Perturbative Harmonic Generation

Perturbative Harmonic Generation is a process whereby laser light of frequency ω and photon energy ħω can be used to generate new frequencies of light. The newly generated frequencies are integer multiples nħω of the original light's frequency. This process was first discovered in 1961 by
Franken et al., using a ruby laser

Ruby laser

A ruby laser is a solid-state laser that uses a synthetic ruby crystal as its gain medium. The first working laser was a ruby laser made by Theodore H. "Ted" Maiman at Hughes Research Laboratories on May 16, 1960....

, with crystalline quartz

Quartz

Quartz is the second-most-abundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,...

 as the nonlinear medium

Nonlinear optics

Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...

.

Harmonic

Harmonic

A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

 generation in dielectric solids

Dielectric

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

 is well understood and
extensively used in modern laser physics

Laser science

Laser science or laser physics is a branch of optics that describes the theory and practice of lasers.Laser science is principally concerned with quantum electronics, laser construction, optical cavity design, the physics of producing a population inversion in laser media, and the temporal...

 (see second harmonic generation

Second harmonic generation

An optical frequency multiplier is a nonlinear optical device, in which photons interacting with a nonlinear material are effectively "combined" to form new photons with greater energy, and thus higher frequency...

). In 1967 New et al. observed the first third harmonic
generation in a gas. In monatomic gases it is only
possible to produce odd numbered harmonics for reasons of
symmetry. Harmonic generation in the perturbative (weak field)
regime is characterised by rapidly decreasing efficiency with
increasing harmonic order and harmonics up to the 11th order have
been observed under these conditions
. This behaviour can be
understood by considering an atom absorbing n photons
then emitting a single high energy photon. The probability of
absorbing n photons decreases as n increases, explaining the rapid
decrease in the initial harmonic intensities.

High Harmonic Generation (HHG)

The first High Harmonic Generation (HHG) was observed in 1977 in interaction of intense CO₂ laser pulses with plasma

Plasma (physics)

In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...

 generated from solid targets. HHG in gases, far more widespread in application today, was first observed by McPherson and colleagues in 1987, and later by Ferray et al. in 1988, with surprising results: the high
harmonics were found to decrease in intensity at low orders, as expected,
but then were observed to form a plateau, with the intensity of the harmonics
remaining approximately constant over many orders.
Plateau harmonics spanning hundreds of eV have been measured which
extend into the soft x-ray

X-ray

X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...

 regime. This plateau ends abruptly at a position called
the High Harmonic Cut-off.

Properties of High Harmonics

High harmonics have a number of interesting properties. They are a
tunable table-top source of XUV/Soft X-rays, synchronised with the
driving laser and produced with the same repetition rate. The
harmonic cut-off varies linearly with increasing laser intensity up
until the saturation intensity I_sat where harmonic generation
stops. The saturation intensity can be
increased by changing the atomic species to lighter noble gas

Noble gas

The noble gases are a group of chemical elements with very similar properties: under standard conditions, they are all odorless, colorless, monatomic gases, with very low chemical reactivity...

es
but these have a lower conversion efficiency so there is a balance
to be found depending on the photon energies required.

High harmonic generation strongly depends on the driving laser
field and as a result the harmonics have similar temporal and
spatial coherence properties. High harmonics are often generated
with pulse durations shorter than that of the driving laser. This
is due to phase matching and ionization

Ionization

Ionization is the process of converting an atom or molecule into an ion by adding or removing charged particles such as electrons or other ions. This is often confused with dissociation. A substance may dissociate without necessarily producing ions. As an example, the molecules of table sugar...

. Often harmonics are only
produced in a very small temporal window when the phase matching
condition is met. Depletion of
the generating media due to ionization also means that harmonic
generation is mainly confined to the leading edge of the driving
pulse.

High harmonics are emitted co-linearly with the driving laser and
can have a very tight angular confinement, sometimes with less
divergence than that of the fundamental field and near Gaussian
beam profiles.

Semi-classical approach to describe HHG

The maximum photon energy producible with high harmonic generation is given by the cut-off of the harmonic plateau. This can be calculated classically

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

 by examining the
maximum energy the ionized electron can gain in the electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

 of the laser. The cut-off energy is given by,



where U_p is the ponderomotive energy

Ponderomotive energy

In strong field laser physics, the term Ponderomotive Energy refers to the cycle averaged quiver energy of a free electron in an E-field as given by,U_p=e^2E_a^2/4m\omega_0^2...

 from the laser field and I_p is the
ionization potential

Ionization potential

The ionization energy of a chemical species, i.e. an atom or molecule, is the energy required to remove an electron from the species to a practically infinite distance. Large atoms or molecules have a low ionization energy, while small molecules tend to have higher ionization energies.The property...

.

This derivation of the cut-off energy is derived from a
semi-classical calculation. The electron is initially treated quantum
mechanically as it tunnel ionizes

Tunnel ionization

Tunnel ionization is a process in which electrons in an atom pass through the potential barrier and escape from the atom . In an intense electric field, the potential barrier of an atom is distorted drastically...

 from the parent atom, but then its subsequent dynamics
are treated classically. The electron is assumed to be born into
the vacuum

Vacuum

In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

 with zero initial velocity, and to be subsequently accelerated
by the laser beam's electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

.

Half an optical cycle after ionization, the electron will reverse direction as the
electric field changes, and will accelerate back towards the parent nucleus. Upon returning to the parent nucleus it can then emit bremsstrahlung

Bremsstrahlung

Bremsstrahlung is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into a photon because energy is conserved. The term is...

-like radiation during a recombination
process with the atom as it returns to its ground state

Stationary state

In quantum mechanics, a stationary state is an eigenvector of the Hamiltonian, implying the probability density associated with the wavefunction is independent of time . This corresponds to a quantum state with a single definite energy...

.
This description has become known as the recollisional model of high harmonic generation
.

Some interesting limits on the HHG process which are explained by this model show that HHG will only occur if
the driving laser field is linearly polarised. Ellipticity on the laser beam

Laser

A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

causes the returning electron to miss the parent nucleus. Quantum
mechanically, the overlap of the returning electron wavepacket
with the nuclear wavepacket is reduced. This has been observed
experimentally, where the intensity of harmonics decreases rapidly
with increasing ellipticity. Another effect which limits the
intensity of the driving laser is the Lorentz force

Lorentz force

In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...

. At
intensities above 10¹⁶ Wcm⁻² the magnetic component of
the laser pulse, which is ignored in weak field optics, can become
strong enough to deflect the returning electron. This will cause
it to 'miss' the parent nucleus and hence prevent HHG.