Physics:Attosecond physics

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Short description: Study of physics on quintillionth-second timescales
High harmonic generation in krypton. This technology is one of the most used techniques to generate attosecond bursts of light.

Attosecond physics, also known as attophysics, or more generally attosecond science, is a branch of physics that deals with light-matter interaction phenomena wherein attosecond (10−18 s) photon pulses are used to unravel dynamical processes in matter with unprecedented time resolution.

Attosecond science mainly employs pump–probe spectroscopic methods to investigate the physical process of interest. Due to the complexity of this field of study, it generally requires a synergistic interplay between state-of-the-art experimental setup and advanced theoretical tools to interpret the data collected from attosecond experiments.[1]

The main interests of attosecond physics are:

  1. Atomic physics: investigation of electron correlation effects, photo-emission delay and ionization tunneling.[2]
  2. Molecular physics and molecular chemistry: role of electronic motion in molecular excited states (e.g. charge-transfer processes), light-induced photo-fragmentation, and light-induced electron transfer processes.[3]
  3. Solid-state physics: investigation of exciton dynamics in advanced 2D materials, petahertz charge carrier motion in solids, spin dynamics in ferromagnetic materials.[4]

One of the primary goals of attosecond science is to provide advanced insights into the quantum dynamics of electrons in atoms, molecules and solids with the long-term challenge of achieving real-time control of the electron motion in matter.[5]

The advent of broadband solid-state titanium-doped sapphire based (Ti:Sa) lasers (1986),[6] chirped pulse amplification (CPA)[7] (1988), spectral broadening of high-energy pulses[8] (e.g. gas-filled hollow-core fiber via self-phase modulation) (1996), mirror-dispersion-controlled technology (chirped mirrors)[9] (1994), and carrier envelop offset stabilization[10] (2000) had enabled the creation of isolated-attosecond light pulses (generated by the non-linear process of high harmonic generation in a noble gas)[11][12] (2004, 2006), which have given birth to the field of attosecond science.[13]

The current world record for the shortest light-pulse generated by human technology is 43 as.[14]

In 2022, Anne L'Huillier, Paul Corkum, Ferenc Krausz were awarded with the Wolf prize in physics for their pioneering contributions to ultrafast laser science and attosecond physics. This was followed by the 2023 Nobel Prize in Physics, where L'Huillier, Krausz and Pierre Agostini were rewarded “for experimental methods that generate attosecond pulses of light for the study of electron dynamics in matter.”

Introduction

File:HHBreath.webm

Motivation

The natural time scale of electron motion in atoms, molecules, and solids is the attosecond (1 as= 10−18 s). This fact is a direct consequence of quantum mechanics.

Indeed, for simplicity, consider a quantum particle in superposition between ground-level, of energy [math]\displaystyle{ \epsilon_0 }[/math], and the first excited level, of energy [math]\displaystyle{ \epsilon_1 }[/math]:

[math]\displaystyle{ |\Psi\rangle=c_g|\psi_g\rangle+c_e|\psi_e\rangle }[/math]

with [math]\displaystyle{ c_e }[/math] and [math]\displaystyle{ c_g }[/math] chosen as the square roots of the quantum probability of observing the particle in the corresponding state.

[math]\displaystyle{ |\psi_g(t)\rangle= |0\rangle e^{-\frac{i\epsilon_0}{\hbar} t} \qquad |\psi_e(t)\rangle =|1\rangle e^{-\frac{i\epsilon_1}{\hbar}t} }[/math]

are the time-dependent ground [math]\displaystyle{ |0\rangle }[/math] and excited state [math]\displaystyle{ |1\rangle }[/math] respectively, with [math]\displaystyle{ \hbar }[/math] the reduced Planck constant.

The expectation value of a generic hermitian and symmetric operator,[15] [math]\displaystyle{ \hat{P} }[/math], can be written as [math]\displaystyle{ P(t)=\langle\Psi|\hat{P}|\Psi\rangle }[/math], as a consequence the time evolution of this observable is:

[math]\displaystyle{ P(t)=|c_g|^2\langle0|\hat{P}|0\rangle+|c_e|^2\langle1|\hat{P}|1\rangle+2c_ec_g\langle0|\hat{P}|1\rangle\cos\left(\frac{\epsilon_1-\epsilon_0}{\hbar}t \right) }[/math]

While the first two terms do not depend on time, the third, instead, does. This creates a dynamic for the observable [math]\displaystyle{ P(t) }[/math] with a characteristic time, [math]\displaystyle{ T_c }[/math], given by [math]\displaystyle{ T_c=\frac{2\pi \hbar}{\epsilon_1-\epsilon_0} }[/math].

