RP Photonics

Encyclopedia … combined with a great Buyer's Guide!


Instantaneous Frequency

Definition: temporal derivative of the oscillation phase divided by 2π

German: instantane Frequenz, Momentanfrequenz

Categories: general optics, light pulses

Formula symbol: ν, νi

Units: Hz

How to cite the article; suggest additional literature

The instantaneous frequency is a useful concept for describing non-monochromatic signals. It is defined as

instantaneous frequency

i.e. essentially as the temporal derivative of the oscillation phase φ. (Without the factor 1/2π, one would have the instantaneous angular frequency.) In contrast to a Fourier frequency, the instantaneous frequency is generally a time-dependent frequency. The instantaneous frequency of a sinusoidal signal is constant and equals the oscillation frequency, as expected.

chirped pulse
Figure 1: Electric field of a strongly up-chirped pulse, where the instantaneous frequency increases with time.

The concept of the instantaneous frequency is particularly important in the context of frequency noise and phase noise, but it is also applied to chirped optical pulses (Figure 1), which have a time-dependent instantaneous frequency. The basic idea is intuitive, actually more than that of Fourier frequencies. The same concept is used e.g. in music: music scores essentially specify notes as time intervals for which the instantaneous frequency has a certain value (corresponding to the pitch of one voice, and disregarding overtones). However, the concept can become problematic for complicated signals, e.g. for white noise. In the context of lasers, the instantaneous frequency can be easily defined for single-frequency lasers, whereas for multimode lasers one would first have to separate the different frequency components (with some filtering technique) before retrieving their instantaneous frequencies. The instantaneous frequency is also useful in the context of chirped optical pulses, where the instantaneous frequency varies during the pulse.

Note that the Fourier spectrum of an oscillating signal (in optics, the optical spectrum) does not represent the probability distribution of instantaneous frequencies, and that the linewidth measured from such a spectrum is not an r.m.s. (root mean squared) value of the instantaneous frequency. The relation between instantaneous frequency and Fourier frequency is significantly more subtle than that.

The time dependence of the instantaneous frequency can sometimes be estimated from a spectrogram. However, a plot of instantaneous frequency versus time generally does not carry the same information.

Measuring the Instantaneous Frequency

The instantaneous frequency of an electronic signal (e.g. a beat note) can be obtained using a phase-locked loop (PLL), containing a voltage-controlled oscillator (VCO) and phase discriminator in a feedback system which forces the VCO to follow the oscillation of the input signal. The input signal of the VCO can be used as a measure of the instantaneous frequency.

The basic idea of this approach can also be utilized in the form of a software phase tracker, which evaluates the instantaneous frequency of an already recorded signal. This approach is simple to implement but has some flaws, particularly the limited bandwidth and delayed response. Significantly more powerful, but also more complicated techniques utilize fast Fourier transforms.


[1]Spotlight article of 2007-10-11: "Understanding Fourier Spectra"

(Suggest additional literature!)

See also: frequency noise, chirp, linewidth, spectrograms, Spotlight article 2007-10-11
and other articles in the categories general optics, light pulses

How do you rate this article?

Click here to send us your feedback!

Your general impression: don't know poor satisfactory good excellent
Technical quality: don't know poor satisfactory good excellent
Usefulness: don't know poor satisfactory good excellent
Readability: don't know poor satisfactory good excellent

Found any errors? Suggestions for improvements? Do you know a better web page on this topic?

Spam protection: (enter the value of 5 + 8 in this field!)

If you want a response, you may leave your e-mail address in the comments field, or directly send an e-mail.

If you enter any personal data, this implies that you agree with storing it; we will use it only for the purpose of improving our website and possibly giving you a response; see also our declaration of data privacy.

If you like our website, you may also want to get our newsletters!

If you like this article, share it with your friends and colleagues, e.g. via social media: