Boris Lukiyanchuk Laser - matter interactions · Laser - matter interactions Nonresonant processes...
Transcript of Boris Lukiyanchuk Laser - matter interactions · Laser - matter interactions Nonresonant processes...
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