Power Spectral Density

When the spectral distribution of optical power e.g. of some laser source or other light source is measured e.g. with an optical spectrum analyzer (e.g. a spectrograph), the result is usually given as some spectral quantities. For example, one may specify the radiant flux per unit optical frequency or wavelength, which is called the spectral flux. In other cases, one specifies a spectral intensity (radiant intensity per unit optical frequency or wavelength), or some other spectral quantities. The mentioned quantities are all quantities from the field of radiometry; similar spectral quantities are also used in photometry.

Radiant spectral quantities can in principle also be interpreted as power spectral densities. In some cases, they are mathematically calculated as the power spectral density of some kind of amplitude, e.g. involving statistical models. In many other cases, they are calculated from a total optical power in conjunction with a known spectral shape, or directly measured e.g. with some kind of optical spectrometer.

Optical power spectral densities are also required for visualizing e.g. pulse shapes with spectrograms. In that case, power spectral densities are calculated for limited time intervals by applying Fourier transforms to sections of data as extracted using some window function.

In the case of noise powers, a PSD always refers to averaged power levels related to intervals of noise frequency (rather than optical frequency). Such noise PSDs can occur in the context of any optical or electrical signals. They can be used not only in conjunction with optical power (→ intensity noise), but also e.g. for phase noise, frequency noise, noise of a pulse duration, pulse energy, or timing jitter, e.g. of a mode-locked laser Similarly, PSDs can apply to electrical voltages or currents.

PSDs can be defined via the squared modulus of the Fourier transform of the quantity of interest, but the straightforward approach works only for functions which have significantly non-zero values only in a finite time interval. For the frequently encountered case of fluctuations around a long-term mean value (e.g., concerning the power or phase fluctuations of a continuous-wave laser), a definition such as

may be used for the (two-sided) PSD of a variable X(t). Here, the integral is restricted to a finite time interval (thus obtaining convergence of the integral), and the mean squared modulus of the Fourier transform is divided by the length T of the time interval. Finally, the limit for large time intervals T is calculated. This definition is conceptually clear, but not always convenient (in particular not for analytical calculations). Therefore, it is common to use the Wiener–Khinchin theorem (or Wiener–Khintchin theorem), which states that

where

is the autocorrelation function of X(t).

In any case, it is important to note that power spectral densities are statistical measures, which can be estimated from real data by averaging over the results from many measurements. Taking a single measurement trace gives only a very rough estimate of the PSD.

Power spectral densities can be specified as one-sided functions of only positive frequencies, or as (two times smaller) two-sided functions of positive and negative frequencies. Optical power densities are usually one-sided and can be measured e.g. with an optical spectrum analyzer. Noise PSDs are mostly one-sided in the engineering disciplines, but often two-sided in physics. Noise power densities can be measured with electronic spectrum analyzers or calculated from data recorded in the time domain. They are specified e.g. in dBc/Hz (dB relative to the carrier in a 1-Hz bandwidth) for the relative intensity noise, or in rad²/Hz for phase noise. Sometimes, square roots of such power densities are specified, with the peculiar units of e.g. rad per square root of a hertz.

The variance of a quantity (e.g. the optical phase) in a certain range of noise frequencies is given as an integral over the (one-sided) PSD:

The square root of such a value is a root-mean-squared (r.m.s.) value. Note, however, that such integrals do not always converge (e.g. when the PSD has a singularity at f = 0). In the case of phase noise, such a divergence is related to a finite linewidth. Noise integrals are also often used for calculating a signal-to-noise ratio.

Unfortunately, the measurement or calculation of power spectral densities is prone to many errors. Some frequent mistakes are:

• confusing one-sided and two-sided PSDs, or not making clear which ones are used
• inappropriate settings of electronic spectrum analyzers, e.g. concerning detector mode and averaging method
• the failure to apply certain correction factors (e.g. for the effective noise bandwidth) to data obtained with electronic spectrum analyzers
• the failure to remove artifacts by proper windowing, when PSDs are calculated from time-domain data

Adequate training concerning mathematical foundations, physical effects, and details of electronic spectrum analysis is required for dealing correctly with power spectral densities in laboratory environments.

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This encyclopedia is authored by Dr. Rüdiger Paschotta, the founder and executive of RP Photonics Consulting GmbH. How about a tailored training course from this distinguished expert at your location? Contact RP Photonics to find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, training) and software could become very valuable for your business!