Abstract

A coupled power theorem is developed which is applied to radial line (disk) optical dielectric waveguides. This theorem is utilized to develop transverse-electric and transverse-magnetic orthogonality relations for modes propagating in perfect disk guides. Because azimuthal variation is easily controlled, the disk guide can be employed as a device for coupling energy from a single source to a large number of optical fiber guides.

© 1977 Optical Society of America

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References

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  1. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  2. R. B. Adler, Proc. IRE 40, 339 (1952).
    [Crossref]
  3. S. E. Miller, Bell Syst. Tech. J. 33, 661 (1956).
    [Crossref]
  4. M. Abramowitz and I. A. Stegun, Nat. Bur. Stand. (U. S.) Appl. Math. Ser. 55, 360 (1964).

1964 (1)

M. Abramowitz and I. A. Stegun, Nat. Bur. Stand. (U. S.) Appl. Math. Ser. 55, 360 (1964).

1956 (1)

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1956).
[Crossref]

1952 (1)

R. B. Adler, Proc. IRE 40, 339 (1952).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Nat. Bur. Stand. (U. S.) Appl. Math. Ser. 55, 360 (1964).

Adler, R. B.

R. B. Adler, Proc. IRE 40, 339 (1952).
[Crossref]

Miller, S. E.

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1956).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Nat. Bur. Stand. (U. S.) Appl. Math. Ser. 55, 360 (1964).

Stratton, J. A.

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

Bell Syst. Tech. J. (1)

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1956).
[Crossref]

Nat. Bur. Stand. (U. S.) Appl. Math. Ser. (1)

M. Abramowitz and I. A. Stegun, Nat. Bur. Stand. (U. S.) Appl. Math. Ser. 55, 360 (1964).

Proc. IRE (1)

R. B. Adler, Proc. IRE 40, 339 (1952).
[Crossref]

Other (1)

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

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Figures (5)

FIG. 1
FIG. 1

(a) Basic geometry of the disk waveguide. (b) Disk optical waveguide as a coupler to several optical fibers.

FIG. 2
FIG. 2

Cylindrical shell used to develop the coupled power theorem.

FIG. 3
FIG. 3

Power density as a function of sampling radius (based on data for TE modes, with n1 = 2. 23, n3 = 1. 393, and t = 2. 0 × 10−6 m, P = 1 W).

FIG. 4
FIG. 4

Typical k0kr diagram (based on data for TE modes, with n1 = 2. 23, n3 = 1. 393, and t = 2. 0 × 10−6 m).

FIG. 5
FIG. 5

Power density as a function of thickness (based on data for TE modes, with n1 = 2. 23, n3 = 1. 393, r = 0. 010 m, P = 1 W).

Equations (30)

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M ¯ = × ψ ,
N ¯ = ( 1 / τ ) × M ¯ ,
L ¯ = ψ ,
2 ψ + k 2 ψ = 0 ,
E r = 1 r ψ ϕ , H r = 1 j ω μ 2 ψ r z , E ϕ = ψ r , H ϕ = 1 j ω μ 1 r 2 ψ ϕ z , E z = 0 , H z = 1 j ω μ ( 2 ψ z 2 + k 2 ψ ) ,
H r = 1 r ψ ϕ , E r = 1 j ω 2 ψ r z , H ϕ = ψ r , E ϕ = 1 j ω 1 r 2 ψ ϕ z , H z = 0 , E z = 1 j ω ( 2 ψ z 2 + k 2 ψ ) .
ψ = A R ( r ) Φ ( ϕ ) Z ( z ) .
R ( r ) = ( 2 π k r r ) 1 / 2 exp ± j ( k r r n π 2 π 4 ) .
P ν μ = S 1 ( Ē ν × H ¯ μ * ) · n ̂ d S .
P ν μ + P μ ν * = P 0 ,
P ν μ = S ( Ē ν × H ¯ μ * ) · n ̂ d S S 0 ( Ē ν × H ¯ μ * ) · n ̂ d S ,
P ν μ = V [ H ¯ μ * · × Ē ν Ē ν · × H ¯ μ * ] d V S 0 ( Ē ν × H ¯ μ * ) · n ̂ d S .
P μ ν * = V [ E μ * · × H ¯ ν H ¯ ν · × Ē μ * ] d V + S 0 ( H ¯ ν × Ē μ * ) · n ̂ d S .
P ν μ = j ω V { Ē ν · Ē μ * μ H ¯ ν · H ¯ μ * } d V S 0 ( Ē ν × H ¯ μ * ) · n ̂ d S ,
P μ ν * = j ω V { Ē ν · E μ * μ H ¯ ν · H ¯ μ * } d V + S 0 ( H ¯ ν × Ē μ * ) · n ̂ d S .
P ν μ + P μ ν * = S 0 { H ν × E μ * E ν × H μ * } · n ̂ d S = S 1 { H ¯ ν × Ē μ * Ē ν × H ¯ μ * } · n ̂ d S .
P ν μ + P μ ν * = P 0 ,
P ν μ = S 1 ( ψ ν × · ψ μ ) · r ̂ 1 d S S 1 ( ψ ν × · ) ψ μ * · r ̂ 1 d S .
P ν μ + P μ ν * = 0
P μ ν * = S 1 ( ψ ν × ) × ( ψ μ * × ) · r ̂ 1 d S .
P ν μ + P μ ν * = ( k r ν 2 j ω R μ * ( r ) R μ * ( r ) k r μ 2 j ω R ν ( r ) R ν ( r ) ) S 1 ψ ν × ψ μ * μ ( z ) d S .
T ν μ TE = S 1 ψ ν × ψ μ * μ ( z ) d S = 0 ν μ
S 1 ψ ν × ψ μ * d S = 0 .
P ν μ + P μ ν * = ( k r ν 2 j ω R μ * ( r ) R μ * ( r ) k r μ 2 j ω R ν ( r ) R ν ( r ) ) S 1 ψ ν ψ μ * ( z ) d S ,
T ν μ TM = S 1 ψ ν ψ μ ( z ) d S = 0 ν μ .
P = 1 2 Re ( S 1 ( Ē ν × H ¯ ν * ) · r ̂ 1 d S ) ,
I = 1 2 Re [ ( Ē ν × H ¯ ν * ) · r ̂ 1 ] .
I = [ Z ( z ) Φ ( ϕ ) ] 2 P ( 1 + δ n , 0 ) ( t + { 1 / [ k r 2 ( 3 / 0 ) k 0 2 ] } 1 / 2 Q · 1 π r .
Q = { 1 TE modes ( 3 / 0 ) k 0 2 [ 1 + ( 3 / 1 ) ] k r 2 ( 3 / 0 ) k 0 2 TM modes .
I 1 / r t .