High Harmonic Generation and Attosecond Light Pulses

High harmonic generation and attosecond light pulses

Thierry Ruchon November 30, 2016

Contents

1 Summary of the lecture3

2 Short light pulses3 A Usefulness of short light pulses ...... 3 1 Working principle of a camera...... 3 2 Time scales...... 4 3 Measurement of fast phenomena: why attosecond pulses?...... 4 B What does it take to make attosecond pulses?...... 5

3 High Harmonic Generation in gases (HHG)5 A Three step model...... 6 1 General description...... 6 2 Atomic units...... 7 3 Tunnel ionization ...... 8 4 Excursion of the electron in the continuum...... 9 B Elements of the strong eld approximation (SFA)...... 10 C Macroscopic aspects (not treated in class)...... 12 1 Introduction to phase matching ...... 12 2 Gouy phase ...... 13 3 Neutral dispersion...... 14 4 Electronic dispersion ...... 14 5 Dipole phase...... 15 6 Typical exemples ...... 15 D Summary...... 16

4 Attosecond pulse measurement 18 A Measurement principle...... 18 B Reminder about SPIDER ...... 18 C Attosecond pulse measurements with RABBIT ...... 20 D Reminder about FROG ...... 24 E Attosecond pulse measurements with the FROG-CRAB technique ...... 26

1 5 A few attosecond sources 28 A Sources based on ultrashort lasers...... 28 B Sources based on long laser pulses (20-50 fs)...... 31 C Sources with non typical shapes...... 33 1 Source with even harmonics ...... 33 2 Sources with a spatial spreading...... 34

6 Conclusion 35

2 1. Summary of the lecture

1 attosecond (1 as) = 10−18s

. • By focusing an intense laser in a gas target one may observe the generation of a discrete spectrum of high harmonics of the laser.

• It is an extremely nonlinear laser-matter interaction process → Phase matching plays a crucial role.

• We can shape this radiation to form light pulses with attosecond duration.

• One can measure the temporal prole of the pulse since 2001.

• We start using them for applications in atomic and molecular physics.

2. Short light pulses

A. Usefulness of short light pulses

1. Working principle of a camera A camera is essentially based on three parts: an optical imaging device, a sensor on which the image will be printed, and in between, a mechanical shutter (cf. Fig.1). To take a sharp picture, the mechanical shutter opens during a time lapse all the shorter as the object moves quickly, in order to freeze its movement. As the shutter is an object of macroscopic dimensions, the minimum opening time is naturally limited to a fraction of a millisecond in practice.

Fig. 1: Schematic of a camera. Left picture = opening time ...... right picture = opening ......

For an object out of which we want to resolve a detail of typical dimensions L, which is moving at speed v, it typically requires an opening time on the order of t = L/v. Numerical application to the macroscopic domain:

v = 10 m/s, d=1cm→ t= 1ms (1)

3 2. Time scales Time and size scales in nature

• Biology : s-ms and sizes of the order of a mm to a fraction of a mm. For example the case in ionic channels in cells.

• Solid state physics : µs-ns, sizes of the order of µm. Example: computer processors with GHz frequencies, i.e. switching times under 10−10s. Exemple 2: collective eect of atomic bunches..

• Chemistry : Chemistry is mainly the study of the reorganization of the atoms going from one molecule to another. Size of an atom of the order of a fraction of a nm. Mass ' 10−27 g. Period of vibration / rotation in the range of ps / hundreds of fs. Femtochemistry = measurement and control of atomic rearrangements during a reaction.

• Atomic and molecular physics : Control of electrons. Mass ' 10−31g, size.. hummm. Period on the rst orbit of the Bohr atomic model 150 as = 150 10−18s. Conclusion: The lighter and smaller the objects, the quicker they move. The typical associated time scales get smaller as the size decreases, falling in the attosecond range for electrons.

3. Measurement of fast phenomena: why attosecond pulses? General principles for the measurement of fast phenomena

• Series of camera / ultrafast camcorders → Movements up to a few 10-100 µs.

• Stroboscopic approach: The scene is placed in the dark and light is ashed briey: frozen movement.

• From a stroboscop to a pump-probe movie: receipe 1. Have a phenomenon that you can trigger at will at specic times with a rst pump light pulse. Ex: a photochemical reaction. 2. Shine it with a second probe ultrashort light pulse. 3. Temporal resolution of the order of the longer pulse.

Conclusion: To hope accessing electron dynamics, one must have attosecond pulses at hand, which will eventually be used in pump-probe schemes.

4 Spectre Profil temporel Phase #2

Phase #1

Amplitude Axe des fréquences Axe des temps Fig. 2: Fourier transforms: exemples

B. What does it take to make attosecond pulses? As an exemple, let us take a gaussian spectrum with a at spectral phase. In the time domain, a gaussian is also obtained. From Cauchy-Schwarz inequality, the temporal and spectral widths are connected by

h¯ ∆E(eV )∆t ≥ 4 ln 2 = 1.8 1015eV.s (2) e Numerical Application: How wide the spectrum must be to obtain a pulse no longer that 100 as:18 eV What spectral range should we target?Width of the visible spectrum= 1.5 eV → Inevitably in the XUV. And if the phase is not controlled? It does not yield any pulse. Conclusion: one needs

1. An XUV source

2. A spectrum several tens of eV's wide

3. A well controlled spectral phase.

3. High Harmonic Generation in gases (HHG) In the 80's, the shortening of the duration of laser pulses made achievable extremely high intensities on target, in the1014 W/cm2. It opened the door to the strong eld regime: the laser eld becomes comparable to the Coulomb eld of atoms or molecules. Two phenomena are discovered on the Saclay plateau: above threshold ionization (ATI, P. Agostini et al., 1979) and high harmonic generation HHG, A. L'Huillier et al., 1979, and simultaneously in Chicago). In the rst case, a gas target submitted to a strong laser eld emits electrons

5 having energy matching an integer number of photons energies. In the second case, photons having energies odd multiples of the energy of the laser photon are emitted. The spectrum can be very large (see Fig.3) in the XUV. Generally, two zones of the spectrum can be distinguished: the low energy side of the spectrum, where the amplitude of the harmonics decreases slightly with the order, which is called plateau, and the high energy side, where the decay is fast, called the cuto. The question that remains unresolved is: is the phase controlled? An indication is provided by the generation process (three-step model) which is detailed below.

A. Three step model

1. General description An atom is submitted to an oscillating laser eld generally infrared, extremely intense. At the maxima of the eld, the Coulomb potential is so distorted that it becomes possible for the most weakly bound electrons, to tunnel out into the continuum. The electronic wave packet emitted is then guided by the eld and can, under certain conditions, pass close to the parent ion later, with a high speed. It then has a nonzero probability to recombine with the ion, emitting its kinetic energy in the form of photons. So to summarize, three successive

) Plateau r C i 7 t 10 o /

s u

n p

o u t

o r e h 6

p 10 ( l a n g i

S 105 30 40 50 60 70 80 Or dr e Har monique

Fig. 3: Typical harmonic spectrum

temps 0 T/2 T

Fig. 4: Three step model of high harmonic generation.

