, 645 (1920).
[CrossRef]

1910 (1)

D. Hondros and P. Debye, Ann. Physik 32, 465 (1910).
[CrossRef]

1897 (1)

Rayleigh, Phil. Mag. 43, 125 (1897).
[CrossRef]

Abele, M.

M. Abele, Nuovo cimento 5, 274 (1948).
[CrossRef]

Astrahan, M. M.

R. E. Beam, M. M. Astrahan, W. C. Jakes, H. M. Wachowski, and W. L. Firestone, Northwestern University Report ATI 94929, Chap. V (1949).

Beam, R. E.

R. E. Beam, M. M. Astrahan, W. C. Jakes, H. M. Wachowski, and W. L. Firestone, Northwestern University Report ATI 94929, Chap. V (1949).

Coate, G. T.

J. F. Reintzes and G. T. Coate, Principles of Radar (McGraw-Hill Book Company, Inc., New York, 1952), 3rd ed., Chap. 8, p. 609.

Corson, J. R.

J. R. Corson, S. P. Mead, and S. A. Schelkunoff, Bell System Tech. J. 15, 310 (1936).
[CrossRef]

Debye, P.

D. Hondros and P. Debye, Ann. Physik 32, 465 (1910).
[CrossRef]

Firestone, W. L.

R. E. Beam, M. M. Astrahan, W. C. Jakes, H. M. Wachowski, and W. L. Firestone, Northwestern University Report ATI 94929, Chap. V (1949).

Hicks, J. W.

E. Snitzer and J. W. Hicks, J. Opt. Soc. Am. 49, 1128 (1959), Abstract TB36; H. Osterberg, E. Snitzer, M. Polanyi, R. Hilberg, and J. W. Hicks, ibid. 49, 1128 (1959), Abstract TB37.

Hondros, D.

D. Hondros and P. Debye, Ann. Physik 32, 465 (1910).
[CrossRef]

Jakes, W. C.

R. E. Beam, M. M. Astrahan, W. C. Jakes, H. M. Wachowski, and W. L. Firestone, Northwestern University Report ATI 94929, Chap. V (1949).

King, D. D.

S. P. Schlesinger and D. D. King, I.R.E. Trans. M.T.T.-6, 291 (1958).

Mead, S. P.

J. R. Corson, S. P. Mead, and S. A. Schelkunoff, Bell System Tech. J. 15, 310 (1936).
[CrossRef]

Rayleigh,

Rayleigh, Phil. Mag. 43, 125 (1897).
[CrossRef]

Reintzes, J. F.

J. F. Reintzes and G. T. Coate, Principles of Radar (McGraw-Hill Book Company, Inc., New York, 1952), 3rd ed., Chap. 8, p. 609.

Schelkunoff, S. A.

J. R. Corson, S. P. Mead, and S. A. Schelkunoff, Bell System Tech. J. 15, 310 (1936).
[CrossRef]

S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1943), p. 425.

Schlesinger, S. P.

S. P. Schlesinger and D. D. King, I.R.E. Trans. M.T.T.-6, 291 (1958).

Schriever, O.

O. Schriever, Ann. Physik 63, 645 (1920).
[CrossRef]

Snitzer, E.

E. Snitzer and J. W. Hicks, J. Opt. Soc. Am. 49, 1128 (1959), Abstract TB36; H. Osterberg, E. Snitzer, M. Polanyi, R. Hilberg, and J. W. Hicks, ibid. 49, 1128 (1959), Abstract TB37.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), Chap. V.

Wachowski, H. M.

R. E. Beam, M. M. Astrahan, W. C. Jakes, H. M. Wachowski, and W. L. Firestone, Northwestern University Report ATI 94929, Chap. V (1949).

Wegner, H.

H. Wegner, Air Material Command Microfilm ZWV/FB/RE/2018, R8117F831.

Ann. Physik (2)

D. Hondros and P. Debye, Ann. Physik 32, 465 (1910).
[CrossRef]

O. Schriever, Ann. Physik 63, 645 (1920).
[CrossRef]

Bell System Tech. J. (1)

J. R. Corson, S. P. Mead, and S. A. Schelkunoff, Bell System Tech. J. 15, 310 (1936).
[CrossRef]

I.R.E. Trans. (1)

S. P. Schlesinger and D. D. King, I.R.E. Trans. M.T.T.-6, 291 (1958).

J. Opt. Soc. Am. (1)

E. Snitzer and J. W. Hicks, J. Opt. Soc. Am. 49, 1128 (1959), Abstract TB36; H. Osterberg, E. Snitzer, M. Polanyi, R. Hilberg, and J. W. Hicks, ibid. 49, 1128 (1959), Abstract TB37.

Nuovo cimento (1)

M. Abele, Nuovo cimento 5, 274 (1948).
[CrossRef]

Phil. Mag. (1)

Rayleigh, Phil. Mag. 43, 125 (1897).
[CrossRef]

Other (5)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), Chap. V.

