# Step-Index Planar Waveguides

This part is a continuation from the previous tutorial - Wave Equations for Optical Waveguides.

A step-index planar waveguide is also called a slab waveguide. We have used it in the waveguide modes tutorial with the approach of ray optics to illustrate an intuitive picture and some basic characteristics of the wave behavior in a waveguide.

In this tutorial, the important characteristics of a slab waveguide are discussed, beginning with solution of the wave equations developed in the wave equations for optical waveguide tutorial.

The structure and parameters of the three-layer slab waveguide under discussion are shown in figure 4 below.

Normalized Waveguide Parameters

The mode properties of a waveguide are commonly characterized in terms of a few dimensionless normalized waveguide parameters. The normalized frequency and waveguide thickness, also known as the V number, of a step-index planar waveguide is defined as

$$$\tag{46} V = \frac{2 \pi}{\lambda} d \sqrt{n_1^2 - n_2^2} = \frac{\omega}{c} d \sqrt{n_1^2 - n_2^2}$$$

where d is the thickness of the waveguide core.

The propagation constant β can be represented by the following normalized guide index:

$$$\tag{47} b = \frac{\beta^2 - k_2^2}{k_1^2 - k_2^2} = \frac{n_\beta^2 - n_2^2}{n_1^2 - n_2^2}$$$

where nβ = cβ/ωβλ/2π is the effective refractive index of the waveguide mode that has a propagation constant β.

The measure of the asymmetry of the waveguide is represented by an asymmetry factor a,  which depends on the polarization of the mode under consideration. For TE modes, we have

$$$\tag{48} a_E = \frac{n_2^2 - n_3^2}{n_1^2 - n_2^2}$$$

For TM modes, we have

$$$\tag{49} a_M =\frac{n_1^4}{n_3^4} \cdot \frac{n_2^2 - n_3^2}{n_1^2 - n_2^2}$$$

Note that for a given asymmetric structure, aM > aE. For symmetric waveguides, n3 = n2 and aE = aM = 0.

Mode Parameters

For a guided mode, k1 >  β > k2 > k3. Therefore, positive real parameters h1γ2, and γ3 exist such that

$$$\tag{50} k_1^2 - \beta^2 = h_1^2$$$ $$$\tag{51} \beta^2 - k_2^2 = \gamma_2^2$$$ $$$\tag{52} \beta^2 - k_3^2 = \gamma_3^2$$$

In correlation with the discussions in the waveguide modes tutorial leading to (7), it can be seen from (50) that h1 = k1cosθ, which has the meaning of the transverse component of the wavevector in the core region of a refractive index n1.

For a guided mode, the transverse components of the wavevectors in the substrate and cover regions given by h2 = (k22 -β2)1/2 and h3 = (k32 -β2)1/2, respectively, are purely imaginary because β > k2 > k3. The field of the guide mode has to decay exponentially in the transverse direction in the substrate and cover regions with γ2 = |h2| and γ3 = |h3| being the decay constants in these regions.

For a substrate radiation modek1 > k2 > β k3. Then h2 can be chosen to be real and positive, and (51) is replaced by $$$\tag{53} k_2^2 - \beta^2 = h_2^2$$$while (52) is still valid in this case.

For a substrate-cover radiation modek1 > k2 > k3 > β. Then both h2 and h3 are real and positive. In this case, in addition to replacing (51) with (53), (52) is replaced by $$$\tag{54} k_3^2 - \beta^2 = h_3^2$$$

The transverse field pattern of a mode is characterized by the transverse parameters h1γ(or h2), and γ3 (or h3). Because k1k2, and k3 are well-defined parameters of a given waveguide, the only parameter that has to be determined for a particular waveguide mode is the longitudinal propagation constant β. Once the value of β is found, the parameters associated with the transverse field pattern are completely determined. Therefore, a waveguide mode is completely specified by its β.

Alternatively, if any one of its transverse parameters, such as h1 for most instances, is determined, the value of its β is also determined, by (50), and the mode is completely specified also. As will be seen in the following, this approach is commonly taken for solving the normal modes of a waveguide.

