Laser - matter interactionsBoris Lukiyanchuk

Singapore, 12 November 2018

Lecture 4.

Laser - matter interactions

Nonresonant processes Resonant processes

Physical Processes

Chemical Processes

Vapor PlasmaProcesses

Plasmonics Photonics

NonlinearOptics

Resonant Chemistry

Lecture 4. Nonlinear optics with inertial nonlinearities

If = const then Maxwell equationa, e.g. wave equation, 𝚫𝑬 + 𝒌𝟎𝟐 𝑬 = 0, are linear.

In linear optics, fields follow the superposition principle.

Why = const ?

Hendrik Lorentz

1853 – 1928Paul Drude

1863 – 1906

Drude–Lorentz

model 1905Drude

model

1900

1902

Arnold

Sommerfeld

1868 – 1951

Drude–Sommerfeld

model 1937

1967

Hans Bethe

1906 – 2005

𝒎(𝒅𝟐𝒙

𝒅𝒕𝟐+ 𝜸

𝒅𝒙

𝒅𝒕+ 𝝎𝟎

𝟐 𝒙) = 𝑒𝐸Equation for electron motion E( ) exp( )t i t

𝒙 = −𝒆/𝒎

𝝎𝟐−𝝎𝟎𝟐+𝒊𝜸𝝎

𝑬, D = E, = 1+ 4 P , P = 𝑵𝒆ex

= 1 −𝝎𝒑𝟐

𝝎𝟐−𝝎𝟎𝟐+𝒊𝜸𝝎

where the plasma frequency 𝝎𝒑𝟐 =

4 𝒆𝟐 𝑵𝒆

𝒎

For free electrons 𝝎𝟎 → 𝟎

Drude model and

improvements

The main source of nonlinearity is anharmonicity. Linear term 𝝎𝟎𝟐 𝒙 is force from parabolic

potential . 𝑭 ∝ −𝒅𝑼

𝒅𝒙where 𝑼 ∝ 𝒙𝟐. Anharmonicity yields contribution of 𝒙𝟑 in potential

and nonlinear term in polarizability. In general nonlinear optical phenomena can be described by a Taylor series expansion of the dielectric polarization density

P(t) = 𝝌 𝟏 𝑬 𝒕 + 𝝌 𝟐 𝑬𝟐 𝒕 + 𝝌 𝟑 𝑬𝟑 𝒕 + … 𝑬 ~ 𝑬𝒂𝒕 ≅ 𝟏𝟎𝟖 − 𝟏𝟎𝟗 𝑽/𝒄𝒎when

If we consider only a second-order nonlinearity and assume field E(t)

is made up of two components at frequencies ω1 and ω2

𝑷𝑵𝑳 = 𝝌 𝟐 𝑬𝟐 𝒕

The other sources of nonlinearity:

1) Effect of dissipative term 𝜸𝒅𝒙

𝒅𝒕. Here collision frequency γ can be expressed as ,st nv

where is momentum-transport cross section, σ (v, θ) is

differential cross section, 𝒏𝒔 is average concentration of scatters. Thus, 𝜸𝒅𝒙

𝒅𝒕~ 𝒆− 𝒊 𝟐𝝎 𝒕.

0

sin)cos1(),(2 dvt

2) Lorentz force F = e v H ~ 𝒆− 𝒊 𝟐𝝎 𝒕

3) The Schwinger limit, where the vacuum itself is expected to become nonlinear

Julian Schwinger

1918 – 1994

1965

A Feynman diagram for photon–photon scattering; one

photon scatters from the transient vacuum charge

fluctuations of the other.

J. Schwinger, On Gauge Invariance and Vacuum Polarization, Phys. Rev. 82, pp. 664–679 (1951).

where "c.c." stands for

complex conjugate.