6 steps:

1. Step 1: Tunnel ionization

2. Step 2: Excursion in the continuum

3. Step 3: Recombination.

In the following, we will successively describre these steps. To make computations easier, we will use the atomic units that are recalled in the following.

2. Atomic units The following is largely copied from Lene Raymond Abdul notes and Vasile Chis notes. The atomic units make the fundamental electron properties all equal to one atomic unit.

(me = 1, e = 1, h¯ = h/2π = 1, a0 = 1...)

Table 1: Basic quantities for the atomic unit system (from Lene Raymond Abdul notes).

Quantity Unit Physical signicance Value in SI-units −31 Mass me Electron mass 9.109 38 ×10 kg Charge e Absolute value of electron charge 1.602 18 ×10−19 C Angular momentum ¯h = h/2π Planck's constant divided by 2π 1.054 57×10−34 Js −10 −1 Electrostatic constant 4π0 4π times the permittivity of free space 1.112 65×10 Fm

Table 2: Derived quantities for the atomic unit system (from Lene Raymond Abdul notes).

Quantity Unit Physical signicance Value in SI-units 2 2 −11 Length a0 = 4π0¯h /mee Bohr radius for atomic hydrogen 5.291 77 ×10 m Energy 2 2 Twice the ionisation potential of 4.359 75 −18 J Eh =h ¯ /mea0 ×10 atomic hydrogen −17 Time τ0 =h/E ¯ h Time required for electron in rst 2.418 88 ×10 s Bohr orbit to travel one Bohr radius 6 −1 Velocity v0 = a0/τ0 = 1/α c Magnitude of electron velocity in 2.187 69×10 ms rst Bohr orbit Angular frequency v0 Angular frequency of electron in 6.579 69 15 s−1 ω0 = 2πa ×10 0 rst Bohr orbit Electric eld strength e Strength of the Coulomb eld ex- 5.142 21 11 Vm−1 F0 = (4π )a2 ×10 0 0 perienced by an electron in the rst Bohr orbit of atomic hydrogen

7 3. Tunnel ionization Total potential of the electric eld:

1 V (x) = − + E x (3) |x| 0

Ionization will be completely opened when the barrier is fully lowered below the fundamental energy of the electron, that is to say, assuming x < 0, ∂V 1 = − + E (4) ∂x x2 0 √ √ The maximum of the barrier is obtained at xm =−1/ E0 and writes V (xm) =−2 E0. It is equal to the ionization potential for 2 . This value of the electric eld is called Esat =Ip /4 the saturation eld (Esat). In practice, it is more convenient to express it as a light pulse intensity. Which reads, in S.I units cε I = 0 E2 (5) sat 2 sat Numerical application for noble gases

Ne 8.7 1014 W/cm2 Ar 2.5 1014 W/cm2 Xe 0.9 1014 W/cm2

Table 3: Saturation intensities in noble gases

Second parameter to take into account: it is necessary that the electron has time to cross the barrier while it is lowered. The estimation will be carried out in the tutorial next week. The result is r I τ ' p (6) 2E2 The criterion is: is this time short or long compared to the laser period. It reads: s s τ Ip Ip (7) γ = = 2 2 ∝ T0 2E T0 2Up

where Up is the ponderomotive energy associated to the eld. It is dened in atomic and SI units as I e2E2 U = = 0 (8) p 4ω2 4mω2

γ is called the Keldysh parameter. We are in the tunnel regime if γ is small. For example, for

8 14 2 10 argon with an intensity of the order of 10 W / cm , a 800 nm laser, we get E0 ' 4 10 V/m,

or else Up = 12 eV and γ =0 .8. It is rather in the tunnel regime. The ponderomotive potential associated with the eld is the energy that the laser eld can communicate an electron in the oscillating (see calculation in the tutorial).

4. Excursion of the electron in the continuum The electron in the continuum evolves under the eect of the laser eld in a which, as a rst approximation may be described using Newton's equation:

mx¨ = −eE(t), (9)

We will study more thoroughly the solutions of 60 in the coming tutorial next week. The main result is that electrons born at dierent times in the laser eld evolve along dierent trajectories and pass by the ionic core, where they may recombine emitting a photon, at given times with given energies. This is summarized in Fig.5. The main point is that it is all deterministic: a given time of birth corresponds to a given energy at a given time of recombination. More precisely, srutinizing Fig.5, it is observed that two trajectories, recombining at dierent times have the same energy. These are so called the short and the long trajectories. While these are experimentally observable (see how below), higher order trajectories, which theoretically exist, have never been reported. These two rst trajectories are so called the short and the long trajectories. From this plot, one may deduce the emission time as a function of the energy of the photon (g.5) Two behaviors may be identied: n o x i 2500 s s Energy i

m 2000 é ’ d 1500 s

0 T/2 temps p

m 1000 e

T 10 20 30 40 50 Ordre harmonique Fig. 5: Temps d'excursion obtenus par le modèle classique à trois étapes.

• The plateau range, where the group delay dispersion (GDD) is constant (group delay (GD) linear)

• The cuto range, where the GDD is zero (constant GD). The second important observation is that all energies are not obtained. There is a maximum which is obtained numerically (cuto law)

9 Ecoupure = (1.3)Ip + 3.17Up (10)

This is one of the most famous and important rule when dealing with HHG. It says that the more intense the laser eld is, the higher the energies of the harmonics may be. The same applies with the gas target: higher ionization potential gases allow to extend the cuto. Numerical application : use the previous HHG conditions and deduce how many harmonics

may be observed. We have Up = 12eV so the cuto is about 40 eV above the Ip. With a photon of 1.5 eV, it yield a cuto about harmonic 27 above the treshold, i.e. H37 for argon. However there is a limit given by the saturation of ionization (see below sectionC). The second alternative to get higher cuto is to change the wavelength towards longer wavelength. This is a very active current research area, where HHG is studies with lasers in the 3-4 µm wavelength range. This is very attractive since it is a parabolic law. However the yield is dramatically reduced due to the increase spreading of the wavepacket. Note that the last question raised is implicitly resolved: there is a well dened phase relationship between successive harmonics. The phase should be controlled, although not at.

