Università di Pisa
Dipartimento di Fisica "Enrico Fermi"
Corso di Laurea Specialistica in Scienze Fisiche
Ultrashort Surface Plasmon
Generation by Rotating Wavefronts
Candidato: Relatore:
Francesco Pisani Dr. Andrea Macchi
Contents
1 Introduction 5 2 Theory 9 2.1 Surface Plasmons . . . 9 2.2 Grating coupling . . . 15 2.3 Plasmonics . . . 172.4 Wave Front Rotation . . . 20
2.5 Generation of High Harmonics . . . 22
3 Simulation 29 3.1 MEEP . . . 29
3.2 Pulse implementation . . . 31
3.3 Grating implementation . . . 35
4 Results 37 4.1 Effects of the grating parameters . . . 37
4.2 Resonance width . . . 42
4.3 Rotation parameter dependency . . . 47
4.4 Absolute phase . . . 55
5 Conclusions 57 5.1 Further study . . . 57
Bibliography 59
Chapter 1
Introduction
The first laser was built by Theodor H. Maiman in 1960 [1]. Since then photonics, the study of how the light (photons) can be used to transport information or inter-act with materials, has been enriched with various and advanced research areas. A variety of light sources with higher efficiency or intensity and different wavelengths have been developed: light-emitting diodes (LEDs) [2], terahertz quantum-cascade lasers [3], frequency combs [4]. Materials have been tailored in order to obtain pro-prieties not reproducible with common materials: photonic crystal waveguides [5], perfect lenses [6], cloaking devices [7] and so on. In this thesis we are manly inter-ested in two other important and active areas: the generation of ultra-short laser pulses and plasmonics.
Ultra-short laser refers to a laser pulse with a duration of few cycles. These pulses are typically generated with a Ti:Sa laser having a wavelength of ∼ 800 nm and the oscillation period of ∼ 2.67 fs, thus the duration of an ultra-short pulse is of a few femtoseconds. Nowadays lasers with a duration in the fs range and a peak power up to few MWs are compact tabletop systems accessible with relative ease (e.g. from Thorlabs, Onefive, Imra and other companies) and the research went forward to the attosecond range. The possibility to generate and control pulses in the attosecond range is of great interest, allowing the study of phenomena with very fast time scale, e.g. the molecular chemical processes [8] or the electrons motion in an atom.
Another fundamental characteristic of a laser is its intensity. Thanks to the Chirped Pulse Amplification (CPA), laser pulses can reach extreme power and in-tensity values at the focus far beyond the ionization threshold of all materials [9]. With these high-intensity laser pulses it is possible to study non linear phenomena, e.g. the generation of high harmonics. In order to generate a laser pulse with a duration in the attosecond range a higher frequency is needed, thus the coherent emission of phase-locked harmonics can be a powerful instrument to achieve such purpose [10].
In the last decades plasmonics has been developing in parallel to photonics. Plasmons are collective oscillations of the electrons in the plasma. In particular we are interested in the coupling, under the right condition, of a laser pulse on a surface, leading to the excitation of a Surface Plasmon (SP). We will refer as SPs to those electromagnetic surface waves that propagate on an interface between two mediums with opposite dielectric constants. They can propagate on the sur-face for hundred of micrometers (depending on the materials used) and they are evanescently confined in the perpendicular direction. The decay length in vacuum is of the order of half a wavelength. SPs have various interesting applications: the possibility to concentrate the light under the diffraction limit and to guide it on a surface makes the SP suitable for nanometric photonic circuits [11] and energy concentration [12]. Also the increase in the energy transferred to the surface is of great interest e.g. for higher efficiency solar cells [13] as well as for high intensity applications such the generation of plasmas or electrons and ions acceleration (see section 2.3 for more details). Moreover ultra-short SPs can be powerful instru-ments to study phenomena with fast time scale, e.g. the effects of the vibration of a lattice on the electric and thermal transport [14] or the ultra-fast surface-enhanced Raman spectroscopy [15]. Ultra-short SPs can also find application in photonic circuits, where they are employed as bits of information. Their duration and velocity of propagation are thus of fundamental relevance [16]. Also when the duration of the SP is of only a few cycles, the envelope of the pulse changes significantly in one oscillation. Therefore the effects of the absolute phase of the field and the carrier envelope become important and can be studied with various methods [17].
In this thesis we will propose a technique of possible relevance to plasmonics applications in the ultrashort regime. This technique was inspired from two works concerning the generation of high order harmonics ([10] and [18]) that we will de-scribe in section2.5. We will use a pulse with wave fronts rotating in time to excite a SP. The aim is to generate a SP with a very short duration (few femtoseconds). This technique represents a linear method to generate ultra-short pulses: the SP generated will have the same frequency of the laser pulse and an energy propor-tional to it. We will see in section 2.1 that the generation of the SP with a laser pulse occurs only at a given angle of incidence (resonance angle), defined by the normal to the surface and the wave fronts. Since we will use a pulse with the wave fronts rotating in time, the angle of incidence will be different for different sections of the pulse. Thus the excitation of the SP will occur only for a small portion of the pulse, i.e. we will select only a part of the laser pulse to generate the SP and we will obtain a shorter pulse. In order to test the effectiveness of this technique
7 we made some simulations using the open source code "MEEP" (see chapter 3).
This thesis is structured as follows. In chapter 2 we will give a description of the two phenomena that are involved in our work: the Surface Plasmon and the Wave Front Rotation. In chapter 3 we will describe the code we used for the simulations, explaining its strength and weakness points, as well as some of its limitations in our work. In chapter 4 we will report the results of our simulation campaign and describe the influence of various effects on the duration of the SP.
We will find that with this technique we can obtain a SP with a duration of ∼ 5 fs from a laser pulse of ∼ 30 fs, with a wavelength of 800 nm. That is the shortest duration we managed to achieve and we will see that is independent from the duration of the laser pulse, thus it represents the limit of this technique. We will also find a correlation between the absolute phase of the SP’s field and the phase of the laser pulse’s field that excites it: changing the absolute phase of the laser pulse’s electric field of a given amount corresponds in a change of the same amount in the phase of the SP’s electric field, making this technique suitable also for applications involving the absolute phase with phase-stabilized laser pulses [19].
Chapter 2
Theory
In this chapter we will describe the physical phenomena involved in our work. In the first part we will give a detailed description of the Surface Plasmon, as well as some applications in which they are involved. Then we will describe the phenomenon of the Wave Front Rotation: we will explain how a pulse with the wavefront rotating in time can be generated and how a rotating wavefront pulse may excite an ultrashort SP. We will conclude describing two works that inspired this thesis concerning the generation of high harmonics form both solid and gaseous targets.
2.1
Surface Plasmons
Surface Plasmons (SPs) are electromagnetic excitations propagating on an inter-face between two mediums with dielectric permittivity of opposite sign, e.g. a metal and vacuum. In this thesis we shall refer to Surface Plasmons only as those modes that propagates on the surface. These surface waves are collective modes of the free electrons in the metal. In the direction perpendicular to the surface they are confined, i.e. the field is evanescent on both sides of the interface. In this chapter we want to explain in detail how to excite a SP using an electromagnetic wave, i.e. a laser light. Therefore we will find the dispersion relation starting from the Maxwell’s equations. We will discuss the problem of coupling an electromag-netic wave with the SP and explain the grating coupling method used in this work. In order to obtain the resonance condition we need to find the dispersion rela-tion of the SP. The interested reader can find more details in [20].
