R Weissenbilder
email: robin.weissenbilder@fysik.lth.se

S Carlstr Öm

L Rego

C Guo

C M Heyl

P Smorenburg

E Constant

C L Arnold

A L'huillier

How to optimize high-order harmonic generation in gases

High-order harmonic generation (HHG) in gases leads to short-pulse extreme ultraviolet (XUV) radiation useful in a number of applications, for example, attosecond science and nanoscale imaging. However, this process depends on many parameters and there is still no consensus on how to choose the target geometry to optimize the source efficiency. Here, we review the physics of HHG with emphasis on the macroscopic aspects of the nonlinear interaction. We analyze the influence of medium length, pressure, position of the medium and intensity of the driving laser on the HHG conversion efficiency (CE), using both numerical modelling and analytical expressions. We find that efficient high-order harmonic generation can be realized over a large range of pressures and medium lengths, if these follow a certain hyperbolic equation. The spatial and temporal properties of the generated radiation are, however, strongly dependent on the choice of pressure and medium length. Our results explain the large versatility in gas target designs for efficient HHG and provide design guidance for future high-flux XUV sources.

Key points:

• The physics of high-order harmonic generation in gases is reviewed with emphasis on the macroscopic aspects.

• The medium length, gas pressure, position of the medium relative to the laser focus and intensity of the driving laser influence the HHG conversion efficiency.

• Efficient high-order harmonic generation can be realized over a large range of pressures and medium lengths, if these follow a certain hyperbolic equation.

• The spatial and temporal properties of the generated radiation depend on the choice of pressure and medium length.

• Different gas target designs for HHG are suitable depending on the required properties of the XUV radiation and flexibility of experimental setup.

Website summary:

We study the macroscopic aspects of high-order harmonic generation in gases. A high conversion efficiency can be achieved over a large range of pressures and medium lengths, following a hyperbolic equation.

Our results provide design guidance for future high-flux extreme ultraviolet sources.

Introduction

Extreme Ultraviolet light sources based on high-order harmonic generation in gases [START_REF] Mcpherson | Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases[END_REF][START_REF] Ferray | Multiple-harmonic conversion of 1064 nm radiation in rare gases[END_REF] are becoming ubiquitous in many areas of science, from atomic and molecular physics to condensed matter physics, as well as more applied topics, such as coherent imaging [START_REF] Zürch | Real-time and sub-wavelength ultrafast coherent diffraction imaging in the extreme ultraviolet[END_REF] and microscopy [START_REF] Schmidt | Time-resolved two photon photoemission electron microscopy[END_REF] . The radiation consists of a train of extremely short light bursts, in the attosecond range, allowing for outstanding temporal resolution. HHG has opened the field of attosecond science, capturing ultrafast electron dynamics in matter [START_REF] Krausz | Attosecond physics[END_REF] . The spatial properties of HHG sources are also interesting for some applications since the radiation is spatially-coherent over a broad spectral range. During many years, amplified femtosecond titanium-sapphire lasers have been the "standard" laser for HHG. Recently, there is an increased diversity of HHG sources driven by a variety of lasers ranging from high energy lasers at low repetition rate, with up to hundreds of mJ energy per pulse [START_REF] Coudert-Alteirac | Micro-focusing of broadband high-order harmonic radiation by a double toroidal mirror[END_REF][START_REF] Kühn | The ELI-ALPS facility: the next generation of attosecond sources[END_REF][START_REF] Kleine | Soft x-ray absorption spectroscopy of aqueous solutions using a table-top femtosecond soft x-ray source[END_REF][START_REF] Tsatrafyllis | The ion microscope as a tool for quantitative measurements in the extreme ultraviolet[END_REF][START_REF] Fu | High efficiency ultrafast water-window harmonic generation for single-shot soft x-ray spectroscopy[END_REF] , to high average power lasers, based upon optical parametric amplification or simply high-power oscillators, with pulse energies in the µJ range or below [START_REF] Rothhardt | Absorption-limited and phase-matched high harmonic generation in the tight focusing regime[END_REF][START_REF] Comby | Cascaded harmonic generation from a fiber laser: a milliwatt xuv source[END_REF][START_REF] Mikaelsson | A high-repetition rate attosecond light source for time-resolved coincidence spectroscopy[END_REF] . Also, high-power, compact, HHG sources based on post-compressed, ytterbium-doped femtosecond lasers [START_REF] Boullet | High-order harmonic generation at a megahertz-level repetition rate directly driven by an ytterbium-doped-fiber chirped-pulse amplification system[END_REF][START_REF] Hädrich | High photon flux table-top coherent extreme-ultraviolet source[END_REF] are becoming increasingly interesting for industrial applications. Finally, the use of lasers in the mid infrared range allows the generation of high energy photons, in the soft X-ray range [START_REF] Sheehy | High harmonic generation at long wavelengths[END_REF][START_REF] Popmintchev | Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers[END_REF] . HHG sources can thus be vastly different, with parameters such as peak power or repetition rate varying by several orders of magnitude [START_REF] Heyl | Introduction to macroscopic power scaling principles for high-order harmonic generation[END_REF] .

The HHG efficiency depends both on the response of a single atom to a strong laser field [START_REF] Krause | High-order harmonic generation from atoms and ions in the high intensity regime[END_REF][START_REF] Corkum | Plasma perspective on strong field multiphoton ionization[END_REF][START_REF] Lewenstein | Theory of high-harmonic generation by low-frequency laser fields[END_REF] and, as in any coherent nonlinear optical process, on the phase matching between the waves emitted by individual atoms in a macroscopic medium [START_REF] L'huillier | Higher-order harmonic generation in xenon at 1064 nm: The role of phase matching[END_REF][START_REF] Brabec | Intense few-cycle laser fields: Frontiers of nonlinear optics[END_REF][START_REF] Kazamias | Global optimization of high harmonic generation[END_REF][START_REF] Gaarde | Macroscopic aspects of attosecond pulse generation[END_REF][START_REF] Popmintchev | The attosecond nonlinear optics of bright coherent x-ray generation[END_REF][START_REF] Constant | Optimizing high harmonic generation in absorbing gases: Model and experiment[END_REF] . The single atom response depends on the driving laser intensity, wavelength [START_REF] Popmintchev | Ultraviolet surprise: Efficient soft x-ray high-harmonic generation in multiply ionized plasmas[END_REF] , and the atomic species, and can, to some extent, be boosted by multicolor schemes, at the cost of more complex optical setups [START_REF] Kim | Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field[END_REF][START_REF] Brizuela | Efficient high-order harmonic generation boosted by below-threshold harmonics[END_REF] . The macroscopic response, on the other hand, can be optimized by choosing an appropriate focusing geometry, medium length and pressure. Quasi-phase matching setups using complex gas target designs have also been developed [START_REF] Christov | Quasi-phase matching of high-harmonics and attosecond pulses in modulated waveguides[END_REF][START_REF] Hareli | Phase matching and quasi-phase matching of high-order harmonic generation-a tutorial[END_REF] . The versatility of HHG sources can be understood, partly, by a rather simple scaling principle: A given configuration for HHG can be scaled up (or down), if certain relations between input energy, focusing geometry, atomic density and medium length are conserved [START_REF] Heyl | Scale-invariant nonlinear optics in gases[END_REF][START_REF] Nefedova | Development of a high-flux xuv source based on high-order harmonic generation[END_REF] . Similar conversion efficiencies can thus be achieved using quite different laser systems, with energy per pulse differing by several orders of magnitude [START_REF] Takahashi | Generation of 10-µj coherent extreme-ultraviolet light by use of high-order harmonics[END_REF][START_REF] Takahashi | Generation of high-energy high-order harmonics by use of a long interaction medium[END_REF][START_REF] Rudawski | A high-flux high-order harmonic source[END_REF][START_REF] Hädrich | Exploring new avenues in high repetition rate table-top coherent extreme ultraviolet sources[END_REF] .

A remaining question is: given a certain laser energy, how should the focusing geometry, medium length and atomic density be chosen in order to maximize the HHG efficiency? Although HHG sources have been around for more than 30 years, there is no consensus regarding optimal medium design, and current setups vary from high pressure gas jets [START_REF] Paul | Observation of a train of attosecond pulses from high harmonic generation[END_REF] , to low pressure cells [START_REF] Spielmann | Generation of coherent x-rays in the water window using 5-femtosecond laser pulses[END_REF] , semi-infinite cells [START_REF] Sutherland | High harmonic generation in a semi-infinite gas cell[END_REF] , or capillaries [START_REF] Rundquist | Phase-matched generation of coherent soft x-rays[END_REF] , just to cite the most common ones.

In this work, we investigate the dependence of the HHG conversion efficiency and spatial and temporal properties on the medium length and pressure, with the help of numerical simulations and analytical derivations. After a tutorial review of the main physics behind HHG, we present simulations based on solving the timedependent Schrödinger equation (TDSE) and a wave propagation equation, including dispersion, absorption and ionization, for the 23 rd harmonic in argon and the 69 th harmonic in neon, using 810 nm, 22 fs full width at half maximum (FWHM) laser pulses. We find that for any given focusing geometry, efficient HHG can be achieved over a wide range of pressures, as long as the medium length is appropriately chosen. The relation between pressure and length, corresponding to a high conversion efficiency, follows a hyperbolic equation, which is derived analytically. However, the choice of pressure (or length) strongly influences the temporal and spatial properties of the emitted radiation. Our results, which can easily be extended to other harmonics, laser wavelengths and atomic species, explain why the achieved conversion efficiencies are often comparable despite the large variety of design, ranging from gas jets to capillaries, making HHG such a robust technique. This allows us to give a recommendation on gas target design and geometry for optimizing the HHG flux depending on the application requirements. High conversion efficiencies may be achieved at very high laser intensities, in the overdriven regime, where the driving pulse is strongly reshaped, and the harmonic structure is partly lost [START_REF] Cassou | Enhanced high harmonic generation driven by high-intensity laser in argon gas-filled hollow core waveguide[END_REF][START_REF] Rivas | Propagation-enhanced generation of intense high-harmonic continua in the 100-ev spectral region[END_REF][START_REF] Major | Propagation-assisted generation of intense few-femtosecond high-harmonic pulses[END_REF][START_REF] Johnson | High-flux soft x-ray harmonic generation from ionization-shaped few-cycle laser pulses[END_REF][START_REF] Major | High-harmonic generation in a strongly overdriven regime[END_REF] . Here, we limit ourselves to intensities in the medium below 10 15 Wcm -2 , where conventional phase matching dominates.

