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Power Spectral Density

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Acronym: PSD

Definition: optical power or noise power per unit frequency interval

German: spektrale Leistungsdichte

Categories: fluctuations and noise, optical metrology

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In optics, power spectral densities (also sometimes just called power densities) occur basically in two different forms:

In the following, both types of quantities are discussed.

Optical Power Spectral Densities

When the spectral distribution of optical power e.g. of some laser source is measured e.g. with an optical spectrum analyzer (spectrometer), the result is usually given either as a power spectral density (e.g. in units of mW/nm or dBm/nm, with dBm = dB relative to 1 mW), or as a power for a given measurement bandwidth.

Concerning optical power spectral densities, there is a wide range of quantities which are related to each other. Essentially two different issues have to be observed:

Planck's law

Figure 1: Power spectral density of black bodies at various temperatures according to Planck's law, plotted referring to frequency intervals.

Planck's law

Figure 2: Same as Figure 1, but referring to wavelength intervals. Note that this changes the position of the maximum. For 6000 K, for example, the maximum is at 483 nm, corresponding to 621 THz, whereas the upper graph shows the corresponding maximum at 353 THz.

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.

Noise Power Spectral Densities

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.

intensity noise spectrum

Figure 3: Intensity noise spectrum of a solid-state laser. Here, the power spectral density relative to that of shot noise is displayed.

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

definition of PSD

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

Wiener--Khinchin theorem


autocorrelation function

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:

variance from 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:

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.

See also: optical spectrum, noise specifications, intensity noise, phase noise, laser noise
and other articles in the categories fluctuations and noise, optical metrology

Dr. R. Paschotta

This encyclopedia is authored by Dr. RĂ¼diger Paschotta, the founder and executive of RP Photonics Consulting GmbH. Contact this distinguished expert in laser technology, nonlinear optics and fiber optics, and find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, or staff training) and software could become very valuable for your business!

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