Abstract

A model is introduced that facilitates an easy scheme to design planar multilayered waveguides. The basis of this model is a field related vector that follows simply shaped trajectories as a function of the depth coordinate in the waveguide. Its diagram provides qualitative insight into the effects upon guided modes of a change in the number of layers, in their phase thicknesses and in the state of polarization. In addition, the method offers the possibility of studying the corresponding effects upon field components inside the waveguide.

© 1989 Optical Society of America

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References

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  1. J. Chilwell, I. Hodgkinson, “Thin-Films Field-Transfer Matrix Theory of Planar Multilayer Waveguides and Reflection from Prism-Loaded Waveguides,” J. Opt. Soc. Am. A, 1, 742–753 (1984).
    [Crossref]
  2. Yi-Fan Li, J. W. Y. Lit, “General Formulas for the Guiding Properties of a Multilayer Slab Waveguide,” J. Opt. Soc. Am. A, 4, 671–677 (1987).
    [Crossref]
  3. J. F. Revelli, “Mode Analysis and Prism Coupling for Multilayered Optical Waveguides,” Appl. Opt. 20, 3158–3167 (1981).
    [Crossref] [PubMed]
  4. L. M. Walpita, “Solutions for Planar Optical Waveguide Equations by Selecting Zero Elements in a Characteristic Matrix,” J. Opt. Soc. Am. A, 2, 595–602 (1985).
    [Crossref]
  5. S. Ruschin, G. Griffel, A. Hardy, N. Croitoru, “Unified Approach for Calculating the Number of Confined Modes in Multilayered Waveguide Structures,” J. Opt. Soc. Am. A, 3, 116–123 (1986).
    [Crossref]
  6. Yi-Fan Li, J. W. Y. Lit, “Contribution of Low-Index Layers to Mode Number in Multilayer Slab Waveguides,” J. Opt. Soc. Am. A, 4, 2233–2239 (1987).
    [Crossref]
  7. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  8. H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Adam Hilger Ltd., Bristol, 1986).
    [Crossref]
  9. For guided modes other authors call this kind of condition dispersion relation, characteristic equation, eigen value equation, eigen mode equation, or (characteristic) mode equation.
  10. The formulas obtained in this way are comparable with Eq. (7) of Y.-F. Li and J. W. Y. Lit [6]; ϕj equals their hjdj in layers where nj > β; ψj,+ equals their −ϕj+1, but their ϕj+1 is limited to the interval [0,π) while our |ψj,+| may amount to many times 2π.
  11. P. K. Tien, “Light Waves in Thin Films and Integrated Optics,” Appl. Opt., 10, 2395–2413 (1971).
    [Crossref] [PubMed]
  12. P. K. Tien, R. Ulrich, “Theory of Prism-Film Coupler and Thin-Film Light Guides,” J. Opt. Soc. Am., 60, 1325–1337 (1970).
    [Crossref]
  13. H. Oltmans, “Het Ontwerp van Planaire Meerlaags Golfgeleiders met Overdrachtmatrices,” Master Thesis, in Dutch (Delft U. Technology, Department of Applied Physics, 1988).
  14. χj−1,+ as defined here, is comparable with ψj in Eqs. (9b) and (14b) of Y.-F. Li and J. W. Y. Lit (2), and in Eq. 3(c) of Y.-F. Li and J. W. Y. Lit (6), if our ψj−1,+ is in Region I. If our ψj−1,+ is in Region II, their ψj equals our χj−1,+ + 1/2 πi.
  15. T. Tamir (Ed.), Integrated Optics (Springer-Verlag, Berlin, 1979).

1987 (2)

1986 (1)

1985 (1)

1984 (1)

1981 (1)

1971 (1)

1970 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Chilwell, J.

Croitoru, N.

Griffel, G.

Hardy, A.

Hodgkinson, I.

Li, Yi-Fan

Lit, J. W. Y.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Adam Hilger Ltd., Bristol, 1986).
[Crossref]

Oltmans, H.

H. Oltmans, “Het Ontwerp van Planaire Meerlaags Golfgeleiders met Overdrachtmatrices,” Master Thesis, in Dutch (Delft U. Technology, Department of Applied Physics, 1988).

Revelli, J. F.

Ruschin, S.

Tien, P. K.

Ulrich, R.

Walpita, L. M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Other (7)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Adam Hilger Ltd., Bristol, 1986).
[Crossref]

For guided modes other authors call this kind of condition dispersion relation, characteristic equation, eigen value equation, eigen mode equation, or (characteristic) mode equation.

The formulas obtained in this way are comparable with Eq. (7) of Y.-F. Li and J. W. Y. Lit [6]; ϕj equals their hjdj in layers where nj > β; ψj,+ equals their −ϕj+1, but their ϕj+1 is limited to the interval [0,π) while our |ψj,+| may amount to many times 2π.