The article is organized as follows: In Section 2, we give an overview of ionization by strong laser fields and the single atom response of high-order harmonic generation. In Section 3, we discuss the macroscopic aspects of high-order harmonic generation in gases, and develop a one-dimensional analytical model to describe phase matching. In Section 4, we present the results of our simulations and compare them to the predictions of the model. In Section 5, we give two examples of how to optimally choose the geometry and gas target, and we discuss the range of parameters for which our guidelines are valid. We conclude with a recommendation in Section 6.

Atoms in strong laser fields

Strong field ionization

HHG is intimately related to ionization in strong laser fields, which initiates the single atom response, and at the same time limits the macroscopic yield when a too high density of free electrons in the medium prohibits phase matching and modifies the laser propagation. Fig. 1(a) shows ionization rates, Γ, as a function of laser intensity, I, for Ar and Ne for a wavelength of 810 nm. Results obtained by solving the TDSE numerically in the single

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active electron approximation 48 (dashed line) compare well with calculations using the so-called Perelomov-Popov-Terent'ev (PPT) [START_REF] Perelomov | Ionization of Atoms in an Alternating Electric Field[END_REF] formula (solid line). The deviation at high intensity in the Ar case corresponds to depletion, not included in PPT. The steps shown using the PPT approximation are due to channel closings related to the increase of the ionization energy in strong laser fields. The well known Ammosov-Delone-Krainov (ADK) [START_REF] Ammosov | Tunnelling ionization of complex atoms and of atomic ions in an alternating electromagnetic field[END_REF] approximation, which is the low frequency limit of PPT, valid in the tunneling limit, can also be used to calculate ionization rates, though the results (not shown) deviate more than PPT from those obtained by solving the TDSE. For few cycle pulses, where sub-cycle dynamics can be important [START_REF] Geissler | Phase-matched high-order harmonic generation in the nonadiabatic limit[END_REF][START_REF] Kroon | Attosecond pulse walk-off in high-order harmonic generation[END_REF][START_REF] Hernández-García | Group velocity matching in high-order harmonic generation driven by mid-infrared lasers[END_REF][START_REF] Rudawski | Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate opcpa-system[END_REF] , the Yudin-Ivanov (YI) ionization rate [START_REF] Yudin | Nonadiabatic tunnel ionization: Looking inside a laser cycle[END_REF] , which is an extension to the cycle averaged PPT, can be used. In Fig. 1(b) we present the ionization degree (η fe ) obtained after half of the laser pulse, as function of laser intensity, for a Gaussian pulse with 22 fs pulse duration FWHM. The ionization degree at the peak of the pulse, assumed centered at t = 0, is given by

η fe = 1 -exp - 0 -∞ Γ[I(t)]dt .
(1)

Single atom response of High-order harmonic generation

The physics of HHG is well described using a semi-classical three-step model [START_REF] Corkum | Plasma perspective on strong field multiphoton ionization[END_REF][START_REF] Kulander | Time-Dependent Studies of Multiphoton Processes[END_REF] , or equivalently a quantum mechanical description based on the Strong-Field-Approximation [START_REF] Lewenstein | Theory of high-harmonic generation by low-frequency laser fields[END_REF] . Briefly, the first step of the process is ionization by a strong low-frequency field, as described in Section 2.1. The second step is the classical motion of the electron in the continuum driven by the laser field. Most of the electron trajectories do not come back to the nucleus, hence contribute to the partial ionization of the medium. For the trajectories coming back, there is a small probability for recombination back to the ground state. This third step, which is the inverse of the photoionization process described below in Section 3.6, is accompanied by the emission of extremely short, attosecond, XUV light pulses. Interference between attosecond pulses emitted twice per laser cycle, with a sign flip between consecutive pulses, leads to the emission of odd-order harmonics of the laser light. There are mainly two electron trajectories leading to a given XUV photon energy, originating at different tunneling times. These trajectories are referred to as "short" or "long", depending on the electron excursion time.

The three step model successfully predicts the cut-off energy for high-order harmonic generation, equal to

E cut-off = I p + 3.17U p . ( 2
)
I p is the ionization energy and U p = e 2 |E 0 | 2 /(4m e ω 2 ) is the ponderomotive energy, where e, m e are the electron charge and mass, E 0 is the laser field strength and ω is the laser frequency. In Fig. 1(c) we present the variation of the 23 rd harmonic yield in Ar (blue) and the 69 th harmonic yield in Ne (red) as a function of driving field intensity. The single-atom data are obtained by solving the time-dependent three-dimensional Schrödinger equation in Ar and Ne, for a large number of peak intensity values (about five thousand) [START_REF] Schafer | Numerical Methods in Strong Field Physics[END_REF] . Both curves present rapid variations as a function of intensity, resulting from interference between different electron trajectories [START_REF] Lewenstein | Rings in above-threshold ionization: A quasiclassical analysis[END_REF] . The variation of the harmonic yields with intensity can be approximated by an I n power law with different exponents for the cut-off region and plateau [START_REF] Kulander | Theory of high-order processes in atoms in intense laser fields: Introduction[END_REF][START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF] . The harmonic yield varies rapidly in the cut-off region, n ≈ 16.5 for Ar and n ≈ 40 for Ne (dashed black lines), and much slower in the plateau, n ≈ 2.6 for Ar and n ≈ 5.5 for Ne (solid black lines).

3 Macroscopic aspects of high-order harmonic generation

Propagation of high-order harmonics and geometrical scaling

Using the slowly-varying envelope and paraxial approximations [START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF] , the propagation of the q th harmonic field can be described by a wave equation,

∇ 2 ⊥ E q -2ik q ∂E q ∂z = -µ 0 q 2 ω 2 P q e -i(qk 1 -k q )z , (3)
where the harmonic field and polarization are related to the slowly varying envelopes as E q exp[i(qωtk q z)] + c.c. and P q exp[i(qωtqk 1 z)] + c.c. The symbol ∇ 2 ⊥ refers to double differentiation with respect to the transverse directions x and y (z denoting the propagation direction) and µ 0 is the vacuum permeability. The wavevector of the fundamental field is k 1 = n 1 ω/c, and the wavevector of the q th harmonic field is k q = n q qω/c, where c is the speed of light and n 1 and n q are the refractive indices at frequencies ω and qω respectively. For propagation distances small compared to the Rayleigh length of the driving laser, diffraction, i.e. the first term of Eq. (3), may be neglected, resulting in a one-dimensional wave equation. Integrating the one-dimensional equation for a homogeneous medium with length L and excluding the effect of reabsorption yields

|E q | 2 ∝ sin(∆kL/2) ∆k 2 , (4)
where ∆k = qk 1k q is the phase mismatch between the polarization field at frequency qω, induced by the response of the medium to the fundamental field, and the generated harmonic field. This highlights the importance of minimizing ∆k to optimize the conversion efficiency. Eq. ( 4) can be generalized to include diffraction and focusing, assuming a power law (not necessarily within lowest-order perturbation theory) for the single atom response [START_REF] L'huillier | Theoretical aspects of intense field harmonic generation[END_REF] . Under the transformation (x, y, z, ρ) → (ηx, ηy, η 2 z, ρ/η 2 ), where η is a scaling factor 33 , we have

∇ 2 ⊥ → ∇ 2 ⊥ η 2 , ∂ ∂z → ∂ η 2 ∂z , P q → P q η 2 , (5)
so that Eq. (3) remains invariant. Scaling z implies to scale the medium length by the same factor. Such a scaling also requires to keep the same laser intensity distribution in the medium, and therefore to scale the input (and output) energies by η 2 . This geometrical scaling allows us to discuss the efficiency of HHG independently of the focusing geometry. The results presented below will thus be scaled by the Rayleigh length z R .

Phase mismatch in HHG

In HHG in gases, there are four contributions to the phase mismatch between the fundamental field and the generated q th harmonic field. These contributions are due to the dispersion in the neutral medium ∆k at , the presence of free electrons ∆k fe , the influence of the laser focusing ∆k foc and finally the so-called dipole phase contribution ∆k i . The dipole phase is due to the single atom response [START_REF] Lewenstein | Rings in above-threshold ionization: A quasiclassical analysis[END_REF] , and depends on the electron trajectory in the continuum (short or long). In this section, we discuss the phase mismatch along the propagation axis due to these four contributions, neglecting the contribution of ions [START_REF] Popmintchev | Ultraviolet surprise: Efficient soft x-ray high-harmonic generation in multiply ionized plasmas[END_REF] ,

∆k = ∆k at + ∆k fe + ∆k foc + ∆k i . ( 6
)

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The dispersion in the neutral medium is equal to

∆k at = (n 1 -n q ) qω c . ( 7
)
At the fundamental frequency, n 1 > 1 while at the harmonic frequencies above the ionization threshold n q < 1, so that this contribution is positive. We introduce the polarizability α q = 2ϵ 0 (n q -1)/ρ at frequency qω, where ϵ 0 is the vacuum permittivity and ρ is the atomic density of the medium. Eq. ( 7) can be written as a function of the polarizabilities (α 1 ) and (α q ) as

∆k at = qωρ 2ϵ 0 c (1 -η fe ) α 1 -α q , ( 8
)
where η fe is the ionization degree in the medium. For long wavelengths, the polarizability at the fundamental wavelength (α 1 ) can be approximated by the static polarizability (α 0 ), and the ionization degree can be assumed to be small (η fe ≪ 1), so that

∆k at ≈ qωρ 2ϵ 0 c α 0 -α q > 0. ( 9
)
The variation of α 0α q as a function of photon energy is plotted in Fig. 2(a) in Ar and Ne. In the energy region shown, the refractive index in Ar and Ne is below 1 and α 0α q ≥ 0. For a guided geometry, e.g. a hollow core capillary, the phase mismatch should also include the effect of the mode dispersion in the wave guide [START_REF] Rundquist | Phase-matched generation of coherent soft x-rays[END_REF] , leading to an additional phase mismatch

∆k wg = -q u 2 nm λ 4πa 2 , ( 10
)
where a is the radius of the wave guide, λ is the laser wavelength, and u nm are constant factors of the propagating modes, given by the m th root of the equation J n-1 (u nm ) = 0 61 , where J n (x) is the Bessel function of the first kind of order n. Typically HHG in guided geometries use the first order mode, for which u 11 = 2.4.

