Abstract

The power transmitted by surface waveguide modes propagating along a semi-infinite dielectric fiber is determined. For a plane wave normally incident on the end of a perfectly circular fiber only the HE1m modes are excited. Of these modes the majority of the power is contained in the HE11 mode. Far above the cutoff conditions for the HE13 mode, approximately 99.8% of the incident power is contained in the HE11, HE12, and HE13 modes.

© 1966 Optical Society of America

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References

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  1. E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961).
    [Crossref]
  2. N. S. Kapany and J. J. Burke, J. Opt. Soc. Am. 51, 1067 (1961).
    [Crossref]
  3. E. Snitzer, J. Opt. Soc. Am. 49, 1128 (1959).
  4. J. Enoch, J. Opt. Soc. Am. 53, 71 (1963).
    [Crossref]
  5. G. Biernson and D. Kinsley, IEEE Trans. Microwave Theory Tech. MTT-13, 345 (1965).
    [Crossref]
  6. G. Goubau, Proc. IRE 40, 865 (1952).
    [Crossref]
  7. G. N. Watson, Theory of Bessel Functions (Cambridge University Press, London, 1949).

1965 (1)

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

1963 (1)

1961 (2)

1959 (1)

E. Snitzer, J. Opt. Soc. Am. 49, 1128 (1959).

1952 (1)

G. Goubau, Proc. IRE 40, 865 (1952).
[Crossref]

Biernson, G.

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

Burke, J. J.

Enoch, J.

Goubau, G.

G. Goubau, Proc. IRE 40, 865 (1952).
[Crossref]

Kapany, N. S.

Kinsley, D.

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

Snitzer, E.

E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961).
[Crossref]

E. Snitzer, J. Opt. Soc. Am. 49, 1128 (1959).

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge University Press, London, 1949).

IEEE Trans. Microwave Theory Tech. (1)

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

J. Opt. Soc. Am. (4)

Proc. IRE (1)

G. Goubau, Proc. IRE 40, 865 (1952).
[Crossref]

Other (1)

G. N. Watson, Theory of Bessel Functions (Cambridge University Press, London, 1949).

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Figures (4)

Fig. 1
Fig. 1

Relative power of the surface modes HE11, HE12, HE13 versus the normalized frequency. Curves for HE11, HE12, and HE13 are for finite excitation; He11 and HE12 are for infinite excitation. Dashed curves represent an approximation for small dielectric differences.

Fig. 2
Fig. 2

Field pattern of the HE11 mode far from cutoff.

Fig. 3
Fig. 3

Semi-infinite dielectric rod excited by an infinite current sheet. I: infinite current sheet consisting of electric currents K and magnetic currents M.

Fig. 4
Fig. 4

Semi-infinite dielectric rod excited by finite current sheet. F: finite current sheet at end of rod.

Equations (73)

