Laser - matter interactionsBoris Lukiyanchuk

Singapore, 22 October 2018

Lecture 2.

Laser - matter interactions

Nonresonant processes Resonant processes

Physical Processes

Chemical Processes

Vapor PlasmaProcesses

Plasmonics Photonics

NonlinearOptics

Resonant Chemistry

Lecture 2.

Plasmonics and Photonics

The physical reason for diffraction limit is related to loss of evanescent waves in far-field.

DIFFRACTION LIMIT

Ernst Abbe

1840 –1905

The resolution limit of the microscope

𝑑 =

2 𝑁𝐴(1873)

NA = n sin = n sin[arc tan (𝐷

2 𝑓)] n

𝐷

2 𝑓

Helmholtz states this formula

was first derived by Lagrange,

𝑑 = 𝐾𝑁𝐴

, K = 0.5 (Abbe), K = 0.473 (Sparrow), K = 0.515 (Houston), K = 0.61 (Rayleigh)

Let the field radiated by some sources distribution in the half-space z < 0 is known

in the plane z = 0.

zyxAAAkA ,,,02

dydxyikxikyxAA yxk

exp0,,

yxzyxk dkdkzikyikxikAzyxA

exp2

1,,

2

-4 -2 0 2 4

-4

-2

0

2

4

z

x

d

dz and

1z

dor

2

dz

2D diffraction: A = A(x,z)

2

2sin

2,0

2,10,

dk

dkA

dx

dxxA

x

xk

xzxk dkzikxikAzxA exp

2

1,

kkatkki

kkatkkkkk

xx

xx

xz

,

,

22

2222

where

Width of Fourier spectrum

kkkdk xx sin,2

Diffraction angledkd

2At the distance z the beam increase width by z

dz

One should compare

Waves with kx > k (evanescent wave) disappeared during propagation

1) Geometrical optics 2) Fresnel diffraction

3) Fraunhofer diffraction 1z

dor

2

dz

1z

dor

2

dz

There are three characteristic regions:

The characteristic distance presents the Rayleigh length.2dRz

The Uncertainty Principle“The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.”

Werner Heisenberg, 1927

Limitation for the light focusing

2 xpx

2 kp

2

4

xpx

222

zx kkk

Remember, k is 3D vector Damping wave in z direction

ikz

Then22 kkx

With big no limitation for spatial coordinate x

To overcome diffraction limit one should provide fast decay of z wave component

1901 –1976Werner Heisenberg

1932 Doctoral students:

Felix Bloch, Robert Milliken,

Isidor Rabi, John van FleckUgo Fano, Rudolf Peierls,

George Placzek, John C. Slater,

Edward Teller, Victor Weisskopf,

Carl Weizsäcker

Classical optics has limitations related to diffraction

0.98

0.99

1

1.01

1.02

z zf0.04

0.02

0

0.02

0.04

r r0

0

2500

5000

7500

10000

I I0

0.98

0.99

1

1.01

1.02

z zf

22

0

2

20

zfr

rexp

zf

Iz,rI

Distribution of intensity for focused Gaussian beam

where r0 is radius of Gaussian beam

2

0

22

12

r

z

fz

zzf

zf is focus distance, is diffraction parameter

dzfz

zd is diffraction length20rkdz

2kk is wavevector

2w0

Surface electromagnetic wave

0,120

21,01

1

zforkkkzkxki

eHH

A surface H wave can be propagated along a plane boundary between two media whose permittivitiesare of opposite signs. (Problem to §88 of “Electrodynamics of continuous media” in L.L. book)

0,220

22,02

2

zforkkkzkxki

eHH

This field fulfills Maxwell equation and boundary condition for continuity of magnetic field. Condition of continuity for tangent component of electrical field yields relation:

21

212

0

2

kk

Let us consider vacuum 11

2

22

0

2

1

kk

"ii

p

20

2

2

12

From Drude formula follows

When 0 = 0, p 2

2

2 1

p and k -> at sp = p/2 (surface plasmon resonance)

Ritchie, R. H. Plasma losses by fast electrons in thin films. Phys. Rev. 106, 874–881 (1957)

Arnold Sommerfeld1868-1951

Doctoral students

Heisenberg, Pauli,

Debye, Bethe,

Pouling, Rabi,

von Laue

84 nominations

The scientific investigation of plasmonic effects began as early as 1899 with theoretical studies by Arnold

Sommerfeld and experimental observations of plasmonic effects in light spectra by Robert Wood in 1902.

icurlH1( )

k0e

1

=E1( )

Robert W. Wood1868 – 1955

Sommerfeld, 1899

W. L. Barnes, A. Dereux & T. W. Ebbesen, Surface plasmon subwavelength optics, Nature 424, 824 (2003)

Effect of dissipation

Periodic texturing of the metal surface can lead to the formation of

an SP photonic bandgap

Robert W. Wood

1868 – 1955

Wood's Anomalies

Grating formula sin (θn) = sin (θ) + nλ/d,

θ - angle of incidence, θn - angle of diffraction

- 1

0

+1

+2 Rayleigh`s waveRayleigh

1842 – 1919nλ/d = − sin(θ) ± 1, n = ±1, ±2, ±3...