Evolution of the angular probability density of the superposition between 1s and 2p state in hydrogen atoms. The color bar indicates the angular density (orientation of the wavepacket) as a function of the polar angle from 0 to π (x-axis), at which one can find the particle, and time (y-axis).

As a consequence, for energy levels in the range of [math]\displaystyle{ \epsilon_1-\epsilon_0 \approx }[/math] 10 eV, which is the typical electronic energy range in matter,[5] the characteristic time of the dynamics of any associated physical observable is approximately 400 as.

To measure the time evolution of [math]\displaystyle{ P(t) }[/math], one needs to use a controlled tool, or a process, with an even shorter time-duration that can interact with that dynamic.

This is the reason why attosecond light pulses are used to disclose the physics of ultra-fast phenomena in the few-femtosecond and attosecond time-domain.[16]

Generation of attosecond pulses

To generate a traveling pulse with an ultrashort time duration, two key elements are needed: bandwidth and central wavelength of the electromagnetic wave.[17]

From Fourier analysis, the more the available spectral bandwidth of a light pulse, the shorter, potentially, is its time duration.

There is, however, a lower-limit in the minimum duration exploitable for a given pulse central wavelength. This limit is the optical cycle.[18]

Indeed, for a pulse centered in the low-frequency region, e.g. infrared (IR) [math]\displaystyle{ \lambda= }[/math]800 nm, its minimum time duration is around [math]\displaystyle{ t_{pulse}=\frac{\lambda}{c}= }[/math]2.67 fs, where [math]\displaystyle{ c }[/math] is the speed of light; whereas, for a light field with central wavelength in the extreme ultraviolet (XUV) at [math]\displaystyle{ \lambda= }[/math]30 nm the minimum duration is around [math]\displaystyle{ t_{\rm pulse}= }[/math]100 as.[18]

Thus, a smaller time duration requires the use of shorter, and more energetic wavelength, even down to the soft-X-ray (SXR) region.

For this reason, standard techniques to create attosecond light pulses are based on radiation sources with broad spectral bandwidths and central wavelength located in the XUV-SXR range.[19]

The most common sources that fit these requirements are free-electron lasers (FEL) and high harmonic generation (HHG) setups.

Physical observables and experiments

Once an attosecond light source is available, one has to drive the pulse towards the sample of interest and, then, measure its dynamics.

The most suitable experimental observables to analyze the electron dynamics in matter are:

  • Angular asymmetry in the velocity distribution of molecular photo-fragment.[20]
  • Quantum yield of molecular photo-fragments.[21]
  • XUV-SXR spectrum transient absorption.[22]
  • XUV-SXR spectrum transient reflectivity.[23]
  • Photo-electron kinetic energy distribution.[2]

File:Pump-probe techniques in physics.ogv The general strategy is to use a pump-probe scheme to "image" through one of the aforementioned observables the ultra-fast dynamics occurring in the material under investigation.[1]

Few-femtosecond IR-XUV/SXR attosecond pulse pump-probe experiments

As an example, in a typical pump-probe experimental apparatus, an attosecond (XUV-SXR) pulse and an intense ([math]\displaystyle{ 10^{11}-10^{14} }[/math] W/cm2) low-frequency infrared pulse with a time duration of few to tens femtoseconds are collinearly focused on the studied sample.

At this point, by varying the delay of the attosecond pulse, which could be pump/probe depending on the experiment, with respect to the IR pulse (probe/pump), the desired physical observable is recorded.[24]

The subsequent challenge is to interpret the collected data and retrieve fundamental information on the hidden dynamics and quantum processes occurring in the sample. This can be achieved with advanced theoretical tools and numerical calculations.[25][26]

By exploiting this experimental scheme, several kinds of dynamics can be explored in atoms, molecules and solids; typically light-induced dynamics and out-of-equilibrium excited states within attosecond time-resolution.[20][21][23]

Quantum mechanics foundations

Attosecond physics typically deals with non-relativistic bounded particles and employs electromagnetic fields with a moderately high intensity ([math]\displaystyle{ 10^{11}-10^{14} }[/math] W/cm2).[27]

This fact allows to set up a discussion in a non-relativistic and semi-classical quantum mechanics environment for light-matter interaction.