H. Oltmans, “Het Ontwerp van Planaire Meerlaags Golfgeleiders met Overdrachtmatrices,” Master Thesis, in Dutch (Delft U. Technology, Department of Applied Physics, 1988).

χj−1,+ as defined here, is comparable with ψj in Eqs. (9b) and (14b) of Y.-F. Li and J. W. Y. Lit (2), and in Eq. 3(c) of Y.-F. Li and J. W. Y. Lit (6), if our ψj−1,+ is in Region I. If our ψj−1,+ is in Region II, their ψj equals our χj−1,+ + 1/2 πi.

T. Tamir (Ed.), Integrated Optics (Springer-Verlag, Berlin, 1979).

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Figures (11)

Fig. 1
Fig. 1

Multilayer stack consisting of J planar films lying between the semi-infinite media 0 and J + 1.

Fig. 2
Fig. 2

Examples of p and F inside cover (medium 0) and substrate (medium J + 1), as functions of xox and xxJ, respectively.

Fig. 3
Fig. 3

Examples of pj,+ and pj,−, showing the relation between them. Here, υj+1/υj = 0.636.

Fig. 4
Fig. 4

Construction of p and F inside layer j, and of pj−1,+ and Fj−1 from pj,+, for the case nj > β.

Fig. 5
Fig. 5

Hybrid field vector diagram of a TE2 mode in a two-layer waveguide.

Fig. 6
Fig. 6

Asymptotes and hyperbolas over which the endpoint of pj−1,+ moves for changing ϕj. Points on different hyperbolas, which occur on one line through the origin, have equal χ values. Ia, Ib, IIa and IIb denote the four sections between the asymptotes.

Fig. 7
Fig. 7

χ, given in π units, at several points on some hyperbolas, and one p vector, with ψ, at such a point. The arrows next to the hyperbolas denote directions of increasing χ.

Fig. 8
Fig. 8

Hybrid field vector representations of a coupling layer, with ψj,− in Region Ia (case a) and in Region IIa (case b). In the figure, j and j − 1 denote the numbers of the interface planes.

Fig. 9
Fig. 9

Hybrid field vector diagram of the TE2 mode in the four layer guide given by Table IV. Numbers in parentheses denote the layer numbers; the other numbers denote the interfaces.

Fig. 10
Fig. 10

Truncated parabolic n-profile of a graded-index waveguide.

Fig. 11
Fig. 11

Field vector diagram of the TEo guided mode in the graded-index waveguide, given by Fig. 10, for λV = 632.8 nm. p is drawn for J = 15 and for J → ∞. The numbers correspond to numbers of layer interfaces.

Tables (5)

Tables Icon

Table I Definition of the Polarization Dependent Parameter γj and the Field Amplitude Components U and V

Tables Icon

Table II Polarization Dependent Quantities g and υj

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Table III Data of a Two-Layer Planar Waveguide and a Second-Order TE Guided Mode

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Table IV Data of a Four-Layer Waveguide

Tables Icon

Table V Results of Numerical Calculations for Zero-Order Modes of the Graded-lndex Guide, Presented in Figs. 10 and 11

Equations (77)