We start from Maxwell’s equations: ∇ · E = 4πρ ∇ · B = 0 ∇ × E = −1 c∂tB ∇ × B = 1 c 4πJ + ∂tE (2.1)
Taking the curl of the third equation and the time derivative of the fourth equation we get ∇ × ∇ × E = −1 c2 4π∂tJ + ∂t2E (2.2) We assume an harmonic dependence for the electric and magnetic fields. Using ∇ × ∇ × E = ∇(∇ · E) − ∇2E the equation becomes
∇(∇ · E) − ∇2E = −1
c2 4πiωJ − ω 2E
(2.3) The electric current is related to the electric field via the conductivity: J = σE. We can define then the dielectric constant εr = 1 +
4πiσ
ω to obtain the equation: ∇(∇ · E) − ∇2E = −ω
2
c2εrE (2.4)
We consider a homogeneous medium with free electrons (like a metal or a plasma) described by the dielectric function:
ε(ω) = 1 − ω
2 P
ω2+ iγω. (2.5)
Where ωP is the plasma frequency. For simplicity we assume the dielectric constant
to be real (γ = 0), neglecting dissipation, and we assume the material to be cold, neglecting the motion of the ion population. Solving eq.(2.4) with the value of the dielectric constant given by eq.(2.5) we find that for ω < ωP an electro-magnetic
(EM) wave cannot propagate through the medium, i.e. it is evanescent. Instead, for ω > ωP, two kinds of waves can be excited: electromagnetic transverse waves
(∇ · E = 0), with the dispersion relation ω(k) = pω2
P + k2/c2 and electrostatic
2.1. SURFACE PLASMONS 11
Figure 2.1: Representation of the system used: the two mediums are described by the dielectric constant ε1,2. The SP is propagating at the interface in the x direction and it
is evanescent in the z direction.
We now want to solve the eq.(2.4) in a medium with a discontinuity in the dielectric constant: we assume the z > 0 region to be filled by a medium with dielectric constant ε2 and z < 0 with by a medium with dielectric constant ε1
(fig.2.1).
We look for solution characterized by ∇ · E = 0, so we can simplify eq.2.4:
∇2E − ω 2
c2εrE = 0 (2.6)
In this configuration, the SP propagates on the interface between the two mediums. We are looking for particular solutions characterized by B · k = 0 (TM modes) and ∇ · E = 0, i.e. the volume charge density ρ = 0, thus only the surface charge density is allowed and it will appear on the interface (z = 0).
Without loss of generality we can take x as propagation’s direction and we can write E(r, t) = E(z)eikxxe−iωt, with k
x the component of the wave vector in the x
direction. Equation (2.6) reduces to ∂2E ∂z2 + (εr ω2 c2 − k 2 x)E = 0. (2.7)
A similar equation can be found for B. Using Maxwell’s curl equations with the assumption ∂
∂zEy = −iωBx ∂zEx− ikxEz = iωBy ikxEy = iωBz ∂zBy = i ωεr c2 Ex ∂zBx− ikxBz = −i ωεr c2 Ey ikxBy = −i ωεr c2 Ez (2.8)
We can find a self-consistent solution assuming a transverse magnetic (TM) wave, i.e. with only Ex, Ez and By. For this mode, the system reduces to
Ex = −i c2 ωεr ∂By ∂z Ez = − kxc2 ωεr By (2.9)
And the wave equation becomes ∂2B
y
∂z2 + (k 2
0εr− kx2)By = 0. (2.10)
To find the dispersion relation we need to solve the equations in both regions. We are searching for solutions that are confined at the interface so we try solutions with an exponential decay in the z direction.
For z > 0, By(z) = A2eikxxe−k2z Ex(z) = iA2 c2 ωε2 k2eikxxe−k2z Ez(z) = −A2 kxc2 ωε2 eikxxe−k2z, (2.11)
2.1. SURFACE PLASMONS 13 For z < 0 By(z) = A1eikxxek1z Ex(z) = −iA1 c2 ωε1 k1eikxxek1z Ez(z) = −A1 kxc2 ωε1 eikxxek1z. (2.12) Where k2 1,2 = kx2 − ω2
c2ε1,2 is the component of the wave vector perpendicular to
the interface, representing the inverse of the decay length. Continuity of By, Ey and εEz at the interface requires that
A1 = A2 k2 k1 = −ε2 ε1 (2.13) We see from these conditions that ε2 and ε1 must have opposite signs in order to
obtain evanescent waves in the z direction.
From the equations for kj we finally get the dispersion relation for the SP:
kx2 = ω 2 c2 ε1ε2 ε1+ ε2 (2.14) The condition for the dielectric constants becomes ε1ε2
ε1+ ε2
> 0in order to obtain a real wave vector. Since we have ε1ε2 < 0, we must chose ε1 and ε2 such that
ε1+ ε2 < 0.
A possible choice is ε2 = 1 and ε2 = 1 −
ωp2
ω2, with 2 −
ωp2
ω2 < 0, which represent a
metal-vacuum interface.
In this case the dispersion relation becomes
k2x= kSP2 = ω 2 c2 1 − ω 2 p ω2 2 − ω 2 p ω2 (2.15) Fig.2.2 shows the dispersion relation of a SP at the interface between a Drude metal, i.e. a medium described by the dielectric constant (2.5), and vacuum (εr =
1).
from the vacuum region. The excitation occurs when the matching between the component of the beam’s wave vector parallel to the surface k|| = k sin θ =
ω c sin θ and the SP’s wave vector is achieved: kSP = k||, where θ is the incidence angle.
We can see (Fig.2.2) that the light cone ω = k||c(dotted lines) always lies on the
left of the dispersion relation of the SP. Since there is no matching between the SP’s and light’s wave vectors, it is not possible to excite SP with a laser beam at a flat metal-vacuum interface.
Figure 2.2: Dispersion relation of SP at the interface between vacuum and metal. The dotted line represents the light line ω = k||c. We can see that there is no matching, i.e.
no intersection, between the light line ω = k||c/ sin θ and the dispersion relation of the
2.2. GRATING COUPLING 15
2.2
Grating coupling
Various methods have been developed to solve the matching problem and excite SPs. A widely used one is grating coupling. A metal surface with a periodic structure, a grating (Fig.2.4), is used instead of the flat interface. With the periodic medium, the wave equation has periodic coefficients and its solutions are given by the Floquet-Bloch theorem: the dispersion relation is folded in the first Brillouin zone (Fig.2.3) and the matching condition between the light beam’s wave vector and the SP’s wave vector becomes:
ω
c sin θ = kSP ± n 2π
d (2.16)
where d is the grating period. For any given angle θ a set of discrete solutions for the SP’s wave vector kSP always exists in this case.
Figure 2.3: Left frame: the replicas of the SP dispersion relation are folded in the first Brillouin zone (Image taken from [21]). Right frame: the red dots represent the possible SP excited by a light beam, i.e. when k sin θ = kSP .
The solutions of the equation (2.16) can be found substituting the expression of kSP from the eq.(2.15):
ω c sin θ = ω c s 1 − ω2 p/ω2 2 − ω2 p/ω2 ± n2π d (2.17) If we take ωP >> ω we obtain sin θ ' 1 ± nλ d (2.18)
We can see in the right frame of fig.2.3 the possible solutions, i.e. the inter-sections between the light line and the SP dispersion relation. Both in eq.(2.18) and in fig.2.3we assumed that the dispersion relation of the SP is the same as the flat interface case. This assumption is an approximation: the dispersion relation changes as the grating’s grooves became deeper, affecting the matching condition. Even band gaps can appear (more details can be found in [22]). However, usu-ally shallow grooves are used and the variations to the dispersion relation remain small. In fact, it can be shown that the contribution of the surface’s roughness in the dispersion relation is only of the second order in the groove’s depth h, i.e. kSP(h, w) = kSP(w) + O
h2
d2 [23], therefore eq.(2.16) is a good approximation.