Atoms

Resolution of time dependent Schrödinger equation in an electromagnetic field

The time evolution of a single electronic wave function in an atom, [math]\displaystyle{ |\psi(t)\rangle }[/math] is described by the Schrödinger equation (in atomic units):

[math]\displaystyle{ \hat{H}|\psi(t)\rangle=i\dfrac{\partial}{\partial t}|\psi(t)\rangle \quad (1.0) }[/math]

where the light-matter interaction Hamiltonian, [math]\displaystyle{ \hat{H} }[/math], can be expressed in the length gauge, within the dipole approximation, as:[28][29]

[math]\displaystyle{ \hat{H}=\frac{1}{2}\hat{\textbf{p}}^2+V_{C}+ \hat{\textbf{r}}\cdot\textbf{E}(t) }[/math]

where [math]\displaystyle{ V_C }[/math] is the Coulomb potential of the atomic species considered; [math]\displaystyle{ \hat{\textbf{p}}, \hat{\textbf{r}} }[/math] are the momentum and position operator, respectively; and [math]\displaystyle{ \textbf{E}(t) }[/math] is the total electric field evaluated in the neighbor of the atom.

The formal solution of the Schrödinger equation is given by the propagator formalism:

[math]\displaystyle{ |\psi(t)\rangle=e^{-i\int_{t_0}^{t}\hat{H}dt'}|\psi (t_0)\rangle \qquad(1.1) }[/math]

where [math]\displaystyle{ |\psi (t_0)\rangle }[/math], is the electron wave function at time [math]\displaystyle{ t=t_0 }[/math].

This exact solution cannot be used for almost any practical purpose.

However, it can be proved, using Dyson's equations[30][31] that the previous solution can also be written as:

[math]\displaystyle{ |\psi(t)\rangle=-i\int_{t_0}^{t}dt'\Big[ e^{-i\int_{t'}^{t}\hat{H}(t'')dt''}\hat{H}_I(t')e^{-i\int_{t_0}^{t'}\hat{H}_0(t'')dt''}|\psi(t_0)\rangle \Big]+e^{-i\int_{t_0}^{t}\hat{H}_0(t'')dt''}|\psi(t_0)\rangle \qquad(1.2) }[/math]

where,

[math]\displaystyle{ \hat{H}_0=\frac{1}{2}\hat{\textbf{p}}^2+V_{C} }[/math]

is the bounded Hamiltonian and

[math]\displaystyle{ \hat{H}_I=\hat{\textbf{r}}\cdot\textbf{E}(t) }[/math]

is the interaction Hamiltonian.

The formal solution of Eq. [math]\displaystyle{ (1.0) }[/math], which previously was simply written as Eq. [math]\displaystyle{ (1.1) }[/math], can now be regarded in Eq. [math]\displaystyle{ (1.2) }[/math] as a superposition of different quantum paths (or quantum trajectory), each one of them with a peculiar interaction time [math]\displaystyle{ t' }[/math] with the electric field.

In other words, each quantum path is characterized by three steps:

  1. An initial evolution without the electromagnetic field. This is described by the left-hand side [math]\displaystyle{ \hat{H}_0 }[/math] term in the integral.
  2. Then, a "kick" from the electromagnetic field, [math]\displaystyle{ \hat{H}_I(t') }[/math] that "excite" the electron. This event occurs at an arbitrary time that uni-vocally characterizes the quantum path [math]\displaystyle{ t' }[/math].
  3. A final evolution driven by both the field and the Coulomb potential, given by [math]\displaystyle{ \hat{H} }[/math].

In parallel, you also have a quantum path that do not perceive the field at all, this trajectory is indicated by the right-hand side term in Eq. [math]\displaystyle{ (1.2) }[/math].

This process is entirely time-reversible, i.e. can also occur in the opposite order.[30]

Equation [math]\displaystyle{ (1.2) }[/math] is not straightforward to handle. However, physicists use it as the starting point for numerical calculation, more advanced discussion or several approximations.[31][32]

For strong-field interaction problems, where ionization may occur, one can imagine to project Eq. [math]\displaystyle{ (1.2) }[/math] in a certain continuum state (unbounded state or free state) [math]\displaystyle{ |\textbf{p}\rangle }[/math], of momentum [math]\displaystyle{ \textbf{p} }[/math], so that:

[math]\displaystyle{ c_{\textbf{p}}(t)=\langle\textbf{p}|\psi(t)\rangle=-i\int_{t_0}^{t}dt'\langle \textbf{p}|e^{-i\int_{t'}^{t}\hat{H}(t'')dt''}\hat{H}_I(t')e^{-i\int_{t_0}^{t'}\hat{H}_0(t'')dt''}|{\psi(t_0)}\rangle +\langle \textbf{p} |e^{-i\int_{t_0}^{t}\hat{H}_0(t'')dt''}|\psi(t_0)\rangle \quad (1.3) }[/math]

where [math]\displaystyle{ |c_{\textbf{p}}(t)|^2 }[/math]is the probability amplitude to find at a certain time [math]\displaystyle{ t }[/math], the electron in the continuum states [math]\displaystyle{ |\textbf{p}\rangle }[/math].

