Power Spectral Density
In optics, power spectral densities (also sometimes just called power densities) occur basically in two different forms:
- optical power spectral densities, defined as the optical power per optical frequency (or wavelength) interval, e.g. specified in mW/THz or mW/nm
- noise power densities, defined as the power spectral density of the fluctuations of a quantity such as an optical power or phase, where the frequency argument refers to a noise frequency (rather than to an optical frequency)
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:
- In any case, there is some kind of energy, power, intensity or the like, which is distributed over some range of wavelengths or frequencies. Correspondingly, the integrated quantity has units of joules, watts, watts per square centimeter, etc.
- Optical power spectral densities can be related either to fixed intervals of optical frequency (measured e.g. in terahertz), or to intervals of wavelength (e.g. in nanometers). If the integrated quantity has units of watts, the resulting power density is measured e.g. either in watts per terahertz or in watts per nanometer. Great care is required for the conversion of such quantities, because the conversion factor is wavelength-dependent: for infinitesimal frequency and wavelength intervals, the conversion must be done according to dν = (c / λ2) dλ. This shows that at shorter wavelengths, each nanometer is worth more terahertz. A consequence of this is that the peak position of a spectrum with large bandwidth can significantly depend on whether the power spectral density refers to frequency or wavelength intervals. Confusion arises if such issues are overlooked or power spectral densities are not properly specified. See the Figures 1 and 2 as an illustration.
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.
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
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.
This encyclopedia is authored by Dr. Rüdiger 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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