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Dr. Christopher S. Baird

When does a light beam have only a single frequency?

Category: Physics      Published: May 8, 2014

spectrum of monochromatic wave
Even a monochromatic wave contains a spread of frequencies because of its finite lifetime. The top image shows a light wave consisting of a sine wave contained in a Gaussian envelope (to ensure a finite lifetime), and the bottom image shows the spectrum of frequencies contained in this light wave. The fact that the spike in the spectrum has a non-zero width indicates that it contains several frequencies. Public Domain Image, source: Christopher S. Baird.

A light beam never has exactly one frequency. Even a single bit of light (a photon) never has exactly one frequency. It is fundamentally impossible for a photon to have exactly one frequency. Certain beams of light, such as laser beams, can get very close to having one frequency, but can never have exactly one frequency. Said another way, every physical beam of light has a spread of frequencies. When a light beam has a very small spread of frequencies, we often call it "monochromatic". The word monochromatic is not meant to imply that there is exactly one frequency in the light. Rather, it is meant to imply a very narrow range of frequencies contained in the light such that, for many practical purposes, we can approximate the light as only containing one frequency.

The fewer frequencies that there are contained in a light beam, the closer it gets to having exactly one frequency, and the better we can use the light to probe materials. For this reason, laser designers are continually working to make their lasers emit light with ever fewer frequencies. The amount of frequencies contained in a monochromatic light beam is characterized by its "spectral linewidth". When you take a certain light beam, measure the amount of power it contains at different frequencies, and plot the results, you get the beam's frequency spectrum.  Sunlight has a very complicated spectrum. In contrast, monochromatic light's spectrum is a sharp spike centered at the dominant frequency. The narrower this spike, the fewer frequencies are contained in the light. The spectral linewidth is actually the width of this spike. In this way, a light beam with exactly one frequency would have a spectral linewidth of zero, meaning that the spike in its frequency spectrum is infinitely thin. But such a case cannot happen in real life.

There are many things that can contribute to a range of frequencies being present in a monochromatic light beam (called "linewidth broadening"). Noise such as from thermal fluctuations can contribute to linewidth broadening. But even if all external broadening effects are removed, there is one effect that can never be removed: lifetime broadening.

In classical (non-quantum) electrodynamics, light is described as a physical wave in the electromagnetic field. Because electromagnetic waves obey the superposition principle (meaning that two waves at the same point add together linearly to give the total wave), they strictly obey the principles of Fourier analysis. Fourier analysis is the branch of mathematics that deals with representing any function as a sum of single frequency waves (sine waves). Using Fourier analysis, we can mathematically determine the frequency spectrum of a wave directly from its shape in time. Or we can go the other way and mathematically determine the light's wave shape in time using the measured frequency spectrum. We can mathematically go back and forth between the wave shape of the light as a function in time, and its corresponding frequency spectrum, which can be thought of as the wave shape in frequency space. Time and frequency therefore are conjugate variables. As such, Fourier analysis tells us that the closer a wave gets to a perfect sine wave in time, the closer it gets to an infinitely thin spike in frequency space. But there's a complication: a mathematically perfect sine wave is infinitely long, i.e., has an infinite lifetime. Note that when we talk about the "lifetime" of a light beam in this context, we do not mean that the light is going to die or decay into something else. We simply mean the time it takes for the entire beam to pass you by. A beam with an infinite lifetime would be passing you at all moments in time, stretching back into the infinite past and forward into the infinite future.

Fourier analysis applied to classical electrodynamics therefore tells us that in order to have a light beam with exactly one frequency, we would have to have a light beam that is infinitely long. Since infinitely long light beams don't exist in the real world, neither do exactly-single-frequency light beams. A light beam that has a basic sine wave shape but only lasts 10 seconds is actually not a perfect sine wave. Mathematically, it is a wave packet envelope containing a sine wave. In order to construct the wave packet shape (i.e. the beginning and ending edges of the light beam), you need more than one frequency. Therefore, assuming that the light beam has a basic sine wave shape within the lifetime when it exists, its spectral linewidth is ultimately determined by its lifetime. In other words, even assuming that the electrons in a laser had a potentially infinite transition time (which they don't), you would still have to turn on your laser at some moment in time and turn it off at some moment in time. The very act of turning on and off your laser means that the light beam it creates has a finite lifetime, and therefore a non-zero linewidth. Ultimately, the age of the universe places an upper limit on the lifetime of a light beam, and therefore a lower limit on its spectral linewidth. The age of the universe is the ultimate fundamental limit on how few frequencies a light beam can contain. In practice, this limit is far out of reach. The lifetime of electrons transitioning in a laser, and therefore the lifetime of the light beam it creates, is on the order of microseconds to nanoseconds. When all other linewidth-broadening effects are removed, only lifetime broadening remains, and the spread of frequencies becomes the "natural linewidth".

When we add the principles of modern quantum electrodynamics to the discussion above, the concepts are very much the same. The time-frequency limitations placed on a light beam by Fourier Analysis become the Heisenberg Uncertainty Principle in quantum mechanics. One form of the Heisenberg Uncertainty Principle is:

ΔE Δth/4π

In the above equation, ΔE is the uncertainty or spread in the energy of a wave, Δt is the lifetime of the wave, and h is Plank's constant (sometimes h/2π is written as hbar). For a photon, its energy E and frequency f are related trivially by: E = hf. Inserting this relation into the above equation, the Heisenberg Uncertainty Principle becomes:

Δf Δt ≥ 1/4π

In the above equation, Δf is the spread of frequencies in the photon (its natural linewidth as discussed above), and Δt is still the lifetime of the wave. This quantum mechanics equation tells us essentially the same thing that classical electrodynamics and Fourier analysis told us: a light beam with zero linewidth would have to have an infinite lifetime, and therefore does not exist.

Topics: Fourier Analysis, Heisenberg Uncertainty Principle, frequency, lifetime, light, light beam, monochromatic, photon, quantum