Guided TE Modes

The fields of a TE mode are obtained by solving (30) for $$\mathcal{E}_y$$ and by using (32) and (33) for $$\mathcal{H}_x$$ and $$\mathcal{H}_z$$, respectively [refer to the wave equations for waveguides tutorial].

The boundary conditions require that $$\mathcal{E}_y$$, $$\mathcal{H}_x$$, and $$\mathcal{H}_z$$ be continuous at the interfaces at x = ±d/2 between layers of different refractive indices. From (32) and (33), it can be seen that this is equivalent to requiring $$\mathcal{E}_y$$ and $$\partial\mathcal{E}_y /\partial x$$ be continuous at these interfaces.

For a guided mode, we have use  h1γ2, and γ3 defined above for the transverse field parameters in the core, substrate, and cover regions, respectively. The solutions of (30) and the requirement of the boundary conditions yield the following mode field distribution:

\tag{55} \hat{\mathcal{E}_y} = C_{TE} \, \left\lbrace \begin{align} & \cos(h_1d/2 - \psi) \exp[\gamma_3 (d/2 - x)], \qquad x > d/2 \\ & \cos(h_1 x - \psi), \qquad \qquad \qquad \qquad -d/2 < x < d/2 \\ & \cos(h_1d/2 + \psi) \exp[\gamma_2 (d/2 + x)], \qquad x < - d/2 \\ \end{align} \right.

and the following eigenvalue equations:

$\tag{56}\tan h_1 d = \frac{h_1 (\gamma_2 + \gamma_3)}{h_1^2 - \gamma_2\gamma_3}$

and

$\tag{57}\tan2\psi = \frac{h_1 (\gamma_2 - \gamma_3)}{h_1^2 + \gamma_2\gamma_3}$

To normalized the mode field, we apply the normalization relation of (44) to the field in (55) [refer to the wave equations for waveguides tutorial]. This procedure yields

$\tag{58} C_{TE} = \sqrt{\frac{\omega\mu_0}{\beta d_E}}$

where

$\tag{59} d_E = d + \frac{1}{\gamma_2} + \frac{1}{\gamma_3}$

is the effective waveguide thickness for a guided TE mode.

Guided TM Modes

The fields of a TM mode are obtained by solving (34) for $$\mathcal{H}_y$$ and by using (35) and (36) for $$\mathcal{E}_x$$ and $$\mathcal{E}_z$$, respectively [refer to the wave equations for waveguides tutorial]. Note that for the step-index waveguide considered here, dε/dx = 0 in each waveguide layer except at the boundaries. The boundary conditions require that $$\mathcal{H}_y$$, $$\epsilon\mathcal{E}_x$$, and $$\mathcal{E}_z$$ be continuous at the interfaces at x = ±d/2 between layers of different refractive indices. Note that $$\mathcal{E}_x$$ is not continuous because it is the electric field component normal to the interfaces where discontinuities in ε occur. Similarly, $$\partial \mathcal{H}_y /\partial x$$ is not continuous at the interfaces. Rather, it is $$\epsilon^{-1}\partial\mathcal{H}_y /\partial x$$, or $$n^{-2}\partial\mathcal{H}_y /\partial x$$, that is continuous at the interfaces. Therefore, the boundary conditions are simply that $$\mathcal{H}_y$$ and $$n^{-2}\partial\mathcal{H}_y /\partial x$$ are continuous at the interfaces.

For a guided TM mode, the solutions of (34) [refer to the wave equations for waveguides tutorial] and the requirement of the boundary conditions yield the following mode field distribution:

\tag{60} \hat{\mathcal{H}}_y = C_{TM} \, \left\lbrace \begin{align} &\cos(h_1 d/2 - \psi)\exp[\gamma_3(d/2 - x)], \qquad x > d/2 \\ &\cos(h_1 x - \psi), \qquad \qquad \qquad \qquad -d/2 < x < d/2 \\ &\cos(h_1 d/2 + \psi)\exp[\gamma_2 (d/2 + x)], \qquad x < -d/2 \end{align} \right.

and the following eigenvalue equations:

$\tag{61}\tan h_1 d = \frac{(h_1 /n_1^2)(\gamma_2 /n_2^2 + \gamma_3 /n_3^2)}{(h_1 /n_1^2)^2 - \gamma_2\gamma_3 /n_2^2 n_3^2}$

and

$\tag{62}\tan 2\psi = \frac{(h_1 /n_1^2)(\gamma_2 /n_2^2 - \gamma_3 /n_3^2)}{(h_1 /n_1^2)^2 + \gamma_2\gamma_3 /n_2^2 n_3^2}$

To normalize the mode field, we apply the normalization relation of (45) [refer to the wave equations for waveguides tutorial] to the field in (60). This procedure yields

$\tag{63}C_{TM} = \sqrt{\frac{\omega\epsilon_0 n_1^2}{\beta d_M}}$

where the effective waveguide thickness for a guided TM mode is

$\tag{64}d_M = d + \frac{1}{\gamma_2 q_2} + \frac{1}{\gamma_3 q_3}$

and

$\tag{65} q_2 = \frac{\beta^2}{k_1^2} + \frac{\beta^2}{k_2^2} - 1$ $\tag{66} q_3 = \frac{\beta^2}{k_1^2} + \frac{\beta^2}{k_3^2} - 1$

Modal Dispersion

Guided modes have discrete allowed values of β. They are determined by the allowed values of h1 because β and h1 are directly related to each other through (50). Because γ2 and γ3 are uniquely determined by β through (51) and (52), respectively, they are also uniquely determined by h1. In terms of the normalized waveguide parameters, we have

$\tag{67}\gamma_2^2 d^2 = \beta^2 d^2 - k_2^2 d^2 = V^2 - h_1^2 d^2$$\tag{68}\gamma_3^2 d^2 = \beta^2 d^2 - k_3^2 d^2 = (1 + a_E)V^2 - h_1^2 d^2$

Therefore, there is only one independent variable h1 in the eigenvalue equations, (56) for TE modes and (61) for TM modes. The solutions of (56) yield the allowed parameters for guided TE modes, while those of (61) yield the parameters for guided TM modes. A transcendental equation such as (56) or (61) is usually solved graphically by plotting its left- and right-hand sides as a function of h1d while using (67) and (68) to replace γ2 and γ3 by expressions in terms of h1d. The solutions yield the allowed values of β, or the normalized guide index b, as a function of the parameters a and V. The results for the first few guided TE modes are shown in figure 5 below. For a given waveguide, a guided TE mode has a larger propagation constant than the corresponding TM mode of the same order:

$\tag{69} \beta_m^{TE} > \beta_m^{TM}$

However, for ordinary dielectric waveguides where $$n_1 - n_2 \ll n_1$$, the difference is very small. Then figure 5 can be used approximately for TM modes with a = aM.

For a given waveguide, the values of aE and aM, as well as those of d and $$n_1^2 - n_2^2$$, are fixed. Then, β is a function of optical frequency ω because V depends on ω. Figure 6 below illustrates a typical relation between β and ω for guided modes of different orders.

Comparing β, k1, and k2 in figure 6, its is seen that the propagation constant of a waveguide mode has a frequency dependence contributed by the structure of the waveguide in addition to that due to material dispersion. This extra contribution also causes different modes to have different dispersion properties, resulting in the phenomenon of modal dispersion. Polarization dispersion also exists because TE and TM modes generally have different propagation constants. Polarization dispersion is very small in weakly guiding waveguides where $$n_1 - n_2 \ll n_1$$.

Example

An asymmetric slab waveguide is made of a polymer layer of thickness d =  1 µm deposited on a silica substrate. At 1 µm optical wavelength, n1 = 1.77 for the polymer guiding layer, n2 = 1.45 for the silica substrate, and n3 = 1 for the air cover. Find the propagation constants of the guided TE and TM modes of this waveguide. Plot the mode field distributions.