1904

Gothic stained glass rose window of Notre-Dame de Paris. The colours were achieved by colloids of gold nano-particles.

How ancient is plasmonics??

Stained glass window atSüleymaniye Mosque.

Propagating surface plasmon polaritons E, H ∝ 𝐸𝑥𝑝[𝑘𝑥 − 𝜔𝑡]Localized plasmon polaritons E, H ∝ 𝐸𝑥𝑝[−𝜔𝑡] F[kx]

Plasmon is the quantum of plasma oscillation with energy oscillation E = ħωp and

lifetime τ = 2/.

Plasmon is not an electron but a collection of electrons.

A combined excitation consisting of a surface plasmon and a photon is called a

surface plasmon polaritons SSP.

Long-range plasmons

In the case of a thick metal film on a dielectric substrate twoindependent SPP modes exist related to different dielectricconstants of the media adjacent to metal interfaces. Thesemodes will degenerate if the film is in a symmetricalenvironment. If a metal film is thin enough so that theelectromagnetic interaction between the interfaces cannot beneglected, the SPP dispersion is significantly modified andcoupling between SPP modes on different interfaces of thefilm must be considered. The interaction of surface polaritonmodes removes the degeneracy of the spectrum and thesurface plasmon frequencies split into two branchescorresponding to symmetric (low-frequency mode) andantisymmetric (high frequency mode) field distributionsthrough the film. For large SPP wave vectors the frequency ofthese modes can be estimated from

Propagation length 1/ImkThis leads to a very longpropagation length of suchSPP modes, called the long-range SPPs.

Boundary conditions: continuity of tangential components electric and magnetic fields

E1 =E2, H1 =H2

The 3D wave equation permits separation of variables in a number of coordinate systems:Cartesian, spherical, cylindrical, parabolic,cylindrical parabolic, spheroidal, toroidal, conical,hyperbolical… (see P. M. Morse and H. Feshbach “Methods of Theoretical Physics”)

DE+ k2E = 0

Mie theoryLaplacian in spherical coordinate systems

For arbitrary shape one can use numerical solution

Final result for scattered fields

cosPrkB

rk

cosE m

e

m

s

r

1

12

1

,sin

cosPrkBisincosPrkB

rk

cosE m

mm

e

m

s

1

11

,sincosPrkBisin

cosPrkB

rk

sinE m

mm

e

m

s

1

11

,cosPrkB

rk

sinH m

m

m

ms

r

1

12

1

,sincosPrkBisin

cosPrkB

rk

siniH m

mm

es

1

11

0

,sin

cosPrkBisincosPrkB

rk

cosiH m

mm

es

1

11

0

),(h )(1

2

1

2

11

2

1

1

iNJHh

0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

q

a1

ε = - 1(R), - 2(G), - 3(B)

0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

q

b1

ε = - 1(R), - 2(G), - 3(B)

max aℓ

2=1, max b

ℓ

2=1

For e <0 and q <<1 bÐ

<< aÐ

Qℓ

max( )=

2 2ℓ +1( )q2

Inverse hierarchy of plasmon resonances

Tribelsky M.I., Luk'yanchuk B. S.

Anomalous light scattering by small particles

Phys. Rev. Lett. 97, 263902 (2006)

q =2pa

l

Trajectories of the first three surface plasmon

resonances with = - 2 (dipole), - 1.5 (quadrupole) and

- 4/3 (octupole) versus size parameter q.

Surface plasmon resonances for an Al particle of 30 nm as a function of incident light wavelength λ.

Octupole Quadrupole Dipole

Normalized extinction, scattering and absorption cross

sections, for an Al particle of 30 nm as a function of incident

light wavelength λ.

