# Transform Limit

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: a limit for the time–bandwidth product of an optical pulse

Alternative term: Fourier transform limit

In ultrafast optics, the transform limit (or Fourier limit, Fourier transform limit) is usually understood as the lower limit for the pulse duration which is possible for a given optical spectrum of a pulse. A pulse at this limit is called *transform limited*. The condition of being at the transform limit is essentially equivalent to the condition of a frequency-independent spectral phase (which leads to the maximum possible peak power), and basically implies that the time–bandwidth product is at its minimum and that there is no chirp. The minimum time–bandwidth product depends on the pulse shape, and is e.g. ≈ 0.315 for bandwidth-limited sech^{2}-shaped pulses and ≈ 0.44 for Gaussian-shaped pulses. (These values hold when a full-width-at-half-maximum criterion is used for the temporal and spectral width.)

For a given pulse duration, transform-limited pulses are those with the minimum possible spectral width. This is important e.g. in optical fiber communications: a transmitter emitting close to transform-limited pulses can minimize the effect of chromatic dispersion during propagation in the transmission fiber, and thus maximize the possible transmission distance.

Many mode-locked lasers, particularly soliton mode-locked lasers, are able to generate close to transform-limited pulses. During propagation e.g. in transparent media, phenomena such as chromatic dispersion and optical nonlinearities can cause chirp and thus can lead to non-transform-limited pulses. Such pulses may be brought back to the transform limit (and thus temporally compressed) by modifying their spectral phase, e.g. by applying a proper amount of chromatic dispersion. This is called *dispersion compensation*. For not too broad spectra, compensation of second-order dispersion is often sufficient, whereas very broad spectra may require compensation also of higher-order dispersion in order to approach the transform limit.

## More to Learn

Encyclopedia articles:

Blog articles:

- The Photonics Spotlight 2007-10-11: “Understanding Fourier Spectra”
- The Photonics Spotlight 2008-06-13: “Heisenberg's Uncertainty Principle and the Transform Limit”

## Questions and Comments from Users

2023-01-18

Are the terms *Fourier-limited linewidth* and *lifetime-limited linewidth* exactly the same?

The author's answer:

No, they are not. Fourier-limited linewidths can occur in situations where no kind of lifetime is involved. However, in cases where we have spontaneous emission from a system with a limited upper-state lifetime, the emission linewidth may be Fourier-limited.

2023-03-22

Does the width of the pulse refer to the width at which the pulse amplitude is <$1/e$> the maximum value, or to the pulse duration (FWHM)?

The author's answer:

In the context of the time–bandwidth product, one usually uses FWHM.

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2022-05-12

Do GDD, TOD and even higher order dispersion terms for a Fourier-limited pulse become zero? So does that mean that only zero and first order components exist in a transform-limited pulse?

The author's answer:

GDD and TOD are generally not considered to be properties of a pulse, but rather of optical components through which pulses may propagate.

However, you can analyze the frequency dependence of the spectral phase of a pulse in the same way as you can analyze the frequency-dependent phase changes introduced by a dispersive optical element. Then, you will indeed find that the second and higher order components in the Taylor expansion vanish for a transform-limited pulse.