It has to be noticed that the solution of eq.(2.18) can be a negative angle if λ
d > 1. In this case the SP propagates in the opposite direction with respect to the the component of the beam’s wave vector parallel to the surface, i.e. kSP = −k||.
Figure 2.4: A triangularly shaped grating used to couple the incident radiation (Iinc)
with the surface. A portion of the radiation is reflected back (Iref l) while the remaining
excites the SP. The grating on the right is used to decouple the SP allowing it’s analysis via the decoupled radiation (Idec).
Fig.2.4 shows the representation of an experimental set-up with two gratings: the one on the left is used as a coupler to excite the SP; the one on the right is used as a decoupler. The incident beam (Iinc) is coupled by the first grating,
then the SP propagates on the surface for hundreds of micrometers, depending on the material and wavelength used. The SP can also radiate light with the same principle: as the light beam can excite the SP thanks to the grating, even the inverse process can occur. The dispersion relation of the SP matches the light line thanks to the grating and light is radiated. This phenomenon can be used to analyze the SP proprieties using a second grating as a decoupler [24]. If the grating is placed far enough from the incidence point, the reflected light can be neglected, therefore a detector can be placed above it collecting only the decoupled light Idec.
2.3. PLASMONICS 17 light Iref l and the decoupled light Idec: when the excitation occurs a portion of the
beam’s energy is transferred to the SP, causing a decrease of Iref l. The SP travels
on the surface until it reaches the second grating where it radiates light, causing an increase of Idec.
2.3
Plasmonics
The SP’s excitation, proprieties and applications are studied in plasmonics. Plas-monics is a very active research area with many applications still growing in num-ber. As an example the possibility to concentrate the light below the diffraction limit leads to an enhancement of the electric field that is not reachable with conven-tional optics [12]. Also the possibility to manipulate the light below the diffraction limit with a precise tailoring of the surface offers the potential to develop waveg-uides of sub-wavelength dimension [25]. These nanometric plasmonics elements can be implemented in nanophotonic circuits reducing their dimension and there-fore representing the future of nanotechnology [11]. The coupling between the incident light and the SP effectively enhances the light trapped on the surface, suggesting possible applications also for photovoltaic solar cells [13].
In fig.2.5 we show some of the mentioned applications. On the top left the energy concentration is shown in the tip of a metal tapered waveguide: the SP’s velocity of propagation decreases and eventually tends to zero as the SP reaches the tip. This phenomenon leads to an accumulation of the energy on the tip and to the generation of extreme fields [26]. In the top right a simplified design for a solar cell is shown. The metallic grating on the bottom couples the solar light with the SP, trapping more energy in the solar cell. On the bottom left we can see a waveguide for a plasmonic circuit. In the bottom right the SP excited by a laser in a Kretschmann configuration (see below) induces the Raman scattering of the molecules on the surface. This technique called Surface-enhanced Raman spectroscopy, has been recently used to detect a single molecule on the surface [27]. The highly concentrated electromagnetic radiation due to the SP enhances the number of photons that invest the molecule and thus augments the probability of the interaction via Raman scattering.
One of the key problems in plasmonics is the excitation of the SP by an ex-ternal driver. Due to their hybrid optical-electronic nature the SP can be excited by either photons or electrons. Many coupling methods have been developed to excite the SP, e.g. in fig.2.5, on the bottom right, the prism is necessary to achieve the coupling between the laser pulse and the SP. The Kretschmann or Otto
con-Figure 2.5: Pictures of some applications of SPs. In the top left we can see the energy density of a SP focusing on a metal tapered strip (taken from [28]). On the top right a possible scheme for the light trapping in a solar cell is illustrated (taken from[29]). In the bottom left some pictures of a waveguide for a SP are reported (taken from [30]). On the bottom right the Kretschmann configuration is used for the detection of molecular cells using the Surface Enhanced Raman Scattering.
2.3. PLASMONICS 19 figurations [31] are two commonly used methods in plasmonics in which the laser propagates through a dielectric medium, therefore they are used with low intensity lasers.
Recently, with the advent of the ultra-high intensity lasers with high-contrast ratio, there has been interest in extending plasmonics to the "high field" regime. In the "high field" regime, with the use of terawatt lasers, every dielectric medium crossed by the beam becomes ionized, therefore other coupling methods that avoid the propagation through the medium are needed. In the top right panel of fig.2.5
one of the most used is shown: the "grating coupling". The structure of the surface can be tailored to obtain the excitation condition (see section 2.2).
Some works in the "high field regime" have been previously reported, involving the generation of x-rays [32], with strong limitations in the laser intensity: the problem with the first ultra-high intensity laser was the intensity of the prepulse. The prepulse is an incoherent radiation generated by the amplified spontaneous emission when the active medium is pumped. With ultra-high intensity lasers the intensity of the prepulse may be high enough to ionize the target. The generated plasma has the time to expand before the laser pulse reaches it, erasing the shape of the target and preventing the effective interaction of the pulse with the surface. Only in the past few years some technique, e.g. double plasma mirror [33], have been implemented to reach a contrast ratio high enough to allow the use of these lasers in plasmonics experiments.
High field plasmonics is a very young research field that has been little explored. The main idea was to use the plasmonic resonance to enhance the absorption of the laser energy in a solid target, leading to a higher heating and thus to the generation of denser and hotter plasmas. This effect has been indirectly verified measuring an increase in the energy of the ions emitted by the target [34] and by the observation of x-ray emission from the plasma [32]. More interesting applica-tion still to explore are the acceleraapplica-tion of electrons [35] and the generation of high harmonics [18].
2.4
Wave Front Rotation
An ultra-short laser beam can show the presence of spatio-temporal coupling, i.e. it is not possible to separate the spatial component from the temporal component: E(x, y, t) 6= E0S(x, y)T (t). The wave front rotation (WFR) is an example of this
coupling.
This phenomenon consists in the rotation of the wave fronts in a electromagnetic wave. A laser pulse with rotating wavefront can be either generated deliberately for some applications, or incidentally in optical system including dispersive elements; in the latter case, the effect can be detrimental for, e.g. the pulse duration and the focus dimension increase, as we will explain later.
The WFR often occurs in high intensity laser experiments. The laser pulse in such experiments is amplified with a technique called Chirped Pulse Amplifica-tion (CPA): the ultra-short laser pulse is stretched in time by a system of gratings, then, after being amplified, it is compressed again with another system of gratings. We will see that a small misalignment in the gratings can induce the rotation of the wave fronts.
The WFR can be achieved by focusing a pulse with the wave fronts tilted with respect to the direction of propagation [36]. Such pulse can be generated in many ways: using a prism or a system of misaligned gratings. In example, consider the system of fig.2.6: the first part separates the beam’s components of different wavelengths, producing a spatial chirp. In the second part, the resulting beam travels through a dispersive medium. The longer wavelengths travel faster than the shorter ones and the result is a light beam with the wave fronts tilted.
Figure 2.6: The system on the left produces spatially a chirped pulse. Longer wavelengths propagate faster in the dispersive medium on the right [37] and a pulse with tilted fronts is generated.
2.4. WAVE FRONT ROTATION 21 more details) E(xi, t) = E0exp h − 2t − ξxi τi 2 − 2x 2 i w2 i + iωLt i , (2.19)
where xi is the transverse coordinates, wi and τi are the beam’s diameter and
duration far from the focus. ξ is the pulse-front tilt parameter, as can be seen from the equation it represents the delay between different parts of the beam: tdelay(xi) = ξxi.