Milky Way Central Region Map

Extinction of Interstellar Dust

is related to surface plasmon

resonance of carbon nanoparticles

Carbon

onion

Luk’yanchuk B. S., Luches A., Blanco A., Orofino V.Physical Modelling of the Interstellar Dust

Proc. SPIE, vol. 4070, 154 (2000)

Localized surface plasmon for spherical particle

Fano resonances in plasmonic clusters

Topological optics (singular optics, vortices, and dislocations of the wave front)

M. R. Dennis, K. O'Holleran, and M. J. Padgett, Singular Optics: Optical Vortices and Polarization Singularities, Progress in Optics53, 293 (2009)

J. F. Nye and M. V. Berry, Dislocations in Wave Trains, Proc. R. Soc. Lond. A 336, 165 (1974)

Bohren C.F. How can a particle absorb more than the

light incident on it? Am. J. Phys. 51, 323–327 (1983).

Z. B. Wang, B. S. Luk’yanchuk, et al. Energy flow around a

small particle investigated by classical Mie theory, Phys. Rev. B

70, 035418 (2004)

Formation of optical vortices

Near field effects in light scattering

These effects can be used for manipulation of the energy flow in the nanoscale region

Luk`yanchuk B. S., et. al. Peculiarities of light scattering by nanoparticles and nanowiresnear plasmon resonance frequencies, Journal of Physics: Conference series, 59, 234 (2007)

Luk`yanchuk B.S. et al., Light scattering at nanoparticles close to plasmon resonance frequencies, J. Op. Tech. 73, 371 (2006) Tribelsky M.I., Luk'yanchuk B. S., Anomalous light scattering by small particles, Phys. Rev. Lett. 97, 263902 (2006)

Light vortices

Interference effects at near field: a method to combine “nano-Fano” with “nano-vortices” permits to control a topological

charge on a nanoscale. This yields promising applications inquantum optics.

SingularOptics

Distribution of the Poynting vector vs. size parameter and dissipation

-4 -2 0 2 4-4

-2

0

2

4

`` = 0.05

x/a

y/a

0.01

0.05

0.30

1.62

5.60

q = 0.1

-4 -2 0 2 4-4

-2

0

2

4

`` = 0.05

x/a

y/a

0.01

0.04

0.18

0.75

2.25

q = 1.0

-4 -2 0 2 4-4

-2

0

2

4

`` = 0.05

x/a

y/a

0.01

0.05

0.26

1.29

3.64

q = 0.5

-4 -2 0 2 4-4

-2

0

2

4

`` = 0.25

x/a

y/a

0.01

0.03

0.11

0.38

0.94

q = 0.1

-4 -2 0 2 4-4

-2

0

2

4

`` = 0.25

x/a

y/a

0.01

0.04

0.19

0.80

1.77

q = 0.5

-4 -2 0 2 4-4

-2

0

2

4

`` = 0.25

x/a

y/a

0.01

0.04

0.16

0.64

1.43

q = 1.0

-4 -2 0 2 4-4

-2

0

2

4

`` = 0.5

x/a

y/a

0.01

0.03

0.12

0.41

0.91q = 0.1

-4 -2 0 2 4-4

-2

0

2

4

`` = 0.5

x/a

y/a

0.01

0.07

0.53

1.21

q = 1.0

-4 -2 0 2 4-4

-2

0

2

4

`` = 0.5

x/a

y/a

0.01

0.04

0.14

0.53

1.66

q = 0.5

-6 -4 -2 0 2 4 6-4

-2

0

2

4

4i3i

9

1087

2i

1i 65

4

3

2

x/a

y/a

0.01

0.06

0.38

2.34

7.24

1

Contour plot of

.

0.03 0.02 0.01 0 0.01 0.02 0.03

x

0.1

0.11

0.12

0.13

0.14

0.15

y

Electrical fieldpolarization around thesingular center point(radially polarized light).

B. S. Luk`yanchuk, V. Ternovsky, Light scattering by a thin wire with a surface plasmon resonance: Bifurcations of the Poynting vector field , Phys. Rev. B, vol. 73, 235432 (2006)

Near-field effects

264 sin

8

31

27

64ak

Necessary condition to overcome diffraction limit (fast decay of one field component)

easy to fulfill in near-field region. Four typical situations are considered:

1) Aperture limited beams (near-field scanning optical microscope, NSOM)

2) Evanescent waves at total internal reflection

3) Surface electromagnetic waves (plasmons, polaritons).

4) Light scattering on the tip

E.H. Synge, "A suggested method for extending the microscopic resolution

into the ultramicroscopic region" Phil. Mag. 6, 356 (1928)J.A. O'Keefe, "Resolving power of visible light", J. Opt. Soc. Am., 46, 359

(1956)

λ / 60 resolution in a scanning near field microwave microscope using 3

cm radiation. E.A. Ash and G. Nichols, Nature 237, 510 (1972).

