TE and TM Modes in Symmetric Slab Waveguides

Key Takeaways

• The three types of modes in slab waveguides are guided mode, substrate radiation mode, and superstrate-cover radiation mode.

• Guided modes within a symmetric slab waveguide include TE mode and TM mode.

• In the TE mode of propagation, there is no electric field component in the direction of propagation. In the TM mode, the magnetic field component is absent in the direction of propagation. This makes TM mode and TE mode behave similarly.

Slab waveguides are useful for many applications including photonics

Slab waveguides are simple waveguides that provide light guidance in most photonics and optical circuits. The simplicity in geometry and mathematical expressions—both for guided and radiation modes—make slab waveguides useful for many applications. Apart from optical applications, designers use slab waveguides to model and understand waveguiding properties of other complicated optical waveguides.

Slab waveguide's refractive index varies only in one direction, making them one-dimensional. Slab waveguides support propagation-transverse electric (TE) mode and transverse magnetic (TM) mode. Both TE and TM modes in symmetric slab waveguides behave the same. In this article, we will take a closer look at wave propagation in symmetric slab waveguides, particularly TE and TM modes of propagation.

Symmetric Slab Waveguides

In a slab waveguide with materials with refractive indices n₁, n₂, and n₃ for film, substrate, and superstrate, respectively, the relationship maintained in the slab waveguide is:

n₁ >n₂ >n₃

Slab waveguides are called symmetric slab waveguides when the refractive indexes of the substrate and superstubstrate match.

n₂ = n₃

The waveguiding structure reduces to a core-cladding interface that supports two guided modes of propagation. 

TE and TM Modes of Propagation in Symmetric Slab Waveguides

In a slab waveguide, total internal reflection can occur at either the core-substrate interface or in the core-superstrate (cover) interface. Depending on the interface where the total internal reflection occurs, there are four types of slab waveguide modes.

Guided Mode

In this mode, the total internal reflection is confined only to the core region. Guided mode mitigates leaks leading to less optical losses. The number of guided modes in slab waveguide depends on:

• The core’s thickness.
• The wavelength of the electromagnetic wave.
• The refractive indices of the core. 
•  The refractive indices of the substrate and superstrate. 

TE and TM modes are guided modes in symmetric slab waveguides and follow even or odd field distribution. 

TE and TM Guided Modes

In the TE mode of propagation in a symmetric slab waveguide, there is no electric field component in the direction of propagation. In TM mode, the magnetic field component is absent in the direction of propagation. So, the behavior of both TE mode and TM mode in a symmetric slab waveguide is similar.

If a symmetric slab waveguide is uniform in the y-direction, the electric field becomes uniform in the same direction. The solution for the wave equation or Helmholtz equation can be sinusoidal or shown as the following equation, where and is the longitudinal phase constant:

The solution is sinusoidal when:

The solution is exponential when:

The solution for the wave equation in a symmetric slab waveguide can be shown with the following set of equations, where k and h are the transverse propagation constants:

Substrate Radiation Mode

The total internal reflection spreading between the core-substrate interface leads to substrate radiation mode. Substrate radiation mode is an unguided mode.

Superstrate Radiation Mode

The total internal reflection spreading between the superstrate and core interface is responsible for superstrate-cover unguided radiation mode.

From the modes of slab waveguides discussed, it is clear that TE and TM modes in symmetric slab waveguides are suitable for guided and low-loss propagation. Cadence’s software can help design symmetric and asymmetric slab waveguides for electromagnetic wave propagation.

Subscribe to our newsletter for the latest updates. If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.