When the beam is focused with a lens, its upper part intersects the lens before the lower part, and so the longer wavelengths reach the focus before the shorter ones (Fig.2.7). Every part of the beam has a different direction of propagation, thus, as time goes on, we see in the focus wave fronts with different wave vectors. This effect causes the WFR. The beam’s expression at the focus is [36]
E(xf, t) ∝ exp h − 2t 2 τ2 f − 2x 2 f w2 f i expiϕ(xf, t), (2.20) ϕ = 4 wiξ wfτfτi xft + ωLt,
where xf is the transverse coordinate in the focus position and wf and τf are
the beam’s diameter and duration at the focus. The direction of propagation is β(t) ∼ k⊥(t)/kL, where k⊥(t) = ∂ϕ/∂xf is the transverse component of the wave
vector.
The wave front rotation velocity can be found from: vr = ∂β ∂t = wi2 f τ2 i ξ 1 + (wiξ/τi)2 . (2.21)
Where f is the focal length of the focusing lens. Queré et al. [10] performed some simulations and experiments to generate the WFR: the pulse front tilt was introduced on the laser beam by rotating one prism used to compress the pulse. The amount of pulse front tilt could be adjusted by varying the angle of inclination of the prism. They reached in the focus the value of vr ∼ 15mrad/fs and estimated
a maximum value of vr∼ 30 mrad/fs .
This phenomenon is considered detrimental in the experiments with high in-tensity lasers, because it leads to a reduction of the inin-tensity at the focus [36]. Different components of the laser reach the focus at different time and so the duration of the beam in the focus is increased:
τf =
q τ2
Figure 2.7: Pulse-front tilt becomes wave front rotation in the focus [36]. For the same reason also the beam waist is increased:
wf = r λxf πwi 2 +λxfξ πτi 2 . (2.23)
Both these effects reduce the intensity of the beam in the focus.
2.5
Generation of High Harmonics
In the attempt to generate subfemtosecond laser pulses, the high harmonics rep-resents one of the most promising tools. A lot of effort is put into the study of the generation and properties of the harmonics. Here we briefly describe two methods that inspired our thesis work: high harmonics generated with a WFR pulse and high harmonics generated from a grating target.
While unwarranted WFR can be detrimental for keeping the short duration or the tight focusing of a laser pulse, it can also be induced by purpose for some specific application. An example is the generation of attosecond pulses via the production of high order harmonics in high intensity interaction with solid or gaseous targets.
High harmonics in an atom can be described with a simplified semi-classical three steps model that follows the so called "simple man’s theory" [38]: when a linearly polarized intense laser pulse of frequency ω is focused on an atomic gas jet, the electric field alters the Coulomb potential of the atoms and can induce an electron to tunnel out from the bound state. Then the electron is displaced from the nucleus and accelerated by the field until it eventually falls back to the nucleus emitting a photon (Fig.2.8). The ionization occurs mostly when the amplitude of electric field is at its maximum, i.e. two times every period, thus the spectrum of the harmonics emitted consists of peaks at odd multiples of the fundamental frequency: ω + 2nω.
2.5. GENERATION OF HIGH HARMONICS 23
Figure 2.8: Representation of the three steps model: the electric field deforms the Coulomb field and induces the electron to tunnel out of the bound state; then the elec-tron is accelerated in the vacuum and eventually it recombines on the atom emitting photons. The processes is repeated twice every laser period, resulting in the emission of odd harmonics.
In the temporal domain, the high harmonic signal appears as a train of at-tosecond pulses [39]. This is related to the high harmonics being coherent and phase-locked. Harmonics show interesting proprieties: their divergence angle is less then the fundamental light’s divergence. Approximately
∆θN =
λ0
πw0N
= ∆θ0
N (2.24)
where λ0, w0 and θ0 are respectively the wavelength, waist and divergence angle
of the fundamental light and ∆θN is the divergence angle of the n-th harmonic.
Moreover their spectrum shows a very large bandwidth, that increases with the harmonic order. This large bandwidth is consistent with a pulse with attoseconds duration. One of the problem is to spatially separate every harmonics in order to isolate these attoseconds pulses (more details regard to the high harmonics gener-ation and properties can be found in [40] and references therein).
Querè er al. investigated the generation of harmonics implementing a pulse with WFR [10]. The tilted pulse was generated and controlled with a thin trian-gular prism that could be rotated. Then the pulse was focused on a gas jet where, as described in the previous section, the pulse tilt leads to the rotation of the wave fronts (Fig.2.9) . Without WFR the harmonics were generated in a collinear train of pulses that could not be separated. Instead, due to the WFR, every front in the pulse crosses the gas jet with a different angle, thus the coherent emission of harmonics occurs for different directions, leading to the spatial separation of every attosecond pulse (lighthouse effect).
Figure 2.9: Experimental set-up for the generation of attosecond pulses. A pulse with tilted fronts is focused on a gas jet and generates high harmonics. The rotation of the fronts in the focus spatially separates every attosecond pulse [41].
The intensity of harmonics generated from gasses decreases with the harmonic order (the intensity for the high orders is typically 10−6I
L where IL is the laser
intensity) and it is also limited by the ionization: the electron must form a bound state to emit radiation and generate harmonics, thus the laser intensity must be kept below 1015− 1016 W/cm2 in order to avoid strong ionization of the gas [40].
Therefore, due to the possibility to increase the intensity of the laser and obtain more intense harmonics, there has been interest in the study of high harmonics generation with solid targets with a free electron density, e.g. a metal (although at high intensity also insulator become similarly, due to the strong ionization by the laser field).
If a laser pulse impinges on an metallic target, the high harmonics are gen-erated by a different effect: the radiation can drive, due to the Lorentz force, a collective oscillation of the electrons in the metal at the frequency ω or 2ω de-pending on the polarization (Fig.2.10). If the target is over-dense, i.e. ω < ωP,
it will then behave like an oscillating mirror. The reflected radiation thus shows the presence of the high harmonic spectrum: if the pulse has oblique incidence and P polarization, the oscillation is driven by the electric field with a frequency ω, therefore the high harmonics will have the frequencies ω + nω; if the pulse is
2.5. GENERATION OF HIGH HARMONICS 25 S polarized then the oscillation is driven by the v × B term in the Lorentz force with a frequency 2ω, thus the high harmonics will have the frequencies ω+2nω [42].
Figure 2.10: Representation of the oscillating mirror model: on the left the laser has a P polarization and the harmonics emitted have a frequency ω + nω; on the right the laser has a S polarization, resulting in the emission of only odd harmonics.
The central problem is again the angular separation of the attosecond pulses generated. Querè er al. used a pulse presenting WFR. As in the gas, since the wave fronts are rotating in time, the high harmonics emitted by different sections of the pulse propagate in different directions, allowing a spatial separation of the single attoseconds pulses (Fig.2.11).
A different approach to separate the harmonics was proposed by Chercez et al. [18]. They presented the first experimental evidence of high harmonics generation by a grating target irradiated by a high intensity laser pulse. With a flat target, all the harmonics generated propagate in the direction of reflection. In fig.2.12 on the left panel we reported the space Fourier transform of the light reflected from a flat target in a 2D simulation. We can see that all the harmonics are emitted in the specular direction. On the opposite with a grating target the m-th harmonic is emitted for a discrete set of angles nλ/md = sin θi + sin θmn, where d is the
grating pitch, θi the angle of incidence and n the order of diffraction (central panel
of fig.2.12).
Fedeli et al. [43] also performed a simulation campaign to study the effect of the generation of a SP on the high harmonics emitted. They noticed that the generation of a SP leads to an enhancement of the fields on the surface and thus to an enhancement of the harmonics intensity. In the central panel of fig.2.12, the harmonics intensity is shown for an incidence angle θi different from the angle of
excitation θR of the SP; in the right panel instead θi = θR. We can see that the
intensity of the high harmonics is enhanced by the excitation of the SP.
