Robert D. Grober, Todd Rutherford, and Timothy D. Harris, "Modal approximation for the electromagnetic field of a near-field optical probe," Appl. Opt. 35, 3488-3495 (1996)
A formalism is given in which the optical field generated by a near-field optical aperture is described as an analytic expansion over a complete set of optical modes. This vectoral solution preserves the divergent behavior of the near field and the dipolar nature of the far field. Numerical calculation of the fields requires only evaluation of a well behaved, one-dimensional integral. The formalism is directly applicable to experiments in near-field scanning optical microscopy when relatively flat samples are evaluated.
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Summary of the Cylindrical Solutions to the Vector-Wave Equationa
Transverse Magnetic Field Modes
Transverse Electric Field Modes
Ez = k2ψ
Bz = 0
Ez = 0
Bz = k2ψ
ψn,k(ρ, θ, z > 0, t) = exp(inθ)Jn(kρ)exp(iκz − iωt)
Only solutions that are finite at the origin are considered. For these solutions, Jn(x) is a Bessel function of the first kind, − ∞ < n < ∞, k is the lateral spatial frequency, 0 < k < ∞, κ is the longitudinal spatial frequency, k2 + κ2 = k02 = (ω/c)2, and z > 0. Note that when k > k0, κ → iκ and the modes become evanescent. We obtain solutions for half-plane z < 0 by letting κ → −κ.
Table 2
Summary of the Electromagnetic Fields Calculated by Bouwkamp in the Plane of the Aperture (z = 0) When Driven by Incident FieldElectric Field
a
Electric Field Components
Magnetic Field Components
ρ/a = ξ < 1
Bρ = sin θ
Bθ = cos θ
Ez = 0
ρ/a = ξ > 1
Eρ = 0
Eθ = 0
Bz = 0
For the purposes of this paper, the square-root divergences and the azimuthal symmetries are the important aspects of the solution.
Table 3
Summary of the n = ±1 Basis Set of Table 1 that is Relevant to the Modal Analysis of This Papera
These modes have the same azimuthal symmetry as the solution summarized in Table 2.
Tables (3)
Table 1
Summary of the Cylindrical Solutions to the Vector-Wave Equationa
Transverse Magnetic Field Modes
Transverse Electric Field Modes
Ez = k2ψ
Bz = 0
Ez = 0
Bz = k2ψ
ψn,k(ρ, θ, z > 0, t) = exp(inθ)Jn(kρ)exp(iκz − iωt)
Only solutions that are finite at the origin are considered. For these solutions, Jn(x) is a Bessel function of the first kind, − ∞ < n < ∞, k is the lateral spatial frequency, 0 < k < ∞, κ is the longitudinal spatial frequency, k2 + κ2 = k02 = (ω/c)2, and z > 0. Note that when k > k0, κ → iκ and the modes become evanescent. We obtain solutions for half-plane z < 0 by letting κ → −κ.
Table 2
Summary of the Electromagnetic Fields Calculated by Bouwkamp in the Plane of the Aperture (z = 0) When Driven by Incident FieldElectric Field
a
Electric Field Components
Magnetic Field Components
ρ/a = ξ < 1
Bρ = sin θ
Bθ = cos θ
Ez = 0
ρ/a = ξ > 1
Eρ = 0
Eθ = 0
Bz = 0
For the purposes of this paper, the square-root divergences and the azimuthal symmetries are the important aspects of the solution.
Table 3
Summary of the n = ±1 Basis Set of Table 1 that is Relevant to the Modal Analysis of This Papera