If this probability amplitude is greater than zero, the electron is photoionized.

For the majority of application, the second term in [math]\displaystyle{ (1.3) }[/math] is not considered, and only the first one is used in discussions,[31] hence:

[math]\displaystyle{ a_{\textbf{p}}(t)=-i\int_{t_0}^{t}dt'\langle \textbf{p}|e^{-i\int_{t'}^{t}\hat{H}(t'')dt''}\hat{H}_I(t')e^{-i\int_{t_0}^{t'}\hat{H}_0(t'')dt''}|{\psi(t_0)}\rangle \quad (1.4) }[/math]

Equation [math]\displaystyle{ (1.4) }[/math] is also known as time reversed S-matrix amplitude[31] and it gives the probability of photoionization by a generic time-varying electric field.

Strong field approximation (SFA)

Strong field approximation (SFA), or Keldysh-Faisal-Reiss theory is a physical model, started in 1964 by the Russian physicist Keldysh,[33] is currently used to describe the behavior of atoms (and molecules) in intense laser fields.

SFA is the starting theory for discussing both high harmonic generation and attosecond pump-probe interaction with atoms.

The main assumption made in SFA is that the free-electron dynamics is dominated by the laser field, while the Coulomb potential is regarded as a negligible perturbation.[34]

This fact re-shapes equation [math]\displaystyle{ (1.4) }[/math] into:

[math]\displaystyle{ a_{\textbf{p}}^{SFA}(t)=-i\int_{t_0}^{t}dt'\langle \textbf{p}|e^{-i\int_{t'}^{t}\hat{H}_{V}(t'')dt''}\hat{H}_I(t')e^{-i\int_{t_0}^{t'}\hat{H}_0(t'')dt''}|{\psi(t_0)}\rangle \quad (1.4) }[/math]

where, [math]\displaystyle{ \hat{H}_V=\frac{1}{2}(\hat{\textbf{p}}+\textbf{A}(t))^2 }[/math] is the Volkov Hamiltonian, here expressed for simplicity in the velocity gauge,[35] with [math]\displaystyle{ \textbf{A}(t) }[/math], [math]\displaystyle{ \textbf{E}(t)=-\frac{\partial \textbf{A}(t)}{\partial t} }[/math], the electromagnetic vector potential.[36]

At this point, to keep the discussion at its basic level, lets consider an atom with a single energy level [math]\displaystyle{ |0\rangle }[/math], ionization energy [math]\displaystyle{ I_P }[/math] and populated by a single electron (single active electron approximation).

We can consider the initial time of the wave function dynamics as [math]\displaystyle{ t_0=-\infty }[/math], and we can assume that initially the electron is in the atomic ground state [math]\displaystyle{ |0\rangle }[/math].

So that,

[math]\displaystyle{ \hat{H}_0|0\rangle=-I_P|0\rangle }[/math] and [math]\displaystyle{ |\psi(t)\rangle=e^{-i\int_{-\infty}^{t'}\hat{H}_0dt}|0\rangle=e^{I_Pt'}|0\rangle }[/math]

Moreover, we can regard the continuum states as plane-wave functions state, [math]\displaystyle{ \langle\textbf{r}|\textbf{p}\rangle=(2\pi)^{-\frac{3}{2}}e^{i\textbf{p}\cdot{\textbf{r}}} }[/math].

This is a rather simplified assumption, a more reasonable choice would have been to use as continuum state the exact atom scattering states.[37]

The time evolution of simple plane-wave states with the Volkov Hamiltonian is given by:

[math]\displaystyle{ \langle\textbf{p}|e^{-i\int_{t'}^{t}\hat{H}_{V}(t'')dt''}=\langle\textbf{p}+\textbf{A}(t)|e^{-i\int_{t'}^{t}(\textbf{p}+\textbf{A}(t''))^2dt''} }[/math]

here for consistency with Eq. [math]\displaystyle{ (1.4) }[/math] the evolution has already been properly converted into the length gauge.[38]

As a consequence, the final momentum distribution of a single electron in a single-level atom, with ionization potential [math]\displaystyle{ I_P }[/math], is expressed as:

[math]\displaystyle{ a_{\textbf{p}}(t)^{SFA}=-i\int_{-\infty}^{t} \textbf{E}(t')\cdot \textbf{d}[\textbf{p}+\textbf{A}(t')] e^{+i(I_Pt'-S(t,t'))}dt' \quad (1.5) }[/math]

where,

[math]\displaystyle{ \textbf{d}[\textbf{p}+\textbf{A}(t')]=\langle\textbf{p}+\textbf{A}(t')|\hat{\textbf{r}}|0 \rangle }[/math]

is the dipole expectation value (or transition dipole moment), and

[math]\displaystyle{ S(t,t')=\int_{t'}^{t}\frac{1}{2}(\textbf{p}+\textbf{A}(t''))^2dt'' }[/math]

is the semiclassical action.

The result of Eq. [math]\displaystyle{ (1.5) }[/math] is the basic tool to understand phenomena like:

  • The high harmonic generation process,[39] which is typically the result of strong field interaction of noble gases with an intense low-frequency pulse,
  • Attosecond pump-probe experiments with simple atoms.[40]
  • The debate on tunneling time.[41][42]
Weak attosecond pulse-strong-IR-fields-atoms interactions

Attosecond pump-probe experiments with simple atoms is a fundamental tool to measure the time duration of an attosecond pulse[43] and to explore several quantum proprieties of matter.[40]

Scheme of a strong IR field and a delayed attosecond XUV pulse interacting with a single electron in a single-level atom. The XUV can ionize the electron, which "jumps" in the continuum by direct ionization (blue path in the figure). The IR pulse, later, "streaks" up and down in energy the photo-electron. After the interaction, the electron has final energy which can be subsequently detected and measured (e.g. time-of-flight apparatus). The multi-photon ionization process (red path in the figure) is also possible, but, since it is relevant in different energetic region, it can be disregarded.

This kind of experiments can be easily described within strong field approximation by exploiting the results of Eq. [math]\displaystyle{ (1.5) }[/math], as discussed below.

As a simple model, consider the interaction between a single active electron in a single-level atom and two fields: an intense femtosecond infrared (IR) pulse ([math]\displaystyle{ (\textbf{E}_{IR}(t),\textbf{A}_{IR}(t)) }[/math],

and a weak attosecond pulse (centered in the extreme ultraviolet (XUV) region) [math]\displaystyle{ (\textbf{E}_{XUV}(t),\textbf{A}_{XUV}(t)) }[/math].

Then, by substituting these fields to [math]\displaystyle{ (1.5) }[/math] it results

[math]\displaystyle{ a_{\textbf{p}}(t)=-i\int_{-\infty}^{t} (\textbf{E}_{XUV}(t')+\textbf{E}_{IR}(t'))\cdot \textbf{d}[\textbf{p}+\textbf{A}_{XUV}(t')+\textbf{A}_{IR}(t')] e^{+i(I_Pt'-S(t,t'))}dt' \quad (1.6) }[/math]

with

[math]\displaystyle{ S(t,t')=\int_{t'}^{t}\frac{1}{2}(\textbf{p}+\textbf{A}_{IR}(t'')+\textbf{A}_{XUV}(t''))^2dt'' }[/math].

At this point, we can divide Eq. [math]\displaystyle{ (1.6) }[/math] in two contributions: direct ionization and strong field ionization (multiphoton regime), respectively.

Typically, these two terms are relevant in different energetic regions of the continuum.

Consequently, for typical experimental condition, the latter process is disregarded, and only direct ionization from the attosecond pulse is considered.[31]

Then, since the attosecond pulse is weaker than the infrared one, it holds [math]\displaystyle{ \textbf{A}_{IR}(t)\gt \gt \textbf{A}_{XUV}(t) }[/math]. Thus, [math]\displaystyle{ \textbf{A}_{XUV}(t) }[/math] is typically neglected in Eq. [math]\displaystyle{ (1.6) }[/math].

In addition to that, we can re-write the attosecond pulse as a delayed function with respect to the IR field, [math]\displaystyle{ [\textbf{A}_{IR}(t),\textbf{E}_{XUV}(t-\tau)] }[/math].

Therefore, the probability distribution, [math]\displaystyle{ |a_{\textbf{p}}(\tau)|^2 }[/math], of finding an electron ionized in the continuum with momentum [math]\displaystyle{ \textbf{p} }[/math], after the interaction has occurred (at [math]\displaystyle{ t=\infty }[/math]), in a pump-probe experiments,

with an intense IR pulse and a delayed-attosecond XUV pulse, is given by:

[math]\displaystyle{ a_{\textbf{p}}(\tau)=-i\int_{-\infty}^{\infty} \textbf{E}_{XUV}(t-\tau)\cdot \textbf{d}[\textbf{p}+\textbf{A}_{IR}(t)] e^{+i(I_Pt-S(t))}dt \quad (1.7) }[/math]

with

[math]\displaystyle{ S(t)=\frac{1}{2}|\textbf{p}|^2t+\int_{t}^{\infty}(\textbf{p}\cdot\textbf{A}_{IR}(t')+\frac{1}{2}|\textbf{A}_{IR}(t')|^2)dt' }[/math]

Equation [math]\displaystyle{ (1.7) }[/math] describes the photoionization phenomenon of two-color interaction (XUV-IR) with a single-level atom and single active electron.

This peculiar result can be regarded as a quantum interference process between all the possible ionization paths, started by a delayed XUV attosecond pulse, with a following motion in the continuum states driven by a strong IR field.[31]

The resulting 2D photo-electron (momentum, or equivalently energy, vs delay) distribution is called streaking trace.[44]

Techniques

Here are listed and discussed some of the most common techniques and approaches pursued in attosecond research centers.

Metrology with photo-electron spectroscopy (FROG-CRAB)

Simulation of a streaking trace in Neon. The attosecond pulse duration is 350 as, with central wavelength at the 33 harmonics of an 800 nm laser. The 800 nm pulse, which has the role of streaking up and down the photoelectron trace, has a duration of 7 fs with a peak intensity of 5 TW/cm2.[45]

A daily challenge in attosecond science is to characterize the temporal proprieties of the attosecond pulses used in any pump-probe experiments with atoms, molecules or solids.

The most used technique is based on the frequency-resolved optical gating for a complete reconstruction of attosecond bursts (FROG-CRAB).[43]

The main advantage of this technique is that it allows to exploit the corroborated frequency-resolved optical gating (FROG) technique,[46] developed in 1991 for picosecond-femtosecond pulse characterization, to the attosecond field.

Complete reconstruction of attosecond bursts (CRAB) is an extension of FROG and it is based on the same idea for the field reconstruction.

In other words, FROG-CRAB is based on the conversion of an attosecond pulse into an electron wave-packet that is freed in the continuum by atomic photoionization, as already described with Eq.[math]\displaystyle{ (1.7) }[/math].

The role of the low-frequency driving laser pulse( e.g. infra-red pulse) is to behave as gate for the temporal measurement.

Then, by exploring different delays between the low-frequency and the attosecond pulse a streaking trace (or streaking spectrogram) can be obtained.[44]

This 2D-spectrogram is later analyzed by a reconstruction algorithm with the goal of retrieving both the attosecond pulse and the IR pulse, with no need of a prior knowledge on any of them.

However, as Eq. [math]\displaystyle{ (1.7) }[/math] pinpoints, the intrinsic limits of this technique is the knowledge on atomic dipole proprieties, in particular on the atomic dipole quantum phase.[40][47]

The reconstruction of both the low-frequency field and the attosecond pulse from a streaking trace is typically achieved through iterative algorithms, such as:

  • Principal component generalized projections algorithm (PCGPA).[48]
  • Volkov transform generalized projection algorithm (VTGPA).[49]
  • extended ptychographic iterative engine (ePIE).[50]