With the given parameters of the waveguide, we find that V = 6.378,  aE = 1.07, and aM = 10.5 by using (46), (48), and (49). We also find that k1 = 2πn1/λ = 11.12 µm-1k2 = 2πn2/λ = 9.11 µm-1, and k1 = 2πn3/λ = 6.28 µm-1. To find the propagation constant, the parameter h1 has to be found by solving (56) for a TE mode or (61) for a TM mode. To solve the eigenvalue equations, we take the variable ξh1d and express γ2d  and γ3d  in terms of ξ by using the relations in (67) and (68):

$$\gamma_2 d = (V^2 - \xi^2)^{1/2}$$    and    $$\gamma_3 d = [(1 + a_E)V^2 - \xi^2]^{1/2}$$

Then the eigenvalue equation in (56) for the TE mode can be expressed in terms of a single variable ξ as

$\tan \xi = \xi \frac{(V^2 - \xi^2)^{1/2} + [(1+a_E)V^2 - \xi^2]^{1/2}}{\xi^2 - (V^2 - \xi^2)^{1/2}[(1+a_E)V^2 - \xi^2]^{1/2}}$

and the eigenvalue equation in (61) for the TM mode can be expressed as

$\tan\xi = \xi\frac{n_1^2 n_3^2 (V^2-\xi^2)^{1/2} + n_1^2 n_2^2 [(1+a_E)V^2 - \xi^2]^{1/2}}{n_2^2 n_3^2 \xi^2 - n_1^4(V^2-\xi^2)^{1/2}[(1+a_E)V^2 - \xi^2]^{1/2} }$

These equations yield only discrete eigenvalues for given values of waveguide parameters n1n2n3, V, and aE. They are transcendental equations that have to be solved graphically or numerically. With given waveguide parameters, numerical solution yields two eigenvalues for each of the two equations, indicating two guided TE modes and two guided TM modes. Once the eigenvalues for ξ are found, h1γ2, and γ3 are found. They can be used to find the phase $$\psi$$ from (57) for a TE mode and from (62) for a TM mode. The propagation constant can be found using (50) as $$\beta = (k_1^2 - h_1^2)^{1/2}$$. The effective waveguide thickness can be calculated directly from (59) for a TE mode and from (64) for a TM mode. The numerical results, as well as the confinement factors ΓTE and ΓTM discussed later, are summarized below.

We see from the listed values that a TE mode has a larger propagation constant than a TM mode of the same order, confirming the relation stated in (69). Among all of the modes found for this waveguide, β has the largest value for the TE0 mode and the smallest value for the TM1 mode. Using the mode parameters listed above, the distributions of $$\hat{\mathcal{E}}_y(x)$$ given in (55) for the TE modes and $$\hat{\mathcal{H}}_y(x)$$ given in (60) for the TM modes are plotted in figure 7 below.

Cutoff Conditions

As discussed above, γ2, and γ3 are real and positive for a guided mode, so that the fields of the mode decay exponentially in the transverse direction outside the core region and remain bound to the core. This is equivalent to the condition that θθc2 > θc3 in the ray optics picture illustrated in figure 3 [refer to the waveguide modes tutorial] so that the ray in the core is totally reflected by both interfaces.

Because θc2 > θc3, the transition from a guided mode to an unguided radiation mode occurs when θ = θc2. This corresponds to the condition that β = k2 and γ2 = 0. As can be seen from the mode field solutions in (55) and (60), the fields extend to infinity on the substrate side for γ2 = 0. This defines the cutoff condition for guided modes. The cutoff condition is determined by γ2 = 0, rather than by γ3 = 0, because γ3 > γ2 and γ2 reaches zero first as their values are reduced.

At cutoff, V = Vc. The cutoff value Vc for a particular guided mode is the value of V at the point where the curve of its b versus V dispersion relation, shown in figure 5 above, intersects with the horizontal axis b = 0. From (67) and (68), we find by setting γ2 = 0 that

$\tag{70} h_1 d = V_c \qquad \text{and} \qquad \gamma_3 d = \sqrt{a_E}V_c$

at cutoff. Substitution of (70) and γ2 = 0 in (56) for a guided TE mode yields

$\tag{71}\tan V_c = \sqrt{a_E}$

Therefore, the cutoff condition for the mth guided TE mode is

$\tag{72}V_m^c = \tan^{-1} \sqrt{a_E} + m\pi, \qquad m = 0, 1, 2, ...$

A similar mathematical procedure yields the following cutoff condition for the mth guided TM mode:

$\tag{73}V_m^c = \tan^{-1} \sqrt{a_M} + m\pi, \qquad m = 0, 1, 2, ...$

Because aM aE for a given asymmetric waveguide, the value of Vc for a TM mode is larger than that for a TE mode of the same order.

