Pulse shaping

3.3.1 Introduction

Femtosecond pulse shaping is a technique that allows generation of complex optical wavefronts by manipulating of optical spectrum. Successful application of femtosecond pulse shaping have been demonstrated in various areas from optical fiber communication [115], to nonlinear bioimagining and microscopy , and co-herent control. Due to the lack of instruments capable to modify temporal profile of femtosecond laser pulses in ultrafast scale, pulse shaping is performed in the frequency domain by Fourier transform.

Different pulse shaping methods have been introduced during last decade . Some of them are listed below:

• Fixed phase and amplitude masks;

• Holographic masks[116];

• Liquid crystal SLM [117,118,119];

• Acousto-optic and electro-optic devices[120,121,122];

• movable[123] and deformable mirrors[124,125];

Liquid crystal spatial light modulators (SLM) are the most common devices for femtosecond pulse shaping. They can be used both for independent amplitude and phase modulation, and polarization pulse shaping. The main disadvantage of liq-uid crystal SLM pulse shapers deployment in optimal control experiments is their limitation to UV-visible and IR working spectral regions, while UV is absorbed by liquid crystal molecules.

Development of acousto-optic pulse shapers made them available for amplitude and phase shaping in different regions from deep UV 200 nm up to 2 mm. Spec-tral amplitude and phase pulse shaping are based on diffraction from a traveling acoustic wave. One of the commercial available acousto-optic shapers working in spectral region of 250-400 nm, Dazler [121], has been employed previously for op-timal control experiment of amino-acids discrimination [10]. However significant

limitations are found like two-photon absorption in the crystals and strong spatial-temporal coupling, diffraction lose. Besides these devices have low efficiency of shaping, only 15%of spectrum can be temporally modulated.

To overcome issues related with absorption in the crystals, spectral flexibilities, and also energy through-put and speed, reflective techniques have to be deployed for temporal pulse shaping. Presently there two different techniques exist based on fully reflective approach: deformable mirrors based instruments and MEMS (Mi-cro Electro Mechanical Systems) mi(Mi-cromirror arrays. The first instruments based on deformable mirrors therefore do not allow making sharp phase jumps, which limits their application for coherent control experiments. On the contrary, MEMS based pulse shapers not only capable for phase and amplitude shaping through broad spectral range, but also permit to set spectral phases in time domain of dif-ferent complexities.

In this contribution our group GAP-Biophotonics has been working for many years under development of novel pulse shaper that can be implemented in current optimal control experiments. The review of this instrument is done recently [123].

It can be employed in different spectral regions. Our group has shown the MEMS shaper realization for amplitude modulation in XUV [126], which can be extended further for phase shaping on coherent control experiments on atomic and molecular systems. In this work we demonstrate application of our self-developed MEMS pulse shaper for coherent control experiment for discrimination of peptides and proteins, described in Chapters 4 and 5, 6.

3.3.2 MEMS micromirror array for temporal pulse-shaping

A 1D micromirror array for pulse-shaping dedicated for optimal control exper-iments was developed in the group of GAP-Biophotonics in collaboration with EPFL/STI/IMT-NE/SAMLAB [123]. This device consists of an array of indepen-dent micromirrors capable to simultaneous piston and tilt motion, allowing phase-and amplitude pulse-shaping.

Here is briefly described the shaping device that is used for phase pulse-shaping only. Figure 3.4a shows a scanning electron microscope image of mi-cromirror array with wirebonding connections. Mimi-cromirror array is composed of

100 mirrors with 3 µm gaps. It is 160 × 1000µm long. Each mirror has X-shape springs, tilt actuator, triangular height adapter used for actuator connection and main bar. Motion of the mirrors is generated by applying voltage to the tilt actuators through the wirebond connections.

When parabolic mask is applied to mirror array we observe the white light in-terferometry picture shown in Figure 3.4b. It can be seen that some of the mirrors in the array are not actuated. This might be related to short circuits or incomplete production release during the lithography manufacturing.

200 µm

Figure 3.4:a) Scanning electron microscope (SEM) image of the mirror array; b) White-light interferometry image of MEMS micromirror aray with applied parabolic mask.

3.3.3 Background theory of temporal pulse shaping

Very briefly, temporal pulse-shaping technique can be describe within a linear, time-invariant filter. Let us considerein(t) is an electric field of the input pulse in the time-domain, then the output electric fielde_out(t)will be given by the con-volution function of the input pulse ein(t) and its inpulse response functionh(t) [117]

e_out(t) =e_in(t)∗h(t) = Z

dt⁰e_in(t⁰)h(t−t⁰) (3.7) In the frequency domain, a filter is characterised by frequency response function H(w), the Fourier Transform of itsh(t). Thereby, in frequency domainEout(w),

the result of the electric field passed through a pulse shaper will be given by:

Eout(w) =Ein(w)H(w) (3.8) whereE_out(w),E_in(w)are the Fourier Transforms of its electric field functions in time spaceeout(t),ein(t)respectively. Frequency response function can be also expressed by the following equation:

H(w) =A(w)e^iΨ(w) (3.9)

where the A(w), Ψ(w) are the amplitude and phase frequency response func-tions. Thus Amplitude and phase of the laser pulse can be modulated independently or both depending on the applications.

