# Laser Linewidth and Bandwidth Calculator

Created by Davide Borchia
Reviewed by Steven Wooding
Last updated: Oct 06, 2022

Lasers are often assumed to be nearly monochromatic: with our laser linewidth and bandwidth calculator, you will learn that this is not necessarily true.

• A quick but clear explanation of the operations of a laser device;
• What is the laser linewidth: a definition;
• How to calculate the laser linewidth: the formula for the deviation from monochromaticity;
• The differences between laser linewidth and laser bandwidth;
• How to calculate the laser bandwidth; and
• How to use our laser linewidth and bandwidth calculator.

## Lasers straight to the pointer

Lasers are devices that use the property of light and a medium to obtain emission of directional, monochromatic, and coherent light, which means that:

• Laser light propagates in one direction;
• Most of the photons emitted by a laser have the same wavelength; and
• The emitted photons are in phase.

How do we obtain these properties? The answer lies in the design of laser emitters. In any device, from a pointer to an anti-missile military megawatt laser, an excited medium emits, when stimulated, photons at a specific wavelength. The particular design of a laser's oscillating chamber, with two mirrors on the opposite ends, allows for the emission of a powerful light beam with high spatial and temporal coherence.

Each medium emits at a specific wavelength: a source of energy (for example, an electric field) excites the medium, moving a fraction of the material's electrons to a higher, metastable energy level. One of these electrons will fall to the ground state by chance, emitting a photon. This photon, bouncing back and forth in the amplification chamber, stimulates the relaxation of other electrons. A partially transparent window on one of the mirrors allows the escape of a portion of the amplified light.

💡 You might also be interested in checking out our laser brightness calculator.

## What is the linewidth of a laser?

In the ideal world where there's no air resistance and cows are spherical, lasers emit (if properly engineered) light of a single wavelength. In reality, sources of noise (both technical, due to the design of the device, and quantum, due to the intrinsic nature of the atomic-scale world) cause shifts in the emission of the device.

The amplitude of these shifts defines how much a laser device deviates from monochromaticity. We can quantify this phenomenon with the concept of the spectral linewidth of a laser

We define the linewidth of a laser as the full width at half maximum (FWHM) of the amplitude spectrum of a laser emitter. This definition may not be that clear, so let's analyze it in detail.

A real laser emits a spectrum of frequencies (the optical spectrum). The power of the laser output is distributed on this spectrum — the linewidth measures how much of the laser's output doesn't peak around the fundamental frequency.

Now that you know the definition of laser linewidth, we can learn how to calculate it with the laser linewidth equation.

## How to calculate the linewidth of a laser

The calculations to find the formula for the linewidth of a laser come from the depths of quantum mechanics (even though at the dawn of the field, the derivation of this quantity still used many classical concepts).

The calculation relies on the uncertainty principle associated with energy. Knowing that the Planck constant relates energy and time suggest that the characteristic times of the stimulated emission process significantly affect the output performances of laser devices.

The laser linewidth equation is:

$\Delta\nu = \frac{\pi\ h\ \nu\ \Gamma^2}{P}$

where:

• $\Delta\nu$ — Laser linewidth;
• $h$Planck constant, a reminder that we are dealing with quantum phenomena;
• $\nu$ — Fundamental frequency of the laser;
• $\Gamma$Q-factor of the "cold" laser cavity, a measure of the strength of the damping of the oscillations (also known as cavity linewidth); and
• $P$Power of the laser mode.

🔎 In the formula you can see the expression $h\nu$: this is the Planck's relation with which you can calculate the energy of a photon, and no, it's not a coincidence!

This formula is a modification of the original calculations for the spectral linewidth of a laser by Schawlow and Townes but is developed entirely in a quantum framework. It is considered the best expression for the linewidth.

## Laser linewidth vs. laser bandwidth

Apart from the laser spectral linewidth, each laser has a specific bandwidth. What's the difference?

You can consider the bandwidth as the possible spectrum of the laser's output, while the laser linewidth considers the "occupied" frequencies in the optical spectrum.

The laser bandwidth can't be calculated using a single equation but rather depends on multiple factors, and it isn't easy to model the characteristics of the resonating chamber. The value of the bandwidth is usually given in the device's specification.

