# Monochromaticity: the spectrum of a laser or other light source

We know that the wavelength and therefore the frequency of a light wave is related to the color that we perceive.  A light wave with a single wavelength has a single color; it is monochromatic.  All of the harmonic electromagnetic waves we’ve been writing so far are monochromatic, because they have a single frequency and wavelength.  But no light wave in the real world is ever purely monochromatic.  White light sources, such as lightbulbs, are designed to emit many wavelengths of light, as we generally find this broad-spectrum light more enjoyable – we evolved to make use of sunlight, which has a very broad spectrum of wavelengths.  Laser light has a much narrower spectrum of wavelengths than a lightbulb, but even laser light can’t be purely monochromatic.  An ideal monochromatic wave would have to persist for all time, because even turning the light beam on and off introduces more frequency components.

Non-monochromatic light can be written as a sum or superposition of monochromatic light waves.  Mathematically, we call this Fourier analysis.  Fourier series are sums of time harmonic functions, which can represent any periodic function; Fourier transforms are integrals over time harmonic functions, which can represent any arbitrary function.  This is the perfect excuse for me to link to one of my favorite web apps, the Fourier Series Applet:

which allows you to play around with Fourier Series in a visual way.

The spectrum produced by some light source is a plot of the power spectral density, i.e. the power as a function of frequency/wavelength, per unit frequency/wavelength.  In plain(er) language, if we split a light beam up into monochromatic waves of different wavelengths, the spectrum is the power of each of those monochromatic waves plotted versus their wavelengths.    Here is a link to an awesome slideshow from Popular Mechanics that shows spectra for various light sources (although not lasers):

For comparison, here’s the spectrum of a laser you can buy from Ocean Optics:

https://oceanoptics.com/product/turnkey-raman-lasers/high-resolution-532-nm-laser-spectrum/

Take a close look at the x-axis of the spectra.  You will see that the laser has a much narrower spectrum, although it has finite width.

Physically, it is possible to split up a light beam into monochromatic waves by using a prism.  If we shine a flashlight beam through a prism, and then onto a screen, we will see spots of different colors, some brighter than others.  A bright green spot would mean that the spectrum of the flashlight beam contained a lot of power at wavelengths corresponding to green light.  Below are some fun prism videos:

The power of a wave is related to its magnitude squared (as we’ll see in more detail later).  In order to calculate the spectrum of a beam of light we need to first take the Fourier transform, which tells us the amplitude of each monochromatic light wave composing the beam, and then take the magnitude-squared of those amplitudes to obtain the power (intensity).  The Fourier transform is $\displaystyle E(\omega) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} E(t) e^{i \omega t} dt$

where $E(t)$ is the time-dependent electric field of the beam of light at some fixed position, and $E(\omega)$ will be the amplitude of each monochromatic wave composing the beam of light.  This Fourier transform can be thought of as an ‘inner product’ between the electric field and a monochromatic signal.

The spectrum of a light source is measured using a spectrometer, which is a device that splits out the various wavelength composing the light to be analyzed, similar to the prism we described above.  The spectrum of a truly monochromatic beam would be a delta function (infinitely narrow), but as we’ve seen, all real light sources have finite width spectra.

Amongst non-laser light sources, a light source composed of a single atomic species, like a hydrogen gas, exhibits the narrowest spectrum.  This is because photons are produced by electrons in the atoms making energy transitions between discrete energy levels in the atoms.  Lasers can have even narrower spectra because the process by which they produce light (stimulated emission) makes copies of photons, and each copy has the same wavelength as the original.  However, there is a competing light emission process called spontaneous emission.   Spontaneous emission produces a broader spectrum of light than stimulated emission, and the unavoidable presence of spontaneous emission broadens the overall laser spectrum a bit. We’ll discuss this in detail later.

The frequency or wavelength width of the spectrum of a light source is characterized by the linewidth, $\Delta \lambda$.  This is typically the full-width half-maximum of the spectrum: We can plot spectra vs. frequency or wavelength, but it is typical to plot vs. wavelength for light sources.  If we have the frequency linewidth $\Delta \nu$ and want to obtain the wavelength linewidth $\Delta \lambda$, we might be tempted to try $\Delta \lambda = c / \Delta \nu$, but this is incorrect.  That’s because $\displaystyle \Delta \nu = \nu_{upper} - \nu_{lower} = \frac{c}{\lambda_{lower}} - \frac{c}{\lambda_{upper}}$

and $\displaystyle \Delta \lambda = \lambda_{upper} - \lambda_{lower}$

So the long way to find $\Delta \lambda$ would be to convert $\nu_{upper}$ and $\nu_{lower}$ into $\lambda_{lower}$ and $\lambda_{upper}$, respectively, and then take the difference of those two.  An approximate expression for relatively narrow linewidths is $\displaystyle \Delta \lambda \approx \frac{c}{\nu^2} \Delta \nu$

where $\nu$ is the center frequency of the spectral line.

To wrap up, we’ve described what the spectrum of a light source is, and described the spectra of common light sources.  Lasers generally have much narrower spectra than other light sources.  Why is a narrow-spectrum light source useful?  Well, it’s useful in science in order to exert control of the interaction of the light with matter – with a laser at a certain wavelength, we will only excite certain specific energy transitions in the atoms.  A narrow spectrum light source is useful technologically as a carrier wave for fiber-optic communications.  With several carrier waves closely spaced in wavelength, we can have multiple ‘channels’ that can carry different signals down the same fiber.  This is referred to as ‘wavelength division multiplexing’.