What is a "Fourier transform limited pulse"?

Question

I have some doubts about the definiton of a Fourier transform limited pulse. For example if I consider a generic pulse: $$E(z,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}A(\omega)e^{i(-\beta(\omega)z+\omega t)}d\omega$$ Defining:

• $S(\omega,z)=|\mathscr{F}[E(z,t)]_{\omega,z}|^{2}$

• $\Delta \omega(z)$ as the range of frequencies which correspond at half height of $S(\omega,z)$ for a fixed $z$ (pulse bandwith).

• $\tau_{p}$ (the duration of the pulse) as the range of time which correspond at half height of $|E(z,t)|^2$ at fixed $z$. I'm considering a pulse inside a dispersive medium so its duration depends on the $z$ you are at.

Done these definitions, is it right to say that a pulse is Fourier limited if $$\Delta \omega(z)\tau(z)=\alpha$$ for each $z$, where $\alpha$ is a number which changes according to the type of pulse (Gaussian, ecc...)?

Answer 1

I'm considering a pulse inside a dispersive medium so its duration depends on the z you are.

... then the concept of transform-limited pulse does not hold globally for your setup. Transform-limited pulses are a 1D (generally time-domain) phenomenon, so in your configuration the question "is the pulse transform-limited" would be asked and answered locally and independently at each different point. And, in the presence of dispersion, if the pulse is transform-limited at a given point $z_0$, then it will not be transform-limited at any other point in general.

Generically, given a locally-defined electric field $$ E(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}A(\omega)e^{+i\omega t}d\omega, $$ with spectral amplitude $A(\omega) = |A(\omega)| e^{i\phi(\omega)}$, the pulse is said to be transform-limited if its duration is minimal over the set of pulses that have an identical power spectrum $|A(\omega)|^2$. The reason we use this definition is that the power spectrum $|A(\omega)|^2$ is both

• fixed by the gain profile of the laser gain medium and the details of the cavity, and
• easily measurable by using a conventional spectrometer,

whereas the time duration is

• extremely hard to measure,
• not determined by the laser source, since the introduction of any dispersive optics will affect the pulse duration without affecting the power spectrum, and
• accessible (given enough money, time, and dedication) to experimental modification via a number of pulse-shaping schemes.

For a given laser source, the power spectrum is basically fixed, and therefore so is the bandwidth $\Delta\omega$, and this puts a limit, via the Fourier bandwidth theorem, on the minimal pulse duration that's achievable with your laser source. However, unless you've done a lot of work, the pulse that comes out of your source will not be that short - instead, it will contain chirp and other types of dispersive features which make it longer than that minimal pulse duration. That problem can be fixed by using pulse shapers to introduce additional spectral phases (i.e. additional terms $e^{i\phi_\mathrm{shaper}(\omega)}$ multiplying the spectral amplitude) which cancel out the chirp and other dispersive behaviours to minimize the pulse duration.

The transform-limited pulse duration is the minimal pulse duration that's achievable using this procedure.

If you want to get truly technical, then this also depends on the choice of measure for the duration of the pulse (i.e. choosing the FWHM, as you've done with your $\tau$, or some other measure which e.g. takes into account some pre-defined sensitivity to pre- or post-pulses), but if you're arguing about that then you're well and truly into the weeds by that point.

The concept of a transform-limited pulse is of extreme relevance in on-the-ground experimental situations, where the spectrum of your pulse is some jagged beast instead of some nice smooth spectrum (say, take fig. 2(b) of this paper). To evaluate the transform-limited duration, you basically take a set of reasonably-smooth spectral phases $\phi(\omega)$ that's as expansive as you can, and you select the one that gives you the smallest pulse duration. (And yes, by duration you use the FWHM by default, but really you should use whatever is the best descriptor of the temporal resolution limits in your experiment, which will depend on the process you're using.)

Answer 2

Given a generic pulse: $$E[z,t]=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} A(w)e^{i[\beta (w)z -wt]}dw$$ You have: $$A(w)=\mathscr{F}(E(0,t))_{w}$$ $$S(w)=|A(w)|^2$$ $$S(w)_{FWHM}=\Delta w$$ $$I(z,t)=|E(z,t)|^2$$ $$I(z,t)_{FWHM}=\tau_{p}(z)$$ Fixed $S(w)$ ( and consequently $\Delta w$), which is automatically fixed once you choose a particular laser, you have: $$\tau_{p}(0)=I(0,t)_{FWHM}=|E(0,t)|^2_{FWHM}$$ $$=\vert \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \sqrt{S(w)} e^{-iwt+i\phi}dw \vert ^2_{FWHM}$$ So $\tau_{p}(0)$ depends on $\phi$. If our laser produces a pulse with that partcular $\phi$ for which $\tau_{p}(0)$ is minimum then we call it transform limited pulse. For example if $S(w)$ is fixed as a Gaussian function then this pulse will be T.L. only if $\tau_{p}(0)\Delta w =2.44$.

Finally, for $z>0$ since we are considering a dispersive media then: $$\tau_{p}(z)>\tau_{p}(0)$$ So we are sure that the pulse won't be T.L. anymore.