The dispersion due to free electrons can be expressed as

∆k fe = - qωρ 2ϵ 0 c η fe e 2 m e 1 ω 2 - 1 q 2 ω 2 < 0. ( 11
)
The second term in the parenthesis above, which scales as 1/q 2 , can be neglected for high-order harmonics. When a laser beam goes through a focus, the phase varies (in addition to the usual kz). For a Gaussian beam the phase variation is the Gouy phase shift ζ(z) =tan -1 (z/z R ), where z R is the Rayleigh length. The Gouy phase leads to an increase of the fundamental phase velocity. Neglecting the effect of the focusing of the harmonic beam, the phase mismatch is

∆k foc = q dζ dz = - qz R z 2 + z 2 R < 0. ( 12
)
∆k foc is negative and equal to -q/z R when z = 0. For a guided geometry ∆k foc is equal to zero. Finally, to estimate the phase mismatch ∆k i , we use an approximate expression for the phase accumulated by the electron in the continuum [START_REF] Guo | Phase control of attosecond pulses in a train[END_REF] ,

Φ i = α i I + t pi qω - I p h + γ i I qω - I p h 2 , ( 13
)
where i=s,ℓ for the short and long trajectories respectively, and h is the reduced Planck constant. In the literature, the dipole phase is commonly expressed as Φ i = α i (qω)I 57 . The advantage of Eq. ( 13), where α i I now refers only to the first harmonic above the ionization threshold, is that it gives a simple analytical expression for the frequency dependence of the phase. For the short trajectory α s = 0 and for the long trajectory α ℓ = -0.16αλ 3 /(m e c 3 ) 64 , with α being the fine structure constant. The second term, where t pi is an approximate threshold return time, represents the group delay and does not influence phase matching. The third term describes the group delay dispersion of the 14)] in Ar, with I = 2.5 × 10 14 Wcm -2 and (c) in Ne with I = 5 × 10 14 Wcm -2 for the short (solid) and long (dashed) trajectories, obtained by solving the saddle point equations in the strong field approximation [blue, red] and from the model in Eq. ( 13) [purple, yellow]. attosecond pulse. The quantity γ s is equal to 0.22cm e /(αλ), while γ ℓ = -0.19cm e /(αλ). Fig. 2(b,c) compares the predictions of this model for

∂Φ i ∂I = α i -γ i qω - I p h 2 1 I 2 , (14)
with calculations performed by solving the saddle point equations within the Strong Field Approximation [START_REF] Lewenstein | Rings in above-threshold ionization: A quasiclassical analysis[END_REF][START_REF] Varjú | Frequency chirp of harmonic and attosecond pulses[END_REF] . The good agreement between the two calculations validates the analytical model which will be used throughout.

The dependence of Φ i with intensity, and therefore with z, leads to a phase mismatch

∆k i = ∂Φ i ∂I ∂I ∂z ( 15
)
∆k i changes sign across the focus, being negative when z ≤ 0 and positive when z ≥ 0. For a Gaussian beam characterized by a Rayleigh length z R , and a laser focus at z = 0, Eq. ( 15) becomes

∆k i = - 2zβ i (z) z 2 + z 2 R , (16)
where

β i (z) = α i I(z) - γ i I(z) qω - I p h 2 < 0. ( 17
)
In a guided geometry the intensity is constant and thus ∆k i = 0.

Phase matching on axis

Phase matching amounts to compensating the effect of the fundamental Gouy phase (and to some extent the q th harmonic dipole phase) by choosing appropriately the pressure-dependent refractive indices at frequencies ω and qω for a given ionization degree.

We first consider the case where the nonlinear medium is centered at the focus of the laser, so that ∆k i can be neglected. Introducing ∆κ at = ∆k at /ρ and ∆κ fe = ∆k fe /(ρη fe ), the atomic density required to achieve phase matching (i.e. ∆k = 0) is The dashed black lines correspond to the critical ionization (η mac fe ) for the 23 rd and 69 th harmonic in Ar and Ne respectively. The ionization degree is indicated with the bottom (top) x-axis for Ar (Ne). (b) Multiplication factor f i [see Eq. ( 23)] describing the effect of moving the medium relative to the laser focus for the 23 rd harmonic in Ar and the 69 th harmonic in Ne for the long (dashed) and short (solid) trajectory. The intensities are chosen to be 2.5 ×10 14 W cm -2 , and 5 ×10 14 W cm -2 , respectively in the medium.

ρ match = - ∆k foc ∆κ at + η fe ∆κ fe . ( 18
The atomic density is related to the pressure through the ideal gas law p = ρk B T, where k B is the Boltzmann constant and T is the temperature in Kelvin, often assumed to be the room temperature. Although not always true, e.g. for a gas jet with supersonic expansion, we will here express our results in terms of pressure and not atomic density, assuming that these are related by the ideal gas law. Using ∆k foc = -q/z R , we have,

p match z R = qk B T ∆κ at + η fe ∆κ fe , ( 19
)
where p match is the phase matching pressure. This equation holds when ∆κ at > |η fe ∆κ fe |, requiring that the ionization degree is less than a critical value η mac fe defined as [START_REF] Rundquist | Phase-matched generation of coherent soft x-rays[END_REF][START_REF] Durfee | Phase matching of high-order harmonics in hollow waveguides[END_REF] ,

η mac fe = m e ω 2 e 2 α 0 -α q , ( 20
)
where α 0α q is obtained from Fig. 2(a). The critical ionization degree, η mac fe , corresponds to equal phase velocities of the fundamental and q th harmonic fields. An ionization degree above η mac fe implies that the phase velocity difference changes sign and that phase matching on axis is no longer possible. Eq. ( 19) can be rewritten as a function of the ionization degree as

p match z R = 2m e ωϵ 0 ck B T e 2 (η mac fe -η fe ) . ( 21
)
Fig. 3(a) shows p match z R in mbar cm (or equivalently in Pa m) as a function of the ionization degree in the medium for different harmonic orders in Ar and in Ne. The critical ionization degree, indicated by the dashed lines, is about 6% for the 23 rd harmonic in Ar, while for the 69 th harmonic in Ne it is slightly more than 1.1%.

When the medium is not centered at the laser focus, we take into account ∆k i and generalize Eq. ( 21) to

p match z R = 2m e ωϵ 0 ck B T f i e 2 (η mac fe -η fe ) , ( 22
)
where

f i = z 2 R z 2 + z 2 R 1 + 2zβ i (z) qz R . ( 23
)

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Here, z is the position of the center of the medium relative to the laser focus ( f i = 1 for z = 0). This factor is shown in Fig. 3(b) for the 23 rd and 69 th harmonic in Ar and Ne for intensities I = 2.5 × 10 14 W cm -2 , and I = 5 × 10 14 W cm -2 , respectively, in the center of the medium. While f i remains close to unity for the short trajectory, it displays a strong variation for the long trajectory. Consequently, phase matching will be achieved at different pressures for the two trajectories when the medium is moved relative to the laser focus. In addition, f i becomes negative for the long trajectory when the medium is located after the laser focus, preventing it from being phase matched in conditions where η fe < η mac fe [START_REF] Antoine | Attosecond pulse trains using high-order harmonics[END_REF] . However, in this case, phase matching above the critical ionization might become possible.

Intensity window for efficient HHG

Efficient HHG is limited by the intensity and duration of the driving laser pulse, through the cut-off energy [see Eq. ( 2)], above which the harmonic yield decays rapidly, and through the critical ionization degree [see Eq. ( 20) and Fig. 3(a)], above which phase matching cannot be achieved. This allows us to define an intensity window for efficient HHG. The minimum intensity I mic is the cut-off intensity, obtained from Eq. ( 2),

I mic = m e ω 2 3.17 × 2πα qω - I p h , ( 24
)
where the laser intensity I is related to the field strength E 0 by I = cϵ 0 |E 0 | 2 /2. The index "mic" indicates that this intensity cut-off is related to the microscopic single atom response. Using Fig. 1(b), we can deduce the corresponding ionization degree η mic fe .

(a) For a given laser pulse duration, we introduce I mac , as the peak intensity for which η fe = η mac fe at the peak of the pulse [as shown in Fig. 1(b)]. We indicate the values of I mic and I mac and the corresponding ionization degrees η mic fe and η mac fe for the 23 rd harmonic in Ar and the 69 th harmonic in Ne in Table 1, for a 22 fs FWHM Gaussian pulse. For a given generating gas and harmonic order, as the pulse duration increases, I mac generally decreases, while I mic is constant, thus reducing the intensity window for efficient HHG. This is shown in Fig. 4(a) for the 23 rd harmonic in Ar and the 69 th harmonic in Ne, where the shaded regions indicate the pulse lengths and peak intensities for which efficient, phase matched HHG is possible. Similarly, as the harmonic order increases, I mac generally decreases [see Fig. 3(a)], while I mic increases, thus reducing the intensity window for efficient HHG.

The condition I mic = I mac defines a cut-off harmonic [START_REF] Popmintchev | Extended phase matching of high harmonics driven by mid-infrared light[END_REF] above which on-axis phase matched HHG is no longer possible. This cut-off harmonic can be moved to higher XUV photon energies by using longer driving wavelengths [START_REF] Popmintchev | Extended phase matching of high harmonics driven by mid-infrared light[END_REF] , as shown in Fig. 4(b), at the cost of a lower CE [START_REF] Shiner | Wavelength scaling of high harmonic generation efficiency[END_REF] . Alternatively, as shown in Fig. 4(c), shorter driving pulse lengths can be used [START_REF] Popmintchev | The attosecond nonlinear optics of bright coherent x-ray generation[END_REF][START_REF] Klas | Ultra-short-pulse high-average-power megahertz-repetition-rate coherent extreme-ultraviolet light source[END_REF] . This highlights the importance of using short driving laser pulses for efficient harmonic generation, and longer wavelengths to reach the soft X-ray regime [START_REF] Sheehy | High harmonic generation at long wavelengths[END_REF][START_REF] Popmintchev | Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers[END_REF] .

