Lifshitz formula by spectral summation method

The Lifshitz formula is derived by making use of the spectral summation method which is a mathematically rigorous simultaneous application of both the mode-by-mode summation technique and scattering formalism. The contributions to the Casimir energy of electromagnetic excitations of different types (surface modes, waveguide modes, and photonic modes) are clearly retraced. A correct transition to imaginary frequencies is accomplished with allowance for all the peculiarities of the frequency equations and pertinent scattering data in the complex $\omega$ plane, including, in particular, the cuts connecting the branch points and complex roots of the frequency equations (quasi-normal modes). The principal novelty of our approach is a special choice of appropriate passes in the contour integrals, which are used for transition to imaginary frequencies. As a result, the long standing problem of cuts in the complex $\omega$ plane is solved completely. Inconsistencies of some previous derivations of the Lifshitz formula are traced briefly. For completeness of the presentation, the necessary mathematical facts are also stated, namely, solution of the Maxwell equations for configurations under consideration, scattering formalism for parallel plane interfaces, determination of the frequency equation roots, and others.Comment: The version published in Phys. Rev. A 86, 052503 (2012); criticism of some previous derivations of the Lifshitz formula is diminished (Introduction) and emphasizing the authors achievements (Conclusion) is moderate

Standing waves for acoustic levitation

Standing waves are the most popular method to achieve acoustic trapping. Particles with greater acoustic impedance than the propagation medium will be trapped at the pressure nodes of a standing wave. Acoustic trapping can be used to hold particles of various materials and sizes, without the need of a close-loop controlling system. Acoustic levitation is a helpful and versatile tool for biomaterials and chemistry, with applications in spectroscopy and lab-on-a-droplet procedures. In this chapter, multiple methods are presented to simulate the acoustic field generated by one or multiple emitters. From the acoustic field, models such as the Gor'kov potential or the Flux Integral are applied to calculate the force exerted on the levitated particles. The position and angle of the acoustic emitters play a fundamental role, thus we analyse commonly used configurations such as emitter and reflector, two opposed emitters, or arrangements using phased arrays