*John Weiner and Frederico Nunes*

- Published in print:
- 2017
- Published Online:
- March 2017
- ISBN:
- 9780198796664
- eISBN:
- 9780191837920
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198796664.003.0009
- Subject:
- Physics, Atomic, Laser, and Optical Physics, Condensed Matter Physics / Materials

Chapter 9 introduces the notions of ‘left-handed materials’, negative-index metamaterials, and waveguides, and how they may be used to tailor light flow. The emphasis is on the basic physics of ...
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Chapter 9 introduces the notions of ‘left-handed materials’, negative-index metamaterials, and waveguides, and how they may be used to tailor light flow. The emphasis is on the basic physics of transmission and reflection in periodic stacked layers of alternating dielectric and metallic slabs. This geometry is the simplest implementation of fabricated anisotropic materials with engineered properties of transmission and reflection. Bloch waves, band gaps, band edges, Bragg reflection, and transmission by resonant tunnelling are introduced and developed using transition matrix theory (TMT) as the basic approach. The usefulness of effective medium theory (EMT) is evaluated for the implementation of metamaterials from stacked layers of conventional materials, and an alternative analysis, based on S-matrix theory, is proposed as a more useful design guide.Less

Chapter 9 introduces the notions of ‘left-handed materials’, negative-index metamaterials, and waveguides, and how they may be used to tailor light flow. The emphasis is on the basic physics of transmission and reflection in periodic stacked layers of alternating dielectric and metallic slabs. This geometry is the simplest implementation of fabricated anisotropic materials with engineered properties of transmission and reflection. Bloch waves, band gaps, band edges, Bragg reflection, and transmission by resonant tunnelling are introduced and developed using transition matrix theory (TMT) as the basic approach. The usefulness of effective medium theory (EMT) is evaluated for the implementation of metamaterials from stacked layers of conventional materials, and an alternative analysis, based on S-matrix theory, is proposed as a more useful design guide.

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter sets out basic definitions and features for the differential and integral calculus of c-number and Grassmann functions. Derivatives and integrals of functions of c-number variables are ...
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This chapter sets out basic definitions and features for the differential and integral calculus of c-number and Grassmann functions. Derivatives and integrals of functions of c-number variables are defined, involving c-numbers and their complex conjugates, and distinguishing analytic and non-analytic functions. Integration over the complex plane is defined. Cauchy–Riemann equations and Glauber identities for analytic functions are presented. Derivatives and integrals of functions of Grassmann variables are defined, involving both left and right differentiation and integration, and extended to cover multiple derivatives and integrals. Results for differentiating even or odd functions, product and chain rules for differentiation, rules relating higher-order derivatives, and Taylor series expressions for Grassmann functions are presented. Properties of Grassmann differentials are set out, and key results obtained for full integrals of Grassmann functions and their derivatives. Grassmann integration changes under linear transformations, and important integration-by-parts and Fourier results for Grassmann functions are obtained.Less

This chapter sets out basic definitions and features for the differential and integral calculus of c-number and Grassmann functions. Derivatives and integrals of functions of c-number variables are defined, involving c-numbers and their complex conjugates, and distinguishing analytic and non-analytic functions. Integration over the complex plane is defined. Cauchy–Riemann equations and Glauber identities for analytic functions are presented. Derivatives and integrals of functions of Grassmann variables are defined, involving both left and right differentiation and integration, and extended to cover multiple derivatives and integrals. Results for differentiating even or odd functions, product and chain rules for differentiation, rules relating higher-order derivatives, and Taylor series expressions for Grassmann functions are presented. Properties of Grassmann differentials are set out, and key results obtained for full integrals of Grassmann functions and their derivatives. Grassmann integration changes under linear transformations, and important integration-by-parts and Fourier results for Grassmann functions are obtained.