B. Elements of the strong eld approximation (SFA) To get insights into the last step the recombination step, we now present a quantum version of the three step model. It was proposed by M. Lewenstein in 1993 and is called the strong eld approximation (SFA). It is based on a perturbative method and yields the following expression of the position of the electron while the laser perturbs it:

Z t Z ~∗ i S(~p,ti,t) ~ ~ ~x(t) = −i dti d~p d e h¯ Exuv(ti) · d ~ (11) ~p+eA~(t) ~p+eA(ti) 0 where Z t 2 ~ S(~p,ti, t) = − dt Ip + ~p + eA(t) /2 (12) ti is the action integral. Reading equation 11 from right to left, we successively recognize three

terms corresponding to each step of the three-step model. ti is the dlbisionization time and t is the dlbisrecombination time. The Fourier transform of the position, that is to say, the induced dipole, gives the radiated spectrum. It reads Z +∞ iωqt ~x(ωq) = dt ~x(t)e (13) −∞ The computation of this quantity is not straightforward. In particular the phase varies rapidly, requiring convoluted integration methods. However the stationnary phase principle (Fermat principle) may be applied in such a situation. The derivation may be carried

10 out against each variable successively : ~p,ti, t. Deriving against t, it reads ∂S(~p,t , t) + ω t i q = 0 (14) ∂t which gives 2 ~ ωq = Ip + ~p + eA(t) /2 = Ip + Ek (15)

We here nd the anticipated equation of energy conservation at recombination. Deriving against ~p, requires the 3D-derivation:

~ ∇~p (S(~p,ti, t) + ωqt) = 0 (16) which reads

Z t 2 ~ ~2 − dt Ip + ~p + 2e~p.A(t) + A (t) /2 = 0 (17) ti Z t ~ ~p · (t − ti) + dt eA(t) = 0 (18) ti

Deriving the latter against the recombination time t yields

~p = eA~(t) (19) d~p = −eE~ (t) (20) dt which is nothing but Newton's equation of movement. This last stationary phase condition is thus related to the closure of the trajectories studied above.

Finally, one may write a stationnary phase for the ionization time ti. it reads ∂S(~p,t , t) + ω t i q = 0 (21) ∂t)i which gives 2 ~ Ip + ~p + eA(ti) /2 = 0 (22)

It apparently leads to an impossibility: the ionization potential Ip being positive, this equation would require a negative kinetic energy term to be fullled. This apparent impossibility is linked to the tunnel nature of the process: the electron should not be able to escape either in a classical world. In the quantum world, the kinetic energy may indeed be negative, using imaginary time. This imaginary time required to fulll Eq. 22 may be interpreted as the tunneling time of the electron (see Maquet et al., 2014).

11 This model can be used (without saddle point approximation to plot the most probable electron trajectories. It nicely matches the classical predictions (cf. Fig.6).

Fig. 6: Quantum trajectories (with no SFA approximation), courtesy of Mette Gaarde.

C. Macroscopic aspects (not treated in class)

1. Introduction to phase matching As with any non-linear optical phenomena, phase matching aspects come into play in high harmonic generation (Fig.7). These are crucial and especially dicult to control for HHG, due to the width of the spectrum, the high absorption of the gas involved, and the number of terms that aect the phase. More precisely, HHG will occur in a gas medium close to the

z `1 `q z’

Fig. 7: Phase matching principle. The incoming beam is in red. The XUV eld in blue. focus of a high intensity laser beam. By high intensity, one speaks of an intensity that may partially ionize the medium. Depending on the location in the medium there might be

1. dierent ions/electron vs neutral ratios

2. dierent intensities

12 3. dierent optical phases (Gouy phase + focusing) All of these play a role in phase matching: generally speaking, the conversion will be ecient when the wavevectors will satisfy the momentum conservation rule, i.e.

~ ~ qkf − kq = 0 (23) where index f is used for the wavevector of the fundamental beam and q for the wave vector of harmonic q. Since 1. The optical indices of ions, electrons and neutral are dierent, and their dierences have no reason to be the same for the XUV and IR wavelengths. 2. The process of generation itself is highly dependent on intensity: the XUV photons emitted at dierent IR intensities have dierent phases. 3. The medium is usually of a fraction of mm to a few mm, not always negligible with

respect to the Rayleigh length (zR) of the beam. ~ ~ ~ For each of the processes recalled above, one contribution ∆kxxx = qkf − kq(for process xxx) will arise.

2. Gouy phase

The Gouy phase reads (with the convention ei(ωt−kz+ϕGouy) for the phases).

z ϕGouy = arctan (24) zR where πω2 z = 0 . (25) R λ with ω0 the waist of the laser. The z-dependent phase may be written

−kz + ϕGouy ' − (k + kGouy) z (26)

dϕ 1 1 1 with k = − Gouy = − · ' − (27) Gouy z2 dz 1 + 2 zR zR zR

This term leads to a phase matching condition (q is the harmonic order)

~ 1 ∆kGouy = qkGouy(ωf ) − kGouy(ωq) = − (q − 1). (28) zR

It is a term that is always negative. And naturally does not depend on any ionization/composition of the medium

13 3. Neutral dispersion It arises from the fact that the neutral atoms do not show the same optical indices for harmonic q and the fundamental beam. It should be noted that importantly, even if this part of the dispersion is usually not dominant, the corresponding imaginary part of the optical index, responsible for the absorption by the neutral atoms plays a very important role, as will be discussed in the tutorial next week. The phase mismatch here reads

~ 2πq ∆kn = qkn(ωf ) − kn(ωq) = (n(ωf ) − n(ωq)) (29) λf The refractive index is in general higher for the IR pulse than for the XUV eld, yielding a positive contribution. Note that the index itself, for dilute gases is proportional to the density of gas. Ionization, which depletes progressively the medium as the pulse passes thus may modify this term. It is thus a dynamical term.

4. Electronic dispersion It is quite common to use media that end up, after the IR pulse propagation, with an ionization of a few tens of percent. Some of these electrons result from the non recombining events in the 3-step model. The electronic dispersion becomes non negligible and must be taken into account. The index associated to the electrons reads

r ω2 n (ω) = 1 − p (30) plasma ω2 with the plasma frequency dened as s 2 e Ne ωp = (31) ε0me

16 3 with Ne the electronic density. A typical target density is in the 10 atoms/cm , which gives a plasma frequency way below usual high harmonics. A Taylor expension thus gives

ω2 n (ω) ' 1 − p (32) plasma 2ω2 We then get, for the phase mismatch ! 2πqω2 ~ 2πq p 1 1 (33) ∆ke = qke(ωf ) − ke(ωq) = (n(ωf ) − n(ωq)) = − 2 − 2 2 λf 2λf ωf q ωf

2πqω2 2 ω2 2 ~ p q − 1 p q − 1 (34) ∆ke = − 2 2 = − 2λf ωf q 2cωf q

14 This contribution is always negative. It should be noted that it linearly depends on the electrons density through 2. It thus varies spatially and temporally during the pulse. ωp

5. Dipole phase As seen above, the phase of the emitted dipole depends dramatically on the shape of the electric driving eld through

Z t 2 ~ ϕ = ωqt − dt Ip + ~p + eA(t) /2 (35) ti An intensity dependent phase is thus added. As a rst approximation (cf. Fig.8) it remains linear with the intensity (ϕ = αI) with an α coecient much larger for the long trajectory than for the short. Assuming a quadratic intensity prole of the laser in the radial direction

40 0

courte -5 20 -10 I  ∂  / ϕ ϕ 0 -15 longue ∂ -20 -20 -25

0 1 2 3 10 15 20 25 30 35 Intensity (1014 W/cm2) Ordre

Fig. 8: Dipolar phase due to variations in intensity of the pump laser.

(rst-order approximation of a Gaussian), we note that this phase corresponds to that of a spherical wave: the second trajectory diverges more than the rst. Furthermore, this corresponds to an additional wave vector −α∇~ I which should be consid- ered when the condition of phase matching is written. It has both a radial and longitudinal component. Its sign changes with the position relative to the focus. This let envision dierent behaviors when focusing before or after a focus.