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exp [ i k V ( ± α j x + β z ) i ω t ]
α j = n j cos θ j = n j 2 β 2 ,
β = n j sin θ j .
Z V = μ o ε o .
V + = γ j U + ,
V = γ j U .
( U j 1 V j 1 ) = M j ( U j V j ) j = 1 J ,
M j = ( cos Φ j i γ j sin Φ j i γ j sin Φ j cos Φ j ) = ( cosh ( Φ j i ) 1 γ j sinh ( Φ j i ) γ j sinh ( Φ j i ) cosh ( Φ j i ) ) .
Φ j = k V d j α j ,
M = j = 1 J M j .
V o = γ o U o ,
V J = γ J + 1 U J .
( 1 γ o ) U o = M ( 1 γ J + 1 ) U J .
X ( β ) = γ o m 11 + γ o γ J + 1 m 12 + m 21 + γ J + 1 m 22 = 0 ,
υ j = i g γ j ( n j β ) ,
υ j = g γ j ( n j < β ) .
V ˜ = g V ,
( U j 1 V ˜ j 1 ) = ( m 11 j m 12 j g g m 21 j m 22 j ) ( U j V ˜ j ) .
ϕ j = | Φ j |
M ˜ j = ( cos ϕ j sin ϕ υ j υ j sin ϕ j cos ϕ j ) ( n j β ) ,
M ˜ j = ( cosh ϕ j sinh ϕ j υ j υ j sinh ϕ j cosh ϕ j ) ( n j < β ) .
j = 1 J M ˜ j ,
M ˜ = ( m ˜ 11 m ˜ 12 m ˜ 21 m ˜ 22 ) = ( m 11 m 12 g g m 21 m 22 ) ,
V ˜ o = υ o U o ,
V ˜ J = υ J + 1 U J .
( 1 υ o ) U o = M ˜ ( 1 υ J + 1 ) U J .
X ˜ ( β ) = g X ( β ) = υ o m ˜ 11 + υ o υ J + 1 m ˜ 12 + m ˜ 21 + υ J + 1 m ˜ 22 = 0 .
F = ( U , V ˜ ) ,
p = ( U , V ˜ υ ) = ( p cos ψ , p sin ψ ) .
p j , = ( U j , V ˜ j υ j ) = ( p j , cos ψ j , , p j , sin ψ j , ) , ( x = x j ) p j , + = ( U j , V ˜ j υ j + 1 ) = ( p j , + cos ψ j , + , p j , + sin ψ j , + ) .
p j 1 , + = L j B j p j , + .
L j = ( cos ϕ j sin ϕ j sin ϕ j cos ϕ j ) ( n j β ) ,
L j = ( cosh ϕ j sinh ϕ j sinh ϕ j cosh ϕ j ) ( n j < β ) ,
B j = ( 1 0 0 υ j + 1 υ j ) .
p j , = B j p j , + .
p 0 , = B o L 1 B 1 L 2 B 2 B j 1 L j B j B j p J , + .
p 0 , = ( 1 1 ) U o ,
p J , + = ( 1 1 ) U J .
ψ o , = π 4 ( mod π ) ,
ψ J , + = π 4 ( mod π ) ,
υ j tan ψ j , = υ j + 1 tan ψ j , + .
δ j = ψ j , + ψ j , ,
δ j = ψ j , + arctan ( υ j + 1 υ j tan ψ j , + ) m j π .
π 2 < δ j π 2 .
δ j = arctan { ( 1 η j ) tan ψ j , + 1 + η j tan 2 ψ j , + } ,
η j = υ j + 1 υ j .
η j TM η j TE = ( n j n j + 1 ) 2 ,
p j 1 , + = L j p j , .
ψ j 1 , + = ψ j , ϕ j ( n j > β ) ,
p j , = p j 1 , + = p j .
ϕ j = k V ( x j x ) α j .
ψ j 1 , + = ψ j , + δ j ϕ j .
π 4 + δ o + ϕ 1 + δ 1 + ϕ 2 + + δ J 1 + ϕ J + δ J = π 4 + m π .
m = ψ J , + ψ o , π 1 2 .
2 k V n 1 d 1 cos θ 1 2 Φ 10 2 Φ 12 = 2 m π ,
Φ 10 = arctan ( υ o υ 1 ) , Φ 12 = arctan ( υ 2 υ 1 ) .
Φ 10 = ψ o , + = π 4 δ o , Φ 12 = ψ 1 , m π = π 4 δ 2 .
π 4 + δ o + ϕ 1 + δ 2 = π 4 + m π ,
Λ j 2 = π k V α j ,
ψ j , ψ j 1 , + + = ϕ j .
d j = ψ j , ψ j 1 , + + k V α j .
p j , = ( h j cosh χ j , , h j sinh χ j , ) ( Region I ) , p j , = ( h j sinh χ j , , h j cosh χ j , ) ( Region II ) ,
| V ˜ j υ j | | U j | Region I , | V ˜ j υ j | > | U j | Region II .
χ j 1 , + = χ j , + ϕ j ( n j < β ) .
| h j | = p j | cos 2 ψ j , | .
p = ( p cos ψ , p sin ψ ) = ( h cosh χ , h sinh χ ) ( Region I ) , p = ( p cos ψ , p sin ψ ) = ( h sinh χ , h cosh χ ) ( Region II ) .
tanh χ = tan ψ ( Region I ) , cotanh χ = tan ψ ( Region II ) .
Δ j = ψ j , ψ j 1 , + ,
Δ j = ψ j , arctan { tanh [ artanh ( tan ψ j , ) + ϕ j ] } m j π ( Region I ) , Δ j = ψ j , arctan { cotanh [ arcotanh ( tan ψ j , ) + ϕ j ] } m j π ( Region II ) ,
π 2 < Δ j π 2 .
U 2 h 2 V ˜ 2 h 2 υ 2 = ± 1 ,
π 4 + δ o + ϕ 1 + δ 1 + + δ j 1 + Δ j + δ j + + δ J 1 + ϕ J + δ J = π 4 + m π .
ϕ j = artanh ( tan ψ j 1 , + + ) artanh ( tan ψ j , ) ( Region I ) , ϕ j = arcotanh ( tan ψ j 1 , + + ) arcotanh ( tan ψ j , ) ( Region II ) ,
d j = ϕ j k V β 2 n j 2 ( n j < β ) .
x t 1 x t 2 k V n 2 ( x ) β 2 d x = π 2 + m π .
S δ + S Δ + S ϕ = π 2 + m π ,
S δ = j = 0 J δ j ,

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