Graded-Index Planar Waveguides

This is a continuation from the previous tutorial - Symmetric Slab Waveguides.

In the previous two tutorials, we have considered slab waveguides that have step-index profiles. In this tutorial, we consider graded-index planar waveguides, which do not have the piecewise-constant index profiles of step-index waveguides. Two types of graded-index planar waveguides, shown in figure 12 below, are of practical interest.

One is the smooth graded-index waveguide, which has a smooth index profile across its entire structure in the x direction, as shown in figure 12(a). The other is the step-bounded graded-index waveguide, which has a graded-index profile on one side but is bounded by a step-index boundary on the other side of its core, as shown in figure 12(b).

A graded-index planar waveguide can be either asymmetric or symmetric. For the purpose of general discussion, we consider the index profiles shown in figure 12 with the index at the peak of the profile in the waveguide core being n₁ and the indices of the substrate and the cover far away from the core of the waveguide being n₂ and n₃, respectively, where n₁ > n₂ > n₃. The waveguide core is the region between the two points x = a and x = b defined by n(a) = n(b) = n₂ within which an index n(x) larger than both n₂ and n₃ can be found.

The discussions in the wave equations for waveguides tutorial regarding planar waveguides apply to graded-index planar waveguides as well. Many of the general qualitative conclusions obtained in the step-index planar waveguides tutorial are also valid for graded-index planar waveguides. The modes of graded-index planar waveguides are still either TE or TM. It is also true that the fundamental mode is TE₀ and that \(\beta_m^{TE} > \beta_m^{TM}\).

The guided modes can be found by solving (30) [refer to the wave equations tutorial]

\[\tag{94}\frac{\partial^2\mathcal{E}_y}{\partial x^2}+[k^2(x)-\beta^2]\mathcal{E}_y=0\]

 for \(\mathcal{E}_y(x)\), followed by using (32) and (33) to obtain \(\mathcal{H}_x(x)\) and \(\mathcal{H}_z(x)\), respectively. Similarly, the guided TM modes can be found by solving (34),

\[\tag{95}\frac{\partial^2\mathcal{H}_y}{\partial x^2}+[k^2(x)-\beta^2]\mathcal{H}_y=\frac{1}{\epsilon}\frac{\text{d}\epsilon}{\text{d}x}\frac{\partial\mathcal{H}_y}{\partial x}\]

for \(\mathcal{H}_y(x)\), followed by using (35) and (36) to obtain \(\mathcal{E}_x(x)\) and \(\mathcal{E}_z(x)\), respectively.

For a graded-index waveguide, the propagation constant

\[\tag{96}k(x)=\frac{\omega}{c}n(x)=\frac{2\pi}{\lambda}n(x)\]

is a spatially varying function of x. Therefore, (94) and (95) cannot be readily solved analytically. Though such second-order ordinary differential equations can be solved numerically, here we are interested in gaining physical insight into the waveguide characteristics without resorting to complete numerical solutions. For this purpose, we consider the common situation where the term on the right-hand side of (95) is very small compared with \(\partial^2\mathcal{H}_y/\partial{x^2}\) so that it can be neglected for (95) to take the form of (94) approximately. We then find that these equations have the form of the Schrodinger equation in quantum mechanics. Approximate solutions can be obtained using the Wentzel-Kramers-Brillouin (WKB) approximation developed in quantum mechanics for solving the Schrodinger equation of a general graded potential.

The central concept of the WKB method is to realize the fact that a guided mode can be established if it forms a standing wave pattern in the transverse x direction that has oscillatory spatial variations in a certain region within the waveguide core but has decaying fields away from the core in the substrate and cover, as illustrated in Figure 13 below.