Using (46), we can write

$\tag{74}V_m^c = \frac{2\pi}{\lambda_m^c}d\sqrt{n_1^2 - n_2^2} = \frac{\omega_m^c}{c}d\sqrt{n_1^2 - n_2^2}$

where $$\lambda_m^c$$ is the cutoff wavelength, λc, and $$\omega_m^c$$ is the cutoff frequencyωc, of the mth mode. The mth mode is not guided at a wavelength longer than $$\lambda_m^c$$, or a frequency lower than $$\omega_m^c$$.

For given waveguide parameters, (72) and (73) can be used to determine the cutoff wavelengths of TE and TM modes, respectively, from (74). For a given optical wavelength, they can be used to determine the waveguide parameters that allow the existence of a particular guided mode. For given waveguide parameters and optical wavelength, they can be used to determine the number of guided modes for the waveguide. Therefore, for a given optical wavelength and a waveguide with a given V number, the total number of guided TE modes is simply

$\tag{75}M_{TE} = \left[ \frac{V}{\pi} - \frac{1}{\pi}\tan^{-1}\sqrt{a_E} \right]_{\text{int}}$

while that of the guided TM mode is

$\tag{76}M_{TM} = \left[ \frac{V}{\pi} - \frac{1}{\pi}\tan^{-1}\sqrt{a_M} \right]_{\text{int}}$

where []int means the nearest integer larger than the value in the bracket.

A waveguide with M = 1 that supports only the fundamental TE0 and/or TM0 mode is called a single-mode waveguide. A waveguide that also suppors any number of high-order modes is a multimode waveguide.

Example

For the waveguide given in the previous example, verify that there are exactly two guided TE modes and two guided TM modes. If the thickness d of the polymer layer is reduced without changing the index profile of the structure, which among these four modes gets cut off first? At what value of d is it cut off?

We already find that V = 6.378, aE = 1.07, and aM = 10.5 for the waveguide with d = 1 µm and other parameters given in the previous example. Therefore, by applying (75) and (76), we find that

$M_{TE} = \left[ \frac{6.378}{\pi} - \frac{1}{\pi} \tan^{-1} \sqrt{1.07} \right]_{\text{int}} = [1.77]_\text{int} = 2$

and

$M_{TM} = \left[ \frac{6.378}{\pi} - \frac{1}{\pi} \tan^{-1} \sqrt{10.5} \right]_{\text{int}} = [1.63]_\text{int} = 2$

verifying that there are exactly two guided TE modes and two guided TM modes.

From (72) and (73), we learn that (1) for TE and TM modes of the same mode number, the TM mode has a larger value of Vc because aM > aE; and (2) among modes of the same polarization, a higher-order mode has a larger value of Vc. Therefore, among all guided modes found in a waveguide, the highest-order TM mode gets cut off first when the V value is reduced. For the problem under consideration, the TM1 mode is cut off first as d is reduced so that the value of V is reduce. Using (73) we find that the waveguide does not support the TM1 mode when

$V < V_1^c = \tan^{-1}\sqrt{10.5} + \pi = 4.413$

This condition yields d < 0.69 µm, by using (46), for the TM1 mode to be cut off from the waveguide. For V = 4.413 corresponding to d = 0.69 µm, MTE = [1.15]int = 2. Therefore, the TE1 mode is still supported when TM1 reaches its cutoff point.

Mode Confinement

The mode confinement factor, Γmode, of a guided mode is defined as the fraction of its power in the core region. In an active waveguide, such as that in a semiconductor laser or in a waveguide amplifier, the core guiding region is where the optical gain is, whereas the substrate and cover regions are usually passive media without an optical gain. Only the fraction of power in the core region sees a gain, and the effective gain of a given mode is proportionally reduce. Therefore, the confinement factor is very important in assessing the effective gain of an active optical waveguide for a particular guided mode.

