1. Department of Electrical and Computer Engineering
Nano-Photonics (2)
W. P. Huang
Department of Electrical and Computer Engineering
McMaster University
Hamilton, Canada

2. Agenda Optical Properties of Metals
Classical Drude model for free electrons
Modifications due to band-tranistions for bound electrons
Modifications due to quantum size effects
Confinement and resonance of light at nano-scale
Scattering of Light by Metal Particles
Surface plasma polariton resonators
Light-matter interaction in nano-crystals
Optical properties of nano-crystals

3. Optical Properties of Metal

4. Basic Relations for Refractive Indices
Complex Refractive Index
Relative Dielectric Constant
Relationship between Dielectric Constant and Refractive Index If If
then
then

5. Microscopic Models Models Light Matter Classical: Dipole Oscillator
Classical EM Wave
Classical Atoms
Semi-classical:
Inter/intra-band transition
Quantum Atoms
Quantum:
Photon and Atom Interaction
Quantum Photons

6. Element Copper Under Different Magnifications

7. The Atomic Structure
~100 pm

8. Models For Atoms Bohr Model:
The energy of the electron is proportional to its distance from the nucleus. It takes energy to move an electron from a region close to the nucleus to one that is further away.
Only a limited number of regions with certain energies are allowed. As a result, the energy of the electron in a hydrogen atom is quantized.
When light is absorbed an electron jumps from a lower energy state to a higher energy state. The energy of this radiation is equal to the difference between the energies of the two states.
Light is emitted when an electron falls from a higher energy state into a lower energy state, the energy of the radiation is equal to the difference.

9. Nature of Electrons in Atoms
Electron energy levels are quantized
Energy for transition can be thermal or light (electromagnetic), both of which are quantized resulting in “quantum leap”
Electrons arranged in shells around the nucleus
Each shell can contain 2n2 electrons, where n is the number of the shell
Within each shell there are sub-shells
3rd shell: 18 electrons 3d
Sub-shells
2nd shell: 8 electrons 3p 3s

10. Classical Model of Atoms
Classical Model: Electrons are bound to the nucleus by springs which determine the natural frequencies
Bound Electrons (insulators, intrinsic semiconductors)
Restoring force for small displacements: F=–kx
Natural frequency
Natural frequencies lie in visible, infrared and UV range
Free Electrons (metals, doped semiconductors)
k=0 so that natural frequency=0

11. Atoms and Bounds • One atom, e.g. H. E + • Two atoms: bond formation ?
Every electron contributes one state
• Equilibrium distance d (after reaction)

12. Formation of Energy Bands
~ 1 eV
• Pauli principle: Only 2 electrons in the same electronic state (one spin & one spin )

13. Energy Band Characteristics
Empty
outer orbitals
Partly filled
valence orbitals
Filled
Inner
shells
Distance between atoms
Energy
Outermost electrons interact
Form bands
Distance between atoms determines whether we have an insulator, a metal or a semiconductor
Electrons in inner shells do not interact
Do not form bands

14. Band Diagrams & Electron Filling
Energy
Empty band
Empty band
Full band
Gap ( ~ 1 eV)
Semiconductor
Empty band
Full band
Gap ( > 5 eV)
Insulator
Partially full band
Metal
Electrons filled from low to high energies till we run out of electrons

15. Color of Metals Silver Gold Empty band
Partially full band
3.1 eV (violet)
1.7 eV (red)
2.4 eV (yellow)
Gold
Only colors up to yellow absorbed and immediately re-emitted; blue end of spectrum goes through, and gets “lost”
Energy
Empty band
3.1 eV (violet)
2.4 eV (yellow)
> 3.1 eV
1.7 eV (red)
Partially full band
All colors absorbed and immediately re-emitted; this is why silver is white (or silvery)

16. Optical Processes in Metals
Macroscopic Views:
The field of the radiation causes the free electrons in metal to move and a moving charge emits electromagnetic radiation
Microscopic Views:
Large density of empty, closely spaced electron energy states above the Fermi level lead to wide range of wavelength readily absorbed by conduction band electron
Excited electrons within the thin layer close to the surface of the metal move to higher energy levels, relax and emit photons (light)
Some excited electrons collide with lattice ions and dissipate energy in form of phonons (heat)
Metal reflects the light very well (> 95%)

17. Drude Model: Free Carrier Contributions to Optical Properties
Paul Drude
( )
A highly respected physicist, who performed pioneering work on the optics of absorbing media and connected the optical with the electrical and thermal properties of solids.
Bound electrons
Conduction electrons

18. Low Frequency Response by Drude Model
If  << 1:
Constant of Frequency, Negligible at low Frequency
Inverse Proportional to ω, Dominant at Low Frequency
At low frequencies, metals (material with large concentration of free carriers) is a perfect reflector

19. High Frequency Response by Drude Model
If  >> 1:
Plasma Frequency:
(about 10eV for metals)
As the frequency is very high
At high frequencies, the contribution of free carriers is negligible and metals behaves like an insulator

