lect5 - 1.138J/2.062J/18.376J, WAVE PROPAGATION Fall, 2006...

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1.138J/2.062J/18.376J, WAVE PROPAGATION Fall, 2006 MIT Notes by C. C. Mei CHAPTER FIVE MULTIPLE SCATTERING BY AN EXTENDED REGION OF INHOMOGENEITIES In this chapter we shall treat two types of extended inhomogeneities: (i) periodic and (ii) random. 1 Waves in a periodic medium References: C Kittel. Introduction to Solid State Physics . Wiley 1966 A. Yariv & P. Yeh. Optical Waves in Crystals . Wiley 1984 K Inoue, & K, Ohtaka, Photonic Crystals (Physics, Fabrication and Applications) , Springer 2004 J-M. Lourtioz, H Benisty, V. Berger, J-M G´ erard, D Maystre & A A.Tchelnokov, Photonic Crystals (Towards Nanoscale Photonic Devises) Springer, 2006. J. D. Joannopoulos, R. D. Meade & J. N. Winn, Photonic Crystals , Pinceton, 1995. −−−−−−−−−−−−− Propagation of light or sound wave is of long standing interest in several branches of basic and appied physics, from old disciplines such as x ray di±raction in crytallography, to the modern science of photonic crystals. Many problems in natural environment also involve wave propagaion in periodic media. For example, nearly periodic sand bars are freqeuently found in shallow seas outside the surf zone; their presence changes the wave climate near the coast. The technology of remote-sensing, either by underwater sound or by radio waves from a satellite, depends on our understanding of scattering by the wavy sea surface. 1
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Figure 1: Bragg resonance due to contructive interference In a periodic medium, the pheonomenon of Bragg resonance holds a special position. Let us explain it for the one-dimensional case of monochromatic water waves passing over a long stretch of parallel bars on an otherwise horizontal seabed. In nature, the bars are usually of much smaller amplitude than the wave depth. Most waves can pass over them without singi±cant reflection, except when the wavelength is an integral multiple of the bar period. In Figure1, the special case where λ =2 λ bar is sketched. Upon encountering each bar crest, every incident wave crest is mostly transmitted and slightly reflected. At a given bar crest, the height of the reflected wave crest is the sum of in±ntely many left-going crests. We called it the n th crest if its reflection is originated at the n -th bar crest on the right. Because of the 1 : 2 ratio, each crest has traveled the distance of 2 bar = since its ±rst passage over the bar crest in question. As a consequence, they are all in phase with one another, hence the sum of the reflected wave intensity is very high. We treat ±rst waves in an elastic laminate of in±nite extent. Borrowing the existing knowledge in solid-state physics, we discuss the relation between Bragg resonance and band gaps in the dispersion relation. We then treat the more practical case of ±nite extent in order to study the tranmission and reflection. The asymptotic method of 2
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multiple scales is applied to derive the coupled-mode equations for the slowly varying envelopes of incident and scattered waves.
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lect5 - 1.138J/2.062J/18.376J, WAVE PROPAGATION Fall, 2006...

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