H. Wegner, Air Material Command Microfilm ZWV/FB/RE/2018, R8117F831.

S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1943), p. 425.

R. E. Beam, M. M. Astrahan, W. C. Jakes, H. M. Wachowski, and W. L. Firestone, Northwestern University Report ATI 94929, Chap. V (1949).

J. F. Reintzes and G. T. Coate, Principles of Radar (McGraw-Hill Book Company, Inc., New York, 1952), 3rd ed., Chap. 8, p. 609.

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Figures (6)

Fig. 1
Fig. 1

Construction to show the equivalence of the waveguide condition that at cutoff υph = c/n2 with the geometrical optics conditions that propagation occurs only if the angle of incidence of the wave on the fiber wall exceeds the critical angle for total internal reflection. The wave normal is given by p. S1 and S2 are two equiphase surfaces separated by λ/n1, and λg is the guide wavelength.

Fig. 2
Fig. 2

Typical curves of the frequency ν versus 1/λg for mode propagation in a dielectric waveguide. Each mode is represented by a line which is confined to the region between the lines whose slopes are c/n2 and c/n1. At the frequency ν the TE01 mode has a guide wavelength of λg, phase velocity υph and group velocity υgroup.

Fig. 3
Fig. 3

Field plot in the core for the TE02 mode far from cutoff and for a small difference in indexes of refraction of the core and cladding.

Fig. 4
Fig. 4

Field plot in the core for the HE12 mode far from cutoff and for a small difference in indexes of refraction of the core and cladding.

Fig. 5
Fig. 5

Field plot in the core for the EH11 mode far from cutoff and for a small difference in indexes of refraction of the core and cladding.

Fig. 6
Fig. 6

Field plot in the core for the HE21 mode far from cutoff and for a small difference in indexes of refraction of the core and cladding.

Tables (2)

Tables Icon

Table I Summary of cutoff conditions. The Bessel function of order n and argument u is given by Jn(u), and 1 and 2 are the dielectric constants for core and cladding. P gives the relative amount of Hz to Ez in a mode (see text for exact definition).

Tables Icon

Table II Summary of conditions far from cutoff.

Equations (61)