SNOM - D.W. Pohl, W. Denk, M. Lanz, Appl. Phys. Lett. 44, 651 (1984)

Basic problem a small transmission efficiency

H. A. Bethe, “Theory of Diffraction by Small Holes”,

Phys. Rev. 66, 163 (1944)

Hans Albrecht Bethe

Nobel Prize 1967Nature 391, 667 (1998)

Sharp peaks in transmission are observed at

wavelengths as large as ten times the diameter

of the cylinders. At these maxima the

transmission efficiency can exceed unity (when

normalized to the area of the holes), which is

orders of magnitude greater than predicted by

standard aperture theory.

Transmission properties of a single metallic slit in a metallic screen

Contour plot of the electric field intensity

J. Bravo-Abad, L. Martın-Moreno, F. J. Garcıa-Vidal

PHYS. REV. E 69, 026601 (2004)

Plasmonic lens

F. J. Garcıa-Vidal, L. Martın-Moreno,H. J. Lezec, T. W. Ebbesen

Appl. Phys. Lett. 83, 4500 (2003)

Enhancement by hole shaping

F. Chen, A. Itagi, J. A. Bain, D. D. Stancil, T. E. Schlesinger,

L. Stebounova, G. C. Walker, B. B. Akhremitchev

Appl. Phys. Lett. 83, 3245 (2003)

C - shaped Al aperture

Seagate plans to use C-shaped design for HAMR project.

Expected size of focus is about 37 nm for Ag and = 655 nm

Expected transmission about 10-15 %.

From presentation of Dieter Weller, DSI, June 4, 2004

Probably C-shaped design is not optimalI-shaped design

K. Tanaka, M. Tanaka, Optimized computer-aided design of I-

shaped subwavelength aperture for high intensity and small spot

size, Optics Communications 233, 231 (2004)

Field enhancement at metal tips

Field enhancement with Si tip and Si tip with Au nanoparticle

From Wang Z.B. et al, 2006

R. M. Roth et al.

Opt. Exp. 14, 2921 (2006)

15 nm nanolines

From S. M. Huang et al, J. Appl. Phys., 91, 3268 (2002)

SUPER RENS In 2000 Tominaga group suggested to place thin silver oxide layer (AgOx)

J. H. Kim, D. Buechel, T. Nakano, J. Tominaga,

N. Atoda, H. Fuji, Y. Yamakawa

Magneto-optical disk properties enhanced by a

nonmagnetic mask layer

Appl. Phys. Lett. 77, 1774 (2000)

Chemical decomposition leads to formation of Ag

nanoclusters

2 AgO -> Ag2O + 1/2 O2

Ag2O -> 2 Ag + 1/2 O2

Silver nanoclusters has plasmon resonance

Link between plasmonics and photonics: Origin of surface states at condensed matter interfaces

Bloch waves. A wavefunction ψ is a Bloch wave if it has the form:

Felix Bloch

1905 – 1983

Bloch became Heisenberg's first graduate student, and gained his doctorate in 1928.

His doctoral thesis established the quantum theory of solids, using Bloch waves to

describe electrons in periodic lattices. The underlying mathematics was previously

discovered by Émile Mathieu (1868), George William Hill (1877), Gaston

Floquet (1883), and Alexander Lyapunov (1892).

1952

Émile Mathieu

1835-1890

Mathieu's differential equation

Such equation arises e.g. in the quantum mechanics for singular potential V(x) ~ 𝑟−4.

Mathieu's equation admits a complex valued solution of form y = Exp(i k x ) F(a, h, x)

where F is a complex valued function which is periodic in x with period π. The unique

solution of the Mathieu equation is given in terms of Mathieu`s functions: Mathieu cosine

C(a, q, x) and Mathieu sine S(a, q, x). The general solution is a linear combination of these

functions. A special case is

See e.g. http://mathworld.wolfram.com/MathieuFunction.html

Igor Tamm

1895 –1971

1958

Tamm states and Shockley states

I.Tamm, On the possible bound states

of electrons on a crystal surface.

Phys. Z. Sowjetunion. 1, 733 (1932).

William Shockley

(1910 – 1989)

1956

Bulk states

Surface statesW. Shockley, On the Surface States

Associated with a Periodic Potential.

Phys. Rev. 56, 317 (1939)

ProblemsProblem 1. A surface H wave can be propagated along a plane boundary between two media whose permittivities are of opposite signs. Consider media with and .

Solution.