Our work takes inspiration from the works of Querè et al. and Fedeli et al.: we will use a pulse with WFR to excite an ultra-short SP on a grating. Our goal is to generate a SP with the shortest duration possible by controlling both the rotation velocity and the waist of the beam: the matching condition found in section 2.2
determines an exact angle , between the wave vector and the surface’s normal, for which the SP is excited (eq.(2.18)). Without the WFR the SP is excited for the whole duration of the incident pulse. Instead, using a pulse with WFR, since the wave fronts are rotating in time, the effective angle of incidence, i.e. the angle be-tween the wave fronts and the surface’s normal, changes every wave front (fig.2.11) and the matching condition will be satisfied only for a brief period of time. This leads to the excitation of the SP for a duration shorter than the duration of the laser pulse. We will see in chapter4that, due to the finite dimension of the beam, the excitation of the SP is achieved within a width value of the incidence angle, therefore for higher values of the rotation velocity we will generate shorter SPs.
Figure 2.11: A pulse with WFR is reflected by a plasma mirror. The harmonics generated propagates in different direction and form single attoseconds pulses [44].
2.5. GENERATION OF HIGH HARMONICS 27
Figure 2.12: Three pictures showing the Space Fourier transform of the light reflected and diffracted by a solid target in a 2D simulation. Left panel: the high harmonics are generated from the reflection of a laser pulse on a flat target. We can see that all the harmonics propagates in the specular direction. Central panel: the harmonics are generated from a grating target and emitted in different directions according to the diffraction law. Right panel: the intensity of the harmonics is enhanced by the excitation of the SP on the target. Image adapted from [43].
Chapter 3
Simulation
In this chapter we will characterize the fundamental elements of our simulations. In the first section we will briefly explain the operating principle of the code we used and some of its features. In the second section we will describe the expression of the laser pulse: how it is implemented and how the pulse is "injected" in the simulation box. We will also discuss some of the difficulties we encountered and how we managed to solve them. In the last section we will describe the implementation of the grating target.
3.1
MEEP
MEEP (MIT Electromagnetic Equation Propagation) is an open-source code avail-able online at http://ab-initio.mit.edu/wiki/index.php/Meep. More details
about MEEP’s feature and implementation can also be found in [45]. Here we are going to briefly summarize how the code works and which are some of his strength or weakness points.
MEEP is a code that solves Maxwell’s equations using a finite-difference time-domain (FDTD) algorithm. The simulation box is divided into a grid called "Yee lattice", where the electric and magnetic field are located. Each component of the fields are staggered by half a space step in the space-domain. As an example we can see from fig.3.1, a two dimensional "Yee lattice". In two dimension, referring for example to a transverse electric mode, the y component of the electric field are calculated at the center of the top and bottom edges of the grid while the x component at the left and right edges. The z component of the magnetic field is calculated in the center of the grid.
The electric and magnetic field are also staggered by half a time step in the time-domain so if Bzis located at t−∆t/2, x, y, Exwill be located at t, x, y+∆y
Figure 3.1: Yee lattice in two dimensions with a transverse electric or transverse magnetic polarization. The magnetic and electric fields are staggered in space and time by half a space and time step.
and Ey at t, x + ∆x, y.
The Maxwell’s equations are discretized using a central-difference approxima-tion of the partial derivatives both in space and time:
∂tf (t, x, y) = f t + ∆t/2, x, y − f t − ∆t/2, x, y ∆t , (3.1) ∂xf (t, x, y) = f t, x + ∆x/2, y − f t, x − ∆x/2, y ∆x , (3.2) ∂yf (t, x, y) = f t, x, y + ∆y/2 − f t, x, y − ∆y/2 ∆y . (3.3)
The equations are then solved with a "leapfrog" method: the electric field at a time t is calculated from the electric field in t − ∆t and from the spatial curl of the magnetic field in t − ∆t/2. In the same way one calculates the magnetic field in t + ∆t/2. With this method the algorithm has a second-order accuracy in the time and space steps.
The FDTD method is a very versatile code to simulate the Maxwell’s equa-tions. As a time-domain code, when a broadband pulse is used, e.g. an ultra-short laser pulse, the response for every frequency can be obtained in a single simulation.
3.2. PULSE IMPLEMENTATION 31 Any material can be defined in the simulation box as long as the permeability, per-mittivity and conductivity are specified: since the motion of electrons and ions is not simulated, the electromagnetic time response of the medium must be explicitly modelled. The electric and magnetic fields are determined in all the simulation box and the output can be used to produce an animation of the simulation, al-lowing data processing also while the simulation is ongoing. The "Yee lattice" must be defined in all the simulation box with a resolution good enough to resolve the smallest scale of the phenomenon, thus a great computational effort might be required to evolve and store the values of the electric and magnetic fields in all the computation domain. The grid box must be truncated with the right bound-ary condition. In our simulation we used absorbing boundaries around the box, employing the perfectly matched layers (PML), i.e. a fictitious materials that ab-sorbs, without any reflection, electromagnetic waves at all frequencies and angle of incidence. Periodic boundary condition can also be implemented.
3.2
Pulse implementation
In our work we simulate in a two dimensional (2D) Cartesian geometry a Gaussian beam with wave fronts rotating in time. The expression that describes the spatial profile of a 2D Gaussian beam propagating with an angle θ in the (x, y) plane is [46] EGauss(r, z) = 2w2 0 πw(z)2 1/4 exph− r 2 w(z)2 − ikz − ik r2 2R(z) + iψ(z) i , (3.4) where r is the radial distance from the beam’s center and z is the distance from the focus (x0, y0): r = (y − y0) cos θ − (x − x0) sin θ ; z = (x − x0) cos θ − (y − y0) sin θ .
Defining the waist of the beam w0 and the Rayleigh length zR =
πw2 0 λ we get w(z) = w0 r 1 + z zR 2
the radius of the beam, R(z) = zh1 +zR
z 2
i the radius of curvature of the beam’s wave fronts and ψ(z) = 0.5 arctan
z zR
is the Gouy phase.
Notice that the expression for a two dimensional Gaussian beam is different from the one for a three dimensional beam. In particular the normalization responsible
for the conservation of the energy flux for a 3-D beam is w0/w(z), and the Gouy
phase is arctan (z/zR).
Adding the temporal dependence we obtain the expression of the Gaussian pulse with rotating wave fronts:
EWFR(r, z, t) = EGauss(r, z) exp h − (t − t0) 2 τ2 i
exph− iωLt + i(t − t0)rξ
i
, (3.5) where τ is the pulse duration and ωL is the laser frequency.
The last term describes the rotation of the fronts. As already mentioned in section
2.4 this equation is valid only near the focus, so we choose (x0, y0) to be near to
the grating coupler.
We chose the rotation phase to be zero when the beam is at maximum (t = t0), so
that the wavefront corresponding to the peak of the pulse will be perpendicular to the direction of propagation defined by θ. With this adjustment it will be easier to define the angle of incidence on the grating used to excite the SP.
In this work we will use the parameters of a typical femtosecond laser: the wavelength λ = 800 µm and the duration at half maximum (FWHM) τ ∼ 6 − 30 fs. The rotation parameter ξ and the waist of the beam w0 will be varied to study
the effects on the SP.
We made some simulations varying the value of the rotation parameter from ξ = 0 to ξ = 0.8 and measuring the rotation angle for every case. Fig.3.2 shows on the left panel the field of a simulated pulse with the rotation parameter ξ = 0.6. the rotation of the wave fronts is highlighted by the rotation angle ∆β. As already stated in eq.(2.21) we expect a linear dependence of the rotation velocity vr = ∆β/TL with the rotation parameter ξ, where TL= 2.67fs is the laser period.
We verified the theoretical prediction with a linear fit (Fig.3.2, right panel) and we found the relation between vr and the rotation parameter ξ to be vr ' 37.7 ξ
mrad/fs. These value can be used with eq.(2.21) to relate the experimental set-up with the simulated pulse.
