Structure of the course

1
Metamaterials in optical domain
2
Nonlinear nano-optics

• Structural nonlinearity in compound systems
• Raman surface scattering enhancement
• Quantum Mechanical metamaterials one possible
  way for loss compensation

3
Structural nonlinearity in compound systems
e
e-
e
e-
Quadrupole and magnetic dipole moments being
proportional to the second order of charge
coordinates becomes in some cases a nonlinear
function of the electric field. This, in turn,
leads to nonlinear response even without any
nonlinearity in both dielectric host or metal
inclusions.
4
Raman surface scattering enhancement
5
Problem statement
Raman signal

• manufacturing of field enhancing
• structures ?
• why? Raman signal is very sensitive
• to local electric field
• electromagnetic description required

active molecule
local field
6
Basics of the electrodynamic description
scattered wave

• interested in the scattered field
• (i.e., local field, near field)
• ? enhancement factors
• far field observables ( without any
  evanescent decaying
• contributions) ? R,T
• which amount of the incident wave
• is scattered (radiated) and
• which is absorbed ? cext(w), csca(w)

material
I1
A1
incident wave
I0
7
Basics of the electrodynamic description

• physical description in the framework of the
  electrodynamic formalism
• no sources , no external currents, ,
  no magnetization

wave equation in time domain (FDTD)
wave equation in frequency domain (FMM)
8
Outline
(1) Basics of the electrodynamic description (2)
Presentation of numerical methods (3) Analytical
considerations (4) Application to manufactured
SERS geometries (5) Results (6) Summary
9
Numerical methods - FDTD

• the idea of the finite difference time domain
  (FDTD)

spatial discretization
y
temporal discretization
E(t)
monitor (T)
calculation window (boundary conditions)
t
launch field excitation
monitor (R)
x
10
Numerical methods - FMM

• FMM Fourier Modal Method / RCWA rigorous
  coupled wave
• analysis
• Maxwell solver based on the fourier expansion of
  periodic structures
• near fields (1D simple, 2D hard!) and spectra
  (method of choice)
• in house code (available _at_ IFTO/ C. Rockstuhl)
  and commercial
• solutions (RSoft, Unigit)
• memory limited calculation

• M. G. Moharam, T. K. Gaylord Stable
  implementation of the rigorous
• coupled-wave analysis for surface-relief
  gratings enhanced transmittance
• matrix approach JOSA A, 12, 5, 1077 (1995)
• M. G. Moharam, T. K. Gaylord, Formulation for
  stable and efficient
• implementation of the rigorous coupled-wave
  analysis of binary gratings
• JOSA A, 12, 5, 1058 (1995)

11
Analytical considerations

• main interesting theory for a complete
  description ? quasistatics

• the dimensions should be much
• smaller than the wavelength
• an accurate and exact description
• is only possible for spherical particles
• Mie 1906
• the electric field evolution along the
• particle is weak ? constant field inside
• electrostatic description of the particle
• Kreibig, Vollmer 1995,
• Bohren, Huffman 1998,
• Landau, Liefshitz 1983

d
E(x)
x
l
E(x)




-
-
-
-
12
Analytical considerations

• for prolate shaped inclusion (cigar shape) a2
  a3

Side 2 (90 right)
Side 3 (90 up)
Side 1
a1
a3
a1
a2
a2
a3

• the polarizability contains the dispersive as
  well as the geometric properties in
• terms of eamb(w), einc(w) and the depolarization
  factor L

13
Analytical considerations

• for oblate shaped inclusion (disc shape) a2
  a3

Side 2 (90 right)
Side 3 (90 up)
Side 1
a2
a3
a2
a1
a1
a3

• for oblate structures the only the
• depolarization factor changes
• L is not dependent on the absolute
• dimensions, see x

14
Analytical considerations

• with the polarizability of the particle various
  properties beyond neff(w) and
• eeff(w) can be determined
• (a) cross sections

extinction (intrinsic absorption)
scattering
absorption (complete losses)
(b) reflection and transmission spectra
Fresnel equations (Matrix formalism)
the number density h has to be known
15
Analytical considerations
(c) field enhancement factor
intensity enhancement ? G(w)2
enhancement main axis polarization main axis
enhancement short axis polarization main axis
Barber et al 1983
(d) field distribution (only dipole approximation)
scattered electric field of a dipole with the
polarizability a(w) Stratton 1941, Bohren,
Huffman 1998
16
Application to SERS geometries - FDTD
17
Application to SERS geometries - FDTD
18
Application to SERS geometries - FDTD
19
Application to SERS geometries - FDTD
20
Application to SERS geometries - FDTD
21
Application to SERS geometries - FDTD
22
Application to SERS geometries - Analytical

• R,T from quasistatic calculations

• calculation of R,T
• from neff thickness
• with matrix algorithm

• exact numerical
• spectra (FMM)
• ? spectral agreement
• for resonance
• wavelength, not width
• (damping)

23
Application to SERS geometries - Analytical

• e from quasistatic calculations

• Lorentz shaped
• resonances occurring at
• the plasmonic resonances
• of the effective medium
• (analytical calculation)

• the same resonances
• can be obtained by
• rigorous numerical calc.s
• (FMM param. retrieval)

24
Application to SERS geometries - Analytical

• effect of geometrical manipulations on spectral
  resonance frequencies

• (1) manufacturing
• angle variation

• (2) structure height
• variation

25
Application to SERS geometries - Analytical

• field enhancement factors in the quasistatic
  regime

• dramatic field enhancements predicted (due to
  the reduced bandwidth of the
• resonance!)
• even for major axis polarization the field
  enhancement is extremely
• increased
• a lot of SERS papers (during the 1980s) making
  use of these formulas

26
Application to SERS geometries - Analytical

• rhomb vs. ellipsoid or quasistatics (dipole) vs.
  numerics major axis

• near field distribution
• in terms of electric dipoles
• factor 120 in Ex, 90
• in Ey (quantitative result!)

Ex
Ex

• FDTD near field
• distribution for excitation
• along major rhomb
• axis

Ex
Ex
27
Application to SERS geometries - Analytical

• rhomb vs. ellipsoid or quasistatics (dipole) vs.
  numerics minor axis

• the same for the minor
• axis excitation
• factor 2 for Ex and 3 for
• Ey
• ? smaller enhancement

Ey
Ey

• FDTD results for minor
• axis polarization

Ey
Ey
28
Measurements

• rhombs, rectangles and hybrid material SERS
  structures spectral response

29
Quantum Mechanical metamaterials one possible
way for loss compensation
30
What it is the difference between density matrix
approach and classical harmonic oscillator model?
Harmonic oscillator model
Density matrix
31
Dielectric constant for optically active
materials