CONSERVATION LAWS FOR THE INTEGRATED DENSITY OF STATES IN ARBITRARY QUARTERWAVE MULTILAYER NANOSTRUC

1
CONSERVATION LAWS FOR THE INTEGRATED DENSITY OF
STATES IN ARBITRARY QUARTER-WAVE MULTILAYER
NANOSTRUCTURES

• Sergei V. Zhukovsky

Laboratory of NanoOptics Institute of Molecular
and Atomic Physics National Academy of Sciences,
Minsk, Belarus
zhukovsky_at_imaph.bas-net.by
2
Presentation outline

• Introduction
• Quarter-wave multilayer nanostructures
• Conservation of the transmission peak number
• Transmission peaks and discrete eigenstates
• Clearly defined boundary limitation
• Conservation of the integrated DOM
• Density of modes
• Analytical derivation of the conservation rule
• Summary and discussion

3
Introduction

• Inhomogeneous media are known to strongly modify
  many optical phenomena
• However, there are limits on the degree of such
  modification, called conservation or sum rules
• e.g., Barnett-Loudon sum rule for spontaneous
  emission rate
• These limits have fundamental physical reasons
  such as causality requirements and the
  Kramers-Kronig relation in the above mentioned
  sum rule.

• Wave propagation
• Spontaneous emission

• Planck blackbody radiation
• Raman scattering

Stephen M. Barnett, R. Loudon, Phys. Rev. Lett.
77, 2444 (1996)
4
Introduction

• In this paper, we report to have found an
  analogous conservation rule for the integrated
  dimensionless density of modes in arbitrary,
  quarter-wave multilayer structures.

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Quarter-wave multilayer structures

• A sample multilayer

A quarter-wave (QW) multilayer is such
that where N is the number of layers ?0 is
called central frequency
The QW condition introduces the central frequency
?0 as a natural scale of frequency normalization
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Quarter-wave multilayer structures

• The QW condition has two effects on spectral
  symmetry
• 1. Spectral periodicity with period equal to 2?0
  ( )
• 2. Mirror symmetry around odd multiples of ?0
  within each period ( )

Transmission
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Binary quarter-wave multilayers

A binary multilayer contains layers of two types,
labeled 1 and 0. These labels are used as binary
digits, and the whole structure can be identified
with a binary number as shown in the figure.
Periodic
Random
Fractal
S. V. Gaponenko, S. V. Zhukovsky et al, Opt.
Comm. 205, 49 (2002)
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Transmission peaks and eigenstates

• Most multilayers exhibit resonance transmission
  peaks
• These peaks correspond to standing waves (field
  localization patterns), which resemble quantum
  mechanical eigenstates in a stepwise potential.
• That said, the peak frequencies can be looked
  upon as eigenvalues, the patterns themselves
  being eigenstates.
• Thus, the number of peaks per unit interval can
  be viewed as discrete density of electromagnetic
  states

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Conservation of the number of peaks
Numerical calculations reveal that in any
quarter-wave multilayer the number of
transmission peaks per period equals the number
of quarter-wave layers

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Conservation of the number of peaks

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Conservation of the number of peaks

The number of peaks per period equals 8 for all
structures labeled by odd binary numbers from
12910100000012 to 25510111111112 This leads
to an additional requirement
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Clearly defined boundary condition

• Note that the number of peaks is conserved only
  if the outermost layers are those of the highest
  index of refraction
• Otherwise, it is difficult to tell where exactly
  the structure begins, so the boundary is not
  defined clearly.
• This is especially true if one material is air,
  in which case a layer loss occurs.

Material 0 is air
101015 layers
101104 layers
Otherwise
10101
10110
This boundary is unclear
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Non-binary structures
If the clearly defined boundary condition
holds, the number of transmission peaks per
period is conserved even if the structure is not
binary

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Density of modes

• Transmission peaks vary greatly in sharpness
• One way to account for that is to address density
  of modes (DOM)
• The strict DOM concept for continuous spectra is
  yet to be introduced
• We use the following definition
• t is the complex transmission D - total thickness

Transmission / DOM
Normalized frequency
Transmission / DOM
Normalized frequency
J. M. Bendickson et al, Phys. Rev. E 53, 4107
(1996)
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DOM and frequency normalization

• DOM can be made dimensionless by normalizing it
  to the bulk velocity of light in the structure
• N0 and N1 being the numbers, and N0 and N1 the
  indices of refraction of the 0- and 1-layers in
  the structure, respectively, and D being the
  total physical thickness
• Frequency can be made dimensionless by
  normalizing to the above mentioned central
  frequency due to quarter-wave condition

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Integrated DOM conservation

• Numerical calculations confirm that the integral
  of dimensionless DOM over the interval 0, 1 of
  normalized frequencies always equals unity
• This conservation rule holds for arbitrary
  quarter-wave multilayer structures.

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Analytical derivation - part 1

• Though first established by numerical means, this
  conservation rule can be obtained analytically.
• Substitution of normalization formulas yield
• The effective wave vector k is related to ? by
  the dispersion relation
• Again, t is the complex transmission, and D is
  the total physical thickness of the structure

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Analytical derivation - part 2

• In the dispersion relation, ? is the phase of
  transmitted wave. Since the structures are QW, no
  internal reflection occurs at even multiples of
  ?0. Therefore,
• Here, D(opt) is the total optical thickness of
  the structure
• Then, after simple algebra we arrive at which is
  our conservation rule if we take into account the
  above mentioned mirror symmetry.

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Summary and discussion - part 1

• We have found that a relation places a
  restriction on the DOM integrated over a certain
  frequency region.
• This relation holds for any (not necessarily
  binary) QW multilayer.
• The dependence ?(?) itself does strongly depend
  on the topological properties of the multilayer.
• Therefore, the conservation rule obtained appears
  to be a general property of wave propagation.

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Summary and discussion - part 2

• The physical meaning of the rule obtained
  consists in the fact that the total quantity of
  states cannot be altered, and the DOM can only be
  redistributed across the spectrum.
• For quarter-wave multilayers, our rule explicitly
  gives the frequency interval over which the DOM
  redistribution can be controlled by altering the
  structure topology

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Summary and discussion - part 3

• For non-QW but commensurate multilayers, i.e.,
  when there is a greatest common divisor of
  layers optical paths ( ), the structure can be
  made QW by sectioning each layer into several
  (see figure).
• In this case, there will be an increase in the
  integration interval by several times.

Optical path

• For incommensurate multilayers, this interval is
  infinite. Integration is to be performed over the
  whole spectrum.

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Acknowledgements

• The author wishes to acknowledge
• Prof. S. V. Gaponenko
• Dr. A. V. Lavrinenko
• Prof. C. Sibilia
• for helpful and inspiring discussions

References
1. Stephen M. Barnett, R. Loudon, Phys. Rev.
Lett. 77, 2444 (1996) 2. S. V. Gaponenko, S. V.
Zhukovsky et al, Opt. Comm. 205, 49 (2002) 3. J.
M. Bendickson et al, Phys. Rev. E 53, 4107 (1996)