6. Typical exemples To give some more insight into phase matching, we here put all these contributions in a typical case. First the ionization rates are shown in Figure9. It shows that a non negligible fraction of the medium is ionized before the peak of intensity, thus modifying the phase matching conditions. It may be noted that this behavior, here shown in time, is just the same in space at a given time if the pulse prole is also gaussian in space. When putting together the phase mismatch vectors, it is observed that the phase matching

15 condition is quite naturally observed when placing the jet after the focus (Fig. 10). On the contrary, when placing the jet before the focus, it is extremely hard to phase match the harmonics on axis (Fig. 10). However, these can be phase matched o-axis. We may also note that the α necessary for fullling the phase matching condition are just opposite: it should be small when the jet is after the focus, large when placed before. Consequently, the short (resp. long) trajectory is favored in the rst (resp. second) situation. It should be nally noted that we have not discussed at all the reabsorption of the harmonics by the gas (imaginary part of the optical index of the neutral). This plays a major role in saturating the HHG process to eciencies of about 10−5 − 10−6.

D. Summary The three step model and its quantum variations show that High Harmonic Generation process yields an XUV spectrum, broad and coherent. Its features are rstly determined, at a microscopic level, by the ponderomotive potential associated to the laser eld and the saturation intensity of the gas used as a target. Macroscopic aspects, extremely important, must also be taken into account. These modify both the amplitude and phase of the XUV eld obtained at the atomic level.

x 1013 )

2 10 m c I

/ 6

W 2 ( ) . b r σ a . u ( 100 1.1 1014 14

s 90 1.3 10 n 14 o 1.5 10 i 80 % 1.7 1014 70 -60 Temps (fs) 60 Fig. 9: Ionization rates as a laser passes through an argon gas target (I: intensity of the laser, σ: cross section of ionization, % ions: percentage of remaining neutrals.

16 On axis ~ Gouy kq ∆~k ~ n+e αq I q~k1 ∆~k − ∇

Off axis

Fig. 10: Phase matching conditions.

17 Fig. 11: Principle of an intensity autocorrelator. Copied from www.rp-photonics.com.

4. Attosecond pulse measurement

A. Measurement principle Due to the high frequency of the electric eld of light pulses, all sensors used to detect them are integrating over time (e.g. photodiodes, MCP . . . ). To get the temporal proles of light pulses, the measurement chain must then include at least one non-stationary lter and one stationary lter in order to slice them temporally. For instance, for an autocorrelator used in the visible range (Fig. 11), two replica of the pulse to be characterized are delayed by τ and overlapped in a nonlinear medium (usually a doubling crystal). They create a signal (second harmonic) which reads

Z +∞ Z +∞ I(τ) = dt |E(t)E(t − τ)|2 = dt |I(t)| |I(t − τ)| (36) −∞ −∞ This signal naturally highly depends on the temporal overlap of the two pulses. The lter is therefore non-stationary. However, this kind of measurement does not give access to the spectral phase of the pulse. It just yields the pulse duration when one assumes a certain pulse shape. To get the actual spectral phase of a pulse, and hence its temporal shape, rened autocorrelation techniques, putting at play modied replica of the pulse to be characterized, are required. Two main approaches, FROG and SPIDER techniques, are currently used in the visible/IR domain. We will rst detail the SPIDER technique which was adapted to the attosecond domain giving the RABBIT technique, and second the FROG technique which, adapted to the attosecond domain, gave the FROG-CRAB technique. The rst one, RABBIT, is very popular to measure attosecond pulse trains while the second, which is more demanding and not as direct, is used to retrieve any attosecond pulse structure.

B. Reminder about SPIDER SPIDER is an acronym for Spectral Phase Interferometry for Direct Electric-Field Recon- struction.

18 Shear Ω 1/τ Frequency χ(2) λ

Time (2) λ χ λ Delay τ Fig. 12: Principle of the SPIDER technique. The frequency components of the pulse to be characterized are compared in pairs in the spectral domain.

The principle of SPIDER is to create two replicas of the pulse to be characterized and compare, two by two its spectral components through interferences between the replicas (Fig. 12). Let us consider a pulse to be characterized which electric eld reads E(ω). Two iωτ replicas of it are prepared, and delayed by a time τ, E1(ω) and E2(ω) = E1(ω)e . Moreover,

we have at hand a highly chirped pulse in the same frequency domain E3(ω). Both replicas

are mixed with the elongated pulse E3(ω) in a doubling crystal. Due to the delay τ, the two replicas mix up with dierent part of the chirped pulse, hence with dierent frequencies. The doubled spectra obtained are thus slightly shifted spectrally, by a quantity Ω. Apart from this shift, they are just copies of one another and thus read 0 and 0 0 iωτ . E1(ω) E2(ω) = E1(ω − Ω)e These two quasi doubled replicas are brought to interfere on an optical spectrometer. For sake of demonstration, let us rst consider that the delay is zero. The total signal at a given frequency in the spectrometer, which is the coherent sum of the two corresponding doubled frequency components of the replica ( 0 0 2), can be maximum or |E1(ω) + E1(ω − Ω)| minimum depending on the relative phase of the two components concerned. This reads out on the spectrum envelope which may show a bump or a dip. In principle, just comparing the envelope of the sum of the pulses to the direct spectrum of one replica is enough to get the spectral phase derivative ∆ϕ(ω) ∂ϕ(ω) . However, the spectra are usually not very regular, Ω ' ∂ω making this determination cumbersome. If we now consider that the delay is non zero, a modulation of the interference pattern will appear and make the phase dierences show up on a carrier frequency.

iωτ 2 S(ω) = E1(ω) + E1(ω − Ω)e (37) = I(ω) + I(ω − Ω) + pI(ω)I(ω − Ω) cos (φ(ω) − φ(ω − Ω) + ωτ)) (38)

The spectrum is now modulated with the frequency τ, and the phase of these fringes gives an approximation of ∂ϕ(ω) . Since the measurement of the spectrum is straightforward, this ∂ω is exactly what is needed to retrieve the pulse temporal prole.

19 HHG Filtres

IR Spectro d’électrons

Fig. 13: Principle of the RABBIT measurement.