𝑷𝑵𝑳 = 𝝌 𝟐 /4 [𝑬𝟏𝟐 𝒆− 𝒊 𝝎𝟏 𝒕+ 𝑬𝟐

𝟐 𝒆− 𝒊 𝟐𝝎𝟐 𝒕

+ 2 𝑬𝟏 𝑬𝟐 𝒆− 𝒊 (𝝎𝟏+𝝎𝟐)𝒕

+ 2 𝑬𝟏 𝑬𝟐∗ 𝒆− 𝒊 (𝝎𝟏−𝝎𝟐)𝒕

2ω1, 2ω2, Second harmonic generation

Sum-frequency generation𝝎𝟏 +𝝎𝟐

Difference-frequency generation𝝎𝟏 −𝝎𝟐

0 Optical rectification

Peter A. Franken

1928 –1999

Bloembergen

1920 – 2017

Discovery of second harmonic generation in 1961

The theoretical basis

were first described

in monograph

"Nonlinear Optics“

(1965)1981

Arthur Schawlow

1921 – 1999

Co-inventor of maser

with Charles Townes

1981

Phase matchingConstructive interference, and therefore a high-intensity

𝛚𝟑 field, will occur only if phase-matching condition

is fulfilled𝝌 𝟑 nonlinear optics effects

Optical Kerr effect, intensity-dependent refractive index (a 𝝌 𝟑 effect).

•Self-focusing, an effect due to the optical Kerr effect (and possibly higher-order nonlinearities)

caused by the spatial variation in the intensity creating a spatial variation in the refractive index.

•Kerr-lens mode locking (KLM), the use of self-focusing as a mechanism to mode-lock laser.

•Self-phase modulation (SPM), an effect due to the optical Kerr effect (and possibly higher-order

nonlinearities) caused by the temporal variation in the intensity creating a temporal variation in the

refractive index.

•Optical solitons, an equilibrium solution for either an optical pulse (temporal soliton) or spatial

mode (spatial soliton) that does not change during propagation due to a balance

between dispersion and the Kerr effect (e.g. self-phase modulation for temporal and self-focusing for

spatial solitons).

Third-harmonic generation (THG), generation of light with a tripled frequency (one-third the wavelength),

three photons are destroyed, creating a single photon at three times the frequency.

Kerr effect (discovered in 1875)

John Kerr

1824 – 1907

In the electro-optic Kerr effect, a change in refractive index is proportional to

the square of the electric field, ∆𝑛 ∝ 𝐸2 . Later, in 1893 Friedrich

Pockels discovered a linear effect, ∆𝑛 ∝ E (Pockels effect). Kerr also

demonstrated in 1877 a linear effect for magnetic field, ∆𝑛 ∝ H (magneto-

optic Kerr effect).

A.K. Mohapatra, et al., A giant electro-optic effect using

polarizable dark states. Nature Physics 4, 890 (2008)

A giant dc electro-optic effect on the basis of polarizable

(Rydberg) dark states

Magneto-optic Kerr effect (MOKE)

Woldemar Voigt

1850 – 1919

Voigt effect ∆𝑛 ∝ 𝐻2

was discovered in 1898.

Lorentz transformations

(1909) were first examined

by Voigt in 1887.

D = 𝜀 𝐄

g is the gyration vector

Gurgen Askaryan

1928 – 1997

Self-focusing 1960

A plane stationary light beam in a medium with non-linear refractive index is

described by the equation (see e.g. Talanov V. I., Self-focusing of wave beams innonlinear media. JETP Lett 2(5), pp.138-140 (1965).

Here E is the complex envelope of the electric field; it is assumed that the refractive

index is given by the formula . This equation can be reduced toa standard dimensionless form

𝒏 = 𝒏𝟎 + 𝜹𝒏𝒏𝒍 IEI2

Vladimir Talanov

1933 -

The inverse-scattering problem method was discovered

by Gardner, C.S., Greene, J.M., Kruskal, M.D. and Miura,

R.M., Method for solving the Korteweg-deVries

equation. Phys. Rev. Lett. 19, 1095 (1967). – about 4500

citations. Inverse scattering method is universal. It can be

applied to many nonlinear problems, see e.g.

https://arxiv.org/pdf/1803.08261 (solution of the Sine-Gordon Equation)

This equation can be solved exactly by the inverse-scattering problem method: V. E. Zakharov, A. B.

Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation

of waves in nonlinear media, Soviet Physics JETP 34 (1), pp. 52-69 (1972)- more 5300

citations..

Vladimir

Zakharov

1939 -

The Dirac

Gold Medal

2003

Martin Kruskal

1925 – 2006

nonlinear Schrödinger

equation

Self-focusing (continuation)

Light passing through a gradient-index lens is focused as in a convex

lens. In self-focusing, the refractive index gradient is induced by the light

itself. Several mechanisms produce variations in the refractive index

which result in self-focusing: the main cases are Kerr-induced self-

focusing and plasma self-focusing. Self-focusing occurs if the

radiation power is greater than the critical power

where α is a constant which depends on the initial spatial

distribution of the beam. For Gaussian beam α ≈ 1.8962. For silica

the critical power is Pcr ≈ 2.8 MW..