As discussed in the waveguide modes tutorial, the condition for such a standing wave pattern to be formed is that the total phase shift in a round-trip transverse passage be an integral multiple of 2π. This condition is given in (7) [refer to the waveguide modes tutorial] for a step-index planar waveguide. A similar, but more complicated, condition can be obtained for a graded-index planar waveguide without going through the detailed WKB analysis by simply modifying (7). The key point is to realize that the first term, 2k₁dcosθ, in (7) is the round-trip phase shift through the oscillatory region in the x direction. We notice that \(k_1\cos\theta=h_1=(k_1^2-\beta^2)^{1/2}\) for a step-index waveguide and that d is the range of the oscillatory region in the waveguide. For a graded-index waveguide, k₁ has to be replaced by k(x), and the oscillatory region is where k(x) > β so that the following function

\[\tag{97}p^2(x)=k^2(x)-\beta^2\]

has positive values. Therefore, as seen in figure 13, the oscillatory region for a given mode of propagation constant β is the range bounded by the two turning points x = x₂ and x = x₃, where p(x₂) = p(x₃) = 0 for k(x₂) = k(x₃) = β. In the oscillatory region, where x₂ < x < x₃, we have p²(x) > 0 so that a real, positive square root \(p(x)=(k^2(x)-\beta^2)^{1/2}\) exists. In the evanescent regions, where x < x₂ and x₃ < x, p²(x) < 0 and its square roots are purely imaginary.

From these discussions, it is clear that the following condition can be obtained for the guided modes of a graded-index planar waveguide:

\[\tag{98}2\displaystyle\int\limits_{x_2}^{x_3}[k^2(x)-\beta^2]^{1/2}\text{d}x=2m\pi-\varphi_2-\varphi_3,\qquad{}m=0,1,2,...,\]

where x₂ and x₃ are the roots of \(p^2(x)=k^2(x)-\beta^2=0\). Because x₂, x₃, φ₂, and φ₃ are all functions of β, this equation has to be solved self-consistently for β with a given integer m for the mth-order mode. The phase shifts, φ₂ and φ₃, are polarization dependent as well as mode dependent; therefore, TE and TM modes have different solutions, thus slightly different values of β, for the same value of m in (98). They are

\[\tag{99}\varphi_2=-2\tan^{-1}\left[\frac{\beta^2-k^2(x_2^-)}{k^2(x_2^+)-\beta^2}\right]^{1/2}\qquad\text{and}\qquad\varphi_3=-2\tan^{-1}\left[\frac{\beta^2-k^2(x_3^+)}{k^2(x_3^-)-\beta^2}\right]^{1/2}\]

for a TE mode of propagation constant β, and

\[\tag{100}\varphi_2=-2\tan^{-1}\frac{k^2(x_2^+)}{k^2(x_2^-)}\left[\frac{\beta^2-k^2(x_2^-)}{k^2(x_2^+)-\beta^2}\right]^{1/2}\qquad\text{and}\qquad\varphi_3=-2\tan^{-1}\frac{k^2(x_3^-)}{k^2(x_3^+)}\left[\frac{\beta^2-k^2(x_3^+)}{k^2(x_3^-)-\beta^2}\right]^{1/2}\]

for a TM mode of propagation constant β.

The values of the phase shifts are in the range of -π < φ₂, φ₃ < 0 for all guided modes. At x = x₂, where we have assumed that the index grading is smooth for both types of graded-index waveguides shown in figure 12, \(\beta^2-k^2(x_2^-)=k^2(x_2^+)-\beta^2\) as \(x_2^-\) and \(x_2^+\) approach the turning point x₂ infinitesimally from the left and right, respectively. Therefore, we find that φ₂ = - π/2 from (99) and (100) for any guided TE or TM mode. Following the same reasoning, we also find that φ₃ = - π/2 for any guided TE or TM mode for the smooth graded-index waveguide shown in figure 12(a), which has smooth index grading at x₃. For the step-bounded graded-index waveguide shown in figure 12(b), however, the value of φ₃ cannot be so generalized but is a function of \(k_1=k(x_3^-)\) and \(k_3=k(x_3^+)\), with k₁ > k₃, because the turning point x₃ is located at the abrupt index step.