C. Attosecond pulse measurements with RABBIT The previous technique is dicult if not impossible, to implement in the XUV domain. Indeed it requires

1. To prepare replicas, which is most conveniently achieved using beamsplitters

2. To have non linear crystals,

3. To have high enough a ux to induce non linear eects.

In the XUV domain, neither beamsplitters, nor non linear crystals exist. And HHG ux remain usually modest. The turn around for these diculties is to make a cross correlation between the XUV pulse to be characterized and an IR synchronous eld in a gas target (Fig. 13). The non linear medium is here a well known atomic gas that gets ionized by the attosecond XUV eld. More precisely, the Strong Field Approximation formalism may be used to describe the measurement. The probability to get an ionized electron with energy ω when the two elds (XUV and IR) are present with a delay τ reads (see previous lecture)

∞ Z S(t,τ,~p) ~ ~ ~ i a(~pf , τ) = −i dt d ~pf + eA (t) · Exuv(t − τ)e h¯ (39) −∞ with Z ∞ 2 ~ S(t, τ, ~p) = dt Ip + ~p + A(t) /2 (40) ti As for HHG, this expression has a straightforward interpretation in two steps: dipole transition to a continuum state at time t, and second, a phase modulation of the electronic wave packet emitted by the infrared eld. The latter is just given by the integral action phase

accumulated by the electron with energy Ip + Ek. To make it clear, let us rst consider that the IR eld is zero. In that case we get

v2 S = I t + t (41) p 2m

~ ~ ~ a(~pf , τ) = TF (d.Exuv(t)) ' TF (Exuv(t)) (42)

20 where the later approximation considers a at cross section of the ionizing gas and TF denotes the Fourier transform. The electronic distribution is here simply a copy of the XUV spectrum. This is the principle of an electron spectrometer. To be more specic, a standard way to implement this technique is to use a Time-of- Flight Magnetic Bottle Electron Spectrometer (TOF-MBES, see Fig. 14). It is composed of a sensitive region (left in Fig. 14) where a high magnetic eld (' 1 T) exists and where some gas is injected (usually a noble gas). The XUV eld will ionize this gas. The emitted electrons, which can ll a 4π sr solid angle, are collected by the magnetic eld and driven through a hole in a ight tube. There is a smooth transition from the region with a high magnetic eld towards this ight tube where a low magnetic eld is set. Guided by the magnetic eld, the electrons then y away over a distance L, which, depending on the required spectral resolution, ranges from tens of centimeters to meters. Finally these electrons are detected on a micro channel plate detector with a nanosecond time resolution. Considering a gas with an ionization potential Ip and an XUV photon with energy EXUV , the electrons leave the

high eld region with a kinetic energy Ek = EXUV − Ip. They will therefore take a time L t = q , 2Ek m

where m is the electron mass to reach the detector. Depending on their energy they will reach the detector at dierent times. This is the working principle of a Time of Flight (TOF) spectrometer. In practice the arrival times span a few hundred nanoseconds to a few microseconds. It can thus be resolved by fast acquisition card/oscilloscopes having a GHz band pass. Recording the amount of electrons versus time gives a measurement of the number of XUV photons vs. energy, provided the ionization cross section is known. We now consider the case when the dressing IR eld is also present. The vector potential associated to this eld may be approximated by

~ ~ A = A0 cos(ω0t).

The phase in Eq. (40) gets modulated as:

p2 S(t,τ,~p) f iφ (~p ,t) i =ei/h¯(Ip+ 2m )te IR f e h¯ (43) with Z ∞ 1 0 ~ 0 2 ~ 0 2 φIR(~pf , t, z) = − dt 2e~pf · A(t , z)) + e A(t , z) (44) 2mh¯ t

We will not proceed with the full calculation which is a bit technical 1. However, it may be noticed that the total phase from Eqs. (44) and(43) shows 0, ω and 2ω components due

1A full derivation is available in http://dx.doi.org/10.1051/uvx/201301014.

21 Elec trons V XU IR

Fig. 14: Principle of RABBIT: Time-of-Flight Magnetic Bottle Electron Spectrometer. s s e s n d n o o t e r i t o g c h r e l p e é n s E e d

e Nombre i

g d’électrons r e n E qmin qmax Fig. 15: Photoelectron spectrum ionized by a harmonic source in the presence of an infrared dressing eld.

to the presence of the A~(t0, z) eld. When entered in Eq. (39), it will modify the Fourier transform, making sidebands appear. This is very similar to frequency modulation in radio.

More precisely, when the phase of a signal at frequency fq is temporally modulated at

frequency fmod, sidebands appear symmetrically in its spectrum around fq, at frequencies fq ± pfmod, where p are positive and negative integers (Fig. 15 ). The mathematics are given by the Jacobi-Anger expansion (http://en.wikipedia.org/wiki/Jacobi-Anger_expansion):

∞ iz cos(ωt) X inωt e = Jn(z)e . (45) −∞ where z is the modulation depth, which in our case is set by the amplitude of the dressing IR eld.

22 Let us consider that we have an harmonic spectrum generated through HHG, with fre- th quencies of the (2q+1) harmonic given by f2q+1 = (2q + 1)fIR where fIR is the frequency of the driving laser. Let us also consider that the modulation depth is small, so that the expansion in Eq. (45) can be restrictited to n = ±1 and n=0. Let us nally consider that

the dressing beam is at frequency fmod = fIR. The integrand in Eq. (39) then reads, for harmonic 2q + 1 and only considering the linear A-term in the action integral

0 i2π(2q+1)fIRt+ϕ2q+1 iz(A0) cos(2πfIR(t−τ)) (46) E2q+1e · e .

where 0 is the spectral amplitude of harmonic and its spectral phase. From E2q+1 2q + 1 ϕ2q+1 harmonic (2q+1), we thus get sidebands at frequencies SB , which f2q+1 = (2q + 1)fIR ± fIR include in particular a sideband at frequency SB . It will have a phase given by f2q+1 = 2qfIR ϕ2q+1 −2πfIRτ. The same applies for the harmonic just below: we get a sideband at frequency SB , with a phase . We thus come up with f2q−1 = (2q − 1)fIR + fIR = 2qfIR ϕ2q−1 + 2πfIRτ the situation where sidebands from two neighboring harmonics end up at the same spectral location. They will therefore interfere with one another. The reasoning developed for the SPIDER technique applies: if the two neighboring harmonics are in phase, this sideband should be strong. On the contrary, if they are out of phase, it should be zero. In summary, alike in SPIDER, by shearing the spectrum we make neighboring spectral components talk to one another and get in a position to retrieve their relative phase through interferences. The intensity of the sideband reads

q+1 p (47) Isb ∝ Iq+2 + Iq + Iq+2Iq cos (2ωτ + ϕq+2 − ϕq)

with ϕq+2 and ϕq the spectral phases of harmonics q+2 et q and Iq+2, Iq their intensities. Alike spider again, temporal fringes may be introduced by scanning the delay τ between the IR and the XUV attosecond beam, easing the determination of the phase. An exemple of a RABBIT trace (Reconstruction of attosecond beating by interference of two photon transitions) is shown in Fig. 16. The sidebands are clearly identied due to their oscillations. The position of the min/max of successive sidebands informs on the relative phase of the interfering harmonics. It should be noted that

1. It is only valid when there are only odd harmonics of the fundamental

2. It measures an average pulse in the train.

This technique is widely spread in the community. It has the great advantage to be quite straightforward to implement, quick and require a low dressing energy. However, due to the limitations enumerated above, it is not universal. In particular, it does not work for single attosecond pulses. In that case, a variant of the FROG technique is usually implemented.

23 Énergie de photoélectrons (eV) 2 6 10 14 18 12

10 ) s f ( 8 d r a t

e 6 R 4

2 11 13 15 17 19 21 23 Ordre harmonique Fig. 16: Exemple of a RABBIT trace. Abcissa is the energy of photoelectrons, the vertical axis is the delay between the XUV beam and the dressing IR beam. Continuous lines correspond to harmonics while oscillating ones are sidebands. The phase of these sidebands is reminiscent of the group delay of the harmonics.