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E z = A n J n ( λ 1 r ) cos ( n θ + φ n ) exp { i ( h z ω t ) } ,
H z = B n J n ( λ 1 r ) cos ( n θ + ψ n ) exp { i ( h z ω t ) } .
λ 1 2 = k 1 2 h 2 , λ 2 2 = h 2 k 2 2 ;
E r = i ( h / k 2 h 2 ) [ ( E z / r ) + ( μ ω / h ) ( 1 / r ) ( H z / θ ) ] ,
E θ = i ( h / k 2 h 2 ) [ ( 1 / r ) ( E z / θ ) ( μ ω / h ) ( H z / r ) ] ,
H r = i ( h / k 2 h 2 ) [ ( k 2 / μ ω h ) ( 1 / r ) ( E z / θ ) + ( H z / r ) ] ,
H θ = i ( h / k 2 h 2 ) [ ( k 2 / μ ω h ) ( E z / r ) + ( 1 / r ) ( H z / θ ) ] .
A n J n = C n K n ,
A n n h u 2 J n sin ( n θ + φ n ) + B n μ ω u J n cos ( n θ + ψ n ) = C n n h w 2 K n sin ( n θ + φ n ) D n μ ω w K n cos ( n θ + ψ n ) ,
B n J n = D n K n ,
A n k 1 2 μ ω u J n cos ( n θ + φ n ) B n n h u 2 J n sin ( n θ + ψ n ) = C n k 2 2 μ ω w K n cos ( n θ + φ n ) + D n n h w 2 K n sin ( n θ + ψ n ) .
η 1 = [ J n ( u ) / u J n ( u ) ] , η 2 = [ K n ( w ) / w K n ( w ) ] .
( η 1 + η 2 ) ( k 1 2 η 1 + k 2 2 η 2 ) n 2 h 2 ( 1 / u 2 + 1 / w 2 ) 2 = sin ( n θ + φ n ) sin ( n θ + ψ n ) cos ( n θ + φ n ) cos ( n θ + ψ n ) .
ψ n φ n = ± π / 2 .
( η 1 + η 2 ) ( k 1 2 η 1 + k 2 2 η 2 ) = n 2 h 2 ( 1 / u 2 + 1 / w 2 ) 2 .
P = μ ω h B n cos ( n θ + ψ n ) A n sin ( n θ + φ n ) = n ( 1 / u 2 + 1 / w 2 ) η 1 + η 2 .
E z = J n ( λ 1 r ) F c , E r = i ( h / λ 1 ) [ J n P ( n J n / λ 1 r ) ] F c , E θ = i ( h / λ 1 ) [ P J n ( n J n / λ 1 r ) ] F s , H z = ( h / μ ω ) P J n F s , H r = i ( k 1 2 / μ ω λ 1 ) [ P ( h 2 / k 1 2 ) J n ( n J n / λ 1 r ) ] F s , H θ = i ( k 1 2 / μ ω λ 1 ) [ J n P ( h 2 / k 1 2 ) ( n J n / λ 1 r ) ] F s .
F c = A n cos ( n θ + φ n ) exp { i ( h z ω t ) } , F s = A n sin ( n θ + φ n ) exp { i ( h z ω t ) } .
ξ 1 = J n 1 u J n , ξ 2 = K n 1 w K n .
ξ 1 2 ξ 1 [ k 1 2 + k 2 2 k 1 2 ξ 2 + n ( 2 u 2 + k 1 2 + k 2 2 k 1 2 1 w 2 ) ] + [ k 2 2 k 1 2 ξ 2 2 + ξ 2 n ( k 1 2 + k 2 2 k 1 2 1 u 2 + 2 k 2 2 k 1 2 1 w 2 ) ] = 0.
ξ 2 = 1 / [ 2 ( n 1 ) ] .
ξ 1 = ( 1 / n 1 ) [ k 2 2 / ( k 1 2 + k 2 2 ) ] ,
ξ 1 = n [ ( k 1 2 + k 2 2 ) / k 1 2 ] ( 1 / w 2 ) .
[ u J n 2 ( u ) / J n 1 ( u ) ] = ( n 1 ) [ ( 1 2 ) / 2 ] .
ξ 1 = [ ( k 1 2 + k 2 2 ) / k 1 2 ] ( 1 / w 2 )
ξ 1 = [ 2 k 2 2 / ( k 1 2 + k 2 2 ) ] ln ( 2 / γ w ) .
J 1 ( u ) = 0.
u n m = 2 π ( a / λ ) ( n 1 2 n 2 2 ) 1 2 ,
E z = J n ( λ 1 r ) F c , E r = i h λ 1 [ 1 P 2 J n 1 1 + P 2 J n + 1 ] F c , E θ = i h λ 1 [ 1 P 2 J n 1 1 + P 2 J n + 1 ] F s , H z = h μ ω P J n F s , H r = i k 1 2 μ ω λ 1 [ 1 P h 2 / k 1 2 2 J n 1 1 + P h 2 / k 1 2 2 J n + 1 ] F s , H θ = i k 1 2 μ ω λ 1 [ 1 P h 2 / k 1 2 2 J n 1 1 + P h 2 / k 1 2 2 J n + 1 ] F c .
S z = 1 2 ( E r H θ E θ H r ) .
S z = 1 2 h μ ω k 1 2 λ 1 2 A n 2 [ ( 1 P ) ( 1 P h 2 / k 1 2 ) 4 J n 1 2 + ( 1 + P ) ( 1 + P h 2 / k 1 2 ) 4 J n + 1 2 1 P 2 h 2 / k 1 2 2 J n 1 J n + 1 cos 2 ( n θ + φ n ) ] .
S z J n 1 2 .
k 2 2 h 2 k 1 2 .
ω / h = υ p h = ν λ g = c / n eff ,
c / n 1 υ p h c / n 2 .
υ p h = c / n 2 .
λ = λ / n 1 sin α .
( E z , H z ) J n ( u r / a ) cos n θ ,
E r ( ± 1 ) J n ± 1 ( u r / a ) cos n θ ,
E θ J n ± 1 ( u r / a ) sin n θ .
d r / r d θ = E r / E θ .
d r / r d θ = ± cos n θ / sin n θ .
r = C ( sin n θ ) ± 1 / n ,
r sin θ = constant .
r = C sin θ ,
x 2 + ( y C / 2 ) 2 = C 2 / 4 .
r = C ( sin 2 θ ) 1 2
x y = constant .
u n m = u n 2 , m + n 1 u n 2 , m n 1 2 n 2 2 n 2 2 .
n J n / u = 1 2 ( J n 1 + J n + 1 ) ,
J n = 1 2 ( J n 1 J n + 1 ) ,
J n = ( 1 ) n J n .
J n / u J n = J n 1 / u J n n / u 2 .
K n ( w ) = π / 2 i n + 1 H n ( 1 ) ( i w ) ,
K n / w K n = K n 1 / w K n + n / w 2 .
K 0 ( w ) = ln ( 2 / γ w ) ,
K n ( w ) = ( n 1 ) ! 2 n 1 w 2 for n 1 ,
K 0 / w K 1 = ln ( 2 / γ w ) ,
K n 1 / w K n = [ 2 ( n 1 ) ] 1 for n 2.
K n ( w ) = ( π / 2 w ) 1 2 exp { w } ,
K n 1 / w K n = 1 / w .