Abstract

Slab waveguide tapers with finite cladding thickness are analyzed using a geometrical optics approach, and the results are compared with an exact modal analysis. Iteration formulas are derived for the calculation of the propagation angles of the guided modes and the corresponding changes in the core and cladding thicknesses, assuming that the difference between the refractive indexes of the core and cladding is small. Numerical results are presented and compared with the values obtained by a successive step modal analysis, showing good agreement between the two approaches. This indicates that ray tracing techniques should be useful for the analysis and design of taper couplers.

© 1978 Optical Society of America

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References

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  1. T. Ozeki, B. S. Kawasaki, Electron. Lett. 12, 607 (1976).
    [CrossRef]
  2. B. S. Kawasaki, K. O. Hill, Appl. Opt. 16, 1794 (1977).
    [CrossRef] [PubMed]
  3. T. K. Lim, B. K. Garside, J. P. Marton; unpublished.

1977 (1)

1976 (1)

T. Ozeki, B. S. Kawasaki, Electron. Lett. 12, 607 (1976).
[CrossRef]

Garside, B. K.

T. K. Lim, B. K. Garside, J. P. Marton; unpublished.

Hill, K. O.

Kawasaki, B. S.

B. S. Kawasaki, K. O. Hill, Appl. Opt. 16, 1794 (1977).
[CrossRef] [PubMed]

T. Ozeki, B. S. Kawasaki, Electron. Lett. 12, 607 (1976).
[CrossRef]

Lim, T. K.

T. K. Lim, B. K. Garside, J. P. Marton; unpublished.

Marton, J. P.

T. K. Lim, B. K. Garside, J. P. Marton; unpublished.

Ozeki, T.

T. Ozeki, B. S. Kawasaki, Electron. Lett. 12, 607 (1976).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

T. Ozeki, B. S. Kawasaki, Electron. Lett. 12, 607 (1976).
[CrossRef]

Other (1)

T. K. Lim, B. K. Garside, J. P. Marton; unpublished.

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Figures (5)

Fig. 1
Fig. 1

Geometrical configuration of a slab waveguide taper. The shaded regions are the cladding layers. L is the length of the tapered section; n1, n2, and n3 are, respectively, the refractive indexes of the core, cladding, and the surrounding media; and 2a, and b are the thickness of the core and cladding, respectively.

Fig. 2
Fig. 2

The zigzag wave (ZZW) representation of the core-mode propagation. (a) The full ZZW ABCDEFGH. The dashed line segments CD′, DE, and GH′ are, respectively, the mirror images of the actual ray paths CD, DE, and GH about the yz plane. (b) The reduced zone scheme (RZS). The lower half of the ZZW is represented by its mirror image.

Fig. 3
Fig. 3

An arbitrary period ABC of the ZZW for core-mode propagation. Ω1 is the taper angle of the core; θz1, θz2 are propagation angles at A and C, respectively; θ0 is the angle of incidence at B; a1, aM, and a2 represent core half-thickness at A, B, and C, respectively.

Fig. 4
Fig. 4

(a) ZZW for the cladding-mode propagation employing the reduced zone scheme. Reflections from the core–cladding boundary are neglected under the assumption that 1 − (n2/n1) ≪ 1. (b) An arbitrary period of the ZZW shown in (a). Ω1 and Ω2 are taper angles of core and cladding layers with respect to their own axes of symmetry z and z′; ΩT = Ω1 + 2Ω2 is the over-all taper angle. θz1, θz2 are propagation angles at A and E; θ0, θ1, θ2, θ3, and θ4 are angles of incidence or refraction at B, C, and D; ai and bi (i = 1,2,3,4,5) represents the core half-thickness and cladding thickness at A, B, C, D, and E. To avoid undue complication, only a1 and b1 are labeled explicitly.

Fig. 5
Fig. 5

Variation of the propagation angle θz as a function of the normalized core halfwidth a/λ for several even TE modes M = 14, 28, and 44. Parameters are n1 = 1.52, n2 = 1.50, n3 = 1.00, and f = b/a = 0.5. Both Ω1 = 0.001 and 0.01 are used for the ray optics calculations, but the differences for the two cases are not significant. The values of θ 12 c and θ 13 c indicated by the arrows are the critical propagation angles calculated from Eq. (25). The dash–dot curves are calculated using the approximation that n2 = n1 for the cladding mode propagation.