D. Reminder about FROG FROG is an acronym for Frequency Resolved Optical Gating. The principle is the gating of the pulse to be characterized (E(t)) by a delayed eld (g(t−τ)). In its most intuitive version, g(t − τ) is a proper gate, that slices E(t) in little portions that will be spectrally resolved. Such a spectrum is recorded for a series of time delays τ and we nally get a 2D-map:

Z ∞ 2 −iωt I(τ, ω) ∝ dtE(t) · g(t − τ)e (48) −∞ We may note that the gate should not be too narrow. In the limit case when g(t) = δ(t) Eq. (48) yields Z ∞ 2 −iωt 2 I(τ, ω) ∝ dtE(t) · δ(t − τ)e = |E(τ)| . (49) −∞ This is simply an autocorrelation that does not allow to retrieve a spectral phase. Indeed, as already stated earlier, the point behind any spectral phase measurement is to get the dierent spectral components to talk to one another, i.e. to interfere with one another. In the FROG technique, it is the Fourier transform that plays this role: it mixes up dierent spectral components; provided the gate has selected several of them. In the case when the gate is extremely narrow as treated in Eqs. (49), only one component is selected and thereby not brought to interfere with any other by the Fourier transform. It is thus reasonable that

24 Fig. 17: Principle of a frog measurement. Copied from www.rp-photonics.com. no temporal information is nally retrieved. At variance with SPIDER, it should be notice that here many spectral components are mixed at once with the Fourier transform, while it is only two for the SPIDER. This makes the retrieval more complicated for FROG than for SPIDER. For a proper FROG, the gate should be long enough to select several components. One very popular gate is the pulse itself, squared and the autocorrelation in Eq. (48) is performed using a 3rd order non linearity. It nally yields

Z ∞ 2 −iωt I(τ, ω) ∝ dtEsig(t, τ)e (50) −∞ with 2 Esig(t, τ) = E(t) · |E(t − τ)| . (51)

Introducing the Fourier transform of Esig(t, τ) along τ: Z ∞ ˜ −iΩτ Esig(t, τ) = dΩEsig(t, Ω)e (52) −∞ we get Z ∞ Z ∞ 2 ˜ −iωt−iΩτ I(τ, ω) ∝ dΩ dtEsig(t, Ω)e . (53) −∞ −∞ ˜ This is nothing but a double Fourier transform of Esig(t, Ω) for which we only measure the amplitude squared. Fortunately algorithms exist to solve this equation known as 2D-phase retrieval problem. At rst glance, it may appear surprising that the underlying phase may be retrieved out of an intensity map. The point is that the information is here highly redundant (the inputs are two 1D elds, i.e. 2n points if the discretization is performed over n temporal points while the FROG map may be n×n points). The optimization algorithm thus searches for a solution best tting all these autocorrelations at once and the solution is usually unique2

2The similarity between this problem and Coherent diraction imaging is striking: in both cases, whereas only amplitudes are retrieve in 2D, one is able to reconstruct both the phase and the amplitude of the underlying eld.

25 ˜ Finally, if one is able to retrieve Esig(t, Ω), retrieving E(t) is straightforward: Z ∞ ˜ iΩτ Esig(t, Ω) = dτEsig(t, τ)e (54) −∞ Z ∞ = dτE(t) · g(t − τ)eiΩτ (55) −∞ Z ∞ = E(t) · dτg(t − τ)eiΩτ (56) −∞ = E(t) · G(Ω)eiΩt (57)

with Z ∞ G(ω) = dτg(τ)eiΩτ (58) −∞ We thus have E˜ (t, 0) E(t) = sig (59) G(0) Interestingly the gate need not be an amplitude gate, but can also be a phase gate. The point is that it should simply multiply the signal to be determined by a non stationary value. For the attosecond version, it is a phase gate that is used.

E. Attosecond pulse measurements with the FROG-CRAB technique FROG-CRAB means Frequency Resolved Optical Gating for Complete Reconstruction of Attosecond Beatings. It was proposed in CEA-Saclay by Yann Mairesse and Fabien Quéré. In the literature, it may also appear as the streaking technique. The principle is just that of FROG, but applied in the XUV domain. The setup is the same as for RABBIT. Inspecting Eq. (39) it is clear that the spectrum measured through the cross correlation is a FROG trace, where the phase is modulated. The standard retrival algorithm may thus be applied. This FROG-CRAB technique presents several advantages:

1. It reconstructs the actual pulse or series of pulses prole(s) (not a mean pulse),

2. It also works with continuous spectra, thus for isolated pulses.

Exemple of FROG-CRAB spectrograms are shown in Fig. 18. On the left, the dressing pulse is probably 5 to 7 femtosecond long. Both on the left and right extremities (delays = ±6fs) we observe a continuous spectrum centered around 70 eV. This is the central wavelength of the attosecond pulse. Now, when the IR beam arrives it may completely deplete the spectrum and shift it up or down depending on its phase with respect to the IR. This may be interpreted as a RABBIT at very high IR intensities. The large value of the IR intensity allows the creation of sidebands with high orders (almost a 10 eV shift is observed). However, the description with the RABBIT formalism becomes cumbersome and it is preferable to use the description proposed by Quéré & Mairesse: the photoionization frees an electron in the

26 40

60 35 ) V

e 30

( 70

y 25 g r 80 e

n 20 E 90 15

-6 -4 -2 -0 2 4 6 -1.5 1.0 0.5 0.0 0.5 1.0 1.5 Delay (fs) Delay (fs) Fig. 18: Exemple of cross correlation traces (FROG-CRAB traces) in the case of a single attosecond pulse (left) and a train of attosecond pulses (right). (Adapted from M. Schultze et al., New J. Phys. 9, 243 (2007) and P. Johnsson et al., Phys. Rev. Lett. 95, 013001 (2005) respectively).

IR Vector potential M o ∂p~ ∂A~ p m

0 e

= e n

∂t ∂t t u m

IR Field

(Quéré et al., J. Mod. Opt., 52, 339 (2005)).

Fig. 19: Trajectories in momentum space of electron wave packets injected in the continuum at dierent times during the IR eld.

continuum which is then submitted to the remaining IR electric eld. Alike what we used for the 3-step model describing HHG, the Newton equation of movement may be written, taking advantage of the large value of the A-eld to neglect the ionic core. It gives:

d~p ∂A~ = eE~ = −e (60) dt dt which leads, once integrated and taking into account that at innite times the eld is zero to ~ ~pf = ~pi + eA(ti) (61) A certain energy (momentum) distribution of electrons at time of ionization is just shifted in the spectral domain by a value equal to the vector potential at the time of ionization. This is illustrated in Fig. 19. So what we see in Fig. 18 is just the oscillation of the electrons

27 momentum linked to the changing value of the potential vector at the time of ionization when the delay is scanned. It is also reminiscent of the dressing IR pulse prole. This streaking of the spectrum can be processed to retrieve the pulse prole using standard FROG algorithms. On the right of the gure the same technique is applied to a train of attosecond pulses. Here the harmonic structure remains apparent. It is just superimposed to the streaking. These two techniques, RABBIT and FROG-CRAB are, by far, the most widely spread ones to measure temporal proles of attosecond pulses. However, they both have limited validity domains and a signicant amount of work is currently done to extend them into more powerful techniques.