See also

References

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  6. ↑ "Spectroscopic and laser characteristics of Ti:Al_2O_3". Journal of the Optical Society of America B 3 (1): 125. January 1986. doi:10.1364/josab.3.000125. ISSN 0740-3224. Bibcode: 1986JOSAB...3..125M. 
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  8. ↑ "Generation of high energy 10 fs pulses by a new pulse compression technique". Applied Physics Letters 68 (20): 2793–2795. 1996-05-13. doi:10.1063/1.116609. ISSN 0003-6951. Bibcode: 1996ApPhL..68.2793N. 
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  13. ↑ "The birth of attosecond physics and its coming of age". Physica Scripta 91 (6): 063011. 2016-05-25. doi:10.1088/0031-8949/91/6/063011. ISSN 0031-8949. Bibcode: 2016PhyS...91f3011K. 
  14. ↑ "Streaking of 43-attosecond soft-X-ray pulses generated by a passively CEP-stable mid-infrared driver" (in EN). Optics Express 25 (22): 27506–27518. October 2017. doi:10.1364/OE.25.027506. PMID 29092222. Bibcode: 2017OExpr..2527506G. 
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  19. ↑ "Attosecond soft X-ray high harmonic generation". Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences 377 (2145): 20170468. May 2019. doi:10.1098/rsta.2017.0468. PMID 30929634. Bibcode: 2019RSPTA.37770468J. 
  20. ↑ 20.0 20.1 "Electron localization following attosecond molecular photoionization". Nature 465 (7299): 763–6. June 2010. doi:10.1038/nature09084. PMID 20535207. Bibcode: 2010Natur.465..763S. https://portal.research.lu.se/ws/files/1648850/2063932.pdf. 
  21. ↑ 21.0 21.1 "Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses". Science 346 (6207): 336–9. October 2014. doi:10.1126/science.1254061. PMID 25324385. Bibcode: 2014Sci...346..336C. 
  22. ↑ "Direct mapping of curve-crossing dynamics in IBr by attosecond transient absorption spectroscopy". Science 365 (6448): 79–83. July 2019. doi:10.1126/science.aax0076. PMID 31273121. Bibcode: 2019Sci...365...79K. https://yorkspace.library.yorku.ca/xmlui/handle/10315/37193. 
  23. ↑ 23.0 23.1 "Unravelling the intertwined atomic and bulk nature of localised excitons by attosecond spectroscopy". Nature Communications 12 (1): 1021. February 2021. doi:10.1038/s41467-021-21345-7. PMID 33589638. Bibcode: 2021NatCo..12.1021L. 
  24. ↑ "Novel beamline for attosecond transient reflection spectroscopy in a sequential two-foci geometry". The Review of Scientific Instruments 91 (5): 053002. May 2020. doi:10.1063/5.0005932. PMID 32486725. Bibcode: 2020RScI...91e3002L. 
  25. ↑ "The quantum chemistry of attosecond molecular science" (in en). WIREs Computational Molecular Science 10 (1): e1430. 2020. doi:10.1002/wcms.1430. ISSN 1759-0884. 
  26. ↑ "First-principles calculations for attosecond electron dynamics in solids". Computational Materials Science 194: 110274. 2021. doi:10.1016/j.commatsci.2020.110274. ISSN 0927-0256. 
  27. ↑ "ICAN: The Next Laser Powerhouse". https://www.osa-opn.org/home/articles/volume_24/may_2013/features/ican_the_next_laser_powerhouse/. 
  28. ↑ "Foundations of the Strong-Field Approximation" (in en). Progress in Ultrafast Intense Laser Science III. Springer Series in Chemical Physics. 89. Berlin, Heidelberg: Springer. 2008. pp. 1–31. doi:10.1007/978-3-540-73794-0_1. ISBN 978-3-540-73794-0. 
  29. ↑ "Ionization in intense laser fields beyond the electric dipole approximation: concepts, methods, achievements and future directions". Journal of Physics B: Atomic, Molecular and Optical Physics 54 (9): 094001. 2021-05-05. doi:10.1088/1361-6455/abf731. ISSN 0953-4075. 
  30. ↑ 30.0 30.1 "Anatomy of strong field ionization". Journal of Modern Optics 52 (2–3): 165–184. 2005-01-20. doi:10.1080/0950034042000275360. ISSN 0950-0340. Bibcode: 2005JMOp...52..165I. 
  31. ↑ 31.0 31.1 31.2 31.3 31.4 31.5 (in en) High Power Laser-Matter Interaction. Springer Tracts in Modern Physics. 238. Berlin Heidelberg: Springer-Verlag. 2010. doi:10.1007/978-3-540-46065-7. ISBN 978-3-540-50669-0. Bibcode: 2010hpli.book.....M. https://www.springer.com/gp/book/9783540506690. 
  32. ↑ "Gauge-invariant intense-field approximations to all orders". Journal of Physics B: Atomic, Molecular and Optical Physics 40 (7): F145–F155. 2007-03-15. doi:10.1088/0953-4075/40/7/f02. ISSN 0953-4075. http://dx.doi.org/10.1088/0953-4075/40/7/f02. 