Because the power of a TE mode can be calculated using (38) [refer to the wave equations for waveguides tutorial], the confinement factor for a TE mode in a slab waveguide is given by

$\tag{77}\Gamma_{TE}=\frac{\displaystyle\int\limits_{-d/2}^{d/2}|\mathcal{E}_y(x)|^2dx}{\displaystyle\int\limits_{-\infty}^{\infty}|\mathcal{E}_y(x)|^2dx}=\frac{2\beta}{\omega\mu_0} \displaystyle\int\limits_{-d/2}^{d/2}|\hat{\mathcal{E}}_y(x)|^2dx$

For a TM mode, the power can be calculated using (39), and the confinement factor is given by

$\tag{78}\Gamma_{TM}=\frac{\displaystyle\int\limits_{-d/2}^{d/2}n_1^{-2}|\mathcal{H}_y(x)|^2dx}{\displaystyle\int\limits_{-\infty}^{\infty}n^{-2}(x)|\mathcal{H}_y(x)|^2dx} =\frac{2\beta}{\omega\epsilon_1}\displaystyle\int\limits_{-d/2}^{d/2}|\hat{\mathcal{H}}_y(x)|^2dx=\frac{2\omega\epsilon_1}{\beta}\displaystyle\int\limits_{-d/2}^{d/2}|\hat{\mathcal{E}}_x(x)|^2dx$

Using (55) to carry out the integration in (77) together with (56) and (57) to simplify the expression, it can be shown that

$\tag{79}\Gamma_{TE} = \frac{1}{d_E}\left(d+\frac{1}{\gamma_2}\cdot\frac{1}{1+h_1^2/\gamma_2^2}+\frac{1}{\gamma_3}\cdot\frac{1}{1+h_1^2/\gamma_3^2} \right)$

A similar procedure using (60) in (78) together with (61) and (62) yields

$\tag{80}\Gamma_{TM} = \frac{1}{d_M}\left(d+\frac{1}{\gamma_2q_2}\cdot\frac{1}{1+h_1^2/\gamma_2^2}+\frac{1}{\gamma_3q_3}\cdot\frac{1}{1+h_1^2/\gamma_3^2} \right)$

As discussed earlier and displayed in (69), for guided modes of the same order, the TE mode has a larger propagation constant than the corresponding TM mode, βTE > βTM. Therefore, from (50)-(52), we also have

$\tag{81}h_1^{TE} < h_1^{TM}, \qquad \gamma_2^{TE} > \gamma_2^{TM}, \qquad \text{and} \qquad \gamma_3^{TE} > \gamma_3^{TM}$

for TE and TM modes of the same order. Because q2 and q3 defined in (65) and (66) for a TM mode can be either larger or smaller than unity, the relationship between dE and dM and that between ΓTE and ΓTM for modes of the same order are less straightforward. Indeed, for a given mode order, ΓTE can be either larger or smaller than ΓTM, but the difference between them is small. For modes of the same polarization, however, a low-order mode is more confined than a high-order mode. Therefore, we can only state that

$\tag{82}\Gamma_{TE} \approx \Gamma_{TM}, \qquad \text{but} \qquad \Gamma_m > \Gamma_{m+1}$

The fundamental TE mode has the largest propagation constant but it may or may not have the largest confinement factor. Either the fundamental TE or the fundamental TM mode has the largest confinement factor among all guided modes. The confinement factors for the fundamental TE and TM modes of a symmetric waveguide where n2 = n3 is shown in figure 8 below.

Example

Find the confinement factors for the guided TE and TM modes determined in the first example in this tutorial and compare them among modes of different polarizations and modes of different orders.

With the values of h1γ2, and γ3, as well as those of dE and dM, found and listed in the first example, the confinement factors can be calculated using (79) and (80) for the TE and TM modes, respectively. The results are listed in the last column of the table in the first example. By examining the values of hte confinement factors for different modes, we find the following characteristics.

1. The confinement factors for TE and TM modes of the same order are about the same. There is no clear pattern that indicates whether a TE mode or a TM mode has a larger confinement factor. For example, the TM0 mode has a slightly larger confinement factor than the TE0 mode, but the relationship is reversed between TE1 and TM1 modes.
2. Among modes of the same polarization but different orders, it is clear that the confinement factor decreases as the mode order increases. For example, the TE0 mode has a larger confinement factor than the TE1 mode. The same statement can be made for TM modes.

The next part continues in the Symmetric Slab Waveguide tutorial.