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Bragg reflection waveguides with a matching layer

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Abstract

It is demonstrated that Bragg reflection waveguides, either planar or cylindrical, can be designed to support a symmetric mode with a specified core field distribution, by adjusting the first layer width. Analytic expressions are given for this matching layer, which matches between the electromagnetic field in the core, and a Bragg mirror optimally designed for the mode. This adjustment may change significantly the characteristics of the waveguide. At the particular wavelength for which the waveguide is designed, the electromagnetic field is identical to that of a partially dielectric loaded metallic or perfect magnetic waveguide, rather than a pure metallic waveguide. Either a planar or coaxial Bragg waveguide is shown to support a mode that has a TEM field distribution in the hollow region.

©2004 Optical Society of America

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Figures (9)

Fig. 1.
Fig. 1. Planar and cylindrical Bragg reflection waveguides.
Fig. 2.
Fig. 2. First layer width for v ph=c, normalized by Δ q λ 0 / ( 4 ε 1 1 ) . The layer adjacent to the core has a refractive index of n 1=1.6 and the other material has n 2=4.6.
Fig. 3.
Fig. 3. Planar TM profiles. (a) kxD int=0 low refractive index first (b) kxD int=0 high refractive index first (c) kxD int=π/3 (d) kxD int=3π/4 (e) kxD int=π/2 (metallic-like walls) (f) kxD int=π (magnetic-like walls).
Fig. 4.
Fig. 4. Symmetric TM mode dispersion diagram for planar waveguide with D int=0.3λ0 (left) and cylindrical waveguide with R int=0.3λ0 (right). In both cases the red (dashed) curves are obtained with no design procedure, and the blue (solid) curves correspond to a v ph=c design procedure.
Fig. 5.
Fig. 5. Symmetric TM mode dispersion diagram for planar waveguide with D int=1.0λ0 (left) and cylindrical waveguide with R int=1.0λ0 (right). In both cases the red (dashed) curves are obtained with no design procedure, and the blue (solid) curves correspond to a v ph=c design procedure.
Fig. 6.
Fig. 6. Group velocity dispersion for the v ph=c cylindrical waveguide.
Fig. 7.
Fig. 7. Planar odd TM power profiles: kxD int=π/4 (solid blue) and kxD int=/4 (dashed red).
Fig. 8.
Fig. 8. Planar TEM-TM profiles: higher refractive index first (top) and lower refractive index first (bottom).
Fig. 9.
Fig. 9. Coaxial TEM-TM profiles: the higher refractive index is first at the hollow region outer boundary (top) and the lower refractive index first (bottom).

Tables (2)

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Table 1. Hollow core symmetric modes. The transverse wavenumbers are k x , r = ω 2 / c 2 k z 2 .

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Table 2. Comparison between the waveguide parameters that correspond to the field distributions of Fig. 3 and the analogous TE cases.

Equations (21)

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A 1 / E 0 = ( B 1 / E 0 ) * = 1 2 e j k 1 D int cos ( k x D int ) j k 1 2 ε 1 k x e j k 1 D int sin ( k x D int ) ;
Δ v = π 2 ω 0 2 c 2 ε v k z 2 .
{ E z ( x = D int + Δ 1 ) = 0 Z 1 > Z 2 E z x ( x = D int + Δ 1 ) = 0 Z 1 < Z 2 ,
Δ 1 ( TM ) = { 1 k 1 arctan [ ε 1 k x k 1 cot ( k x D int ) ] Z 1 > Z 2 1 k 1 arctan [ k 1 ε 1 k x tan ( k x D int ) ] Z 1 < Z 2 .
Δ 1 ( TM ) = { 1 k 1 arctan [ ( Z 1 η 0 ω 0 c D int ) 1 ] Z 1 > Z 2 1 k 1 arctan ( Z 1 η 0 ω 0 c D int ) Z 1 < Z 2 .
Δ 1 ( TE ) = { 1 k 1 arctan [ k x k 1 cot ( k x D int ) ] Y 1 > Y 2 1 k 1 arctan [ k 1 k x tan ( k x D int ) ] Y 1 < Y 2 .
{ E z = 0 Z v > Z v + 1 E z r = 0 Z v < Z v + 1 .
U ( TM ) ε 1 k 1 Y 1 ( k 1 R int ) J 0 ( k r R int ) 1 k r Y 0 ( k 1 R int ) J 1 ( k r R int ) ε 1 k 1 J 1 ( k 1 R int ) J 0 ( k r R int ) 1 k r J 0 ( k 1 R int ) J 1 ( k r R int ) ,
Δ 1 ( TM ) = { 1 k 1 arcBess tan 0 ( U ( TM ) ) R int Z 1 > Z 2 1 k 1 arcBess tan 1 ( U ( TM ) ) R int Z 1 < Z 2 .
U ( TM ) = ε 1 k 1 Y 1 ( k 1 R int ) R int 2 Y 0 ( k 1 R int ) ε 1 k 1 J 1 ( k 1 R int ) R int 2 J 0 ( k 1 R int ) .
U ( TE ) 1 k 1 Y 1 ( k 1 R int ) J 0 ( k r R int ) 1 k r Y 0 ( k 1 R int ) J 1 ( k r R int ) 1 k 1 J 1 ( k 1 R int ) J 0 ( k r R int ) 1 k r J 0 ( k 1 R int ) J 1 ( k r R int )
α = 2 ω 0 tan δ W E , clad ( z ) P ( z ) .
E z = E 0 sin ( k x x ) e j k z z ,
E x = j k z k x E 0 cos ( k x x ) e j k z ,
H y = j ω 0 ε 0 k x E 0 cos ( k x x ) e j k z z .
Δ 1 ( TM ) = { 1 k 1 arctan [ ε 1 k x k 1 tan ( k x D int ) ] Z 1 > Z 2 1 k 1 arctan [ k 1 ε 1 k x cot ( k x D int ) ] Z 1 < Z 2 .
E z = E 0 J 0 ( k r r ) e j k z z ,
E r = j η 0 ε r ε r 1 E 0 J 1 ( k r r ) e j k z z ,
H ϕ = j ε r 1 E 0 J 1 ( k r r ) e j k z z .
E r = C η 0 1 r e j k z z ,
H ϕ = C r e j k z z .
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