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The use of few-cycle driving pulses introduces additional limitations to phase matching, for example from CEP-slip, as recently pointed out by R. Klas [START_REF] Klas | Efficiency scaling of high harmonic generation using ultrashort fiber lasers[END_REF] . Furthermore, for very long wavelengths (> 3 µm), the validity of the dipole approximation in the single atom response may break down [START_REF] Ludwig | Breakdown of the dipole approximation in strong-field ionization[END_REF] . Using short pulses with very high intensities, it is possible to produce very high photon energies, also for near infrared wavelengths, through a process called nonadiabatic self-phase matching [START_REF] Geissler | Phase-matched high-order harmonic generation in the nonadiabatic limit[END_REF][START_REF] Seres | Generation of coherent soft-x-ray radiation extending far beyond the titanium l edge[END_REF] . However, recent experimental comparisons of the efficiency favor long driving wavelengths [START_REF] Chevreuil | Water-window high harmonic generation with 0.8-µm and 2.2-µm opcpas at 100 khz[END_REF] . Table 1. Intensity window for efficient generation of the 23 rd harmonic in Ar and 69 th harmonic in Ne, ionization degrees, absorption cross sections and α 0α q .

Ar q = 23 Ne q = 69 I mic (W cm -2 ) 1.0 × 10 14 3.3 × 10 14 I mac (W cm -2 ) 2.1 × 10 14 4.6 × 10 14 η mic fe (%) 0.15 0.16 η mac fe (%) 6.0 1.0 σ abs (cm 2 ) 1.5 × 10 -17 3.8 × 10 -18 α 0α q (A 2 s 4 kg -1 ) 3.0 × 10 -40 5.1 × 10 -41

Phase matching off-axis

So far we have only considered phase matching along the propagation direction. Phase matching can also be achieved in other directions, not parallel to the laser propagation. Phase matching can in general be formulated as a vectorial momentum conservation equation [START_REF] Balcou | Generalized phase-matching conditions for high harmonics: The role of field-gradient forces[END_REF] ,

k k k q = qk k k 1 + K K K i . (25)
Here, k k k 1 * and k k k q represent the wavevectors of the fundamental and q th harmonic fields, which include dispersion and focusing, and K K K i = ∇Φ i describes the effect of the dipole phase. For simplicity we only consider phase matching in the focal plane, so that k k k 1 is parallel to the propagation axis and K K K i is perpendicular to it. Using Eq. ( 14), assuming that the fundamental beam is Gaussian, and that the radial coordinate, r, is much smaller than the beam waist,

K i = ∂Φ i ∂r = - 4πrβ i λz R . ( 26
)
Phase matching can therefore be achieved at certain distances from the optical axis, such that k 2 q = q 2 k 2 1 + K 2 i . This requires that the scalar quantity ∆k = qk 1k q is negative and, using Eq. (26) together with

K 2 i ≈ 2qk 1 |∆k|, that r = z R β i qλ|∆k| 2π . ( 27
)
This off-axis phase matching, which depends on the electron trajectory, is characterized by a ring-like emission at the exit of the medium [START_REF] Salières | Coherence control of high-order harmonics[END_REF] . Phase matching off-axis is easier to achieve, since ∆k ≤ 0. It allows for a higher intensity and ionization degree in the medium than on-axis phase matching.

Reabsorption and other macroscopic effects

Photoionization by XUV radiation is intrinsically linked to HHG. Indeed, the reverse phenomenon, recombination, is an inherent step of the single atom response. In addition, the interaction of XUV light with the medium affects propagation, through dispersion (see Section 3.2) and absorption (real and imaginary parts of the refractive index). Fig. 5(a) shows the absorption cross section, σ, for Ar and Ne in the energy range reached with HHG using a near infrared laser driver. The rapid variation of the cross section in Ar is due to a Cooper minimum at 52 eV [START_REF] Cooper | Photoionization from outer atomic subshells. a model study[END_REF] . Absorption of the harmonic field in the medium can be included by modifying Eq. ( 4) as [START_REF] Constant | Optimizing high harmonic generation in absorbing gases: Model and experiment[END_REF][START_REF] Ruchon | Macroscopic effects in attosecond pulse generation[END_REF] 32). [adapted from E. Constant (1999)].

|E q | 2 ∝ e -L 2L abs cosh L 2L abs -cos(∆kL) ∆k 2 + 1 4L 2 abs , (28) 9
where L abs denotes the absorption length (the inverse of the absorption coefficient at frequency qω, which is equal to ρσ abs , see Section 3.6). Fig. 5(b) shows how the harmonic yield varies as a function of the medium length (in units of L abs ) for different coherence lengths, L c = π/∆k. The yellow line is obtained with L abs = ∞ and L c = ∞. For a given coherence length, for example L c = 4L abs (see violet curve), increasing the medium length beyond 3L abs leads to destructive interference and limits the harmonic yield. Recently, this model has been extended to include the effects of linear density gradients along the propagation direction [START_REF] Major | Extended model for optimizing high-order harmonic generation in absorbing gases[END_REF] . Finally, when an intense laser beam propagates in a partially ionized medium, the radially dependent refractive index, due to the free electrons, leads to defocusing, shifting the focus towards negative z. For the gas densities considered in this work, Kerr induced focusing can be neglected. The peak intensity in the effective focus will then be lower than the peak intensity in vacuum. The reshaping of the pulse can lead to efficient generation through self-guiding of the driving field [START_REF] Kim | Optimization of high-order harmonic brightness in the space and time domains[END_REF][START_REF] Major | Effect of plasma-core-induced self-guiding on phase matching of high-order harmonic generation in gases[END_REF] . Strong defocusing can also extend the phase matching cut-off due fast variations of the driving laser intensity along the propagation axis [START_REF] Sun | Extended phase matching of high harmonic generation by plasma-induced defocusing[END_REF] . To estimate the degree of defocusing, one can introduce a defocusing length, L D , corresponding to a doubling of the diffraction limited beam divergence, defined as [START_REF] Fill | Focusing limits of ultrashort laser pulses: analytical theory[END_REF][START_REF] Leemans | Experiments and simulations of tunnel-ionized plasmas[END_REF][START_REF] Rae | Ionization-induced defocusing of intense laser pulses in high-pressure gases[END_REF] L D = πcϵ 0 m e ωk B T pη fe e 2 = πcϵ 0 m e ωσ abs η fe e 2 L abs .

When η fe = η mac fe , L D = 17L abs for the 23 rd harmonic in Ar and 26L abs for the 69 th harmonic in Ne. When η fe < η mac fe , L D increases and defocusing plays a negligible role in phase matching of HHG.

Simulations of high-order harmonic generation 4.1 Method

We performed numerical calculations based on the method presented in [START_REF] L'huillier | Calculations of high-order harmonicgeneration processes in xenon at 1064 nm[END_REF] , and using tabulated single-atom data obtained by solving the time-dependent three-dimensional Schrödinger equation in Ar and Ne [see Section 2.2 and Fig. 1(c)]. These data serve as input in a propagation code based on the paraxial and slowly-varying envelope approximations (i.e. it is accurate for pulse lengths longer than ≈ 3 cycles). For methods going beyond the slowlyvarying envelope approximation see e.g. [START_REF] Brabec | Intense few-cycle laser fields: Frontiers of nonlinear optics[END_REF][START_REF] Gaarde | Macroscopic aspects of attosecond pulse generation[END_REF][START_REF] Altucci | Beyond the single-atom response in isolated attosecond-pulse generation[END_REF] . We use a fundamental wavelength of 810 nm, a pulse duration of 22 fs, peak intensities ranging from 1.5-4.5 × 10 14 W cm -2 in Ar and 5 × 10 14 W cm -2 in Ne. We vary the medium length, pressure and position relative to the fundamental focus. Our propagation code includes effects of reshaping of the fundamental field due to propagation in a partially ionized medium.

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Figure 6. Simulations of HHG. Conversion efficiency (color scale) as a function of pressure and medium length for the 23 rd harmonic in argon for (a) I = 1.5 ×10 14 W cm -2 , (b) 2.5 ×10 14 W cm -2 and (c) 4.5 ×10 14 W cm -2 , (d) conversion efficiency of 69 th harmonic in Ne for I =5 ×10 14 W cm -2 . The medium is centered at the laser focus. The dashed lines represent the prediction from Eq. ( 32). (e) Normalized 23 rd harmonic intensity in Ar as a function of divergence angle in the far field and (f) as a function of time, with t = 0 at the peak of the fundamental pulse, for the cases of high pressure (blue) and low pressure (red), indicated by the blue and red circles in (b). The average ionization degree on axis in the medium as a function of time is shown for high pressure (dashed blue) and low pressure (dashed red). The horizontal black line indicates the critical ionization degree η mac fe . (g) Conversion efficiency (color scale) for the 23 rd harmonic in Ar as a function of pz R and medium length, with the medium centered at z = -z R and (h) as a function of pz R and medium position, for a medium of length L = 0.2z R . The solid (dashed) white lines indicate how Eq. ( 32) varies for the short (long) trajectory. The intensity at the center of the medium is 2.5×10 14 W cm -2 .

There are two reasons why we use peak intensities that are larger than I mac , thus outside the phase matching intensity window introduced in Table 1 and illustrated in Fig. 4(a). First, the rapid variation of the single atom response with intensity strongly influences the CE, and can to some degree compensate for non-optimal phase matching. Second, HHG is a time-dependent macroscopic phenomenon, which takes place in a certain time interval and volume. The definition of I mac is valid only at the focus and t = 0. While I mic can be considered as an intensity threshold for HHG, I mac is not a strict maximum intensity, but represents the highest local intensity (in time and space) at which efficient phase matching is possible. We use a flat medium density profile, with a short sigmoid edge, i.e. we do not study the effects of varying the slopes of the medium. The reader is referred to reference [START_REF] Major | Extended model for optimizing high-order harmonic generation in absorbing gases[END_REF] for a detailed study of the effect of linear density gradients, at the beginning and end of the generating medium, on the CE.

