Limits on the propagation constants of planar optical waveguide modes H. M. de Ruiter

Technische Hogeschool Eindhoven, Department of Elec­trical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Received 9 October 1980. 0003-6935/81/050731-02$00.50/0. © 1981 Optical Society of America. For any source-free domain υ, bounded by a closed sur­

face s with outward unit normal n, the following frequency-domain power equation holds [the complex time factor exp(jωt) is understood]1:

This expression is applied to a section of a straight dielectric waveguide (Fig. 1), the field being taken to be a single surface wave mode. When the z axis is taken along the direction of the cylinder axis, each mode of the electromagnetic field can be written as

and similar expressions for H, D, and B. In Eq. (2), r is the position vector in a right-handed Cartesian reference frame Oxyz, and rT = xix + yiy. For propagation in the direction of increasing z in passive or lossless media, Re(k2) > 0 and lm(kz) < 0. From Maxwell's equations we then obtain the relations

where V = ixdx + iydy — jkziz and where the constitutive relations d = εe and b = μh have been used.

By applying Eq. (1) to the section z1 < z < z2 rT ε D), the integration with respect to z can be carried out. It leads to a factor [exp(-2αz2) - exp(-2αz1)], where kz=β- jα, with α ≥ 0, β ≥ 0, multiplied by an integral in the transverse plane in both sides of Eq. (1). Since this result must hold for any value of z1 and z2 when z1 < z2 it follows that

where Q is the boundary curve of D with outward normal v. The configuration of the slab waveguide to which Eq. (5)

will be applied next is shown in Fig. 2. It is uniform in the y

direction, i.e., ε = ε(x), μ and for 2-D fields dy ≡ 0. The domain 2) is now taken to have unit length in the y direction and to cover — ∞ < x < ∞ (Fig. 2). On the assumption that ε and μ are bounded at infinity and that the field quantities show a sufficiently strong decay as |x| → ∞, Eq. (5) leads to

For such a waveguide the well-known separation into TE and TM modes can be performed; for TE modes, ex,hy,ez = 0 and for TM modes, hx,ey,hz = 0. With the aid of frequency-domain relation (6), restrictions can be found on the location of the eigenvalues kz leading to. surface wave modes in the complex kz plane.

Since for a dielectric waveguide μ = μ0 everywhere, Eq. (4) yields for TE modes

Substituting (7) in (6), writing ε = ε´ - jε", and separating the resulting equation into real and imaginary parts, we arrive at the equations

Fig. 1. Section of a straight dielectric waveguide bounded by two planes perpendicular to its axis.

Fig. 2. Dielectric slab waveguide and location of the domain 2).

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Fig. 3. Admissible region for the location of the complex propaga­tion coefficient K2 = B - jA of a TE mode in a dielectric slab

waveguide.

where k20 = ω20

ε0μ0. In the following, we shall use the nor­

malized quantities Kz = kz/k0, A = α/k0, B = β /k 0 , and εr = ε/ε0 . Since

Eq. (8) imposes the restriction thatmin(e') < 1AB < max(e"r). Since

Eq. (9) can only be satisfied when A2 - B2 + max(ε'r) > 0. So the region in the complex kz plane in which the normalized propagation coefficient kz of the TE modes of a slab wave­guide can be located is subject to the restrictions,

Figure 3 illustrates this. In Fig. 4, an example of the root loci of some TE modes in a symmetrical step-index slab waveguide surrounded by vacuum is given. Note that the simplicity of relations (10) and (11) is due to the fact that fi is real everywhere in the waveguide; when a similar analysis is performed for a slab waveguide with an x -dependent, though real, permeability, the factors m a x ( 0 in (10) and max(ε'r) in (11) should be multiplied by max(μ r) while the factor min(ε"r) in (10) should be multiplied by min(μr). When

732 APPLIED OPTICS / Vol. 20, No. 5 / 1 March 1981

Fig. 4. Admissible region and root loci of the propagation coefficient of the TE0, TE1, and TE2 modes of a step-index symmetrical slab waveguide embedded in vacuum, for which εr = 2.25 — 2.25; inside the slab. The arrows along the curves indicate the directions of

change of increasing frequency; in this case, min(ε"r) = 0.

μ is complex, similar remarks as in the case of TM modes (discussed below) apply.

For TM modes, the analysis runs in principle along the same lines. However, due to the fact that ε is complex, the bounds resulting from the counterparts of (10) and (11) are of less practical value and are not as smooth as those for TE modes.

Whereas for lossless media the propagation coefficients of surface wave modes are located on a restricted part of the real axis, for lossy media they are located in the complex plane and must be determined with the aid of numerical search proce­dures for complex root location. As these procedures are rather time-consuming, any exclusion of certain domains in the complex plane may result in a reduction of the computing time needed. In particular this applies to the Cauchy-integral technique, in which the derivative of a function divided by the function itself is integrated numerically along a closed contour in the complex plane, yielding 2ΠJ (number of zeros - number of poles) inside the contour. A first choice of the contour then yields the number of roots; the location of each separate root is found by subdivision of the area within the initial contour. The bounds on the positions of the roots enable us to restrict the location of the initial contour to a relatively small part of the complex plane only.

Reference 1. R. F. Harrington, Time-Harmonic Electromagnetic Fields

(McGraw-Hill, New York, 1961), Chap. 1.10, p. 21.