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

Nanomeeting 2003 Institute of Molecular and Atomic Physics 2 of 22 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

Nanomeeting 2003 Institute of Molecular and Atomic Physics 3 of 22 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)]

Nanomeeting 2003 Institute of Molecular and Atomic Physics 4 of 22 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.

Nanomeeting 2003 Institute of Molecular and Atomic Physics 5 of 22 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 nBnB nAnA dBdB dAdA A B

Nanomeeting 2003 Institute of Molecular and Atomic Physics 6 of 22 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

Nanomeeting 2003 Institute of Molecular and Atomic Physics 7 of 22 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)]

Nanomeeting 2003 Institute of Molecular and Atomic Physics 8 of 22 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

Nanomeeting 2003 Institute of Molecular and Atomic Physics 9 of 22 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

Nanomeeting 2003 Institute of Molecular and Atomic Physics 10 of 22 Conservation of the number of peaks

Nanomeeting 2003 Institute of Molecular and Atomic Physics 11 of 22 Conservation of the number of peaks The number of peaks per period equals 8 for all structures labeled by odd binary numbers from = to = This leads to an additional requirement

Nanomeeting 2003 Institute of Molecular and Atomic Physics 12 of 22 “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: layers layers Otherwise: This boundary is unclear

Nanomeeting 2003 Institute of Molecular and Atomic Physics 13 of 22 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:

Nanomeeting 2003 Institute of Molecular and Atomic Physics 14 of 22 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 [J. M. Bendickson et al, Phys. Rev. E 53, 4107 (1996)] Transmission / DOM Normalized frequency

Nanomeeting 2003 Institute of Molecular and Atomic Physics 15 of 22 DOM and frequency normalization DOM can be made dimensionless by normalizing it to the bulk velocity of light in the structure: N 0 and N 1 being the numbers, and N 0 and N 1 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:

Nanomeeting 2003 Institute of Molecular and Atomic Physics 16 of 22 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.

Nanomeeting 2003 Institute of Molecular and Atomic Physics 17 of 22 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

Nanomeeting 2003 Institute of Molecular and Atomic Physics 18 of 22 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.

Nanomeeting 2003 Institute of Molecular and Atomic Physics 19 of 22 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.

Nanomeeting 2003 Institute of Molecular and Atomic Physics 20 of 22 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

Nanomeeting 2003 Institute of Molecular and Atomic Physics 21 of 22 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 Commensurate multilayer 2 3 QW multilayer For incommensurate multilayers, this interval is infinite. Integration is to be performed over the whole spectrum.

Nanomeeting 2003 Institute of Molecular and Atomic Physics 22 of 22 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)