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V 2 = a 2 ω 2 μ 0 ( 1 - 2 ) ,
V 2 = W 2 + u 2 .
P = 4 V 2 / W 2 u 2 ( infinite excitation ) ,
P = 4 W 2 / u 2 V 2 ( finite excitation ) .
M r = - E 0 cos ϕ ,
M ϕ = + E 0 sin ϕ ,
K r = [ - E 0 / ( μ 0 / 0 ) 1 2 ] sin ϕ ,
K ϕ = [ - E 0 / ( μ 0 / 0 ) 1 2 ] cos ϕ ,
E = A q E q + E R ,
H = A q H q + H R ,
( E p × H q ) · î z d a = 0             for             p q
( E R × H q ) · î z d a = 0 ,
M = - î z x ( A q E q + E R ) .
- H p · M d a = [ ( A q E q + E R × H p ) ] · î z d a = A p ( E p × H p ) · î z d a .
E p = E p z + E p t
H p = H p z + H p t ,
E p t = e p e - j h z
H p t = h p e - j h z .
A p = - h p · M d a / ( e p × h p ) · î z d a .
R = r / a
Z = z / a
h = h a = ( V 2 / δ - u 2 ) 1 2
k 1 2 = a 2 ω 2 μ 0 1 = V 2 / δ
δ = ( 1 - 2 ) / 1 = 1 - ˜
˜ = 2 / 1
F 2 = W 2 + u 2 u 2 W 2 ( J 1 u J 1 + K 1 W K 1 )
F 1 = u 2 W 2 W 2 + u 2 ( J 1 u J 1 + ˜ K 1 W K 1 )
V 2 = u 2 + W 2 = a 2 ω 2 μ 0 ( 1 - 2 )
E z = J 1 ( u R ) e - j h Z sin ϕ
H z = ( 1 / μ 0 ) 1 2 ( F 2 h / k 1 ) J 1 ( u R ) e - j h Z cos ϕ
e r = ( j h / 2 u ) [ ( F 2 - 1 ) J 0 ( u R ) + ( F 2 + 1 ) J 2 ( u R ) ] sin ϕ
e ϕ = ( j h / 2 u ) [ ( F 2 - 1 ) J 0 ( u R ) - ( F 2 + 1 ) J 2 ( u R ) ] cos ϕ
h r = ( 1 / μ 0 ) 1 2 ( j k 1 / 2 u ) [ - ( F 1 - 1 ) J 0 ( u R ) + ( F 1 + 1 ) J 2 ( u R ) ] cos ϕ
h ϕ = ( 1 / μ 0 ) 1 2 ( j k 1 / 2 u ) [ ( F 1 - 1 ) J 0 ( u R ) + ( F 1 + 1 ) J 2 ( u R ) ] sin ϕ
E z = K 1 ( W R ) e - j h z sin ϕ
H z = ( 1 / μ 0 ) 1 2 ( h F 2 / k 1 ) K 1 ( W R ) e - j h Z cos ϕ
e r = ( j h J 1 / 2 W K 1 ) [ ( F 2 - 1 ) K 0 ( R W ) - ( F 2 + 1 ) K 2 ( R W ) ] sin ϕ
e ϕ = ( j h J 1 / 2 W K 1 ) [ ( F 2 - 1 ) K 0 ( R W ) + ( F 2 + 1 ) K 2 ( R W ) ] cos ϕ
h r = - ( 1 / u 0 ) 1 2 ( j k 1 / 2 W ) ( J 1 / K 1 ) × [ ( F 1 - ˜ ) K 0 ( R W ) + ( F 1 + ˜ ) K 2 ( R W ) ] sin ϕ
h ϕ = ( 1 / μ 0 ) 1 2 ( j k 1 / 2 W ) ( J 1 / K 1 ) × [ ( F 1 - ˜ ) K 0 ( R W ) - ( F 1 + ˜ ) K 2 ( R W ) ] sin ϕ .
( J 1 u J 1 + ˜ K 1 W K 1 ) ( J 1 u J 1 + K 1 W K 1 ) = ( h k 1 ) 2 ( W 2 + u 2 W 2 u 2 ) 2 ,
F 1 / F 2 = ( h / k 1 ) 2 .
V 2 = u 2 + W 2 .
0 2 π 0 h p · M d a = j E 0 a 2 k 1 ( 1 μ 0 ) 1 2 π [ ( F 1 - 1 ) u 0 1 J 0 ( u R ) R d R + ( J 1 K 1 ) ( F 1 - ˜ ) W 1 K 0 ( W R ) R d R ] .
h · M d a = E 0 j a 2 k 1 π ( 1 μ 0 ) 1 2 [ ( F 1 - 1 ) W 2 + ( F 1 - ˜ ) u 2 u 2 W 2 ] .
( e p × h p ) · î z d a = a 2 0 2 π 0 ( e r h ϕ - e ϕ h r ) R d R d ϕ ,
( e p × h p ) · î z d a = - ( 1 μ 0 ) 1 2 a 2 π h k 1 4 × { ( F 2 - 1 ) ( F 1 - 1 ) A 1 + ( F 2 + 1 ) ( F 1 + 1 ) A 2 u 2 + ( J 1 K 1 ) 2 [ ( F 2 - 1 ) ( F 1 - ˜ ) A 3 + ( F 2 + 1 ) ( F 1 + ˜ ) A 4 W 2 ] } ,
A p = j ( 4 E 0 ) h W 2 u 2 [ ( F 1 - 1 ) W 2 + ( F 1 - ˜ ) u 2 ] [ P 1 + ( J 1 / K 1 ) 2 P 2 ]
P 1 = [ ( F 2 - 1 ) ( F 1 - 1 ) A 1 + ( F 2 + 1 ) ( F 1 + 1 ) A 2 ] / u 2 P 2 = [ ( F 2 - 1 ) ( F 1 - ˜ ) A 3 + ( F 2 + 1 ) ( F 1 + ˜ ) A 4 ] / W 2 .
M = M r + M ϕ             R < 1
= 0             R > 1.
0 2 π 0 h p · M d a = j E 0 a 2 π ( 1 μ 0 ) 1 2 k 1 ( F 1 - 1 u 2 ) J 1 ( u ) ,
A p = - j 4 E 0 h u 2 ( F 1 - 1 ) J 1 ( u ) [ P 1 + ( J 1 / K 1 ) 2 P 2 ] ,
P = A p 2 k 1 h a 2 π 8 ( μ 0 / 1 ) 1 2 [ P 1 + ( J 1 K 1 ) 2 P 2 ] .
E 0 2 = ( 2 / π a 2 ) ( μ 0 / 1 ) 1 2 .
P = 4 k 1 h W 4 u 4 { [ ( F 1 - 1 ) W 2 + ( F 1 - ˜ ) u 2 ] J 1 } 2 [ P 1 + ( J 1 / K 1 ) 2 P 2 ] .
P = 4 k 1 h u 4 [ ( F 1 - 1 ) J 1 ( u ) ] 2 [ P 1 + ( J 1 / K 1 ) 2 P 2 ] .
F 1 F 2 - 1
( F 1 - ˜ ) - 2
( F 1 + ˜ ) 0
h k 1
J n - 1 ( u ) / u J n ( u ) K n - 1 ( W ) / W K n ( W )             n > 0.
P 1 + ( J 1 K 1 ) 2 P 2 = 4 ( W 2 + u 2 u 2 W 2 ) J 1 2 .
F 1 F 1
( F 1 - ˜ ) 0
( F 1 + ˜ ) 2
h k 1
J n + 1 ( u ) / u J n ( n ) - K n + 1 ( W ) / W K n ( W )             n > 0.
P 1 + ( J 1 / K 1 ) 2 P 2 = 4 / u 2 [ ( J 2 / K 2 ) 2 K 1 K 3 - J 1 J 3 ] .
P = A p 2 ( k 1 ) 2 a 2 π 2 ( μ 0 / 1 ) 1 2 ( W 2 + u 2 W 2 u 2 ) J 1 2 ( u ) .
P = 4 V 2 W 2 u 2 = 4 u 2 ( V 2 V 2 - u 2 ) .
P = 4 W 2 / u 2 V 2 = 4 ( 1 / u 2 - 1 / V 2 ) .
P = A p 2 ( k 1 ) 2 a 2 π 2 ( μ 0 / 1 ) 1 2 × { 1 u 2 [ ( J 2 ( u ) K 2 2 ( W ) ) 2 K 3 ( W ) K 1 ( W ) - J 3 ( u ) J 1 ( u ) ] } .