Plasma self-focusingRelativistic self-focusing is caused by the mass increase of electrons travelling at speed

approaching the speed of light, which modifies the plasma refractive index

𝝎𝒑𝟐~𝟏/𝒎

A reference threshold for plasma self-focusing is the relativistic critical power

For an electron density of 1019 cm−3 and radiation at the wavelength of 800 nm, the critical

power is about 3 TW. Applications: laser-driven particle acceleration, laser-fusion schemes and

high harmonic generation.

Filamentation

The laser beam with a smooth spatial profile E (r, t) is affected by modulational instability. The

small perturbations caused by roughnesses and medium defects are amplified in propagation.

This effect is referred to as Bespalov-Talanov instability (V. I. Bespalov, V. I.

Talanov, Filamentary Structure of Light Beams in Nonlinear Liquids, JETP Lett. 3 (12): 307–

310. (1966). Increment of this instability can be found in a framework of nonlinear Schrödinger

equation.

In 1965 N. F. Pilipetskii and A. R. Rustamov (JETP Letters 2, 55 (1965)) discovered that laser

beam produces a few filaments during self focusing.

Modulation instability leads to a number of physical effects, see e.g.

V.E. Zakharov, L.A. Ostrovsky, Modulation instability: The

beginning, Physica D 238, pp. 540–548 (2009). In its simplistic

version, the effect of modulation instability is the result of interaction

between a strong carrier harmonic wave at a frequency ω , and small

sidebands ω ± Ω . This is the particular case of four-wave interaction

(two quanta at ω create at ω + Ω and ω − Ω). Growth of the sidebands

can be treated in terms of amplification of weak modulation imposed

on a harmonic wave.Formation the multi-focus structure,

A.L.Dyshko, V.N.Lugovoi, A. M. Prokhorov,

Self-focusing of intense light beams, JETP Lett.

6 146 (1967)

The moving nonlinear foci, V.N. Lugovoi, A.

M. Prokhorov, Possible explanation of small-

scale filaments of self-focusing, JETP Lett. 7,

117 (1968)

Numerical solution of nonlinear Schrödinger equation

The intervals between

neighboring maxima

are much smaller than

the distance from z = 0

to the first maximum.

Modulation instability can lead to the formation of long-lived standing and

moving nonlinear localized modes of several distinct types such as bright

and dark solitons, oscillons, and domain walls (see R. Noskov, P. Belov, Y.

Kivshar, Scientific Reports 2, 873 (2012).

Schematic of a chain of metallic nanoparticles illuminated by a laser beam

and profiles of nonlinear localized states. (a) Arrows indicate Particle

polarizations for a bright soliton/oscillon. Panels (b), (c) and (d) depict

profiles of the polarizations for a typical soliton/oscillons configuration of

bright and dark forms as well as domain wall, respectively.

An oscillon is a soliton-like phenomenon that occurs in granular and other dissipative media. Oscillons in

granular media result from vertically vibrating a plate with a layer of uniform particles placed freely on top.

Modulation instability can lead to super-continuum generation with severe

spectral broadening of the original pump beam, for example using

a microstructured optical fiber. The result is a smooth spectral continuum..

An optical frequency comb is a laser source whose spectrum consists of a series

of discrete, equally spaced frequency lines. Four-wave mixing is a 𝝌 𝟑 process

where intense light at three frequencies f1, f2, f3 interact to produce light at a

fourth f4 = f1 + f2 + f3 frequency. If the three frequencies are part of a perfectly

spaced frequency comb, then the fourth frequency is mathematically required to

be part of the same comb as well.

John

Hall

1934 -

2005 2005

Theodor Hänsch

1941 -

An alternative variation of four-wave-mixing-

based frequency combs is known as Kerr

frequency comb. Here a single laser is coupled

into a microresonator (such as a microscopic glass

disk that has whispering-gallery modes).

P. Del'Haye et al, Optical frequency

comb generation from a monolithic

microresonator, Nature. 450,: 1214

(2007).

Spontaneous parametric down-conversion

Optical phase conjugation is a nonlinear optical process, to exactly reverse the propagation

direction and phase variation of a beam of light. The reversed beam is called a conjugate beam.

