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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
[email protected]
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
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Introduction
• Inhomogeneous media are known to strongly
modify many optical phenomena:
• Wave propagation
• Spontaneous emission
• Planck blackbody radiation
• Raman scattering
• 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 KramersKronig relation in the above mentioned sum rule.
[Stephen M. Barnett, R. Loudon, Phys. Rev. Lett. 77, 2444 (1996)]
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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 B
dA dB
nA nB
A quarter-wave (QW)
multilayer is such that
nA d A  nB d B  ni di 
i  1, 2,
0
4

c
2 0
,N
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
Transmission
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
Normalized frequency
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Binary quarter-wave multilayers
1010101012=34110
Periodic
1101010012=42510
Random
1100001012=32510
Fractal
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.
[S. V. Gaponenko, S. V. Zhukovsky et al, Opt. Comm. 205, 49 (2002)]
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Transmission peaks and eigenstates
1
Most multilayers exhibit
resonance transmission peaks
Transmission
0.8
0.6
0.4
0.2
1
1.5
Normalized
2
frequency
2.5
3
I nt ens ity, ar b . u nits
8
6
4
2
0
0
2
4
6
8
Structuredepth, mm
5
Intensity, arb. units
4
• 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.
3
2
1
0
0
2
4
6
Structuredepth,mm
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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
Structure 10000001
Structure 10001001
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.5
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1
1.5
2
0
0.5
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1
1.5
2
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Conservation of the number of peaks
Structure 10101001
Structure 10110111
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.5
1
1.5
2
0
Structure 10011001
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.5
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1
1.5
1
1.5
2
Structure 10111011
1
0
0.5
2
0
0.5
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1
1.5
2
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Conservation of the number of peaks
Structure 11100111
Structure 11111111
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.5
1
1.5
2
0
0.5
1
1.5
2
The number of peaks per period equals 8 for all
structures labeled by odd binary numbers
from 12910=100000012 to 25510=111111112
This leads to an additional requirement
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“Clearly defined boundary” condition
Material 0 is air:
10101
5 layers
10110
4 layers
Otherwise:
10101
10110
This boundary is
unclear
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Note that the number of peaks is
conserved only if the outermost
layers are those of the highest
index of refraction:
n1  nN  max  n j , n0  ,
j  2,3,
, N 1
• 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.
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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:
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.5
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1
1.5
2
0
0.5
1
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1.5
2
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Transmission / DOM
Density of modes
3
2.5
2
1.5
1
0.5
Transmission / DOM
0
0
0.5
1
1.5
2
Normalized
frequency
Normalized frequency
3
2.5
2
1.5
• 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:
dk () 1 y x  xy
   

, t  x  iy
2
2
d
D x y
1
0.5
0
0
0.5
1
1.5
2
Normalized
frequency
Normalized frequency
t is the complex transmission; D - total thickness
[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:
        v
(bulk)
,
v
(bulk)
N1n0  N0 n1

Nn1n0
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:
   0 ,
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      
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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:
    d   1
1
0
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:
I      v
2
(bulk )
0
d 
1
0

2 0
0
   v
(bulk )
d  
k  2 0 
k  0
v (bulk ) dk  
• The effective wave vector k is related to  by the
dispersion relation:
tan k   D  y   x    tan ,
t  x  iy  T ei
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,
  20   D (opt)  20 c  2  20  N
0
4
 N
Here, D(opt) is the total optical thickness of the structure
• Then, after simple algebra we arrive at I  2
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
Commensurate
multilayer
2 3
QW multilayer
Optical path
• 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.
• 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)
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