To summarize, for a smooth graded-index waveguide, the eigenvalue equation for the propagation constants of its guided modes can be simply expressed as

\[\tag{101}\displaystyle\int\limits_{x_2}^{x_3}[k^2(x)-\beta^2]^{1/2}\text{d}x=\left(m+\frac{1}{2}\right)\pi,\qquad{m=0,1,2,...}\]

For a step-bounded graded-index waveguide, however, the eigenvalue equation can only be simplified to

\[\tag{102}\displaystyle\int\limits_{x_2}^{x_3}[k^2(x)-\beta^2]^{1/2}\text{d}x=\left(m+\frac{1}{4}\right)\pi-\frac{\varphi_3}{2},\qquad{m=0,1,2,...}\]

where φ₃ takes the form in (99) for a TE mode and that in (100) for a TM mode with \(k(x_3^-)=k_1\) and \(k(x_3^+)=k_3\). For both types of graded-index waveguides at locations away from the immediate vicinity of the turning points, the unnormalized mode fields for a guided mode have the following asymptotic form:

\[\tag{103}\begin{align}&\text{TE}\;\mathcal{E}_y(x)\\&\text{TM}\;\mathcal{H}_y(x)\end{align}\sim\begin{cases}\frac{1}{\sqrt{|p(x)|}}\exp\left[-\displaystyle\int\limits_x^{x_2}|p(x')|\text{d}x'\right],\qquad{x\lt{x_2}}\\\frac{2}{\sqrt{p(x)}}\cos\left[\displaystyle\int\limits_{x_2}^xp(x')\text{d}x'-\frac{\pi}{4}\right],\qquad{x_2\lt{x}\lt{x_3}}\\\frac{(-1)^m}{\sqrt{|p(x)|}}\exp\left[-\displaystyle\int\limits_{x_3}^x|p(x')|\text{d}x'\right],\qquad{x\gt{x_3}}\end{cases}\]

where a factor of (-1)^m is used in the field pattern for x > x₃ to account for the correct phase of the field at x = x₃.

Number of Modes

Graded-index waveguides are often used as multimode waveguides because of their low modal dispersion compared with step-index waveguides. This feature is discussed in great detail for optical fibers, but the concept applies generally to planar waveguides as well. In contrast, single-mode waveguides are preferably step-index waveguides because the step-index geometry allows precise control of the waveguide parameters. In addition, the step-index geometry also conforms with the various junction structures for high-performance optoelectronic devices, which normally requires single-mode characteristics.

The number of guided modes supported by a graded-index waveguide can be found by finding the largest integral value of m for a solution of β from its eigenvalue equation. Because β > k₂ > k₃ for any guided mode, the minimum possible value of β is β = k₂. The turning points for β = k₂, which are x₂ = a and x₃ = b shown in figure 12, are where k(a) = k(b) = k₂ and n(a) = n(b) = n₂. The number of guided modes in a given polarization that are supported by a waveguide is found by adding 1 to the mode number of the highest guided mode because the fundamental mode has a mode number of m = 0. For a smooth graded-index waveguide, we find that for β = k₂, the phase shifts given in (99) and (100) are not - π/4, obtained above for (101) where β > k₂, but are φ₂ = φ₃ = 0 for both TE and TM modes. Then, from (98), the number of guided TE and TM modes supported by a smooth graded-index waveguide as shown in figure 12(a) are

\[\tag{104}M_{\text{TE}}=M_{\text{TM}}=\left[\frac{2}{\lambda}\displaystyle\int\limits_{a}^{b}[n^2(x)-n_2^2]^{1/2}\text{d}x\right]_{\text{int}}\]

where []_int means taking the nearest integer larger than the value in the brackets.