Simulation results

Fig. 6(a)-(c) shows the conversion efficiencies for the 23 rd harmonic in Ar for several peak intensities. The medium is centered at the laser focus. The CE is represented in color as a function of medium length, in units of z R , and pz R , in units of mbar cm (or equivalently Pa m). The CE variation has a hyperbolic shape, with a maximum indicated by the dark red color, below 10 -5 in Ar. The position of the hyperbola does not move significantly with the peak intensity, as shown for Ar in Fig. 6(a)-(c). As the intensity increases from 1.5×10 14 W cm -2 to 2.5 × 10 14 W cm -2 , the maximum CE increases from 2.5 × 10 -6 to 6×10 -6 . Increasing the intensity to 4.5 × 10 14 W cm -2 , however, does not lead to significantly higher conversion efficiencies, and for even higher intensities the CE begins to decrease. Increasing the pressure further outside the range shown in Fig. 6 the maximum CE remains approximately constant, with a peak below 10 -5 . We also perform simulations with different pulse duration (∼15-40 fs). The CE increases,

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approximately as τ -0.571 , however the position of the hyperbola does not change significantly. We expect this to remain true also in the few cycle regime.

Fig. 6(d) shows the conversion efficiency for the 69 th harmonic in Ne at an intensity 5×10 14 W cm -2 . The shape is very similar to that obtained in Ar. The range of pressures which lead to a high CE in the case of a long medium becomes, however, very narrow for higher orders. Our simulations do not include high pressure effects, such as clustering or loss of coherence in the single atom response when the mean free path approaches the electron excursion in the continuum [START_REF] Rothhardt | Absorption-limited and phase-matched high harmonic generation in the tight focusing regime[END_REF][START_REF] Ruf | Inhomogeneous high harmonic generation in krypton clusters[END_REF] , which could lead to a decrease of the CE.

This hyperbolic shape reflects the interplay between phase matching and absorption. At any pressure p, phase matching determines an optimum ionization degree to meet the phase matching condition p = p match in Eq. (22). A sufficient peak intensity ensures that, for a wide range of pressures, this condition is reached at some point in the temporal pulse. As the pressure increases, optimum phase matching is reached at a higher ionization degree [as illustrated in Fig. 3(a)]. At the same time, absorption determines an optimum pressure-length product as shown in Fig. 5(b). If a similar coherence length is reached, a similar optimum pressure-length product is obtained, so that the maximum CE is found on the same hyperbola. A maximum CE following such a hyperbolic relationship between pressure and length has been observed experimentally for a gas jet in [START_REF] Comby | Absolute gas density profiling in high-order harmonic generation[END_REF] .

In Fig. 6, two regimes leading to similar conversion efficiencies can be identified (i) A regime where phase matching is achieved at relatively low pressure, which requires a low ionization degree, i.e. reached early in the pulse. In this case, the neutral atom dispersion approximately cancels the sum of the Gouy phase and dipole phase contributions, i.e. ∆k at + ∆k foc + ∆k i ≈ 0. Phase matching at a low ionization degree requires a medium with low pressure [see Fig. 3(a)]. As a consequence of the low pressure, a long medium length is needed to achieve a high efficiency. In this regime, the phase matching pressure approaches its minimal value p 0 , with

p 0 z R = 2m e ωϵ 0 ck B T f i e 2 η mac fe = 2ϵ 0 ck B T f i ω(α 0 -α q ) , (30)
giving p 0 z R = 28 mbar cm for the 23 rd harmonic in Ar and p 0 z R = 170 mbar cm for the 69 th harmonic in Ne.

For the sake of generality we here include explicitly ∆k i through f i , though in Fig. 6(a)-(d) the medium is centered at z = 0 and ∆k i = 0 ( f i = 1). The CE does not vary much with the medium length, which can be explained by the fact that the process is limited by absorption [see Fig. 5(b)]. However, it strongly depends on the pressure, especially for high harmonic orders, since ∆k ∝ q [see Fig. 6(d)]. The maximum medium length is limited to values not exceeding the Rayleigh length since the dipole response far outside z R is small at the intensities considered here. This, in combination with absorption, leads to a lower HHG signal for lengths larger than z R .

(ii) A regime where HHG takes place at relatively high pressures, requiring ionization degrees close to η mac fe , and across a much shorter and well defined interaction length. In this case, the neutral atom dispersion approximately compensates the free electron contribution, i.e. ∆k at + ∆k fe ≈ 0. Thus much higher pressures are required to achieve ∆k at + ∆k fe + ∆k foc = 0. In this regime the efficiency depends strongly on the medium length but not much on the pressure, provided the pressure is sufficiently high (p ≫ p 0 ).

Analytic form of the hyperbola

To parametrize the hyperbolic shape exhibited by the simulations, we assume that HHG is absorption-limited [START_REF] Constant | Optimizing high harmonic generation in absorbing gases: Model and experiment[END_REF] with L = ςL abs , where ς = 3 is a fit parameter depending on the achieved coherence length [see dashed black line in Fig. 5(b)]:

pz R L z R = ςk B T f i σ abs (31)
This suggests a (partial) explanation in which the medium should not be too short (which will make full coherent build-up impossible) but also not too long (which will not lead to any additional coherent build-up, but leads to deteriorating effects such as decoherence or plasma effects).

Including the phase matching pressure for the case of η fe = 0, introduced in Eq. (30) we propose the following equation for the hyperbola

(p -p 0 )L = ςk B T f i σ abs . ( 32
)

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The factor f i can be chosen to be constant, equal to 1, for a medium centered at the focus. Alternatively, we can use f i = f i (L/2), since the XUV radiation emitted at the end of the medium (z = L/2) has experienced the least reabsorption. This leads to a small correction to the hyperbolic equation, with a slightly tilted "horizontal" branch. Eq. ( 32), represented by the dashed white line in Fig. 6(a)-(d) for ς = 3, agrees with the simulations in both Ar and Ne for harmonics in the plateau region. The hyperbolic behavior, which is the main finding of this work, is remarkably universal, i.e. independent of the peak intensity, harmonic order, generating gas, and focusing geometry through the scaling laws presented in Section 3.1.

Spatial and temporal profile of the emitted XUV radiation

The spatial profile of the emitted XUV radiation is shown in Fig. 6(e) using I = 2.5 × 10 14 W cm -2 for the 23 rd harmonic in Ar for (p, L) = (25 mbar cm, 0.7 z R ) and (p, L) = (170 mbar cm, 0.1 z R ) corresponding to regime (i) and (ii), respectively (indicated in Fig. 6(b) as the red and blue circles). As can be seen, the harmonic is much less divergent in regime (i) being confined to a solid angle of θ √ z R < 2 mrad cm 1/2 . In contrast, the spatial profile in regime (ii) is more irregular with a strong off-axis contribution, possibly due to off-axis phase matching, as discussed in Section 3.5 and/or due to the contribution of the long trajectory. Fig. 6(f) shows the temporal profile of the XUV radiation exiting the medium, together with the average ionization degree on axis. In both regimes the generation takes place mostly prior the peak of the IR pulse. The emission is, however, much more transient in regime (ii) than in regime (i), taking place at a later time when the ionization degree is close to the critical ionization degree, indicated by the horizontal black line [see also Fig. 3(a)] [START_REF] Heyl | Introduction to macroscopic power scaling principles for high-order harmonic generation[END_REF] . The higher ionization degree in regime (ii) is a consequence of a higher average intensity in the shorter medium.

Variation of the medium position

Finally, we vary the position of the medium relative to the laser focus, which is known to affect the conversion efficiency, [START_REF] Salières | Coherence control of high-order harmonics[END_REF][START_REF] Major | Compact intense extreme-ultraviolet source[END_REF] and the position of the harmonic focus [START_REF] Wikmark | Spatiotemporal coupling of attosecond pulses[END_REF][START_REF] Quintard | Optics-less focusing of XUV high-order harmonics[END_REF][START_REF] Hoflund | Focusing properties of high-order harmonics[END_REF] . In Fig. 6(g), we show the CE of the 23 rd harmonic in Ar (color scale) as a function of medium length and pz R for a medium centered at z = -z R , i.e. before the laser focus. The laser intensity at the center of the medium is 2.5×10 14 W cm -2 . We observe two hyperbolic shapes with similar CE, converging to different pressures, approximately 26 mbar cm and 58 mbar cm. The hyperbolic equation depends on the different trajectories through f i [see Eq. ( 23) and Fig. 3(b)]. The equation for the short trajectory (solid white line) follows the simulation results very well, while that of the long trajectory (dashed line) follows the trend of the upper feature at high pressure. The discrepancy in the long trajectory at long medium length could be due to strong off-axis phase matching, requiring ∆k < 0 (see Sec. 3.5), and thus allowing for phase matching at ionization degrees larger than η mac fe , or conversely, at lower pressures. Fig. 6(h) shows how the CE varies as a function of medium position, z and pz R for a constant medium length of L = 0.2z R . The solid (dashed) white line shows how pz R , extracted from Eq. ( 32), varies with z for the short (long) trajectory. At z = 0 the long and short trajectory phase match at the same pressure, since there is no dipole phase contribution, leading to a region of high CE which is confined to a narrow range of pressures. For z > 0, (medium after the laser focus) f ℓ becomes negative, meaning that phase matching can only be achieved for the short trajectory. For z < 0 the CE increases, reaching a maximum at z = -0.75z R (short trajectory) and z = -1.1z R (long trajectory). These maxima correspond to optimized generating positions, such that the laser-and dipole-phase-induced wavefronts compensate each other [START_REF] Wikmark | Spatiotemporal coupling of attosecond pulses[END_REF][START_REF] Quintard | Optics-less focusing of XUV high-order harmonics[END_REF] , which are different for the two trajectories. In the interval -1.25z R < z < -0.5z R the CE displays a double peak structure, with the largest splitting at z = -z R . The regions of high CE agree quite well with the phase matching pressure predictions for the long (dashed line) and short (solid line) trajectories. This analysis suggests that the upper hyperbolic feature in Fig. 6(g) corresponds to phase matching of the long trajectory, while the lower one to that of the short trajectory.

