# Bandwidth

Definition: the width of some frequency or wavelength range

More specific terms: gain bandwidth, resonator bandwidth, modal bandwidth, phase-matching bandwidth

German: Bandbreite

Categories: light detection and characterization, physical foundations

Units: Hz, nm

Formula symbol: <$\Delta \nu$>, <$\Delta \lambda$>

Author: Dr. RĂ¼diger Paschotta

In photonics, the term *bandwidth* occurs in many different cases. The following sections discuss some important cases.

## Bandwidth in Terms of Optical Frequency

In the following cases, bandwidth means the width of a range of optical frequencies:

- A light source can have some optical bandwidth (or linewidth), meaning the width of the optical spectrum of the output. For narrow-linewidth lasers, the bandwidth can be extremely small – in extreme cases below 1 Hz, which is many orders of magnitude less than the mean optical frequency. On the other hand, ultrashort pulses with few-femtosecond pulse durations can have very large bandwidth – easily tens of terahertz.
- An optical bandwidth can be the width of a frequency range which can somehow be handled by an optical element or photonic device. For example, it can be the reflection bandwidth of a mirror, the optical transmission bandwidth of an optical fiber, the gain bandwidth of an optical amplifier, or the phase-matching bandwidth of a nonlinear optical device.

A common definition of spectral width is the full width at half maximum (FWHM), but other definitions are also used. For example some authors use the half width at half maximum (HWHM), which is just half the FWHM.

Optical bandwidth values may be specified in terms of frequency or wavelength. Due to the inverse relationship of frequency and wavelength, the conversion factor between gigahertz and nanometers depends on the center wavelength or frequency. For converting a (small) wavelength interval into a frequency interval, the equation

$$\Delta \nu = \frac{c}{{{\lambda ^2}}}\Delta \lambda $$can be used. (It can be obtained by considering the derivative of <$\nu = c / \lambda$> with respect to <$\lambda$>.) This shows that 1 nm is worth more gigahertz if the center wavelength is shorter.

The optical bandwidth of a light source is strongly related to the temporal coherence, characterized with the coherence time.

Both for passive resonators (e.g. optical cavities) and for the output of oscillators (e.g. lasers), the *Q* factor is the oscillation frequency divided by the bandwidth.

## Bandwidth of Modulations

A bandwidth can also indicate the maximum frequency with which a light source can be modulated, or at which modulated light can be detected with a photodetector.

In the area of optical fiber communications, the term *bandwidth* is also often inaccurately used for the *data rate* (e.g. in units of Gbit/s) achieved in an optical communication system. A more appropriate term would be *data rate* or *data transmission capacity*, avoiding any confusion with optical bandwidth.

Note that the data transmission capacity has only a limited relation to the *optical bandwidth*. Although a large data transmission rate is not possible without a large optical bandwidth, different communications devices can differ substantially in terms of *spectral efficiency*, i.e., concerning what data rate is achievable per megahertz of optical bandwidth.

## Bandwidth of Photodetectors

A photodetector has a limited bandwidth, here meaning the frequency range in which modulations of the optical power can be detected. Typically, that frequency range would start from zero frequency, but in some cases (AC-coupled photodetectors) that is not the case. In the common case of DC-coupled photodetectors, the bandwidth is equated to the maximum detectable modulation frequency according to some criterion. Frequently, one specifies a 3-dB-bandwidth, meaning the frequency where the signal power (proportional to the square of the output voltage or current) is reduced by 3 decibels. That quantity is related to the rise and fall time. If those times are equal, they may be estimated to be 0.35 divided by the 3-dB bandwidth.

Note that when modulation frequencies reach the bandwidth limits, one does not only experience a reduction of responsivity, but also phase changes. That can be problematic, for example, in the context of feedback loops.

See also: optical spectrum, time–bandwidth product, gain bandwidth, coherence time, transform limit, modal bandwidth, bandwidth–distance product, Q factor, phase-matching bandwidth, wavelength, optical frequency, telecom fibers, spotlight 2007-10-11

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2020-04-07

Just a curious question about data transfer. Does the choice of light wavelength give us a cap on how much data we can transfer? And would short wavelengths give us a higher cap? And if so is there an analytical expression for this fundamental limit?

The author's answer:

The center wavelength does not matter, only the width of the used optical frequency range.

The possible transmission bandwidth is the product of the optical bandwidth with the so-called spectral efficiency – which depends on the used modulation format and the achieved signal-to-noise ratio, which is of course influenced by propagation losses, detector noise etc. So I think there is no fundamental limit, but there are practical limits to the achievable spectral efficiency. It is typically of the order of 1 bit/s per Hertz of optical bandwidth. For more details see the article on optical data transmission.