For a step-bounded graded-index waveguide, we find that φ₂ = 0 but \(\varphi_3=-2\tan^{-1}\sqrt{a_E}\) for a TE mode and \(\varphi_3=-2\tan^{-1}\sqrt{a_M}\) for a TM mode,  where a_E and a_M are the asymmetric factors defined in (48) and (49), respectively [refer to the step-index planar waveguides tutorial]. Therefore, the numbers of guided TE and TM modes supported by a step-bounded graded-index waveguide as shown in figure 12(b) are

\[\tag{105}M_{\text{TE}}=\left[\frac{2}{\lambda}\displaystyle\int\limits_a^b[n^2(x)-n_2^2]^{1/2}\text{d}x-\frac{1}{\pi}\tan^{-1}\sqrt{a_E}\right]_{\text{int}}\]

and

\[\tag{106}M_{\text{TM}}=\left[\frac{2}{\lambda}\displaystyle\int\limits_a^b[n^2(x)-n_2^2]^{1/2}\text{d}x-\frac{1}{\pi}\tan^{-1}\sqrt{a_M}\right]_{\text{int}}\]

Example

A planar LiNbO₃ waveguide made by Ti diffusion is a step-bounded graded-index waveguide that has an index profile like the one shown in figure 12(b). The optical axis of the LiNbO₃ crystal, which is negative uniaxial, is lined up with the z axis of this waveguide so that both TE and TM modes see only the ordinary index n_o. At λ = 1.3 μm, n_o = 2.222. We take the index step to be located at x = a = 0. Then n(x) = n₃ = 1 for x > 0. The graded-index profile created by Ti diffusion has the following Gaussian profile:

\[n(x)=n_o+\Delta{n}\text{e}^{-x^2/d^2},\qquad\text{for }x\lt0\]

where d is the diffusion depth of Ti in LiNbO₃ required to define the waveguiding region. The diffusion depth is determined by the Ti diffusion coefficient D and the total time duration Δt for the diffusion process as \(d=\sqrt{D\Delta{t}}\). At a temperature of 1020°C, D = 1.4 x 10^-12 cm² s^-1. Design a single-mode Ti:LiNbO₃ waveguide that supports exactly one TE mode and one TM mode at λ = 1.3 μm.

With the given index profile, the condition for the waveguide to support exactly one TE mode can be found from (105) for M_TE = 1 as

\[1+\frac{1}{\pi}\tan^{-1}\sqrt{a_E}\gt\frac{2\sqrt{2n_o\Delta{n}}}{\lambda}\displaystyle\int\limits_{-\infty}^0\text{e}^{-x^2/2d^2}\text{d}x\gt\frac{1}{\pi}\tan^{-1}\sqrt{a_E}\]

for Δn ≪ 1. Using the identity

\[\displaystyle\int\limits_{-\infty}^0\text{e}^{-x^2}\text{d}x=\int\limits_0^{\infty}\text{e}^{-x^2}\text{d}x=\frac{\sqrt{\pi}}{2}\]

this condition is reduced to

\[1+\frac{1}{\pi}\tan^{-1}\sqrt{a_E}\gt2\sqrt{\pi{n_o}\Delta{n}}\frac{d}{\lambda}\gt\frac{1}{\pi}\tan^{-1}\sqrt{a_E}\]

The condition for the waveguide to support exactly one TM mode can be obtained by replacing a_E with a_M in the above relation. A practical index step, which is controlled by the thickness of Ti deposited on the surface of LiNbO₃ during the diffusion process, is chosen to be Δn = 0.01. For this waveguide, we then have n₁ = n_o + Δn = 2.232, n₂ = n_o = 2.222, and n₃ = 1. We also find that a_E = 88 and a_M = 2188. With these parameters and with λ = 1.3 μm, we find

3.607 μm > d > 1.147 μm for a single TE mode

and

3.674 μm > d > 1.213 μm for a single TM mode.

To satisfy both conditions so that the waveguide supports exactly one TE mode and one TM mode, a good choice for the diffusion depth is d = 2 μm, with the chosen index step of Δn = 0.01. Such a waveguide can be made by Ti diffusion at 1020°C for 8 hours because Δt = d²/D ≈ 8 hours from the given value of D.

The next part continues with the Channel Waveguides tutorial.