Once defined, the pulse is implemented in the code via the "add_volume_source()" function. "add_volume_source()" is a predefined function included in the MEEP’s library. It takes as parameters the expression of the field (3.5) and the position whereto inject the pulse.
The position is defined by two points A and B that identify the vertexes of a rectangular plane in the simulation box (Fig.3.3 top panel). The function "cuts" the simulation box through the line or the plane defined and injects the pulse. We had some problem with the position because of its limitations in a two dimensional simulation: we needed to inject the pulse through a line and, with this definition
3.2. PULSE IMPLEMENTATION 33
Figure 3.2: Left frame: a simulated pulse with wave fronts rotating in time. The pa-rameter used is ξ = 0.6, the angle ∆β = 23 mrad. Right frame: the rotation velocity measured for different simulated pulses in function of the rotation parameter ξ. From the slope of the fitted line we found that vr= 37.7 ξ mrad/fs.
of the position, MEEP allows the user to inject the source only through an hori-zontal or vertical line. We would have preferred to inject the pulse through a line perpendicular to the direction of propagation, in order to avoid deformations in the energy profile.
With this constraint we had to be careful in the injection of pulses with high value of the rotation parameter: for ξ = 0.8 the wave fronts rotate more then 50° within the duration of the pulse (29.5 fs). We observed that if the direction of propagation of the pulse is nearly parallel to the line of the injection some numerical distortions can occur in the pulse’s energy profile. Figure 3.3 shows on the bottom panel the same pulse with or without the distortion induced by the line of injection. In order to avoid or minimize this effect we carefully changed the angle of propagation, as well as the inclination of the target, setting the direction of propagation as perpendicular to generation plane as possible. We also observed that these deformations in the pulse’s energy occur less with shorter pulses. That is because, since the pulse is shorter, the wave fronts rotate less within its duration, and the direction of propagation is always far from being parallel to the line of injection.
Figure 3.3: Top panel: three possible "cuts" in the simulation box. If A and B are aligned we obtain a vertical or a horizontal cut, otherwise the injection takes place in all the plane cut: in every point of the plane MEEP adds the electric field defined by expression of the pulse. Bottom panel: the same pulse with or without the numerical distortion induced by the horizontal cut. We used a pulse with waist w0 = 3λ and a
rotation parameter ξ = 0.8. We managed to reduce the distortion’s effects changing the angle of propagation.
3.3. GRATING IMPLEMENTATION 35
3.3
Grating implementation
MEEP allows us to simulate the behaviour of a medium using some predefined functions. With "structure()" we will define the simulation box: the dimen-sion, the resolution and the frequency independent dielectric constant ε∞. With
"add_susceptibility()" we can define the frequency dependent response of the medium: ε(ω)E = ε∞E + P.
With these two functions we obtain the expression: ε(ω) = ε∞+
w2σ
w2− ω2− iγω . (3.6)
Where w, σ and γ are the three parameters required to describe the frequency dependent response. We want to simulate a Drude metal, described by ε(ω) =
1 − ω
2 P
ω2− iγω, where ωP is the plasma frequency of the material used for the
grat-ing and γ is the intraband dampgrat-ing term. Therefore we choose ε∞= 1, w = 0.01
and σ = ω2 P
w2. With this set of parameters ε(ω) ' 1 −
ω2 P
ω2− iγω.
Most of the metals have a plasma frequency of 5 − 14 eV. We choose to simulate a silver grating with ωP = 9 eV and γ = 0.0228 eV, while for the laser chosen
ωL = 1.55 eV (λ = 800 nm). All the frequencies will be normalized to the laser
frequency.
The target has a grating, where the pulse impinges and the SP couples, and then it ends with a flat surface, where the SP propagates without scattering. The flat surface helps with the visualization of the SP and with the measurements of it’s characteristics: the energy flux and the electric field.
The target position and inclination can be changed in order to satisfy the matching condition without having to modify the incident pulse. With this adjustment it will be easier to take care of the distortion problem mentioned before: once the angle of propagation is chosen, the target is moved and rotated in order to satisfy the matching condition.
Chapter 4
Results
In this chapter we will report some of the results we obtained in the simulation campaign. We will start studying the best geometry of the grating target for efficient SP excitation, analyzing the energy flux and the coupling efficiency. In section4.2 we will study the angular width of the resonance and how it is affected from the grating geometry and from the pulse properties. In section 4.3 we will study the dependency of the SP’s duration with the rotation parameter and the waist of the incident pulse. In the last section we will characterize the SP in the case of its shortest duration. We will also vary the absolute phase (see section4.4) of the incident pulse and check how the SP’s field is affected by this change.
4.1
Effects of the grating parameters
As a first test we tried to reproduce the results of a recent experiment [24] in which the SP was excited on a rectangular grating. A second grating was employed to decouple the SP allowing the measure of the SP’s energy. Using the scattered light, a coupling efficiency > 45% was obtained employing a grating with groove pitch, width and depth of 560, 190, 38 nm respectively. They found an over limit of the coupling efficiency because a portion of the SP’s energy emitted from the decoupling grating might had fallen outside the numerical aperture of the objective they used to collect the light.
Using the same parameters we varied the angle of incidence until we found the maximum in the energy transferred to the SP. In fig.4.1 we can see the laser beam (top panel) impinging on the grating (central panel) and generating the SP (bottom panel). The target was designed to have the rectangular grooves in the first half, were the pulse impinges, and to be flat in the second half, where the energy flux is collected. The grooves are too shallow to be distinguished in the figure, but their presence is revealed by the undulated shape of the SP’s energy
density in the bottom panel.
With an incidence angle θ = −22° the reflected pulse’s energy was ∼ 46% of the incident pulse’s energy, meaning that ∼ 54% of the energy was coupled to the surface. We also collected the SP’s energy flux after the grating coupler and we found that it’s energy was 28% of the incident energy: since the flux is collected at a certain distance from the impinging point, a portion of the SP’s energy is lost due to the radiation re-emitted in the propagation along the grating coupler and due to the dissipation in the medium. It has to be noticed that in this case the propagation of the SP is in the opposite direction with respect to k||, the
compo-nent parallel to the grating of the impinging pulse’s wave vector. This is possible because the group velocity vg = ∂ω/∂kSP can be positive or negative according
to the matching condition (see section 2.2). This test was made to check the re-liability of our code and even with a two dimensional simulation (the experiment was obviously 3D) we found a good agreement with the coupling efficiency of the experiment [24].
After reproducing a known result we decided to test different gratings with different geometries (fig.4.2) in order to find the best coupling efficiency for our simulations. Remembering the matching condition
sin θ ' 1 ± nλ
d (4.1)
valid for ω2
P >> ω2L, we can choose the incidence angle θ to determinate the grating
pitch d (we will use "pitch" referring to the grating spatial period in order to avoid confusion with the laser temporal period). Then with a fine tuning of the incidence angle and of the depth h of the grooves we maximize the energy transferred to the SP. With an incidence angle θ = −25° and a wavelength λ = 800 nm the pitch from eq.(4.1) is d = 0.560 µm. With d fixed, we changed independently h and θ.
Equation (4.1) is an approximation, as already mentioned in section 2.2, not only because of the ω
ωP = 0 approximation ω ωP ' 1 6
, but also because it uses the dispersion relation of the SP in the case of a flat surface. Moreover in the simulations we used a laser beam with finite dimension instead of a plane wave (the effect of the dimension will be discussed more in section 4.2). Therefore it is possible that the incidence angle used it’s not optimal for the pitch chosen. Indeed we found that, for all the four gratings, the angle that maximises the energy of the SP was −24° (Fig.4.5).
