conservation laws for the integrated density of states in arbitrary quarter-wave multilayer...

CONSERVATION LAWS FOR THE INTEGRATED DENSITY OF STATES

IN ARBITRARY QUARTER-WAVE MULTILAYER NANOSTRUCTURES

Sergei V. ZhukovskySergei V. Zhukovsky

Laboratory of NanoOptics

Institute of Molecular and Atomic Physics

National Academy of Sciences, Minsk, Belarus

[email protected]


2 of 22Nanomeeting 2003

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



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)]



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.



Quarter-wave multilayer structures

A sample multilayer:

The QW condition introduces the central frequency 0 as a natural scale of frequency normalization

A quarter-wave (QW) multilayer is such that

where N is the number of layers; 0 is called central frequency

nBnA

dBdA

A B 0

04 2

1,2, ,

cA A B B i in d n d n d

i N



Quarter-wave multilayer structures

The QW condition has two effects on spectral symmetry:

1. Spectral periodicity with period equal to 20 ( );

2. Mirror symmetry around odd multiples of 0 within each period ( )

0 1 2 3 4 5 6Normalized frequency

0.2

0.4

0.6

0.8

1

noissimsnar

TT

rans

mis

sion



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.

1010101012=34110

1101010012=42510

Periodic

Random

1100001012=32510

Fractal

[S. V. Gaponenko, S. V. Zhukovsky et al, Opt. Comm. 205, 49 (2002)]



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 states0 2 4 6 8

Structuredepth, m

0

1

2

3

4

5

y,tisnetnIbra.

stinu

0 2 4 6 8Structuredepth, m

0

2

4

6

8

y,tisnetnIbra

.stinu

1 1.5 2 2.5 3Normalized frequency

0.2

0.4

0.6

0.8

1

noissimsnar

T



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

0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1Structure 10000001

0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1Structure 10001001




0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1Structure 10101001

0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1Structure 10011001

0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1Structure 10110111

0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1Structure 10111011




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

0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1Structure 11100111

0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1Structure 11111111



“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.

1 0max , ,N jn n n n 2,3, , 1j N

Material 0 is air:

101015 layers

101104 layers

Otherwise:

10101 10110 This boundary is unclear



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:

0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1



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

2 2

( ) 1,

dk y x x yt x iy

d D x y

[J. M. Bendickson et al, Phys. Rev. E 53, 4107 (1996)]


0

0.5

1

1.5

2

2.5

3


0

0.5

1

1.5

2

2.5

3

Tra

nsm

issi

on /

DO

MT

rans

mis

sion

/ D

OM

Normalized frequency

Normalized frequency



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:

(bulk) (bulk) 1 0 0 1

1 0

,N n N n

v vNn n

0 ,



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.

1

01d



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

0 0

0

2 2 2(bulk) (bulk) (bulk)1

0 0 0

k

kv d v d v dk

I

tan tan , ik D y x t x iy T e



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.

0

0

(opt) 20 0 42 2 2D c N N

2I



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.




• 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




• 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

Commensurate multilayer

2 3

QW multilayer

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



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)

conservation laws for the integrated density of states in arbitrary quarter-wave multilayer...

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