χ(3) produces a nonlinear polarization field: . It produces waves ω= 3ω1, 3ω2, 3ω3 and ω = ±ω1±ω2±ω3. By choosing conditions ω=ω1+ω2− ω3 and k = k1 + k2 − k3,

this gives a polarization field: 𝑷𝑵𝑳 ∝ 𝝌 𝟑 𝑬𝟏𝑬𝟐𝑬𝟑∗ 𝒆𝒙𝒑 𝒊 ω𝒕 − k 𝒙 + c.c. If k1 = - k2 then

k = k4 = − k3 and 𝑬𝟒(𝒙) ∝ 𝑬𝟑∗ (𝒙). The generated beam amplitude is the complex conjugate of

the signal beam amplitude. Since the imaginary part of the amplitude contains the phase of the

beam, this results in the reversal of phase property of the effect. Note that the constant of

proportionality between the signal and conjugate beams can be greater than 1. This is

effectively a mirror with a reflection coefficient greater than 100%, producing an amplified

reflection. The power for this comes from the two pump beams, which are depleted by the

process. In classical Maxwell electrodynamics a phase-conjugating mirror performs reversal of

the Poynting vector:

𝑷𝑵𝑳 = 𝝌 𝟑 (𝚵𝟏 + 𝚵𝟐 + 𝚵𝟑)𝟑

Klyshko D. N., Penin A. N., Polkovnikov B. F.,

Parametric Luminescence and Light Scattering

by Polaritons, JETP Lett. 11, 05 (1970)David Klyshko

1929—2000

Non inertial nonlinearities

Inertial nonlinearity: Self-action effect related to heating of an absorbing medium

P(t) = 𝝌 𝟏 𝑬 𝒕 + 𝝌 𝟐 𝑬𝟐 𝒕 + 𝝌 𝟑 𝑬𝟑 𝒕 + …

Thermal variation of dielectric permeability can be considered as ε (T) = 𝜺𝟎 −𝝏 𝜺

𝝏 𝑻𝑻

Thus, one should solve the wave equation 𝛁 × 𝛁× 𝑬 +ε (T)𝒄𝟐

𝝏𝟐𝑬

𝝏𝒕𝟐= 𝟎

The quasi-optic approximation: E = 𝒆𝑨 𝝁𝒛, 𝝁𝒓 ∙ 𝒆𝒙𝒑𝒊 [𝝎𝒕 − 𝒌𝒛 − 𝒌𝑺 𝝁𝒛, 𝝁𝒓 ]

Hear z and r are cylindrical coordinates, A is a slowly varying amplitude and S is an addition to

the eikonal, and μ << 1. The ray inclination is given by u = 𝜕𝑆

𝜕𝑟.

Finally one can solve the system of equations

where 𝜚𝑐𝑃 is a specific thermal capacity, 𝜅 is a thermal conductivity coefficient, and δ is an

absorption coefficient.

S.A. Akhmanov, A.P. Sukhorukov, R.V. Khokhlov, R.V.,

Self-focusing and diffraction of light in a nonlinear

medium. Physics-Uspekhi, 10 (5), pp. 609-636 (1968).

S. A. Akhmanov et al, Thermal Self-Actions of Laser

Beam, IEEE J. Quantum Electron. 4, 568 (1968)

Rem Khokhlov

1926 - 1977

Sergei Akhmanov

1929 - 1991

Critical parameter:

Non inertial nonlinearities P > PcrThermal inertial nonlinearity E > Ecr

Chemical inertial nonlinearity

F. V. Bunkin, N. A. Kirichenko, and B. S. Luk'yanchuk, Propagation of laser radiation in a medium with a slow-response chemical nonlinearity, Sov. J. Quantum Electron. 12(4), pp. 435-438 (1982)

The refractive index change as a result of changes in the chemical composition of the medium

We assume 𝑛2 > 𝑛1 and assume for simplicity that 𝜅1 = 𝜅2 = κ = const. Thus, the refractive index is now a function of two variables: the temperature Τ and the concentration c . The equations for the spatial distribution of radiation in a gaseous medium in the geometric-optics approximation take the form:

We shall assume that d𝑛1/dT = d𝑛2/dT = dn/dT = const. Then we have

where the coefficient near dc/dr does not depend on temperature. In gaseous media, we generally

find dn/dT < 0 and thus the first term in the Equation results in thermal defocusing.