The depth of the grooves has been varied for every grating in order to maximise the coupling efficiency. The optimal values we found were 80 nm for the rectangular grating, 160 nm for the sinusoidal grating, 140 nm for the symmetric triangular grating, 150 nm for the asymmetric triangular grating. In fig.4.2 the depth of the
4.1. EFFECTS OF THE GRATING PARAMETERS 39
Figure 4.1: Pictures showing the energy density taken at three different times of the same simulation: in the top panel the laser beam is propagating from the top of the simulation box; in the center panel we can see the coupling between the laser beam and the SP; in the bottom panel the SP is propagating on the surface. The grating used had a groove pitch, width and depth of 560, 190, 38 nm respectively, while the laser pulse had a waist w0 = 4λnm.
grooves was increased to give a better exposition of the geometry.
In 4.3 we report the energy flux of the SP for every grating with the optimal incidence angle. For those simulations we used a pulse with waist w0 = 6λwithout
WFR (ξ = 0). We can see that the energy transferred to the SP is maximised for the symmetric triangular grating. The reflected energy was 39% of the incident energy, implying that 61% of the pulse’s energy was coupled to the surface. The SP’s flux was collected after the grating coupler and we found that the SP’s energy was 38% of the incident pulse’s energy. Since the most efficient grating we tested was the symmetric triangular grating we decided to choose that grating for all the simulations in the following.
(a) Sinusoidal grating. (b) Symmetric triangular grating.
(c) Rectangular grating. (d) Asymmetric triangular grating.
Figure 4.2: Tested gratings. The grooves’ pitch d and the depth h of each grating have been varied in order to maximise the energy transferred to the SP. In this picture h has been increased for a better illustration of the geometry.
4.1. EFFECTS OF THE GRATING PARAMETERS 41
Figure 4.3: Energy flux of the SP for every grating compared with the incident pulse’s flux. The parameters used were ξ = 0 (i.e. no wave front rotation) and w0 = 6λ. The
highest coupling efficiency was achieved with a symmetric triangular grating (61% of the incident pulse energy was coupled to the surface). The energy of the SP generated from a symmetric triangular grating was 38% of the incident pulse’s energy.
4.2
Resonance width
The matching condition (4.1) determines an angle of resonance θR that excites the
SP, therefore a laser pulse impinging on the grating with a different angle should not excite it. Instead the excitation of the SP occurs for a range of angles around the resonance angle. This phenomenon can be caused by two effects involving the finite dimension of the pulse.
The first is the curvature of the wave fronts of the pulse: in a 2D plane wave the wave fronts are straight lines, identifying an exact angle between them and the grating target; instead in a pulse with finite dimension, the wave fronts have a curvature. The angle between them and the target varies with the position (Fig.4.4
top panel), therefore it is possible to excite the SP at different angles of incidence. Even at the focus, where the wave front is straight, the Fourier transform of the pulse is composed by a spectrum of wave vectors with different directions. The angular spread of these wave vectors is of the order of λ/w0, where w0 is the waist
of the pulse, thus this effect is expected to be smaller for bigger waist sizes. The second effect is the approximation of periodic potential: in order to find the resonance angle (4.1) we made the assumption of a periodic potential, fold-ing the dispersion relation in the first Brillouin zone. For a real pulse of finite transverse width the assumption of a perfectly periodic (infinite) medium is ob-viously an approximation: the pulse irradiates only on few pitches. Because of the greater number of spatial periods enlightened, this effect should be smaller for gratings with shorter pitch with respect to gratings with longer pitch (Fig.4.4
bottom panel).
In order to understand which of the effects is the major one, we decided to study the angular profile of the resonance by varying the angle of incidence of a laser pulse without WFR and collecting the energy flux of the SP. The energy flux peak is reported in function of the incidence angle for a pulse with waist w0 = 2λ and w0 = 6λ. We choose three different gratings having a groove pitch
d = 0.56, 0.98, 1.2 µm corresponding to an angle of resonance respectively of −25°, 10°, 30°. Varying the waist size we controlled the effects of the curvature of the wave fronts, while varying the pitch length we examined the importance of the periodicity of the medium.
In fig.4.5we reported the data we obtained. The angular width of the resonance (calculated at full width half maximum) is the same for all the three gratings: for w0 = 6λ the angular width is ∆θ6λ ' 4.7°; for w0 = 2λ, ∆θ2λ ' 10.9°. The
resonance width changed only with different waist, this fact suggests that the width is manly caused by the curvature of the wave fronts.
In order to excite a short SP with a pulse with WFR, a tight resonance is needed, therefore one could use a pulse with an even larger waist. We decided not
4.2. RESONANCE WIDTH 43
Figure 4.4: Top panel: the straight wave fronts of a plane wave form an exact angle between them and the grating target (left), while for a pulse with finite dimension there is an angular spread of wavevectors, hence a spread of the effective angle of incidence (right). Bottom panel: the periodical potential is a better a approximation for a grating with short pitch (left) because, with the same waist, the pulse irradiate more pitches with respect to a grating with long pitch (right).
to exceed a waist of 6λ because of the dimension of the simulation box: with larger waists we would need a bigger box, increasing the computational time required for every simulation. Also we encountered more distortion problems with large waists with respect to the small ones, as we will see in the next section.
A third effect that contributes to the resonance width is the bandwidth of the laser pulse: pulses with short duration present a larger bandwidth with respect to longer pulses (Fig.4.6). Since the angle of excitation is related to the wavelength, a large bandwidth could determine a broader resonance. Therefore we decided to control this effect measuring the resonance width with three laser pulses having a duration respectively of 29.5, 19.7 and 6.0 fs. The longer pulse has a tighter bandwidth ∼ 21 THz than the shorter pulse ∼ 105 THz (fig.4.6 on the left). As before we collected the energy flux peak for different angles of incidence and we reported the data in fig.4.6 on the right. We can see that the bandwidth of the laser pulse is reflected in the width of the resonance: the shortest pulse has the largest resonance width ∼ 12.5°. Thus, in order to obtain a shorter SP with our technique, it would be preferable to use a pulse with a longer duration since the resonance is tighter and the excitation occurs for less time. Instead we will see in the next section that the SPs generated by the three pulses of different duration tend to the same duration as the rotation velocity increases.
4.2. RESONANCE WIDTH 45
Figure 4.5: Angular width of the resonance for a pulse with waist w0 = 2λand 6λ. Every
point represents the maximum of the SP’s energy flux in function of the incidence angle. On the top frame the grating had a pitch d = 0.98 µm, on bottom left d = 1.2 µm and on the bottom right d = 0.56 µm. The width (full width at half maximum) for w0= 6λ
Figure 4.6: On the left the bandwidth of a laser pulse with three different durations is shown. We can see that the shortest laser pulse has the largest bandwidth. On the right the maximum of the SP’s energy flux in function of the incidence angle is plotted. We can see that the pulse with the largest bandwidth has also the widest resonance, that is because the angle of excitation of the SP depends on the frequency of the impinging electromagnetic wave. Thus the pulse with a duration of 6 fs can excite the SP whitin a larger values of angles around the resonance angle (∼ 12.5°) with respect to the pulse with a duration of 29.5 fs (∼ 4.7°).
4.3. ROTATION PARAMETER DEPENDENCY 47
4.3
Rotation parameter dependency
In this section we will use a pulse with WFR to excite the SP. Since the wave fronts are rotating in time, the incidence angle will also change. The excitation of the SP will occur only for the angle within the width of the resonance. For higher values of the rotation parameter, the incidence angle rotates faster and spans all the width of the resonance in a shorter time. Therefore the excitation will occur for a shorter period and the SP will last less.