20. Plasma Frequency in Drude Model
For Free Electrons
At the Plasma frequency
The real part of the dielectric function vanishes
At the Plasma frequency

21. Validation of Drude Model
Measured data and model for Ag:
Drude model:
Modified Drude model:
Contribution of bound electrons
Ag:

22. Dielectric Functions of Aluminum (Al) and Copper (Cu) Drude Model
M. A. Ordal, et.al., Appl. Opt., vol.22, no.7, pp , 1983

23. Dielectric Functions of Gold (Au) and Silver (Ag) Drude Model
M.A.Ordal,et.al., Appl. Opt., vol.22,no.7,pp ,1983

24. Model Parameters
M.A.Ordal,et.al., Appl. Opt., vol.22,no.7,pp ,1983

25. Improved Model Parameters
M.A.Ordal,et.al., Appl. Opt., vol.24,no.24,pp ,1985

26. Dielectric Functions of Copper (Cu) Drude Model with Improved Model Parameters
M.A.Ordal,et.al., Appl. Opt., vol.24,no.24,pp ,1985

27. Limitation of Drude Model
Drude model considers only free electron contributions to the optical properties
The band structures of the solids are not considered
Inter-band transitions, which are important at higher frequencies, are not accounted for
When the dimension of the metal decreases such that the size of the metal particle becomes smaller than the mean free path of the free electrons, the electrons collide with the boundary of the particle, which leads to quantum-size effects

28. Refractive Index of Aluminum (Al)
Band-Transition Peak

29. Classical Lorentz Model
Electron Clouds
e-,m
E L
Ion Core ro
k,  x p =- e x r +
Ion Core
Potential Energy
Repulsion Force
Newton’s 2nd Law
Damping Force
Electric Force
Repulsion Force

30. Atomic Polarizability by Lorentz Model
Define atomic polarizability:
Resonance frequency
Damping term

31. Characteristics of Atomic Polarizability
Response of matter is not instantaneous
-dependent response
• Atomic polarizability:
• Amplitude
smaller 
Amplitude
• Phase lag of  with E:
180
At low frequencies charges can follow. At high frequencies they cannot
Any complex number you can always write as an amplitude times an exponent with a phase factor
smaller 
Phase lag 90

32. Correction to Drude Model Due to Band Transition for Bound Electrons
Brendel-Bormann (BB) Model
Lorentz-Drude (LD) Model

33. Refractive Index of Al from Classical Drude Model

34. Refractive Index of Al from Modified Drude Model Considering Band-Transition Effects

35. Dielectric Functions for Silver (Ag) By Different Models
A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp ,1998

36. Dielectric Functions for Gold (Au) By Different Models
A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp ,1998

37. Dielectric Functions for Copper (Cu) By Different Models
A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp ,1998

38. Dielectric Functions for Aluminum (Al)
A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp ,1998

39. Correction to Drude Model Due to Size Effect
For nano-particles with dimensions comparable to free electron mean-free-path (i.e., 10nm), the particle surface puts restriction to the movement of the free electrons, leading to Surface Damping Effect .
A constant whose value depends on the shape of the particle and close to unity
The Fermi velocity of electrons
The radius of the metal particle

40. Size Effects of Metal Particles

41. Size Effects of Metal Particles

42. Localized SPP

43. Ideal metal particle under static electric field
The original electric field induces surface charge on the metal particle, of which the induced scattering electric field cancels the original field inside the metallic particle and enhances the outer field.

44. Ideal metal particle in quasi-static field
In case of quasi-static field, which means the incident field is slow-varying, the scattering field of induced charge and its movement inside the ball follows the incident field.

45. Nano-metallic particle
If the dimension of the particle is much smaller than the incident wavelength, it can be considered as quasi-static case.
Comparing to the light wavelength in scale of μm, we choose nm scale for the radius of the metallic particle, to obey the quasi-static condition.
In this case, the metallic ball can be seen equivalent to an oscillating electric dipole.

46. Electric Potential by Sub-Wavelength Particle
For the radius of the particle much smaller than the optical wavelength, i.e., a<<, the electric quasi-static approximation is valid.
Governing Equations a
Eo(t) 
in z
out
General Solutions
Boundary Conditions

47. Electric Potential by Sub-Wavelength Particle
Eo(t) a 
in z
out
Induced Dipole
Eo(t)  p z
out

48. Polarizability of Sub-Wavelength Particle
For meal particles in dielectric materials
If the following condition is satisfied,
The SPP resonance is due to the interaction between EM field and localized plasma and determined by the geometric and material properties of the sub-wavelength particle, independent of its size
then we have localized SPP resonance

49. Scattering Field Distribution of Small Metal Particle

50. Electric Field Induced by Sub-Wavelength Particle
Field Inside Weakened for Positive Re(in) and Enhanced for Negative Re(in)

51. Field Radiated by Induced Dipole
Near Field
(Static Field)
Far Field
(Radiation Field)
Intermediate Field
(Induction Field)

52. EM Field by An Electric Dipole

53. Near Field Approximation: Static Field
For kr<<1, the static field dominates

54. Far Field Approximation: Radiation Field
For kr>>1, the radiation field dominates

55. Scattering Cross-Section
Time-Average Power Flow Density
Total Radiation Power
Power Flow Density for External Field
Scattering Cross Section

