*et al*, [Nat. Commun. **4**, 1845 (2013)](http://dx.doi.org/10.1038/ncomms2842). L. L. DeRaad Jr. and K. A. Milton, [Ann. Phys. **136**, 229 (1981)](http://dx.doi.org/10.1016/0003-4916(81)90097-X). P. Gosdzinsky and A. Romeo, [](http://dx.doi.org/10.1016/S0370-2693(98)01164-2). K. A. Milton, A. V. Nesterenko, and V. V. Nesterenko, [](http://dx.doi.org/10.1103/PhysRevD.59.105009). E. K. Abalo, K. A. Milton, and L. Kaplan, [](http://dx.doi.org/10.1103/PhysRevD.82.125007). J. P. Straley, G. A. White, and E. B. Kolomeisky, [](http://dx.doi.org/10.1103/PhysRevA.87.022503). For a review, see – K. A. Milton, [](http://dx.doi.org/10.1088/0305-4470/37/38/R01). J. D. Jackson, *Classical Electrodynamics*, 3rd Ed., Sec. 8.2 & 8.6 (New York: John Wiley, 1999). S. Chowla and A. Selberg, [Proc. Natn’l. Acad. Sc. (USA) **35**, 371 (1949)](http://dx.doi.org/10.1073/pnas.35.7.371). For higher orders of the asymptotic expansion, see – J. McMahon, [Ann. Math. **9**, 23 (1895)](http://dx.doi.org/10.2307/1967501). For the cylindrical cavity modes ($\phi_{p_z,\sigma,n}(z,\rho,\theta)$) of the scalar field inside the circular-cylinder, specially for Dirichlet boundary condition, see – S. Bhattacharyya and J. K. Bhattacharjee, [](http://dx.doi.org/10.1103/PhysRevB.60.R746). For Neumann boundary condition, Bessel zeros ($\alpha_{\sigma,n}$) in $\phi_{p_z,\sigma,n}(z,\rho,\theta)$ are to be replaced by the zeros of the first derivative ($\beta_{\sigma,n}$) of the Bessel functions of the first kind. See – Refs. [@Jackson; @Gosdzinsky] for similar replacement specially for electromagnetic field inside a conducting circular-cylinder. D. C. Roberts and Y. Pomeau, [](http://dx.doi.org/10.1103/PhysRevLett.95.145303); [arXiv:cond-mat/0503757](http://arxiv.org/abs/cond-mat/0503757). T. D. Lee, K. Huang, and C. N. 