3.3.4 Reflective geometry of pulse-shaper

Reflective pulse shaper consists of grating, focusing element and MEMS mi-cromirror array. The input laser pulse first is angularly dispersed into frequency components by the reflective grating. A lens with focal distancef then collimates and focus all the spectral components into the Fourier plane at the back focal plane of the lens, at this position the reflective MEMS device is installed. The back re-flected from the MEMS surface spectral components pass through the same lens, and grating recombines them into a single collimated beam. When MEMS mi-cromirrors are not actuated, this setup acts as a zero-dispersion compressor, which works in 4-f configuration. This means that output beam temporally and spec-trally is identical with the input one. The geometry of reflective pulse-shaper setup is illustrated in Figure 3.5.

When the mirrors are displaced in the Fourier plane, the spectral phaseΨcan be modulated through the relation∆Ψ=^4π_λ∆Z, while the amplitudes of spectral components are staying the same. When the mirrors are tilted, setup acts as an am-plitude binary pulse-shaping. When some micromirrors are tilted by a fixed angle around the axis perpendicular to the dispersion plane of the MEMS (Figure 3.5c), two parallel beams separated in space appear from the set-up. The fist beam made up by the spectral components associated to the deflected (D) mirrors, the second one by the complementary ones, reflected by the undeflected (U) mirror elements.

f f Incoming Outgoing beam beam

b)

c) d)

a)

D U

D + ψ U + ψ ψ

f f

f f f f

∆ z

z y

Beam blocker Beam blocker

Figure 3.5:2F geometry shaper with movable mirror array: a) flat phase mask, no amplitude modification; b) arbitrary phase mask, no amplitude modification; c) flat phase mask, with amplitude modification; d) arbitrary phase mask, with amplitude modification.

Simultaneous displacement and tilting the mirrors will lead to simultaneous phase and amplitude modulation (Figure 3.5d).

The relation between the each micromirror displacement on the MEMS surface and spectral phase for each frequency component is given by:

dΨ(ω) dω = 4π

λ ·dZ(ω)

dω (3.10)

wheredΨ(ω)is the spectral component phase measured in rad,Z - is a coordi-nate of each micromirror in relation to its initial position.

The phase of a pulse can be expressed in terms of a Taylor series:

Ψ(ω) = 4π

λ(a₀+a₁ω+a₂ω²+...) (3.11)

Using inverse Fourier transform resulting electric field can be calculated:

E(t) = 1 2π

Z +∞

−∞ Ein(ω)e^iΨ(ω)e^−iωtdω (3.12) Analysing expression 3.11 we obtain that a linear mask, meaning linear dis-placement of micrormirrors, results in a time delay, which correspond to a pulse position modulation in the time domain. Parabolic phase function applied for the micromirror array will induce respectively temporal chirp in the output beam that can be calculated as:

whereτ² is the square of pulse duration of chirped pulse,τ₀² is the square of pulse duration for unchirped pulse, ^d2_dω^ψ(ω)2 =^8πa_λ² is the chirp coefficient resulted from the Taylor expansion of the phase of the output electric field and central laser wavelength.

3.3.5 Spectral resolution of the pulse shaper

Let us consider a propagation of a Gaussian pulse to different planes within the pulse shaper. The full-width half-maximum (FWHM) of the spot size at the Fourier plane∆x0can be derived as

∆x0 = 2 ln 2cosθi

cosθ_d f λ

π∆x (3.14)

θi is the incident angle into the grating, θ_dis the diffracted angle (assuming diffraction into the m = -1 order),f is the focal length of the focusing element,∆x is a spatial width of the input pulse by the FWHM of the spatial intensity. In case of a linear dispersion spatial coordinateX_kof a frequencyw_kat the Fourier plane will be given by

Figure 3.6:Screen-shot from the program used to calculate the spec-tral resolution and temporal window of a pulse-shaper.

X_k=αω_k (3.15)

where

α= λ²f

2πcdcosθ_d (3.16)

is a spatial dispersion parameter, which describes the proportionality between spa-tial displacement and optical frequency. cis the velocity of light, dis the grating period.

The frequency resolution can be defined by δω= ∆x₀

α = 4 ln 2 cosθ_icd

∆xλ (3.17)

The corresponding time window, which gives the temporal limits for shaping, can be calculated by

T = 4 ln 2

δω = ∆xλ

cosθ_icd (3.18)

The shaping time window is proportional to the number of grating lines irradi-ated by the input beam by the period of an optical cycle. If the grating parameter is fixed, a larger time window can only be obtained by expanding the input beam diameter.

3.3.6 MEMS shaping design in the experiment

Here we describe the pulse-shaping geometry used for optimal control experi-ment presented in Chapters 5 and 6. Pulse shaping set-up consists of UV holo-graphic grating 1200 grooves, spherical mirror 3",f = 762 mm focal length (Ed-mund) and folding mirror to reflect a beam into the shaper. Precompressed income UV pulse resulting from the THG of IR pulse is sent to the grating, than focused through spherical mirror to the MEMS. Outcome beam is reflected from the MEMS mirrors, so that the mirrors are slightly tilted on the vertical axis resulting in the outcome beam bellow the income one. Optical setup of pulse shaper used for the discrimination experiment of proteins is represented on the Figure 3.7. Spectral resolution in this configuration was 0.1 nm/pixel.

For the pulse-shaping setup implemented in the Chapter 4, we used a reflective geometry described in the Figure 3.5, where a spherical lens was used used instead of spherical mirror.

MEMS Mirror Array

100 MM

Spherical

Mirror Folding

Mirror

Diffraction grating 800

124100

200

Figure 3.7:Principal scheme of pulse shaper. Simulation is per-formed usingCode V.

270 nm 272 nm 274 nm

268 nm 266 nm

Relative Intensity MEMS

Array

Figure 3.8:Positions of spectral wavelength’s maximums in the Fourier plane. Simulation is performed usingCode V.