56. Absorption Cross-Section
Time-Average Absorbed Power Density
Power Flow Density for External Field
Polarization Vector
Absorption Cross Section

57. Extinction Cross-Section
Extinction Cross- Section for a Silver Sphere in Air (Black) and Silica (Grey), Respectively

58. Beyond Quasi-Static Limit: Multipole Effects
When radius of the particle increases and becomes large compared with the optical wavelength, the distribution of the induced charge and current as well as the phase change or retardation effect of the field need to be considered.
The distribution of charge and current can be decomposed to two sequences of electric and magnetic multi-poles. First four are showed as below,
As the radius increases, higher order multi-poles occurs in sequence.
Electric and magnetic dipoles and quadrupoles

59. Classical Mie’s Theory: General Formulations
Exact solutions to Maxwell equations in terms of vector spherical harmonics.
The wave equations for the scalar potential
The EM fields are expressed by
The EM fields can be expressed in terms of the scalar potential function

60. Classical Mie’s Theory: Scalar Potential
In spherical coordinates, the incident wave can expand as the series of Legendre polynomials and spherical Bessel functions
The scattered wave and the wave inside the sphere are given by
Even solution:
Odd solution:

61. Classical Mie’s Theory: Expansion Coefficients
Match the boundary conditions on the interface to determine the coefficients,
: the radius of metal particle;
: the wavelength in vacuum;
: the refractive index of the metal particle;
: the refractive index of the matrix.

62. Classical Mie’s Theory: EM Fields

63. Efficiency Factors and Cross-Sections
The efficiency factors and cross sections for extinction, scattering and absorption are related as below, where G is the geometrical cross section of the particle, for instance, for a sphere of radius a,
Due to the fundamental extinction formula,

64. Small-Particle Limit: Static Approximation
The small-particle limit is indicated as,
Under this limit, only one of the Mie coefficients remains non-zero value,

65. Multi-pole Approximation
The Polarizability of a sphere of volume V

66. Absorption Spectra of A Nano-Particle of Small Size
A Single Silver Nano-Particle in Matrix of Index 1

67. Absorption Spectra of Nano-Particle of Large Size
A Single Silver Nano-Particle in Matrix of Index 1

68. Broadening of Absorption Spectrum Due to Quantum Size Effect
Modified Drude Model
Additional Damping Due to Size Effect
Increase of Damping Leads to Broadening as Size of the Particle Decreases

69. Normalized Scattering Cross-Section for a Gold Sphere in Air
Note: Normalized by the radius^6

70. Normalized Absorption Cross-Section for a Gold Sphere in Air
Note: Normalized by the volume V

71. Normalized Scattering and Absorption Cross-Sections for a Silver Sphere in Air

72. Absorption Efficiency of a 20 nm Gold Sphere for Different Ambient Refractive Indices

73. Field Patterns for Different Wavelength-Radius Ratio

74. Effects of Geometrical Shape: Ellipsoid
The Polarizabilities along the principal axes: a3 a2 a1

75. Long axes are equal, a=b>c; disk-shaped
Special Cases
Prolate spheroid
Oblate spheroid
Short axes are equal, a>b=c; cigar-shaped
Long axes are equal, a=b>c; disk-shaped

76. Shift of Resonance Peaks Due to Geometric Shape
Normalized absorption cross-section for a gold ellipsoid in the air
Prolate
Oblate

77. Shift of Resonance Peaks Due to Geometric Shape
Normalized absorption cross-section for a silver ellipsoid in the air
Prolate Oblate

78. Effects of Geometrical Shape

79. Coupling between Spheres in a Particle Chain
In the dipole approximation, there are three SP modes on each sphere, two polarized perpendicular to chain, and one polarized parallel. The propagating waves are linear combinations of these modes on different spheres

80. Split Resonance Frequencies Due to Coupling In the Nano-Particle Chain

81. Propagation Modes along SPP Chain
Calculated dispersions relations for gold nanoparticle chain, including only dipole-dipole coupling in quasistatic approximation [S. A. Maier et al, Adv. Mat. 13, 1501 (2001)]
(L and T denote longitudinal and transverse modes)

82. Summary
Localized SPP Resonance occurs at the frequency in which the negative real part of dielectric constant of the metal is equal to positive real part of dielectric constant for the surround materials
For small particles, the SPP resonance frequency is dependent on the geometrical shape of the particle as well as the material properties of the metal and surrounding material, but independent of the size
As the dimension of the particle increases, the multi-pole effects become important, whereas for ultra-small dimension the surface damping effect is more pronounced
Near-field dipole-dipole coupling is important as an efficient energy transfer mechanism for nano-photonic materials and devices.

83. Project Topics: Choose 1 of 2
Topic A: SPP Waveguides and Applications
Topic B: SPP Resonators and Applications
Requirements:
Write a general review for the working principles and potential applications of SPP waveguides or resonators
Submit your project report in MS word format to the instructor