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# Femtosecond pulse shaping

In optics, Femtosecond pulse shaping refers to various techniques to modify the time profile of an ultrashort pulse from a laser. Pulse shaping can be used to shorten the pulse duration from a pulsed laser, or to generate more complex pulses.

## Introduction

An ultrashort pulse with a well-defined electrical field E(t) can be modified with an appropriate filter acting in the frequency domain. Mathematically, the pulse is Fourier transformed, filtered, and back-transformed to yield a new pulse:

$E'(t) = \mathcal{F}^{-1}\{\mathcal{F}\{E(t)\}(\omega)f(\omega)\}(t).$

It is possible to design an optical setup with an arbitrary filter function f(ω) which can be complex-valued, as long as $|f(\omega)|\le1$. Figure 1 shows how a bandwidth-limited pulse could be transformed into a chirped pulse (with a filter only acting on the phase) or into a more complex pulse (with the filter acting on both phase and amplitude).

## Optical design

### Basic principles

Typically, a pulse shaper is based on the design of a pulse stretcher, but in a 4f configuration such that the stretcher actually neither stretches nor compresses incoming pulses. This is shown in Figure 2. A diffraction grating directs different frequency (wavelength) components into different directions and each frequency component is focused at a particular spot in the focal plane. A second grating and a mirror are used to recombine the different frequency components. The two lenses have a focal length f, and the distance from the center of the first grating to the center of the second grating is 4f.

In the focal plane, undesired frequency components can easily be blocked, which would correspond to a real-valued filter function without any effect on the phase.

To make the filter act on the phase of the frequency components, a slab of transparent material with a varying thickness could be used to introduce an extra frequency-dependent delay, as shown in Figure 3. Although one might think that an extra delay is introduced for longer wavelengths (shown in red), the actual effect is that the pulse as a whole will have a shorter travel distance, mainly because the light has a shorter travel distance to the rightmost grating. This is in agreement with the fact that a Fourier filter f(ω) = exp(iωτ) ('linear phase shift') is equivalent to a time-shift τ.

For a more interesting effect on the pulse shape, a quadratic or higher-order phase shift is needed, as shown in Figures 4 and 5.

### Active filters

The examples above are with static filters. With liquid crystal technology, it is possible to create a filter that can have an arbitrary computer-controlled phase and amplitude spectrum. Multiphoton Intrapulse Interference Phase Scan (MIIPS) is a technique based on the computer-controlled phase scan of Spatial light modulator. Through the phase scan to an ultrashort pulse, MIIPS can not only characterize but also manipulate the ultrashort pulse to get the needed pulse shape at target spot (such as Transform-Limited pulse for optimized peak power, and other specific pulse shapes). This technique features with full calibration and control of the ultrashort pulse, with no moving parts, and simple optical setup.

### Single versus double-pass design

In the configuration shown above, the light hits each of the two gratings twice. In many applications, the back-reflecting mirror on the right is omitted, which makes the design simpler. The main disadvantage of such a single-pass design is that the different frequency components do not overlap in space, as can be seen in Figures 4 and 5. However, for many applications, it is only necessary to separate different frequencies in time by only a few picoseconds, which corresponds to just a few tenths of a millimeter of transversal displacement, which is only a minor concern.

## Related Techniques

• MIIPS Multiphoton Intrapulse Interference Phase Scan, a method to characterize and manipulate the ultrashort pulse.