5. A few attosecond sources The important parameters for an attosecond source are

• duration(Width of the spectrum plus spectral phase)

• central frequency

• short or long train, single attosecond pulse, repetition rate ...

• Flux The control means are

• the target

• the pump beam (energy, λ, polarization. . . )

• the periodicity of the pump

• the duration of the pump

A. Sources based on ultrashort lasers Nowadays, ultrashort lasers may reach durations of 1-2 optical cycles, may it be around 800 nm (T=2.7 fs) or in the MidIR range (' 1.5 − 2µm, T=5-8 fs). With these durations becoming comparable to the electric eld period, an important parameter comes in the game: the carrier envelope phase (CEP). Indeed, it is clear that the number of equivalent maxima, and their value changes dramatically while changing the CEP with such ultra short pulses. In the rst case, the tunnel ionization may occur just once, while it may occur equivalently twice in the other case. This results in the emission of one or two attosecond pulses. An example of attosecond pulses generated with such an ultrashort laser is displayed in Fig. 21. First, it must be noticed that the spectrum is almost continuous. This is easily explained if we get back to the origin of the harmonic structure in the HHG spectra: it was attributed to the repetition of the process every half cycle of the eld. Here, the eld is intense enough to ionize the atom only once and the spectrum has therefore no reason

28 to show an harmonics structure. However, it may be noticed that there are a few residual modulations on the spectrum. This is probably the remaining of a second little attosecond pulse that is not completely killed by the decrease of the eld during one half cycle. In other words, this is the remaining of the harmonic structure that is obtained with long driving pulses. Getting hands on so short lasers is nothing easy. In particular, keeping a stable CEP during several minutes remains a technical challenge that is solved today in just a few laboratories worlwide at the level of energy required. To lift the requirements on the duration of the laser while keeping the generation of a single attosecond pulse, several techniques have been demonstrated. Instead of playing on the rst step of the model (ionization step), the popular polarization gating technique acts on the second step (excursion in the continuum). The point is to tailor the eld so that the electronic wavepacket (EWP) is not driven back to the

t2/τ 2 t2/τ 2 e− sin ω0t e− cos ω0t Fig. 20: Two ultrashort pulses with carrier envelope phases shifted by π/2.

Fig. 21: Exemple of single attosecond pulse obtained in Garching (Germany). (Goulielmakis et al., Science 2008)(zirconium lter). Is that a cosine or sine driving pulse ?

29 ionic core during any half cycles but one. This is achieved by making the polarization of the IR eld variable in time: it starts circular, gets shortly linear in the middle of the pulse and ends circular again. Naturally, when the electric eld is circular the EWP is driven away from the core and does not recollide. To shape the pulse in such a way, the most common approach is to play on the dierence between the group and phase velocity of waves in birefringent crystals.

Propagation of an ultrashort pulse in a transparent birefringent crystal Let us consider an ultrashort gaussian laser pulse which electric eld reads

2 2 ~ ~ −σ t iω0t E(t) = E0e e . (62)

We further consider a birefringent plate of thickness d, with indices for the ordinary and

extraordinary axis given by no and ne. We will use the neutral axis of the crystal as a ~ reference frame and denote Eo(t) (resp. Ee(t)) the projection of E(t) on these axis. The eld once propagated through this plate writes

2 2 iω (t−n d/c) −σ (t−d/vog) e 0 o Eo(t) = E0e (63) 2 2 iω (t−n d/c) −σ (t−d/veg) e 0 e Ee(t) = E0e (64)

with c vg = (65) e,o ∂ne,o ne,o(ω) − ω ∂ω If the thickness is very large we will have a temporal separation of the two components. The polarization is always linear during the pulse but the two pulses have their polarisation at 90 degrees from one another. If the thickness goes to zero, naturally nothing happens: the pulse remains linearly polar- ized. In between these two extreme cases, and if in addition the thickness corresponds to a dephasing between the two axis of half a period (thick quarter wave plate), the polarization is linear, gradually circular and nally linear again. To get to the right combination, a second thin quarter wave plate shifts this to circular, linear and circular again. Although fairly ancient (late 90's), this proposition was only tested successfuly in 2006 by a consortium putting together Milan and Bordeaux's groups. Their results are reproduced in Fig. 23. It shows that

1. There is a large spectral tunability

2. The CEP remains important

3. One may get a single attosecond pulse even in the plateau, where XUV uxes are higher.

30 λ/4, Ordre élevé

λ/4, Ordre faible

Fig. 22: Principle of polarization gating

) 1.0 . u . a ( l a n g i 0.5 s V U ) . X u .

a 1.0 (

0 l a

25 30 35 40 45 50 55 n g i 0.5 Photon energy (eV) s V U

X 0 …1 0 1 2 Time (fs) Fig. 23: Attosecond pulses obtained in Milan (M. Nisoli's group, in collaboration with the group in CELIA Bordeaux) using a polarization gate (aluminum lter).

B. Sources based on long laser pulses (20-50 fs) Without any further ltering (and this is also true for attosecond pulses issued from short lasers described in Sec.5.A) the spectra are naturally dominated by their low energy parts (Fig. 24). And in general, the duration of these harmonics will be femtosecond and longer. It is thus crucial to lter out these harmonics to get an attosecond pulse. One very standard way to achieve this function is to use metallic foils placed on the beam. These usually have a thickness of a hundred to a few hundreds nanometers, making them exaggeratedly fragile. The transmittance of a few common lters is reported in Fig. 25. They show a cut on in a range going from 15 eV to 60 eV, perfectly suited to lter out the low order harmonics while keeping those from the plateau. In addition, due to usual Kramers-Kronig relations, right after the cut on the dispersion shows a negative group delay. This is exactly opposite to the natural group delay of the harmonics, induced by the excursion in the continuum.

31 ) . 0

U 10 .

b -3

r 10 A ( -6

é 10 t i

s -9

n 10 e t

n 3 13 23 33 I ordre harmonique

Fig. 24: Typical harmonic spectrum obtained by a source at 800 nm of 35 fs duration. Fig. 25: Examples of complex transmittances of metallic lters used in attosecond physics.

These lters can thereby also fulll a second function: compressing temporally the pulses. An exemple of such a compression is shown in Fig. 26 for a spectrum generated in argon ltered by an aluminum lter. On the same laser, now generating in neon and using a zirconium lter, one may obtain the completely dierent pulse displayed in Fig. 27. The basis spectrum is naturally larger (see lecture I), allowing more tuning opportunities. In conclusion, using

1 ) . U . A ( é

t 0.5 i 1 s n ) e . t n U I . b r A

0 (

é 170 as t 0.4 i 0.5 s n ) e s t f ( n I

e 0 p u o r g

e 600 nm 0 d -0.4 -400 0 400 d r

a 200 nm t

e Temps (as) R -0.8 No filter 20 30 40 50 Énergie (eV) Fig. 26: (left) Spectrum and group delay for a harmonic radiation unltered or ltered by increasing aluminum thicknesses. (right) corresponding time proles. metallic lters, one may

1. Adjust the spectral band,

32 2. Adjust the group delay.

