Wires and fibres: emitters beside a cylinder and beams through it

The plane-wave problem of the previous section has a single axial wavenumber, fixed by the angle of incidence. Two other sources the calculator offers break that simplicity in instructive ways. A point dipole radiates in every direction at once, so its field contains every axial wavenumber $k_z$ and every order $m$ ; the cylinder's response must be integrated over $k_z$ , and the poles of that integrand are the guided modes of the wire or fibre. A focused beam is a bundle of plane waves filling a cone, and a cylinder under an infrared microscope answers each of them differently; the spectrum the instrument records is the cone average.

A dipole beside a cylinder

Expand the dipole's field in cylindrical waves $e^{im\varphi} e^{ik_z z} H^{(1)}_m(q\rho)$ : unlike the plane wave, which has one $k_z$ , the dipole needs all of them, with $q = \sqrt{k^2 - k_z^2}$ passing from real (waves that propagate away from the axis) to imaginary (waves bound to it) as $|k_z|$ grows past $k$ . For each $(m, k_z)$ the cylinder's transfer matrices give the reflected cylindrical wave at the dipole's position, with the two polarizations coupled whenever $k_z \ne 0$ . The power the dipole radiates, relative to free space, is an integral over $k_z$ and a sum over $m$ of the work the reflected field does on the dipole, the direct analogue of the Chance–Prock–Silbey integral for a dipole near a plane (Klimov and Ducloy give the complete formulas for an atom beside a nanofibre).

The integrand has three kinds of contribution, and PyStratify separates them because they are what an experiment sees separately:

$|k_z| < k_h$ in the host: cylindrical waves that radiate away to the far field;
poles of the reflected amplitude at real $k_z = n_\mathrm{eff} k_0 > k_h$ : the guided modes of the fibre (HE $_{11}$ first, which has no cutoff, then TE $_{01}$ , TM $_{01}$ , HE $_{21}$ , … as the fibre grows), each taking a share of the dipole's power that then travels along the fibre without radiating. In a lossless fibre these poles sit on the real axis and the residues are taken analytically, so the guided fraction does not depend on a quadrature grid; PyStratify checks the modes it finds against the fibre's closed-form dispersion relations (Snyder and Love);
the rest of the evanescent range, which against an absorbing wire deposits power in the metal (the quenching channel) and against a lossless one contributes nothing to the total but shapes the near field.

A dipole oriented radially, azimuthally or axially couples to different orders $m$ and different modes, so the three orientations have different rates, and an isotropic ensemble averages them. Close to a metal wire the quenching rises as $d^{-3}$ , as it does near a plane, but the enhancement at the wire's plasmon is sharper, because a thin wire is a nearly one-dimensional resonator.

The thin metal wire, and where the local model stops

The case-II resonance of a thin wire sits where $\varepsilon(\omega) = -\varepsilon_h$ , that is at $\omega_p/\sqrt{2}$ for a Drude metal in vacuum and $\omega_p/\sqrt{3}$ in a host of $\varepsilon_h = 2$ . Ruppin computed the extinction of a sodium wire one nanometre in radius in two ways: with the local Drude permittivity, which gives exactly this resonance, and with a hydrodynamic (nonlocal) electron gas, in which the pressure of the electron fluid pushes the resonance to higher frequency and lets longitudinal bulk plasmons appear above $\omega_p$ . The calculator reproduces the local curve; the nonlocal correction is a different constitutive relation, and it matters only when a dimension shrinks to a few nanometres or an emitter approaches a surface that closely. For gold and silver particles above about 10 nm the local model is quantitatively right, and the local curve is the one to compare an experiment with first.

Classic paper, computed live

A 1 nm sodium wire

R. Ruppin, Opt. Commun. 190, 205 (2001), doi:10.1016/S0030-4018(01)01063-X. Fig. 1, dashed curve: classical extinction of a sodium wire of radius 1 nm.

