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Demonstration of Optical Resonances in
a Cylinder Shell Lattice of Quantum Dots
Jared Maxson
Slava Rotkin
Lehigh University Department of Physics
J. Maxson, S. V. Rotkin, APS March Meeting 2009
Motivation: Geometries of Self-Assembly
• Order on the nanoscale:
A pretty
A pretty
• Regular geometry is a principle virtue in the self-assembly.
• Questions:
– For optically polarizable elements, what is the effect of lattice geometry
on the system response?
– For plane wave excitation, is a resonant interaction between the optical
field and exciton system possible?
• Our particular geometry:
J. Maxson, S. V. Rotkin, APS March Meeting 2009
Modeling the Hollow Nanoelement Cylinder
• We consider a cylinder-shell lattice of polarizable nanoelements
(nanoparticles, quantum dots)
– Modeled as a lattice of induced, fully coupled point dipoles (the excitons)
written in the second quantization, with a single excitation dipole moment
and a single transition frequency. The lattice constant is assumed to be
much smaller than the wavelength of the light to provide coherent
• Subsequent Hamiltonian is analogous to the classical Coupled
Dipole Method (Purcell and Pennypacker) for the calculation of
the optical response of dielectric with arbitrary geometry:
H  p2 E1  p1 E2
two elements:
E2 ( x1 )  Vˆx1x2 p2 ( x2 )  E0 ( x1 )
Dipole-Dipole Interaction
J. Maxson, S. V. Rotkin, APS March Meeting 2009
Model (Continued)
• The fully coupled Hamiltonian of the system is:
– Greek indices represent Cartesian components, and Roman indices represent
lattice sites. **
– Note: Hamiltonian and interaction propogator are diagonal in the same
• For simplicity, we assume:
– Isotropic polarizability,
– Non-retarded interaction propogator:
However, such assumptions are not critical to the model.
– Could also include: multiple transition frequencies, anisotropic polarizability, fully
retarded dipole interaction and quantized field operator (gives rise to polariton
• We treat the cylindrical radius as a parameter to identify geometry dependent
J. Maxson, S. V. Rotkin, APS March Meeting 2009
Methods of Calculation
• As H is diagonal in the same space as V, diagonalization of the
interaction matrix yields the eigenset of polarization:
• A translationally invariant Hamiltonian may be diagonalized by
rotation to Fourier space:
– However, we consider cylinders of finite length, and thus the lattice is not
fully translationally invariant. We therefore diagonalize numerically,
loosing linear and angular momentum data for each mode.
• Diagonalization with numerical methods is
straightforward but not directly informative.
• How do we classify excitation modes, and
determine possible resonant interaction?
J. Maxson, S. V. Rotkin, APS March Meeting 2009
Response Function and Resonance
• We introduce the quantum mechanical response function for such a system to
quantify resonant behavior.
– Resonant behavior: defined by high spatial and temporal coherence of the
incident light and mode(s) of the lattice.
J. Maxson, S. V. Rotkin, APS March Meeting 2009
Response Function Analysis
• Matrix elements of modes with non-negligible oscillator
strength at a specific peak are plotted together (right).
• We note that at each peak the contributing modes
have identical radius/wavelength dependence, and
are small in number (about 10 in 990).
• By our definition, the transfer of significant oscillator
strength to a limited number of modes is evidence of
resonant interaction—high spatial and temporal
coherence of mode and incident wave.
• As the field is polarized along the axis of the cylinder, we
analyze the angular dependence of pz for those modes
with identical radial dependence:
• Near-perfect sinusoidal
behavior indicates sharp values
of mode angular momenta.
• Mode “families” are modes of
identical angular momentum,
but differing linear momentum.
J. Maxson, S. V. Rotkin, APS March Meeting 2009
Conclusion and Summary
For hollow cylindrical case:
• Formation of the quantum mechanical response function permits
determination of regions of cylinder size and light frequency that yield
maximal system response.
• Sharp values of angular momentum in cylinder excitations are preserved,
though translational symmetry is broken.
– Permits quantitative excitation classification.
• For axial polarization, modes with equal angular momenta are grouped
together at absorption peaks, contributing to the resonant behavior of high
mode selectivity.
In general:
• A second quantization model is proposed for the determination of the optical
response of a strongly coupled polarizable nanoelement lattice (for any
• Methods of determining the eigen-excitations are proposed for both infinite
lattices of high symmetry, and finite lattices of reduced symmetry.
• What’s next? Less general, more specific?
J. Maxson, S. V. Rotkin, APS March Meeting 2009