Discussion

The initial motivation for this work was to find a "recipe" for phase matched, efficient HHG, i.e. to suggest a best choice of geometry, medium length and gas pressure for a given set of laser parameters. We find that there is no unique best solution, but several, with advantages and disadvantages. We illustrate this below with two examples. Given a driving pulse energy E = 1 mJ, duration τ = 22 fs FWHM and wavelength of 810 nm, we want to optimize the number of photons of the 23 rd harmonic in Ar, with (i) and without (ii) the additional requirement of a high spatial quality. Finally, we discuss the influence of the driving wavelength and compare our prediction to existing experimental data [START_REF] Mikaelsson | A high-repetition rate attosecond light source for time-resolved coincidence spectroscopy[END_REF][START_REF] Comby | Absolute gas density profiling in high-order harmonic generation[END_REF][START_REF] Koliyadu | Optimization and characterization of high-harmonic generation for probing solid density plasmas[END_REF][START_REF] Sayrac | Pressure optimization of high harmonic generation in a differentially pumped ar or h2 gas jet[END_REF] .

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High conversion efficiency and spatial quality

A high spatial quality requires to be in regime (i), i.e. on the horizontal branch of the hyperbola [see Fig. 6(e)]. We choose a focal intensity equal to 2.5×10 14 W cm -2 , such that I ≥ I mic over a large volume in the medium. Above this intensity the ionization increases and the CE in regime (i) saturates. Assuming Gaussian optics, the Rayleigh length must be chosen to be

z R = 4 √ ln 2E √ πτλI = 0.042 m, (33)
which requires a focal length f = √ πz R /4λD = 200D, where D is the unfocused beam diameter. The pressure should be approximately 7 mbar according to Eq. ( 30) and Eq. (32). The medium length is not a sensitive parameter in this regime. However, the simulations [see Fig. 6(a-c)] indicate that a length equal to 0.5z R ≈ 2.1 cm is appropriate. The obtained CE is 4 × 10 -6 . The beam is well collimated, with a smooth spatial profile and a divergence less than 1 mrad.

High conversion efficiency

We now target regime (ii), i.e. the vertical branch of the hyperbola. We choose a higher focal intensity, equal to 4×10 14 W cm -2 , to reach an intensity I ≈ I mac over a large volume in the medium. This implies a Rayleigh length equal to z R = 0.026 m, and f = 160D. The pressure-length product is equal to 8 mbar cm [see Eq. ( 31)]. As shown in the simulations, the CE increases only slightly when increasing the pressure while decreasing the medium length. In many experimental situations, either the pressure or the length will be determined by technical limitations (e.g. highest backing pressure or shortest possible medium length in the case of a gas jet). Using a pressure equal to 220 mbar the length of the medium should be of the order of 0.35 mm. The obtained CE is approximately 8×10 -6 . The harmonic beam has a poor spatial quality with many annular spatial structures, and a divergence of the order of 6 mrad [see Fig. 6(e)]. Pressure length hyperbolas for different harmonic orders and driving wavelengths in Ar [9 th harmonic of 515 nm (blue), 33 rd harmonic of 1030 nm (red) and 23 rd harmonic of 800 nm (green)] compared to experimental conditions resulting in optimal XUV photon flux. The data points [1-4] are from refs. [START_REF] Comby | Absolute gas density profiling in high-order harmonic generation[END_REF][START_REF] Koliyadu | Optimization and characterization of high-harmonic generation for probing solid density plasmas[END_REF][START_REF] Sayrac | Pressure optimization of high harmonic generation in a differentially pumped ar or h2 gas jet[END_REF][START_REF] Mikaelsson | A high-repetition rate attosecond light source for time-resolved coincidence spectroscopy[END_REF] respectively.

Influence of the driving wavelength and comparison to experimental data

Finally, we examine the validity of our recommendation as a function of driving wavelength and XUV photon energy, and compare to experimental results reported in the literature. Fig. 7(a) shows the variation of the hyperbola for a few different XUV photon energies at two driving wavelengths (810 nm and 1800 nm). The phase matching pressure varies linearly with the driving wavelength, and does not depend on the harmonic order, but rather on the XUV photon energy, as predicted in Eq. (32). The different position of the hyperbola for 56 eV and 84 eV photon 14/19 energies, relative to that for 29 eV photon energy, reflects the large drop in the absorption cross section [see Fig. 5(a)]. In Fig. 7(b) we indicate experimental results, obtained at different driving wavelengths, focusing geometries and XUV photon energies, together with the corresponding hyperbolas. Unfortunately, to our knowledge, very few studies of HHG report the actual pressure in the generating medium. Interestingly, these results [START_REF] Mikaelsson | A high-repetition rate attosecond light source for time-resolved coincidence spectroscopy[END_REF][START_REF] Comby | Absolute gas density profiling in high-order harmonic generation[END_REF][START_REF] Koliyadu | Optimization and characterization of high-harmonic generation for probing solid density plasmas[END_REF][START_REF] Sayrac | Pressure optimization of high harmonic generation in a differentially pumped ar or h2 gas jet[END_REF] , which agree well with our prediction, span the different phase matching regimes discussed in the present article. Note that Comby et al. [START_REF] Comby | Absolute gas density profiling in high-order harmonic generation[END_REF] (blue squares) report five different measurements.

Conclusion

In summary, we have reviewed the main physical principles of high-order harmonic generation in gases, with emphasis on the macroscopic aspects. Using numerical simulations we have calculated the CE as a function of various parameters (pressure, intensity, medium length, position of medium relative to the laser focus, focusing geometry and trajectory) in Ar and Ne.

Efficient generation follows approximately a hyperbolic shape [(pp 0 )L ∝ 1/σ abs , see Eq. ( 32)]. The pressure must be larger than p 0 [see Eq. ( 30)], while L should be less than approximately a Rayleigh length. The conversion efficiency remains approximately constant along this hyperbola, however, the properties of the harmonics strongly depend on the choice of (p, L). At low pressure, the maximum CE does not depend much on the medium length and the generated XUV radiation is well collimated. In contrast, at high pressure, the maximum CE is relatively pressure-independent, and the generated radiation is much more divergent and emitted more transiently, later in the pulse. Finally, moving the medium approximately one Rayleigh length before the laser focus leads to an enhanced CE. In this case, optimal phase matching of the long and short trajectories occurs at different pressures and lengths, leading to two hyperbolas.

In conclusion, for any given focusing geometry, harmonic order and generating gas, efficient generation is possible for a variety of gas pressures and medium lengths. The best choice of pressure and length depends on whether a high spatial quality, high CE or a flexible experimental setup is prioritized for a specific application. We recommend using a long (on the order of 0.5z R ), low pressure medium if a high spatial and temporal quality of the XUV source is required. To optimize (only) the CE, we recommend a short, high pressure medium. The laser intensity, and focusing geometry, should be chosen so that the intensity in the medium is on the order of a few I mic , depending on the optimization strategy above.

Figure 1 .

 1 Figure1. Atoms in strong laser fields (a) Ionization rates, Γ, as a function of intensity in Ne (red) and Ar (blue), calculated using the Perelomov-Popov-Terent'ev approximation[START_REF] Perelomov | Ionization of Atoms in an Alternating Electric Field[END_REF] (solid) and solving the time dependent Schrödinger equation48 (dashed). The laser wavelength is 810 nm and (b) Ionization degree as a function of intensity in Ne (red) and Ar (blue), calculated using PPT ionization rates for a Gaussian pulse with 22 fs FWHM pulse duration. The ionization degree is given at the time t = 0 corresponding to the peak of the laser pulse. The dashed vertical lines define an intensity window, bounded from below by the cut-off intensity obtained from the three-step model, and from above by a critical intensity (see section 3.4). The dashed horizontal lines show the ionization degree associated with these intensities. (c) Variation of the 23 rd harmonic yield in Ar (blue) and 69 th harmonic yield in Ne (red) with intensity, obtained by solving the TDSE for a driving wavelength of 810 nm. The black lines represent a linear approximation I n with n = 16.5 (Ar) and 40 (Ne) in the cut-off region (dashed) and n = 2.6 (Ar) and 5.5 (Ne) in the plateau region (solid).

Figure 2 .

 2 Figure2. Energy dependent parameters governing phase matching (a) Difference between the static polarizability α 0 and the polarizability α q (see Section 3.2) as a function of photon energy in Ar (blue) and Ne (red). Data from[START_REF] Henke | X-ray interactions: Photoabsorption, scattering, transmission, and reflection at e = 50-30,000 ev, z = 1-92[END_REF] . (b) ∂Φ i /∂I [see Eq. (14)] in Ar, with I = 2.5 × 10 14 Wcm -2 and (c) in Ne with I = 5 × 10 14 Wcm -2 for the short (solid) and long (dashed) trajectories, obtained by solving the saddle point equations in the strong field approximation [blue, red] and from the model in Eq. (13) [purple, yellow].

)Figure 3 .

 3 Figure 3. Phase matching pressure (a) as a function of ionization degree in Ar for different harmonic orders [q = 19, 23, 27 (light blue, blue, purple)] and in Ne for different harmonic orders [q = 61, 69, 77 (dark red, red, yellow)].The dashed black lines correspond to the critical ionization (η mac fe ) for the 23 rd and 69 th harmonic in Ar and Ne respectively. The ionization degree is indicated with the bottom (top) x-axis for Ar (Ne). (b) Multiplication factor f i [see Eq. (23)] describing the effect of moving the medium relative to the laser focus for the 23 rd harmonic in Ar and the 69 th harmonic in Ne for the long (dashed) and short (solid) trajectory. The intensities are chosen to be 2.5 ×10 14 W cm -2 , and 5 ×10 14 W cm -2 , respectively in the medium.

Figure 4 .

 4 Figure 4. Phase matching window (a) Variation of I mac as a function of FWHM pulse duration for λ = 810 nm for the 23 rd harmonic in Ar (blue solid) and 69 th harmonic in Ne (red solid). The horizontal dashed lines indicate the corresponding value of I mic . (b) Variation of the cut-off harmonic (such that I mic = I mac ) as a function of driving wavelength for an 8-cycle FWHM pulse length. (c) Variation of the cut-off harmonic as a function of pulse duration for a driving wavelength of 810 nm in Ar (blue) and in Ne (red). The shaded regions indicate where on-axis phase matched HHG is possible.

