Electromagnetic Eigenmodes in Matter. Van Der Waals-London and Casimir Forces

Apostoł, M.; Vaman, G.

doi:10.2528/pierb09111904

Cited by 5 publications

(5 citation statements)

References 36 publications

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“…The summation of zero-point energies of the discrete modes can be accomplished, obviously, in a straightforward way, while for the summation of such energies appertaining to scattering states, one has to take advantage of the scattering formalism [34]. Preceding derivations of the Lifshitz formula in the framework of the mode-by-mode summation alone or by making use of the scattering formalism only seem somewhat contradictory at first 1 The electromagnetic field is coupled, through the Maxwell equations, with charges and currents, therefore one can take, as a dynamic variable, the local displacement of a continuous charged liquid that describes the free electrons inside the medium [28]. 2 In the general case, the spectrum of the differential operator may be more complicated [32].…”

Section: Introductionmentioning

confidence: 99%

Lifshitz formula by a spectral summation method

Nesterenko

Pirozhenko

2012

Phys. Rev. A

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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 ω 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 ω plane is solved completely. Some subtleties and vague points in previous derivations of the Lifshitz formula are elucidated. 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.1 The electromagnetic field is coupled, through the Maxwell equations, with charges and currents, therefore one can take, as a dynamic variable, the local displacement of a continuous charged liquid that describes the free electrons inside the medium [28]. 2 In the general case, the spectrum of the differential operator may be more complicated [32]. However, in the Casimir calculations such problems do not occur. 3 In physical problems dealing with the plane boundaries, the surface modes and waveguide modes are called sometimes the guided modes [33].

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Section: Introductionmentioning

confidence: 99%

Lifshitz formula by a spectral summation method

Nesterenko

Pirozhenko

2012

Phys. Rev. A

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“…We can see that the polarization is given by P ¼ Àenu. This representation of the electric polarization turned out to be very useful in dealing with electromagnetic phenomena in macroscopic bodies, including reflection, refraction, surface plasmons, and polaritons in a half-space (semi-infinite solid), 38,39 a slab, 40 electromagnetic coupling between two half-spaces 41 or the Mie scattering of the electromagnetic field by a sphere. 42 The displacement u obeys Newton's law of motion.…”

Section: Mutually Polarized Bodiesmentioning

confidence: 99%

A generalization of the dipolar force

Apostoł¹,

Ilie²,

Petrut³

et al. 2012

Journal of Applied Physics

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The static dipolar force is generalized to time-dependent classical distributions of dipoles and electromagnetic fields. This force may exhibit a remarkable resonance character for induced dipoles, related to the pole structure of the polarizabilities. The resonance phenomenon is illustrated for two macroscopic polarizable bodies, with mutually induced polarizations, using the well-known Lorentz-Drude model for the dielectric response with optical dispersion and a characteristic (resonance) frequency. Specifically, the calculations are performed for distances much longer than the dimension of the bodies ("point-like" bodies), but shorter than the characteristic wavelength (sub-wavelength, stationary, near-field regime). The polarizations are induced via a localized external field acting upon only one body. The force is practically vanishing for distinct substances and acquires a non-vanishing value for identical substances. It falls off as the 7-th power of the distance, being reminiscent of the van der Waals-London force. The conditions of validity of this resonance phenomenon are emphasized. Particular cases corresponding to independent external fields or two isolated, interacting bodies (closed system) are also analyzed, with similar conclusions regarding the resonance character of the force. V

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“…We have therefore u 0 lm ∼ j l (k 1 r), where the coefficient is determined from Equation (20). Making use of Equation (B2) we get…”

Section: External Plane Wave the Field Inside The Spherementioning

confidence: 99%

“…We take the second derivative in Equation (20) with respect to r and eliminate the intervening integrals by using Equation (20) and its first derivative with respect to r. Then, we use Equations (B6) and (B7) to get…”

Section: External Plane Wave the Field Inside The Spherementioning

confidence: 99%

“…For such bodies with finite boundaries, we get coupled integral equations for the components of the displacement vector, which we solve by using suitable sets of orthogonal functions. In addition, we have also calculated the eigenmodes of such equations and derived van der Waals-London and Casimir forces acting between a pair of semi-infinite bodies [20]. We apply herein this treatment to an ideal spherical metallic particle, and compute the field everywhere in space, as was the original aim of Mie theory.…”

Section: Introductionmentioning

confidence: 99%

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Plasmons and Diffraction of an Electromagnetic Plane Wave by a Metallic Sphere

Apostoł¹,

Vaman²

2009

PIER

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Abstract-The diffraction of a plane electromagnetic wave by an ideal metallic sphere (Mie's theory) is investigated by a new method. The method represents the charge disturbances (polarization) by a displacement field in the positions of the mobile charges (electrons) and uses the equation of motion for the polarization together with the electromagnetic potentials. We employ a special set of orthogonal functions, which are combinations of spherical Bessel functions and vector spherical harmonics. This way, we obtain coupled integral equations for the displacement field, which we solve. In the nonretarded limit (Coulomb interaction) we get the branch of "spherical" (surface) plasmons at frequencies ω = ω p l/(2l + 1), where ω p is the (bulk) plasma frequency and l = 1, 2, . . .. When retardation is included, for an incident plane wave, we compute the field inside and outside the sphere (the scattered field), the corresponding energy stored by these fields, Poynting vector and scattering cross-section. The results agree with the so-called theory of "effective medium permittivity", although we do not start the calculations with the dielectric function. In turn, we recover in our results the well-known dielectric function of metals. We have checked the continuity of the tangential components of the electric field and continuity of the normal component of the electric displacement at the sphere surface, as well as the conservation of the energy flow and re-derived the "optical theorem". In the limit of small radii (in comparison with the electromagnetic wavelength) the sphere exhibits a series of resonant absorptions at frequencies close to the plasmon frequencies given above. For large radii these resonances disappear.Corresponding author: M. Apostol (apoma@theory.nipne.ro). 98Apostol and Vaman

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Electromagnetic Eigenmodes in Matter. Van Der Waals-London and Casimir Forces

Cited by 5 publications

References 36 publications

Lifshitz formula by a spectral summation method

Lifshitz formula by a spectral summation method

A generalization of the dipolar force

Plasmons and Diffraction of an Electromagnetic Plane Wave by a Metallic Sphere

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