Heating of the medium is described by the heat equation. If the characteristic times of the

processes τ are short compared with the characteristic heat conduction time (τ <<𝐫𝟎𝟐

𝒂𝒕, 𝑟0 is the

effective beam radius, 𝑎𝑡 is the thermal diffusivity), the change in temperature satisfies

where w is the energy released by the chemical reaction; 𝑐𝑝 is the specific heat (at constant

pressure); ρ is the density of the medium (we shall assume that 𝑐𝑝 = const, ρ = const).We consider the reaction kinetics equation

where 𝑘0 and 𝑇0 are constants for the given reaction; 𝑇𝑖𝑛 is the initial temperature of themedium. The initial and boundary conditions are:

We assume parabolic approximation 𝑇𝑖𝑛 = 𝑇𝑖𝑛 𝑟 = 𝑇𝑖𝑛(0)

(1 − 𝑟2/𝑟𝑖𝑛2 ), 𝑐 ≪ 1. Integration yields

T(r, t) = 𝑇𝑖𝑛 𝑟 + 𝒘 𝒄(𝒓,𝒕)

𝒄𝒑; 𝒄(𝒓, 𝒕) = - 𝑐1 𝑟 ln(1 − 𝑡/𝜏(𝑟)); 𝑐1 𝑟 = 𝒄𝒑 𝑇𝑖𝑛

2 (r)/ w𝑇0;

𝜏 𝑟 = 𝑐1 𝑟 /𝑑0 𝑟 ; 𝑑0 𝑟 = 𝑘0 exp(- 𝑇0/𝑇𝑖𝑛 𝑟 )

Thus, we have

The time dependence of the nature of propagation of radiation in the medium is due to the term A dc/dr, this term becoming fundamental as time passes. If A < 0, i.e.,

the thermal change in the refractive index due to the exothermicity of the reaction and the radiation will be defocused. In the opposite case (A > 0), as a result of the chemical reaction, the medium becomes focusing.

As long as the first term in the expression for dN/dr is negligible, the ray path near thebeam axis is readily determined from the geometric-optics approximation using themethod of characteristics:

where r = 𝑟0 f(z) is the instantaneous distance of the ray from the beam axis; 𝑟0 = r (z = 0);

We assumed the approximation c = 𝑑0t is valid. For A > 0 we have

Thus, the beam converges over the distance

Self-interaction of radiation in a medium having a slow-response chemical nonlinearity

We assume that the changes in the reaction product concentration and temperature of the medium are fairly small and the radiation intensity distribution over the cross section at the entrance to the medium (z = 0) has the form

i.e., we shall assume that the beam is initially parallel.

In this case, hear equation and chemical equation take the form

where 𝑑0 = 𝑘0 exp(—𝑇0/Tin). We shall introduce the notation:

Using this notation, our problem take the form

We shall seek a solution of this system in the form:

Equating the coefficients for the same powers of ξ, and eliminating the functions ψ, Η, Β, and D from the resultant system, we obtain the following equations:

Influence of diffraction effects

where k – wave number

The self-similar solution takes the form:

Considering the axial approximation, we obtain

where is the diffraction length of the beam.

On entry of the beam into the medium, we find f = 1. Thus, the self-focusing condition𝑓𝜂𝜂I𝜂=0 < 0 (the focusing effect of the chemical lens is stronger than the diffraction-limited

divergence) takes the form

solving the resultant differential equation, we find

In the particular case of purely chemical self-focusing (β = ν = 0), we have

In other words, the critical parameter is the product of the beam energy and the reaction product concentration :

𝐸 = 𝜋𝑟02𝐼0𝑡

c= 𝑑0 𝑡

F. V. Bunkin, N. A. Kirichenko, B. S. Luk'yanchuk, G. A. Shafeev, Thermokinetic processesinduced by laser radiation in chemically active gaseous media, Sov. J. Quantum Electron.13(7), 892 (1983)

Chemical nonlinearity can be seen with small power of laser light and sufficiently big times.

Passage of a CO2 beam through an SF6 - CF3I mixture

Literature

Y. R. Shen, The Principles of Nonlinear Optics, Wiley 1984

Robert W, Boyd, Nonlinear optics. Elsevier, 2003.

Guang S He, Song H Liu, Advanced Nonlinear Optics, World Scientific 2018

Peter E. Powers, Joseph W. Haus, Fundamentals of Nonlinear Optics

CRC Press, 2017

Home work

Read the paper:

Self-focusing and self-trapping of intense light beams in a nonlinear medium

SA Akhmanov, AP Sukhorukov, RV Khokhlov - Sov. Phys. JETP, 1966