Figure 5 .

 5 Figure 5. Influence of absorption (a) Absorption cross section as a function of photon energy in Ar (blue) and Ne (red). Data from 62 . (b) Harmonic intensity as a function of medium length in units of L abs for different coherence lengths [L c = L abs , 4L abs , 8L abs , and L c ≫ L abs (blue, violet, green, red)]. The yellow line indicates the harmonic intensity calculated without absorption. The dashed black lines indicate at what medium length the maximum harmonic intensity is obtained for the case L c = 4L abs , which is related to the fit parameter ς = 3 introduced in Eq. (32). [adapted from E.Constant (1999)].

Figure 7 .

 7 Figure 7. Validity of hyperbola. (a) Pressure-length hyperbola [see Eq. (32)] for different harmonic photon energies [29 eV (blue), 56 eV (red) and 84 eV (green)] in Ar from two different driving wavelengths [810 nm (solid) and 1800 nm (dashed)]. (b) Pressure length hyperbolas for different harmonic orders and driving wavelengths in Ar [9 th harmonic of 515 nm (blue), 33 rd harmonic of 1030 nm (red) and 23 rd harmonic of 800 nm (green)] compared to experimental conditions resulting in optimal XUV photon flux. The data points[1-4] are from refs.[START_REF] Comby | Absolute gas density profiling in high-order harmonic generation[END_REF][START_REF] Koliyadu | Optimization and characterization of high-harmonic generation for probing solid density plasmas[END_REF][START_REF] Sayrac | Pressure optimization of high harmonic generation in a differentially pumped ar or h2 gas jet[END_REF][START_REF] Mikaelsson | A high-repetition rate attosecond light source for time-resolved coincidence spectroscopy[END_REF] respectively.

Acknowledgements

The authors would like to thank Kenneth J. Schafer and Mette B. Gaarde for help regarding the numerical simulations. The authors acknowledge support from the Swedish Research Council (2013-8185, 2016-04907), the European Research Council (advanced grant QPAP, 884900) and the Knut and Alice Wallenberg Foundation. AL is partly supported by the Wallenberg Center for Quantum Technology (WACQT) funded by the Knut and Alice Wallenberg foundation. L.R acknowledges support from Ministerio de Educación, Cultura y Deporte (FPU16/02591).

Author contributions

Competing interests

The authors declare no competing interests.


domain: sdv phys

text:
Innovative instrumentation for assessing the mechanical properties of skin

Bastien Blanchard 1* , Francis Ehrenfeld 1 , Anthony Laffore 1 , Corinne Nardin 1 , Giuseppe Percoco 2 , Laurent Peno-Mazzarino 2 , Christophe Derail 1 1 Université de Pau et des Pays de l'Adour, E2S UPPA, CNRS, IPREM, Pau, France 2 BIO-EC laboratory, Longjumeau, France *bastien.blanchard@univ-pau.fr

A specific instrument to evaluate the viscoelastic properties of human skin has been developed in this project. This technology combined with the use of ex vivo human skin explants under physiological conditions for several days, should enable to help for the development of products intended for treatments. Indeed, we propose to analyse and evaluate in an objective way:

-the effects of different external stresses (stretch marks, UV rays exposure, chemical exposure, air pollution, injuries, freezing, etc.) -the effects of medical or dermo-cosmetic products (moisturising creams, anti-stretch mark creams, healing dressings, etc.) on the viscoelastic properties of human skin.

➢ Skin from surgical operations. ➢ Skin explants kept alive for several days. C. Skin is glued to the arms of the technology.

➔Mechanical tests can be performed

Progress of the project

✓ A new instrumentation to evaluate the mechanical properties of human skin has been developed.

✓ A characterisation of the mechanical properties of ex vivo human skin explants has been carried out.

✓ The 1 st results obtained demonstrate the device's ability to accurately explore the mechanical properties of ex vivo human skin explants.

Perspectives & Applications

➢ In view of the 1 st results, continue the development of the technology:

-Addition of force sensors to control skin bonding pressure.

-Possibility of adapting the technology to in vivo measurements.

➢ Evaluate the effects of different stresses and wound healing on the mechanical properties of human skin.

V. Conclusion

Reference

  A. Skin is positioned on the sample holder.B. Holder is slid into the mechanical part of the technology.


Nidhi Hegde

Fabien Mathieu

Diego Perino

Diffusion épid émique de chunks en quasi-direct : la taille compte †

Keywords: Pair-à-pair, live, taille des chunks, efficacité

Le problème de la diffusion pair-à-pair en quasi-direct consiste à transmettre un contenu à un ensemble de pairs avec la meilleure qualité possible tout en minimisant le différé, c'est-à-dire le délai d'acheminement de la source aux pairs. Dans un grand nombre de solutions, le contenu est découpé en quantités de taille fixée, les chunks. En supposant que le temps de transfert d'un chunk d'un pair à un autre est uniquement déterminé par la bande passante de l'émetteur, le délai optimal possible est typiquement en c s (log 2 (n)), c étant la taille des chunks, s le débit du contenu diffusé et n le nombre de pairs. Il semble alors naturel de choisir c aussi petit que possible afin de minimiser le délai. Il arrive cependant un point où les latences présentes dans le réseau influent nécessairement sur la diffusion du contenu. Notre objectif est de mettre en évidence les phénomènes qui apparaissent lorsque la taille du chunk rend les effets de latence non négligeables. En se basant sur un mécanisme de diffusion épidémique simple, nous mettons en évidence les effets suivants : -des chunks trop petits empêchent l'algorithme de fonctionner efficacement, et génèrent un taux de pertes important ainsi qu'un gaspillage des ressources réseau ; -à partir d'une certaine taille, les pertes cessent et la quantité de messages de contrôle se stabilise, le délai étant proportionnel à la taille du chunk ; -entre les deux se situe un intervalle de tailles adaptées à la diffusion. Le choix d'une taille précise dépend du compromis à réaliser entre pertes et délai. De plus, nous observons que l'introduction d'un certain parallélisme dans la diffusion permet de déplacer la zone utile, augmentant ainsi la performance de l'algorithme.

Introduction

Peer-to-peer (P2P) live streaming applications have become very popular in the last few years. A key issue is whether the delay and quality requirements can be met by a P2P protocol. Most of P2P live streaming algorithms split the stream into atomic units of data called chunks and a peer can only send chunks it has fully received. In such systems one of the main goals is thus to design an efficient chunk exchange policy.

In addition to the scheduling policy, other specific parameters, like the chunk size, the receiver buffer size, the number of peers to probe and so on, matter too. For the BitTorrent file-sharing protocol it has been shown in [START_REF] Marciniak | Small is not always beautiful[END_REF] that small chunks are not always the best choice. In this paper, we propose to perform a similar study for the P2P live streaming. We focus on the epidemic exchange policy [BMM + 08], where a so-called scheme indicates which chunk a given peer should send to whom. We show that chunk size significantly impacts the performance. In fact, there exists a suitable range of chunk sizes, where the specific choice of the chunk size ultimately depends on the desired delay/loss trade-off. The impact of the number of peers to probe and the number of simultaneous upload connections is considered too.

Methodology

Epidemic diffusion schemes and their behavior have been extensively studied in literature. See [BMM + 08] for a detailed study and references therein. Here, we focus on particular scheme called random peer / latest useful chunk (rp/lu) : a given sender peer chooses a recipient peer uniformly at random among its neighbors, to which it sends its most recent chunk not own by the recipient peer. rp/lu has the advantage of being fairly simple, allowing us to focus on the impact of parameters like the chunk size. Moreover, it is efficient in terms of diffusion rate and delay, and it generates low overhead [BMM + 08].

In our analysis, we used an event based simulator developed by the Telecommunication Networks Group of Politecnico di Torino ‡ . The simulator has been modified to take network latencies, control overhead and parallel upload connections into account.

In our simulations, there are n = 1000 peers and a unique source S. The overlay network is an Erdös-Renyi graph G(n + 1, p). We set p to 0.05, so each node is connected to about 50 neighbors. The source injects a stream of rate s = 0.9Mbps, split into chunks of size c. All peers have the same upload capacity u i = 1.03Mbps, and unlimited download capacity (those are typical bandwidth values used in other papers, see for instance [START_REF] Picconi | Is there a future for mesh-based live video streaming[END_REF]).

Every peer periodically selects a subset m of its neighbors at random (the probe set), and requests their chunk maps. Based on the responses, the peer then transmits the latest useful chunks if any. The overhead required for control message exchange is taken into account (we assume that any control message has the same size c c = 1kb).

When transmitting, a peer fairly shares its upload bandwidth among at maximum m ≤ m peers (randomly selected if m < m), which are served in parallel. If a peer finds less than m suitable recipients because it does not have enough useful chunks, it can upload the chunks faster (since there are less than m active connections), but it may stay idle for a period of time after that (it needs to acquire new chunk maps from newly selected peers).

For chunk and chunk maps transmissions, both transmission and network latency are considered. The network latency values are taken from the Meridian project dataset § . We suppose that every peer has a buffer of size 300 (expressed in chunks) in order to avoid possible losses due to buffer shortage. This implies a buffer size proportional to the chunk size.

Chunk size and performance

When considering a streaming algorithm, a crucial performance metric is the rate/delay/overhead tradeoff achieved by that algorithm, which can be summarized by a (loss,delay,overhead) triplet. Loss can be defined as the probability to miss a chunk, while a possible definition for the diffusion delay is the time needed for a chunk to reach a peer on average. Lastly, we define the overhead as the difference between the bandwidth used by peers (throughput) and the actual data received (goodput).

Experiment

As a first experiment, we analyze loss, delay and overhead as a function of the chunk size. The results are shown in Figure 1, with m = m varying from 1 to 5.

Losses (Figure 1a) Two distinct phases may be observed :

-For large chunks (in our experiment, c greater than a few hundred kilobits), there are no losses.