Since the width of the resonance is smaller for a larger waist we would expect to find the shortest SP for the largest waist. In fact we will see (fig.4.11) that even for small values of the rotation velocity, the optimal waist to excite short SPs will be the largest one.
We made the pulse impinge on the grating, varying both the waist and the rotation parameter. Then we collected the energy flux transported by the SP. The flux is calculated far enough from the incidence point so that the contribution of the scattered light is negligible. In fig.4.7 we report the energy density (left column) and the magnetic field Bz (right panel) of the same simulation at two
different times. The first row shows the laser pulse with WFR impinging on the target. In the second row we can see the SP propagating on the surface. The SP is excited only by the part of the pulse impinging with the resonance angle, in fact we can see in fig.4.8 that the duration of the SP at half maximum (τSP = 8.8 fs)
is remarkably shorter than the pulse’s duration (29.5 fs).
The section of the pulse that generates the SP can also be seen in the reflected light: on the top panel of fig.4.9we can see the energy density of a pulse impinging on the grating. In the bottom panel a "hole" in the energy density of the reflected pulse appears. The decrease in the reflected energy identify the portion of the pulse with the wave fronts forming an angle within the resonance with respect to the surface normal, i.e. the portion that excited the SP. The energy is transferred to the SP and thus misses from the reflection. The parameters used for that simulation were the waist w0 = 4λ and the rotation parameter ξ = 0.2. The
grating used had a pitch d = 560 nm corresponding to an angle of resonance of θR = −24°.
(a) Energy density of the impinging pulse. (b) Magnetic field Bz of the impinging pulse.
(c) Energy density of the SP. (d) Magnetic field Bz of the SP.
Figure 4.7: Pictures taken from a simulation with a laser pulse having a waist w0 = 2λ
and a rotation parameter ξ = 0.6. From the top panels we can see the impinging pulse with the wave fronts rotating in time. In the bottom panels the SP is propagating on the surface of the target.
Figure 4.8: Energy flux of the incident pulse and of the SP normalized. The duration of the SP (8.8 fs) is remarkably shorter than the duration of the laser pulse (29.5 fs).
4.3. ROTATION PARAMETER DEPENDENCY 49
Figure 4.9: Top panel: a laser pulse with WFR impinging on a grating. Bottom panel: a "hole" in the energy density of the reflected pulse. The decrease in the energy identify the portion of the pulse that has excited the SP and thus that lost the most energy. For this simulation we used a pulse with waist 4λ and a rotation parameter ξ = 0.2. The resonance angle for this grating was θR= −24°.
We then changed the waist of the pulse from 2λ to 6λ and for each one we varied the rotation parameter ξ from 0 to 0.8. ξ = 0.8 corresponds to a rotation velocity of ∼ 30 mrad/fs, that is the maximum value reachable in the experimen-tal set-up used by Quéré et al. [10]. The maximum value is determined by the eq.(2.21): vmax
r = wi/f τi where wi and τi are respectively the beam diameter and
duration before the focusing lens with focal length f (see section2.4 and [10]). For this section we decided not to exceed that value in order to produce results that can be replicated in experiments already available. We will increment that value later in section 4.4.
In fig.4.10, we report the energy flux of the SP for three waists we used, w0 =
2λ, 4λ, 6λ, over the whole range of values of the rotation parameter ξ. Fig.4.11 shows, for a better confrontation, the duration at half maximum (FWHM) of every SP simulated. We can see that the duration of the SP decreases for higher values of ξ. The shortest SP excited has a duration of 6.2 fs and was generated by a pulse with waist w0 = 6λand a rotation parameter ξ = 0.6.
As anticipated before the optimal waist, i.e. the one that excites the shortest SPs, is the largest one, since the width of resonance is tighter. We can also see in fig.4.11 that every curve tends exponentially to a limit > 0 fs. This is because the duration of the SP is given by the contribution of two effects: the time the wave fronts need to span all the resonance’s width and the time needed by the SP to travel along the lit zone. The pulse impinges on the grating in an area proportional to the waist and the SP is generated through all the irradiated zone. Thus a SP generated by a waist e.g. w0 = 6λ is spatially longer than one generated by a
waist w0 = 2λ, therefore its time duration is longer. This geometrical contribution
is detrimental for our purpose, because it limits the effective shortening of the rotation: when the rotation velocity is high, the time needed to span all the resonance’s width is less than the time needed by the SP to travel along the irradiated zone and the shortening effect of the WFR exponentially diminishes.
Another effect that has to be noticed is the increase in the duration for ξ = 0.8 for the pulses with larger waists. This effect can be caused by some distortions appearing in the laser beam when either an extreme value of the rotation velocity or a large waist is employed. In fig.4.12 the energy density of the incident pulse shows the presence of some numerical distortions, which are reflected in the SP’s energy flux profile: we can see in fig.4.10that some bumps appear on the curves for the pulse with waist 6λ that can be the cause of the increase in the SP’s duration.
4.3. ROTATION PARAMETER DEPENDENCY 51
Figure 4.10: Energy flux of the SP for three of the waists used in the simulations. The rotation parameter had been varied from ξ = 0 to ξ = 0.8. The duration of the SPs decrease with higher rotation volocity.
Figure 4.11: Duration at half maximum for every SP. The best result is achieved with the pulse having a waist w0 = 6λ, reaching the duration of 6.2 fs for ξ = 0.6. The dotted
line represent the duration of the laser pulse.
Figure 4.12: Energy density of an incident pulse with waist w0= 6λand rotation
param-eter ξ = 0.8. We can see the presence of some distortions caused by a numerical problem in the generation of the pulse.
4.3. ROTATION PARAMETER DEPENDENCY 53 For completeness we decided to make a comparison with the other two gratings used in the previous section: one with pitch d = 1.2 µm and the resonance angle θR = 30° and one with pitch d = 0.56 µm and the resonance angle θR= −25°.
We used a pulse with waist w0 = 2λand we changed the rotation parameter from
0 to 0.8. In fig.4.13 we report the duration of the SP for the three gratings. The curves are quite similar, even if a small difference between the three appears es-pecially for the grating with θR = 30°. Further studies are required in order to
understand if the geometry of the grating could have an important influence on the SP duration.
We compared also, as already done in the previous section, pulses with different durations: 29.5, 19.7 and 6.0 fs, for three waists: 2λ, 4λ and 6λ (Fig.4.14). We can see that the curves tend approximately to the same limit, however, although the SPs generated by the longest pulse have the longest duration for ξ = 0, they become the shortest ones for ξ = 0.8. That is because the resonance function is tighter for longer pulses (see section 4.2), thus the shortening effect due to the WFR leads to the excitation of shorter SPs.
It also has to be noticed that the SPs generated with the pulse having a duration of 6.0 fs and a waist w0 = 2λhave the opposite behaviour: their duration becomes
longer as the rotation parameter ξ increases. That is because the dimension of the focus increases with the rotation parameter (see section 2.4). For those SP, the shortening effect of the WFR is negligible, i.e. the resonance function is too width, thus as the rotation parameter and the dimension of the focus increases, the SP is excited on a wider spatial region and it last longer.
Figure 4.13: Duration of the SP generated with a pulse with waist w0 = 2λ, for three
different gratings. The three curve behave in a similar way. Further studies are required to understand the effects of the grating geometry on the SP’s duration.
Figure 4.14: Duration of the SP generated with a pulse with three different durations: 29.5, 19.7 and 6.0 fs, and for three waists: 2λ, 4λ and 6λ. The shortest SP generated has a duration of 6.2 fs and it was excited by a pulse with duration 29.5 fs, waist w0 = 6λ
and a rotation parameter ξ = 0.6. The effect of the bandwidth of the pulse can be seen for ξ > 0.4, in fact the pulse with duration 6.0 fs, despite having the shortest duration, generates the SPs that last longer.
