Propagation Constant, $\beta$

When working with optical and photonic simulation tools such as RSoft LaserMOD, Photon Design HAROLD, ANSYS Lumerical, or FlexCompute Tidy3D, one of the first quantities that appears is the effective index, $n_{\mathrm{eff}}$.

At some point you’ll encounter the relationship

$$ n_{\mathrm{eff}} = \frac{\beta}{k_0} $$

where $\beta$ is called the propagation constant.

Many texts introduce this equation without spending much time explaining what $\beta$ actually represents. This post attempts to build some intuition.

A Wave Is More Than Something Moving Forwards

When we imagine a wave travelling down a waveguide, it’s natural to focus on the fact that energy is moving forwards.

However, a wave is also oscillating.

If we freeze time and look along the direction of propagation, we might see something like:

 /\      /\      /\
/  \    /  \    /  \
    \  /    \  /
     \/      \/

As the wave moves forwards, it repeatedly cycles through:

peak → zero → valley → zero → peak

This repeating cycle is described by the phase of the wave.

A complete cycle corresponds to an increase in phase of

$$ 2\pi $$

radians.

The Idea of Phase Accumulation

Imagine walking along a waveguide.

As you move forwards, the optical field progresses through more and more oscillation cycles.

We can think of this as the wave accumulating phase.

For example:

  • after a short distance, the field may have completed 10 cycles
  • after a longer distance, it may have completed 100 cycles
  • after a much longer distance, it may have completed 1000 cycles

The key question becomes:

How quickly does the phase accumulate as light travels?

The answer is the propagation constant.

The Propagation Constant

A guided optical mode can be written as

$$ E(z,t) = E_0 \cos(\beta z - \omega t) $$

The quantity $\beta$ determines how rapidly the phase changes with position.

Its units are:

$$ \mathrm{radians/meter} $$

which means:

$\beta$ tells us how many radians of phase are accumulated per metre of propagation.

A larger $\beta$ means the phase rotates more rapidly as light travels.

A smaller $\beta$ means the phase rotates more slowly.

A Useful Mental Model

In practice, I find it helpful to think of $\beta$ as:

optical phase accumulation per metre

Whenever you encounter $\beta$, mentally translate it as:

“How quickly does this optical wave move through its oscillation cycle as it propagates?”

A Runner on a Circular Track

Consider two runners on separate circular tracks.

Runner A completes one lap per minute.

Runner B completes two laps per minute.

After ten minutes:

  • Runner A has accumulated 10 laps
  • Runner B has accumulated 20 laps

The lap count plays a similar role to optical phase.

A larger propagation constant means the wave accumulates more phase over the same distance, just as the faster runner accumulates more laps over the same time.

Free Space and Waveguides

For a plane wave in free space:

$$ k_0 = \frac{2\pi}{\lambda_0} $$

where:

  • $k_0$ is the free-space wave number
  • $\lambda_0$ is the free-space wavelength

Inside a waveguide, things become more interesting.

The mode is confined by the geometry and interacts with multiple materials.

Instead of propagating with $k_0$, the mode propagates with $\beta$:

$$ \beta = n_{\mathrm{eff}} k_0 $$

This immediately leads to:

$$ n_{\mathrm{eff}} = \frac{\beta}{k_0} $$

This is why effective index can be interpreted as:

How much faster the phase accumulates compared to free space.

Why Simulators Report Effective Index Instead of β

Most optical mode solvers actually solve for $\beta$.

Internally, the numerical problem is an eigenvalue equation whose solution produces:

  • the modal field distribution
  • the propagation constant $\beta$

The software then converts this into an effective index:

$$ n_{\mathrm{eff}} = \frac{\beta}{k_0} $$

because values such as

neff = 2.45

are typically easier for humans to interpret than

β = 9.93 × 10^6 rad/m

The effective index is therefore not a completely separate quantity.

It is simply another way of expressing the propagation constant.

Why β Matters

The propagation constant sits at the heart of integrated photonics.

Straight Waveguides

The accumulated phase after travelling a distance $L$ is

$$ \phi = \beta L $$

Double the length and you double the phase accumulation.

Mach-Zehnder Interferometers

Two arms with different lengths accumulate different phases:

$$ \Delta\phi = \beta (L_1 - L_2) $$

The resulting interference pattern is determined entirely by this phase difference.

Ring Resonators

A resonance occurs when the round-trip phase shift equals an integer number of cycles:

$$ \beta L = 2\pi m $$

where $m$ is an integer.

Again, the propagation constant directly determines the device behaviour.

The One-Sentence Intuition

If you remember only one thing, make it this:

The propagation constant $\beta$ tells us how quickly an optical wave accumulates phase as it travels along a waveguide.

Everything else, including effective index, Mach-Zehnder interference, and ring resonances, follows naturally from that idea.

Looking Ahead: Group Index

The propagation constant describes how the phase of a wave moves.

When we start caring about how quickly information, energy, or an optical pulse moves, we need a different quantity: the group index.

That will be the subject of a future post.

[ Disclaimer - I used an LLM to help me write some parts of this blog post.]