Can a Monochromatic Pulse Exist?

A monochromatic wave has one frequency, while a pulse is a wave concentrated into a finite interval of time. These descriptions conflict if taken literally: a wave cannot be both exactly single-frequency and a finite pulse.

In practice, people often call a pulse “monochromatic” when it has a well-defined central frequency and a sufficiently narrow bandwidth. This article explains the distinction and shows how to estimate the bandwidth of a short light pulse.

What “monochromatic” means

An ideal monochromatic electric field can be written as

$$E(t)=E_0\cos(2\pi\nu_0 t+\phi),$$

where $\nu_0$ is its frequency. This oscillation continues for all time; it has no beginning or end. In the frequency domain, its energy is concentrated at exactly $\nu_0$ (idealized as a spectral line).

A pulse is different. Its field is appreciable only for a limited time, often described as a rapidly oscillating carrier inside a slower envelope:

$$E(t)=A(t)\cos(2\pi\nu_0 t+\phi).$$

Here $A(t)$ rises and falls to make the pulse. Changing the amplitude over time necessarily adds frequencies around the carrier frequency $\nu_0$. The pulse’s spectrum is therefore a band centered near $\nu_0$, rather than a single frequency.

Why a finite pulse needs bandwidth

Time and frequency descriptions are Fourier-transform pairs. A signal confined more tightly in time has a more spread-out frequency spectrum. In particular, a nonzero signal cannot be both strictly limited to a finite time interval and strictly limited to one frequency. An exact single-frequency wave lasts forever; an exactly time-limited pulse has a spectrum that extends over a range of frequencies.

This is not a peculiarity of visible light. It applies to sound, radio waves, and other wave signals. For light, a pulse with a central optical frequency can still be called quasi-monochromatic if its bandwidth is small relative to that central frequency. The term is an approximation, not a claim that the pulse contains only one frequency.

A worked bandwidth estimate

For a transform-limited Gaussian pulse, the full width at half maximum (FWHM) of the intensity in time, $\Delta t$, and the FWHM of the intensity spectrum in frequency, $\Delta\nu$, obey

$$\Delta t\,\Delta\nu\geq 0.441.$$

A transform-limited pulse reaches the minimum product. A pulse with additional phase variation, called chirp, can have a larger bandwidth for the same duration.

Suppose a Gaussian light pulse has an intensity FWHM duration of $100$ femtoseconds. Its minimum frequency bandwidth is

$$\Delta\nu\geq\frac{0.441}{100\times10^{-15}\,\text{s}} =4.41\times10^{12}\,\text{Hz} =4.41\,\text{THz}.$$

If its central wavelength is $600\,\text{nm}$, its central frequency is approximately

$$\nu_0=\frac{c}{\lambda_0} \approx\frac{3.00\times10^8\,\text{m/s}}{600\times10^{-9}\,\text{m}} =5.00\times10^{14}\,\text{Hz}.$$

The minimum bandwidth is about $4.41/500\approx0.0088$, or $0.88\%$, of that central frequency. For a narrow band, the corresponding wavelength width is approximately

$$\Delta\lambda\approx\frac{\lambda_0^2}{c}\,\Delta\nu\approx5.3\,\text{nm}.$$

So this pulse is strongly centered near $600\,\text{nm}$, but it is not exactly monochromatic. A much shorter pulse would have a wider frequency range.

When the term is useful

In experiments and engineering, “monochromatic pulse” is often convenient shorthand for a pulse whose bandwidth is narrow enough for the question being studied. For example, if a material’s response changes very little across the pulse’s spectrum, treating the pulse as having one representative frequency may be a good approximation. The approximation becomes less suitable when dispersion, absorption, or another effect varies significantly across the bandwidth.

A repeating train of pulses also needs care in interpretation. An ideal infinite, perfectly repeating train has discrete spectral lines, but each individual pulse is still made from a range of frequencies. The repetition pattern and the spectrum of one pulse are related, but they are not the same thing.

Common mistakes

• Equating carrier frequency with the whole spectrum: The carrier frequency is the center of a pulse’s spectrum, not proof that the pulse contains only that frequency.
• Assuming every pulse has the same bandwidth: Bandwidth depends on pulse duration and shape. The numerical factor $0.441$ applies to Gaussian intensity profiles using FWHM widths; other shapes have different factors.
• Treating the uncertainty relation as an exact equality: The product is a lower bound for the stated Gaussian convention. Chirp or other pulse shaping can make it larger.
• Assuming “monochromatic” always means mathematically exact: In practical contexts, it often means narrow-band to a stated or understood tolerance.