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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 Nanomeeting 2003 Institute of Molecular and Atomic 2 of 22 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)] Nanomeeting 2003 Institute of Molecular and Atomic 3 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 4 of 22 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 Nanomeeting 2003 Institute of Molecular and Atomic 5 of 22 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 ( ) Transmission Transmission 1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 Normalized frequency Nanomeeting 2003 Institute of Molecular and Atomic 6 of 22 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)] Nanomeeting 2003 Institute of Molecular and Atomic 7 of 22 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 Nanomeeting 2003 8 Thus, the number of peaks per unit interval can be viewed as discrete density of electromagnetic states Institute of Molecular and Atomic 8 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 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 Nanomeeting 2003 1 1.5 2 0 0.5 Institute of Molecular and Atomic 1 1.5 2 9 of 22 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 Nanomeeting 2003 1 1.5 1 1.5 2 Structure 10111011 1 0 0.5 2 0 0.5 Institute of Molecular and Atomic 1 1.5 2 10 of 22 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 Nanomeeting 2003 Institute of Molecular and Atomic 11 of 22 “Clearly defined boundary” condition Material 0 is air: 10101 5 layers 10110 4 layers Otherwise: 10101 10110 This boundary is unclear Nanomeeting 2003 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. Institute of Molecular and Atomic 12 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: 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.5 Nanomeeting 2003 1 1.5 2 0 0.5 1 Institute of Molecular and Atomic 1.5 2 13 of 22 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)] Nanomeeting 2003 Institute of Molecular and Atomic 14 of 22 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 N 0 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 , Nanomeeting 2003 Institute of Molecular and Atomic 15 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: 1 0 d 1 This conservation rule holds for arbitrary quarter-wave multilayer structures. Nanomeeting 2003 Institute of Molecular and Atomic 16 of 22 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 Tei Again, t is the complex transmission, and D is the total physical thickness of the structure Nanomeeting 2003 Institute of Molecular and Atomic 17 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, 20 D(opt) 20 c 2 20 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. Nanomeeting 2003 Institute of Molecular and Atomic 18 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 19 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 20 of 22 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. Nanomeeting 2003 Institute of Molecular and Atomic 21 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) Nanomeeting 2003 Institute of Molecular and Atomic 22 of 22