Equations (25)

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a 2 a 1 = 1 tan Ω 1 cot θ z 2 1 + tan Ω 1 cot θ z 1 ,
θ z 2 = θ z 1 + 2 Ω 1 ,
Δ z A C = ( a 1 a 2 ) cot Ω 1 .
Ω 1 = tan 1 [ ( a I a 0 ) / L ] ,
θ 0 = ( π / 2 ) θ z 1 Ω 1 ,
θ z 1 = Ω 1 + sin 1 { [ 1 ( n 2 / n 1 ) 2 ] 1 / 2 } .
1 ( n 2 / n 1 ) 1.
Ω T = Ω 1 + 2 Ω 2
θ 0 = ( π / 2 ) θ z 1 Ω 1 , θ 1 = sin 1 [ ( n 1 / n 2 ) sin θ 0 ] , θ 2 = θ 1 2 Ω 2 , θ 3 = θ 2 2 Ω 2 , θ 4 = sin 1 [ ( n 2 / n 1 ) sin θ 3 ] , θ z 2 = ( π / 2 ) θ 4 Ω 1 .
a 2 a 1 = 1 + f 1 1 + f 2 + tan Ω T cot θ z 1 , a 3 a 2 = 1 + f 2 + tan Ω T tan ( θ 2 + Ω T ) ( 1 + f 3 ) [ 1 + tan Ω T tan ( θ 2 + Ω T ) ] , a 4 a 3 = ( 1 + f 3 ) [ 1 tan Ω T tan ( θ 2 Ω T ) ] 1 + f 4 tan Ω T tan ( θ 2 Ω T ) , a 5 a 4 = 1 + f 4 tan Ω T cot θ z 2 ( 1 + f 5 ) ,
f i = b i / a i ( i = 1,2,3,4,5 ) ,
a 5 a 1 = ( 1 + f 1 1 + f 5 ) [ 1 + f 4 tan Ω T cot θ z 2 1 + f 4 tan Ω T tan ( θ 2 Ω T ) ] × [ 1 + f 2 + tan Ω T tan ( θ 2 + Ω T ) 1 + f 2 + tan Ω T cot θ z 1 ] × [ 1 tan Ω T tan ( θ 2 Ω T ) 1 + tan Ω T tan ( θ 2 + Ω T ) ] .
Ω T = tan 1 [ ( D I D 0 ) / 2 L ] ,
Ω T = tan 1 [ ( 1 + f ) tan Ω 1 ] .
θ 2 = θ 0 2 Ω 2 = ( π / 2 ) θ z 1 Ω T , θ z 2 = θ z 1 + 2 Ω T .
tan ( θ 2 Ω T ) = cot θ z 2 , tan ( θ 2 + Ω T ) = cot θ z 1 ,
a 5 a 1 = ( 1 + f 1 1 + f 5 ) [ 1 tan Ω T cot θ z 2 1 + tan Ω T cot θ z 1 ] .
A 5 A 1 = 1 tan Ω T cot θ z 2 1 + tan Ω T cot θ z 1 ,
R 1 h R 2 ( core modes ) ,
R 2 h R 3 ( cladding modes ) ,
R i = n i k a ( i = 1,2,3 ) ,
cot u = { ( u w ) [ w + s tan h ( w f ) s + w tan h ( w f ) ] ( 0 u R 12 R 1 h R 2 ) , ( u υ ) [ υ + s tan ( υ f ) s υ tan ( υ f ) ] ( R 12 u R 13 R 2 h R 3 ) ,
f = b / a , u 2 = R 1 2 h 2 , υ 2 = R 2 2 h 2 , w 2 = h 2 R 2 2 , s 2 = h 2 R 3 2 , R 12 2 = ( n 1 2 n 2 2 ) ( k a ) 2 , R 13 2 = ( n 1 2 n 3 2 ) ( k a ) 2 .
θ z = tan 1 ( u / h ) .
θ 12 c = tan 1 ( n 1 2 / n 2 2 1 ) 1 / 2 θ 13 c = tan 1 ( n 1 2 / n 3 2 1 ) 1 / 2 .

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