-As the chunk size goes below a certain critical value, losses start to appear, roughly proportional to the logarithm of the chunk size (at least for the chunk range considered). This phenomenum can be explained as follow : the time between two consecutive chunks is c/s, and is therefore proportional to the chunk size c. When c increases (all other parameters being the same), more and more control messages per chunk can be exchanged between peers. This should achieve a proper diffusion, providing that enough bandwidth is available, because a sender peer has enough time to find a neighbor needing a given chunk. On the contrary, when c/s is too small, peers do not have enough time to exchange control messages, resulting in losses. Note that increasing the probe set m slightly improves the performance. Delay (Figure 1b) The diffusion delay is roughly proportional to the chunk size (this is why a linear x-scale is used), and increases with m. Note that the observed delays are proportional to the minimal broadcasting delay of a single chunk when RTT is neglected, which is 1c) The overhead is the difference between the throughput and goodput. Only the curves for m = 1 and m = 5 are displayed for readability. For very small chunks, we have a non-intuitive trend, where as c grows, the goodput increases and the throughput decreases (the overhead decreases faster than the goodput increases). This process slows down so that at some point the throughput increases again. For big enough chunks, the overhead becomes roughly constant (for a given m).

d min = m ln(n) ln(1+m) c s . Overhead (Figure
The intuition behind this is that for very small chunks, losses are high, which, as mentioned earlier, come from the fact that not enough control messages can be sent. Asymptotically, we may imagine that only one control message per sent chunk is issued, resulting in an overhead/goodput ratio of c c c . On the other hand, for chunk big enough, we may expect that a peer can send the required number of messages per sent chunk, that is proportional to the chunk characteristic time c/s. This would result in an overhead ratio proportional to c c s , and thus independent of c (but not of other parameters like the median RTT or m).

Suitable range for c

In light of the study above, we deduce an order of magnitude for suitable chunk size in epidemic live streaming. For the parameters considered here, c should be greater than 0.06 Mb (which corresponds to about 15 chunks per second) and smaller than 0.3 Mb (3 chunks per second) :

-to send the stream at more than 15 chunks per second is good for the delay (which stays roughly proportional to c), but results in both an increase in throughput and a decrease in goodput ; -goodput and throughput are stationary for c greater than 0.3 Mb : using bigger chunks only means longer delay ; -between these values, the choice of c results in a loss/delay trade-off : smaller delay with some losses or greater delay with no loss. The optimal value for c depends then on parameters that will not be discussed here, such as the codec used, the required QoS, etc. We observe that the suitable range for chunk size begins when the chunk characteristic time ( c s ) has the same order of magnitude than the median RTT, and ends an order of magnitude later. In other experiments, we scaled the RTT distribution used in order to observe the evolution of the range with the median RTT. The results showed that the range values are indeed roughly proportional to the median RTT.

Size of the probe set

In the results presented so far, we have assumed that the number of simultaneous exchange chunks, m , is identical to the size of the probe set m. We now consider the impact of probing more peers than the targeted number of simultaneous chunk to send. A larger probe set gives a peer a higher chance of finding receivers collisions (power of choices principle). However, it also increases overhead, and possibly delay.

Figure 2a plots the loss and delay trade-off for various m /m pairs. The scheme is rp/lu, the bandwidth is homogeneous and the chunk size is set to c = 0.15 Mb (middle of the suitable range). The figure shows that using m = m is not optimal, and having a larger probe set, m > m significantly reduces both delay and losses. The delay decreases from about 10 s for the m = m cases, to less than 4 s for m = 1 and m = 3, . . . , 6. Regarding the losses, there are some (m /m) couples for which no loss could be observed in our experiment : 1/3, . . . , 6, 2/5, 6, 3/5, 6, 4/6. This suggests that a consequence of using m < m is a shift of the suitable range for c. In order to verify that understanding, we now set c = 0.035 Mb, which is clearly below the suitable range observed in § 3.2 for m = m . The results are shown in Figure 2b.

We observe that no couple m /m can achieve lossless diffusion for such a small c, however the loss/delay trade-off is clearly improved for some couples : using m /m = 2/6, we get a delay of 1.7 s with a loss probability of about 0.02 %. This suggests that c = 0.035 Mb is definitively within the suitable range for m /m = 2/6. Also note how the relative efficiency of the different m /m values is impacted by the choice of c : for instance, 1/6, which is optimal for c = 0.15 Mb, performs rather poorly for c = 0.035 Mb.

As for the overhead (not displayed in Figure 2), we observed that for a given m, it stayed close to the one observed in Figure 1c going below the suitable range defined in § 3.2, and is thus higher for a small c. So reducing c with m < m can reduce the delay, but it requires more throughput.

Conclusion

We have investigated the dissemination parameters of P2P epidemic live streaming through extensive simulations. We have shown that the chunk size significantly impacts performance and should fall within a given range which is mostly determined by the median RTT of the network and the streamrate.

We have also shown the size of the probe set affect performance of diffusion schemes, and, in particular, a probe set larger than the actual number of concurrent connections may improve loss/delay performance by modifying the suitable chunk size ranges.

  FIGURE 1: Losses, delay and overhead as functions of the chunk size

  FIGURE 2: m /m loss/delay trade-off for two values of c

† This work is supported by the project NAPA-WiNe (FP7-ICT-2007-1, grant 214412).


Book review: Employment in the Informal

Sector in India Anurag Sengupta

Presidency University, Kolkata nrgsengupta@gmail.com

Anagh Chattopadhyay

Indian Statistical Institute, Kolkata anagh72@gmail.com* ABSTRACT-Ishita Mukhopadhyay,Employment in the Informal Sector in India. Springer, 2022, 144 pgs., ₹ 478.80 (Hard Cover). ISBN: 978-981-15-0840-0.

The book "Employment in the Informal Sector in India" is an excellent analysis of the so-called "Informal Employment" in a very lucid style with adequate graphical representations that come off as an easily understandable account for it in India. The book essentially talks about how the informal sector became a sizable share of the employment sector in India. Here the author adequately presents how, post-colonization, small-scale industry and petty production paved the way for employment in the informal sector. She also briefly speaks of the situation of these informal workers during the COVID-19 pandemic. The book also speaks of how the rural population shifted focus from agriculture-based occupations towards these unorganised sectors of unemployment. The author also incorporates the idea of changing labour legislations, with special emphasis on its effect on the regions around Kolkata. She also briefly speaks about the different other exogenous shocks that have impacted this sector significantly.

This book covers various aspects of informal employment in India, starting from its history and spreading to its contemporary role in the development of India today.

What particularly fascinates me in the second chapter is the portrayal of the spread of informal unemployment across the lives of the poor. Also remarkable is the very contemporary issue of COVID 19 and its effect on this sector. The book uses the enthralling expression "Footlooseness" to express the characteristics of this sector, which, in the words of the author, are "mobility and easy entry and exit." The book affirmatively emphasised the growth and spread of the informal sector and its importance even in times of pandemics, trade wars, and financial crises. Its collective presentation on the expansion numbers of this sector for various developed and developing countries is noteworthy. Not only that, it has an exemplary description of the spread of the informal sector in India. In the third chapter, the author tries to assert certain theories to explain the existence of the informal labour market.

The general equilibrium approach being one of them, which talks of the effectiveness of the fragmentation of the labour market and its primary role in the proper allocation of resources. The choice-based approach mentioned here spoke of rural-urban migration, and the most appealing thing here was the explanation asserted for the inclination towards the urban informal sector. The author asserted that the growth of the urban informal sector was due to the firm's need for effective and profit-maximizing solutions. Another important approach stated by the author, called the labour market segmentation approach, effectively explained the long-run stable relationship between the formal and the informal sectors. Moreover, the third chapter eloquently presents the different theories to explain the existence of informal employment and its nature. However, it seems the author couldn't completely clarify their existence and kept it open for interpretation for its readers to find out. The next thing the author seemed to have addressed were some alternative explanations for their existence as offered by different labour studies. Now a really remarkable observation was pointed out here: the increase in this informal sector wasn't in the developed economies but in the developing ones. In fact, this book also points out that the workers in this sector don't seem to be well off. Rather, it seems to be out of compulsion in most cases due to the lack of jobs in the formal, protected sector.

Most case studies in the book refer to developing countries, giving us strong evidence of the nature of the economies where this sector seemed to have thrived.

The author also pointed out why and how the ideas of liberalisation, globalization, and privatisation seemed to have influenced this sector deeply in its spread. The This book, as its title suggests, quite well portrays the informal sector of employment.

However, the effects of several crises that India had to undergo and their effects on this sector are not quite explicitly mentioned. Also, due to the lack of credible information, the absolute effect of COVID 19 on this sector does not seem to be very clear. But the book seems to be a major success in its expression of the subject matter. It touched history as well as the very birth of the term. The author also portrayed the disparity among the genders in this sector. The book had spoken about its close linkage with the formal sector. The author, however, did not explicitly mention the future of the employment situation in this sector. This should have been addressed. But all in all, the book is quite worth a read and is really insightful for people working in and doing research on this sector.

  biggest disadvantage mentioned in the book is the lack of adequate accounting in terms of job losses in this sector. The book also clarifies the dilemma, or at least attempts to clarify the dilemma, about the sector's definition. It also speaks about the evolution and changes in this sector. Also, what is more appealing is that the author also pointed out the different efforts made by the various organisations to transform this sector into a formal one. The informal sector seemed to have served as a "continuum" between the formal sector and the self-employed sector, as mentioned by the author. The book essentially highlights data on informal employment conducted in Kolkata, India. It pointed out that in the entire state of West Bengal, Kolkata had the highest number of employed workers in this sector. Most of them were generally rural immigrants working in different industries like watches, sports equipment, jewelry, etc.The challenging period of COVID actually became highly perilous for the migrant workers from various other states. What was more shocking was the large pool of people who migrated out of West Bengal for work and could be seen coming back to the state. The book also pointed out the case study of the 1992 communal riots and their implications for the urban informal sector. It also has adequate information about the economic viability and job losses due to the communal riots in different parts of Bengal, which might in itself serve as a research topic per se. Another really interesting topic of discussion in the book is about the women in the informal sector and how they suffer from marginality. Although the discussion is brief, it is essentially insightful too. It brushes through various topics such as the wage gap, working conditions, and several stereotypes that women are subjected to in this sector.