Extinction efficiency (log scale) of a 1 nm Na wire in vacuum, light polarized across the wire, from 156 to 545 nm (ω/ωp = 1.4 to 0.4).

Inputs: Drude sodium as printed (ωp = 8.65e15 s⁻¹, γ = 0.01 ωp), as an uploaded table.
Agreement: Resonance at ωp/√2 and peak log₁₀ C_e = 0.65, as in Fig. 1; the paper's solid curve adds a hydrodynamic correction the calculator does not model. 5 of 5 checks within tolerance
Try: Put the wire in a medium of ε = 2 (n = 1.414): the resonance moves to ωp/√3, as in the paper's Fig. 4.

Open in calculator Recomputation script

A fibre under a Cassegrain

An infrared microscope illuminates its sample through a Cassegrain objective: a cone of plane waves between an inner obscuration and the outer numerical aperture, each arriving at a different angle $\zeta$ to a fibre lying in the focal plane. For every angle the oblique-incidence solution of the previous section gives the field transmitted towards the detector; the instrument's reading is the incoherent average over the cone of the power collected within its own aperture, and dividing by the empty-field reading gives an apparent absorbance. Davis, Carney and Bhargava showed that this apparent absorbance of a fibre that absorbs nothing is far from zero: light is refracted and scattered out of the collection cone in a way that depends on wavelength through the size parameter, so a textile fibre or a hair shows a broad, oscillating baseline with dips where the fibre acts as a lens. Real absorption bands ride on this baseline, and a quantitative spectrum needs the baseline computed and removed, not fitted by a polynomial. This is the physics the calculator's focused (Cassegrain) source implements for films, cylinders and spheres; the sphere version is the Mie-type scattering that distorts the infrared spectra of cells, met in the Mie theory section.

Classic paper, computed live

An FT-IR microscope looking at a fiber

B. J. Davis, P. S. Carney and R. Bhargava, Anal. Chem. 83, 525 (2011), doi:10.1021/ac102239b. Fig. 5a: transmission through a non-absorbing fiber of radius 5 µm, n0 = 1.477.

Apparent absorbance of a fiber that absorbs nothing, under a Cassegrain (NA 0.1–0.4) from 4000 to 800 cm⁻¹: the refraction-and-scattering baseline that hides real bands.

Inputs: Fiber n = 1.477 (Fig. 5 caption), radius 5 µm, in air. The paper does not state Fig. 5's aperture; NA 0.4 with a 0.1 obscuration (its Fig. 2) reproduces the shape.
Agreement: Transmission minimum 9.9% near 1120 cm⁻¹, 77% near 2100 cm⁻¹ and 42% near 3000 cm⁻¹, against ~10%, ~76% and ~42% read off Fig. 5a. 6 of 6 checks within tolerance
Try: Double the radius to 10 µm, as in Fig. 5b: the deep minimum near 1100 cm⁻¹ disappears.

Open in calculator Recomputation script

Two parameters of the instrument matter and are often left out of a paper's caption: the numerical aperture, which sets the widest angle in the cone, and the central obscuration of the Cassegrain, which removes the near-normal rays. Both change the baseline's shape; Davis's Fig. 5 is reproduced with the aperture of the same paper's Fig. 2. The calculator exposes both, together with the fibre's radius, so the three can be varied against a measured spectrum.

References

V. V. Klimov and M. Ducloy, Phys. Rev. A 69, 013812 (2004), doi:10.1103/PhysRevA.69.013812
A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983), ch. 12–13, doi:10.1007/978-1-4613-2813-1
R. Ruppin, Opt. Commun. 190, 205 (2001), doi:10.1016/S0030-4018(01)01063-X
B. J. Davis, P. S. Carney and R. Bhargava, Anal. Chem. 82, 3474 (2010), doi:10.1021/ac902067p
B. J. Davis, P. S. Carney and R. Bhargava, Anal. Chem. 83, 525 (2011), doi:10.1021/ac102239b
C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), ch. 8, doi:10.1002/9783527618156