This field fulfills Maxwell equation and boundary condition for continuity of magnetic field.Condition of continuity for tangent x-component of electrical field at z=0 yields relation:

𝐻1 = 0, 𝐻1𝑦 , 0 , 𝐻2= 0, 𝐻2𝑦 , 0 ,

𝐻1𝑦 = 𝐻0𝑒𝑖𝑘𝑥−𝑘1𝑧, 𝑘1= 𝑘2 − 𝑘0

2휀1𝜇1

where

, for z > 0

icurlH1( )

k0e

1

=E1( )

𝐻2𝑦 = 𝐻0𝑒𝑖𝑘𝑥+𝑘2𝑧, 𝑘2= 𝑘2 − 𝑘0

2휀2𝜇2, for z < 0

∆𝐻1 + 𝑘02휀1𝜇1 = 0

∆𝐻2 + 𝑘02휀2𝜇2 = 0

At z = 0 𝐻1𝑦= 𝐻2𝑦 continuity of y-component of magnetic field. Electric field from

𝐸1 = 𝑖𝑘1𝑘0휀1

𝐻1𝑦 , 0, 𝑖𝑘

𝑘0휀1𝐻1𝑦 , 𝐸2= −𝑖

𝑘2𝑘0휀2

𝐻2𝑦 , 0, −𝑖𝑘

𝑘0휀2𝐻2𝑦

𝑘1

𝜀1= -

𝑘2

𝜀2i. e.

𝑘2−𝑘02𝜀1𝜇1

𝜀1+

𝑘2−𝑘02𝜀2𝜇2

𝜀2= 0 𝑘2= 𝑘0

2휀1휀2𝜀2𝜇1−𝜀1𝜇2

𝜀22−𝜀1

2

At 𝜇1= 𝜇2 = 1 𝑘2= 𝑘02 𝜀1𝜀2

𝜀2+𝜀1

Problem 2. Fano resonance is similar to classical resonance in system with two coupled oscillators. Find resonance stationary amplitudes.

Solution.

ω𝟏= 1, ω𝟐 = 1.1, Ω = 0.25, γ𝟏 = 0.1, γ𝟐 = 0.01. Constructive and destructive interference occur

correspondingly at the resonant frequencies shown by arrows. These frequencies are in the vicinity of

the hybrid eigenfrequency. Phases of the first (ϕ1) and second (𝜙2) oscillators, and their difference.

Problem 3. Find the Fano resonance in the directional scattering efficiencies for the forward (FS) and backward scattering (BS) for small q << 1 spherical metallic particle.

Solution.

Equations 𝑄𝐵𝑆 = 0 and 𝑄𝐹𝑆 = 0 yield

Both quantities 𝑄𝐵𝑆 (ε) and 𝑄𝐹𝑆(ε) vanish in the vicinity of the

quadrupole surface plasmon resonance at q = 0.02: ε = - 1.5 + 𝛿,𝛿<<1. One can see large modifications of the vortex structures in

the near field. (see J. Opt. 15, 073001 (2013)

Problem 4. Find the propagation modes of a dielectric-slab waveguide consisting of a dielectriccore and metal clads (see IEEE Journal of Quantum Electronics, Vol. QE-8, No. 2, pp. 206-212,Feb 1972, Optics Letters 8, 383 (1983))

The symmetric and asymmetric

TM propagation modes

SP mode inside the metal-clad

planar waveguide

The guided surface plasmons mode

inside the slit of noble metals.

Jap. J. Appl. Phys. 45, 6974 (2006)

Dispersion equation

Problem 5. Tamm surface states in metals are found as solutions to the one-dimensional single electron

Schrödinger equation

Solution

Shockley states

The band splitting at the edges of the Brillouin zone

B. I. Afinogenov, A. A. Popkova, V. O. Bessonov, B.

Lukyanchuk, A. A. Fedyanin “Phase matching with

Tamm plasmons for enhanced second- and third-

harmonic generation”, Phys. Rev. B 97, 115438

(2018)

Literature

1. Landau, L. D., Bell, J. S., Kearsley, M. J., Pitaevskii, L. P., Lifshitz, E. M., Sykes, J. B.,

Electrodynamics of continuous media. Elsevier, 2013. Problem to §88

2. V. M. Agranovich, V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Exitons,

Springer 1984

3. S. Maier. Plasmonics: Fundamentals and Applications. Springer, 2007

4. W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications,

Springer, 2009

5. S.I. Bozhevolnyi, L. Martin-Moreno, F. Garcia-Vidal, Quantum Plasmonics,

Springer, 2017

Home work

1. Reproduce solutions of the Problems 1 – 5

2. Suggest a new problem and its solution