Reflection and transmission in nonlocal susceptibility models with multiple resonances

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1. The authors derive exact expressions for the electromagnetic reflection and transmission coefficients at the planar boundary of a semi-infinite dielectric whose susceptibility carries multiple spatially dispersive resonances, extending the single-resonance treatment of Halevi and Fuchs. The surface enters through an arbitrary phenomenological reflection coefficient for the polarization waves, so that the various additional boundary conditions proposed in the literature — Pekar, Ting and coworkers, Fuchs-Kliewer, Rimbey-Mahan, Agarwal and coworkers — are recovered as particular values of one parameter.Abstract; Section II C, Reflection and Transmission Coefficients; Conclusions, Section VIII

2. Keeping the material’s spatial dispersion removes the unphysical divergence that the purely local model predicts for the spectral energy density of zero-point and thermal radiation near a surface. Instead of growing as one over the cube of the distance to the boundary, the computed energy density climbs and then saturates to a finite value as the distance goes to zero, confirming for multi-resonance media the result the authors found for one resonance.Section VII, Spectral energy density, the local divergent result Equation 64 and Figure 8 for ZnO at 3.44 electronvolts

3. Every resonance contributes two peaks to the spectral energy density outside the medium — one near the transverse frequency from the s-polarization part of the integral and one near the longitudinal frequency from the p-polarization part — so the three exciton bands of ZnO give six peaks in the spectrum measured 8 nanometres from the boundary. The size of every peak depends strongly on which additional boundary condition is assumed, largest for the Ting condition and smallest for the Pekar condition.Section VII, Figure 9 and the accompanying discussion

4. Multiple spatially dispersive resonances cannot be treated one at a time. Closely spaced bands suppress each other’s peaks — the ZnO peak at the first longitudinal frequency is cut down by the nearby second one until it is no larger than the s-polarization peaks — and for the heavy and light exciton bands of GaAs both single-exciton approximations underestimate the peak energy density, because at short distances the result is dominated by the evanescent-wave contribution where the two models differ most.Section VII, Figures 7, 10 and 11 and the closing paragraph of the section

5. For heavy and light exciton bands the authors give an improved single-band approximation, taking the mean of the square roots of the two band parameters rather than an effective single value; and they extend the model to exciton dispersion relations carrying a linear term in the wave vector, as uniaxial crystals have, where the splitting adds a further peak and turns the CdS spectral energy density into an overall three-peak structure with the crystal axis normal to the surface.Section V B, Heavy and Light Excitons; Section VI and Figure 12; Conclusions, Section VIII

6. The authors name the next extensions of the model: separate transverse and longitudinal susceptibilities, higher-order nonlocal terms, using the multi-band results to identify which additional boundary condition is appropriate for a medium with a complex exciton band structure, and applying the coefficients to the calculation of Casimir self-forces.Conclusions, Section VIII, final paragraph

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How to cite it

R. J. Churchill, T. G. Philbin (2017) Reflection and transmission in nonlocal susceptibility models with multiple resonances doi:10.1103/PhysRevB.95.205406

Where it sits in the curriculum

• Casimir physics and vacuum-force engineering
• What the vacuum is