It may be noted that this group delay control may also be achieved by XUV chirp mirrors. For instance the group of Franck Delmotte and Sébastien De Rossi at Institut d'Optique could design mirrors with a reectivity higher than 10 % compensating for half of the group delay of a typical Argon harmonic source (see Fig. 28). This hold great promises. Indeed, the mirrors are made of alternating layers of several materials with optimized thicknesses to ll a given target function. Varying the thickness and the nature of the material may completely change the mirrors properties, allowing to really tune a mirror to a given source. It is much more exible than the metallic foils that are not tunable at all. The use of lters is however simple and straightforward. Placed in front of an HHG source it yields right away an attosecond pulse train ready for use.

C. Sources with non typical shapes

1. Source with even harmonics The rst question that may be asked is : even harmonics, what for? The answer is fairly straightforward when thinking about streaking experiments (see Fig. 19). Apart from Mag- netic Bottle Time of Flight spectrometers, which just give the kinetic energy distribution of electrons, other detectors exist that directly resolve the momentum in 3D. These are Veloc- ity Map Imaging Spectrometer (VMIS) or coincidence detectors like COLTRIMS (reaction microscope). In that case, when dressing the electrons with an IR eld, it is noticed (Fig. 19)

1 0.8

0.4

0.5 R e t a ) r . 0 d U d . e b

r 0 g A r ( 1 o 0.8 u é t p i e s n ( f e s t ) n

I 0.4 0.5

0 temps 0 40 60 80 100 Énergie (eV) Fig. 27: (top) unltered harmonic radiation. (down) After passing through a zirconium lter.

33 that the electrons are directed towards opposite directions every other cycle. In particular, if one wants to study the rescattering of the electron on the ionic core, the image is somewhat blurred due to the superposition of these two electron wave packets evolving in opposite directions. To avoid this situation, a source with one attosecond pulse per laser period would be ideal. It corresponds to having both odd and even harmonics. As a side eect, it may also be used to gate the attosecond emission: if the periodicity is now T instead of T/2 in standard cases, the pulse duration required to get a single attosecond pulse is now longer than in the standard case. Say instead of requiring a 4 fs laser pulse, it can now be roughly 7 fs. Combining that with the polarization gating technique, Chang and collaborators could even demonstrate single attosecond pulses starting from 20 fs laser pulses. The usual approach to get odd and even harmonics is to play on the ionization step. Superimposing to the main laser a laser with much less energy but twice its frequency, every other half cycle the extremum may be lowered (Fig. 29). The tunnel ionization is thus killed and the attosecond emission occurs just once per cycle. Exemples of HHG spectra obtained with such a eld are shown in Fig. 30. A streaking experiment could be perform on these pulses, conrming the attosecond structure of the emission (Fig. 31).

2. Sources with a spatial spreading This is the attosecond light house eect described in the tutorial last week. One imposes a wavefront rotation at focus leading to slightly dierent directions of propagation for the harmonics emitted at successive half cycles (Fig. 32, top panel). This technique holds promises to generate perfectly synchronized sets of attosecond pulses, which could be used to perform attosecond pump/attosecond probe experiments. It is also extremely promising as a diagnos- tic technique of HHG. As an illustration, the middle and bottom panels in Fig. 32 illustrate

Reflectivity Mirror GD (as) 0.2 M1 500 0.1 0

0.0 -500 500 0.2 M2 0 0.1 -500 0.0 500 0.2 M3 0 0.1 -500 0.0 20 40 60 80 20 40 60 80 Photon energy (eV) Photon energy (eV) Fig. 28: (left) reectivities of three attosecond mirrors calculated and deposited at Institut d'Optique. (Right) Group delay calculated and measured on these mirrors.

34 another gating technique revealed by the attosecond light house eect. The middle panel shows HHG in N2 with a reasonably low intensity while the bottom panel shows the same with a much higher intensity. Clearly the spectrum changes: while in the middle panel one may observe 4-5 attosecond pulses, in the lower panel, just one is still visible. This is due to ionization that is much more rapid in the second case and quickly kills any attosecond emission after the rst ecient cycle. Interestingly, over-driving HHG here yields an isolated attosecond pulse. This is so called the ionization gating approach towards single attosecond pulses.

6. Conclusion The availability of attosecond pulses is ever increasing around the world, although it remains a subject of intense research. If the physics of high harmonic generation is fairly well un- derstood in simple cases, it still reveals unexpected features. For instance, HHG in aligned

ω + (15%) ω =

Fig. 29: Sum of two harmonic elds é t i s n e t n I

Ordre harmonique Ordre harmonique Ordre harmonique Fig. 30: HHG spectra obtained with (left) a 35 fs pulse centered at 800 nm (middle and right) the same pulse on which is superimposed a fraction of second harmonic (400 nm) for two dierent 800 nm / 400 nm phases (Spectra from Lund University, Sweden).

35 molecules is nowadays a hot topic that carries the potential to open new spectrometric methods. It is attracting many research groups, may they be theoretician or experimental- ists. The objective is there to have a method to perform molecular spectroscopy in the time domain with an attosecond resolution, which is the time it takes for the electrons to probe the molecule in the 3-step model. The attosecond light house eect, shortly described in Sec. 5.2 is an illustration of the progress towards ever ner insights into strong eld processes at the half cycle level. As an illustration of recent progress, it was (re) discovered lately that we could make attosecond pulses with circular polarization. Using them, it was demonstrated this year that it is extremely well suited to study chiral species in the gas phase. Along this line, new attosecond beam lines are currently built to transfer standard visible domain techniques to the XUV. For instance, transient absorption spectroscopy receives a lot of interest these days. Another debate that is exciting the community is our ability to measure photoionization times. i.e. the time it takes for an electron to reach a detector after being ionized by a single XUV photon. The interpretation of the experiments carried out lift a serious debate about the status of time in quantum mechanics and about tunneling times. These new lines of research are just possible in laboratories that master the synthesis, measurement and transport of attosecond pulses. These are about 20 worldwide today.

TIR

TIR/2

TIR

Fig. 31: Streaking traces obtaines with (top) and without (bottom) the second harmonic present in the generation laser.

36 b c1 c2

c3 c4

14 2 I1= 2x10 W/cm

14 2 I2= 3.8x10 W/cm

Fig. 32: Spectra obtained when a wavefront rotation is set at the focus of the generating laser. Abcissa is the energy, vertical axis is the harmonic divergence. (top) Generation in neon, 7 fs laser, many shots average. (middle) Generation in N2, single shot, fairly low intensity. (bottom) Generation in N2, single shot, about the saturation intensity. From Kim et al., Nature Physics 2013.