  33. ↑ V Popruzhenko, S (2014-10-08). "Keldysh theory of strong field ionization: history, applications, difficulties and perspectives". Journal of Physics B: Atomic, Molecular and Optical Physics 47 (20): 204001. doi:10.1088/0953-4075/47/20/204001. ISSN 0953-4075. Bibcode: 2014JPhB...47t4001P. https://iopscience.iop.org/article/10.1088/0953-4075/47/20/204001. 
  34. ↑ "Symphony on strong field approximation". Reports on Progress in Physics 82 (11): 116001. November 2019. doi:10.1088/1361-6633/ab2bb1. PMID 31226696. Bibcode: 2019RPPh...82k6001A. 
  35. ↑ University of Kassel. "Physical phenomena in laser-matter interaction". https://www.pks.mpg.de/~lmi07/talk1_lein.pdf. 
  36. ↑ Classical electrodynamics (3 ed.). New York: Wiley. 1999. ISBN 0-471-30932-X. OCLC 38073290. https://www.worldcat.org/oclc/38073290. 
  37. ↑ "Atom-Volkov strong-field approximation for above-threshold ionization". Physical Review A 99 (4): 043411. 2019-04-10. doi:10.1103/physreva.99.043411. ISSN 2469-9926. Bibcode: 2019PhRvA..99d3411M. http://dx.doi.org/10.1103/physreva.99.043411. 
  38. ↑ Bechler A, Ślȩczka M (2009-12-25). "Gauge invariance of the strong field approximation". arXiv:0912.4966 [physics.atom-ph].
  39. ↑ "Intense few-cycle laser fields: Frontiers of nonlinear optics". Reviews of Modern Physics 72 (2): 545–591. 2000-04-01. doi:10.1103/RevModPhys.72.545. ISSN 0034-6861. Bibcode: 2000RvMP...72..545B. 
  40. ↑ 40.0 40.1 40.2 "Attosecond streaking enables the measurement of quantum phase". Physical Review Letters 105 (7): 073001. August 2010. doi:10.1103/PhysRevLett.105.073001. PMID 20868037. Bibcode: 2010PhRvL.105g3001Y. 
  41. ↑ "Attosecond Ionization Dynamics and Time Delays" (in EN). CLEO: 2015 (2015), Paper FTh3C.1 (Optical Society of America): FTh3C.1. 2015-05-10. doi:10.1364/CLEO_QELS.2015.FTh3C.1. ISBN 978-1-55752-968-8. https://www.osapublishing.org/abstract.cfm?uri=CLEO_QELS-2015-FTh3C.1. 
  42. ↑ "The attoclock and the tunneling time debate". Journal of Physics B: Atomic, Molecular and Optical Physics 53 (7): 072001. 2020-03-06. doi:10.1088/1361-6455/ab6b3b. ISSN 0953-4075. Bibcode: 2020JPhB...53g2001K. 
  43. ↑ 43.0 43.1 "Frequency-resolved optical gating for complete reconstruction of attosecond bursts". Physical Review A 71 (1): 011401. 2005-01-27. doi:10.1103/PhysRevA.71.011401. Bibcode: 2005PhRvA..71a1401M. 
  44. ↑ 44.0 44.1 "Attosecond streak camera". Physical Review Letters 88 (17): 173903. April 2002. doi:10.1103/PhysRevLett.88.173903. PMID 12005756. Bibcode: 2002PhRvL..88q3903I. https://nrc-publications.canada.ca/eng/view/accepted/?id=bc8b5b86-6d18-4ce3-bc22-a04ef044bb3d. 
  45. ↑ Vismarra, F.; Borrego-Varillas, R.; Wu, Y.; Mocci, D.; Nisoli, M.; Lucchini, M. (2022). "Ensemble effects on the reconstruction of attosecond pulses and photoemission time delays". Journal of Physics: Photonics 4 (3): 034006. doi:10.1088/2515-7647/ac7991. Bibcode: 2022JPhP....4c4006V. https://iopscience.iop.org/article/10.1088/2515-7647/ac7991/meta. 
  46. ↑ "FROG". Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses. Boston, MA: Springer US. 2003. pp. 101–115. doi:10.1007/978-1-4615-1181-6_5. ISBN 978-1-4613-5432-1. 
  47. ↑ "A new method for accurate retrieval of atomic dipole phase or photoionization group delay in attosecond photoelectron streaking experiments". Journal of Optics 19 (11): 114009. 2017-10-23. doi:10.1088/2040-8986/aa8fb6. ISSN 2040-8978. Bibcode: 2017JOpt...19k4009Z. http://dx.doi.org/10.1088/2040-8986/aa8fb6. 
  48. ↑ "Principal components generalized projections: a review [Invited"] (in EN). JOSA B 25 (6): A120–A132. 2008-06-01. doi:10.1364/JOSAB.25.00A120. ISSN 1520-8540. Bibcode: 2008JOSAB..25A.120K. https://www.osapublishing.org/josab/abstract.cfm?uri=josab-25-6-A120. 
  49. ↑ "Volkov transform generalized projection algorithm for attosecond pulse characterization". New Journal of Physics 18 (7): 073009. 2016-07-06. doi:10.1088/1367-2630/18/7/073009. ISSN 1367-2630. Bibcode: 2016NJPh...18g3009K. http://dx.doi.org/10.1088/1367-2630/18/7/073009. 
  50. ↑ "Ptychographic reconstruction of attosecond pulses" (in EN). Optics Express 23 (23): 29502–13. November 2015. doi:10.1364/OE.23.029502. PMID 26698434. Bibcode: 2015OExpr..2329502L. 

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