We can convert the laser bandwidth starting from the fundamental wavelength of the laser $\lambda_0$ and the width of the wavelength range with the formula:

$\Delta\nu = \Delta\nu_{\text{end}}-\Delta\nu_{\text{begin}}$

Where the two extremes of the frequency range are calculated with:

\begin{align*} \Delta\nu_{\text{end}}&=\frac{c}{\lambda_0-\frac{\Delta\lambda}{2}}\\[1.5em] \Delta\nu_{\text{begin}}&=\frac{c}{\lambda_0+\frac{\Delta\lambda}{2}}\\ \end{align*}

These formulas use the relation between frequency and wavelength: you can learn more at our wavelength calculator.

## How to use our laser linewidth and bandwidth calculator

Our laser linewidth and bandwidth calculator allows you to both:

• Calculate the laser linewidth; and
• Convert between wavelength and frequency bandwidth.

Give us the parameter you know, and we will calculate the result.

Remember that our calculator uses the frequency and not the wavelength, which is more often specified in the datasheet. Convert the values using our calculators (as the wavelength to frequency calculator).

It's time for a neat example to teach you how to use our laser linewidth and bandwidth calculator! We consider a common He-Ne laser with fundamental wavelength $632.8\ \text{nm}$. Consider a laser with power $P=1\ \text{W}$ and cavity linewidth $\Gamma = 1\ \text{GHz}$.

🔎 A wavelength of $632.8\ \text{nm}$ corresponds to a frequency of $473.755\ \text{THz}$.

Input these values, with the correct units, in our calculator. We will apply the laser linewidth formula for you:

\begin{align*} \Delta\nu &= \frac{\pi\ h\ \nu\ \Gamma^2}{P}\\[1em] &= \frac{\pi\ h\ 473.755\ \text{THz}\ 1\ \text{GHz}^2}{1\ \text{W}}\\[1em] & = 0.9862\ \text{Hz} \end{align*}

This is a relatively narrow linewidth: we used the specification of a high-level laser.

What about the bandwidth? Let's test our laser bandwidth calculator with an easy example. Take a blue Helium-Cadmium laser with wavelength $411.6\ \text{nm}$. A good quality pointer has a bandwidth of $1\ \text{nm}$. What is the corresponding frequency bandwidth?

Apply the formula by inputting the known values:

\begin{align*} \Delta\nu&=\frac{c}{\lambda_0-\frac{\Delta\lambda}{2}}-\frac{c}{\lambda_0+\frac{\Delta\lambda}{2}}\\[1.5em] &=\frac{c}{411.6 - \frac{1}{2}} - \frac{c}{411.6 + \frac{1}{2}} \\[1.5em] & = 1.77\ \text{THz} \end{align*}

## FAQ

### What is the linewidth of a laser?

The linewidth of a laser is a measure of the spread of the output's power over a finite range of frequencies. The laser linewidth is defined as the full width at half maximum (FWHM) of the power spectrum of the output.

The laser linewidth is strongly related to the spectral coherence of the output: the smaller the coherence, the cleaner the spectrum, with a strongly peaked output around the fundamental frequency of the device.

### What is a narrow linewidth laser?

A narrow linewidth laser is a device that emits light with a high degree of monochromaticity. To obtain such characteristics, the laser must operate at a single frequency, and external sources of noise should be reduced as much as possible (preventing mode hopping). Finally, the laser design should minimize internal noise sources (e.g., phase noise).

Narrow linewidth lasers have critical applications in medicine and sensing.

### How do I calculate the spectral linewidth of a laser?

To calculate the laser linewidth, we use the following equation:

Δν = (π h v Γ²)/P

1. Multiply the fundamental frequency (v) of the laser by the square of the cavity linewidth (Γ).
2. Multiply the result by the constant πh, where h is the Planck constant.
3. Divide by P, the power of the laser mode.

### Which is the linewidth of a typical laser pointer?

19.7 kHz. Considering a typical red laser pointer, we know the following quantities:

1. The device's power: P = 5 mW.
2. The fundamental wavelength: v = 635 nm.
3. The cavity linewidth: Γ = 10 GHz.

Apply the formula for the linewidth of a laser:

Δν = (π h v Γ²)/P = (π h (472.114 THz) (1 GHz)²)/(0.005 W) = 19.7 kHz

Davide Borchia
Linewidth
Laser frequency (ν)
THz
Cavity linewidth (Γ)
GHz
Power (P)
W
Linewidth (Δν)
Hz
Bandwidth
Fundamental wavelength
nm
Wavelength bandwidth
nm
Frequency bandwidth
THz
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