The Limits of Conventional Optics
Standard focusing with lenses and microscope objectives cannot achieve efficient focusing to the nanoscale. This pages gives a mathematical framework for this.
Materials for Plasmonic Focusing Efficient Optical Coupling to the Nanoscale: PhD Dissertation

There are, of course, several problems implicit in focusing light to deeply subwavelength dimensions. These limits for homogenous media come directly from Maxwell’s Equations. We begin with Maxwell’s Equations^{[i]} in differential form
These equations simplify at optical frequencies, as the optical magnetic response is negligible, making . The free currents, represented by J_{f},cannot respond at optical frequencies, and they too are zero. The assumption of homogenous, isotropic linear media then allows us to eliminate the auxiliary field H. After some simplification, we arrive at:
In the equation above, n represents the index of refraction which is equal to the square root of the relative dielectric constant (e) of the medium. The electric field may now be decomposed into a complete basis set of plane waves^{[ii]}, with k_{i} representing the spatial frequency in the direction i.
This puts a fundamental limit on the achievable spatial wavevector, which is constrained principally by the low indices of refraction in conventional optical materials. In the regime of low loss in the visible band of the spectrum, indices of refraction top out around 1.9 with flint glass^{[iii]}. This limits the maximum wavevector to a spatial frequency of .
As is known in the art, the spot focus in the image plane can be represented as a Fourier transform of spatial frequencies. The above equations then set an upper limit on the frequency which thereby determines the minimum pitch at the image to be greater than l_{p}/2. The use of focusing optics in the regime of Fraunhoffer diffraction^{[iv]} drastically worsens the situation. As is typically the case with diffractive optics, a circular lens is used as the focusing element. This lens forms a circular aperture which acts as a low pass spatial filter with a maximum spatial frequency of nD/2l_{0}d_{i}. Here D represents the diameter of the lens and d_{i} is the distance from the lens to the image plane. With the high spatial frequencies cutoff, the circular optic creates an Airy disk^{[v]} in the image plane, as illustrated in Figures 12(a) and 12(b). The diffraction limited spot is now limited to a minimum diameter of approximately 1.22 l_{p}/NA where the numerical aperture (NA) is defined as n sin q. Although typically worse in practice, this then sets the minimum pitch to greater than 0.65l_{0}. Clearly, focusing to 1nm spot sizes is not possible using conventional focusing techniques at visible frequencies. To fulfill the promise of advanced microscopy and lowpower nonlinear optics, then, a new solution is needed.
Figure 1‑2: (a) Illustrates the diffraction limited spot size due to the clipping of the higher spatial frequencies. (b) shows the field pattern of an Airy disk.
[i] D.J.Griffiths, Introduction to Electrodynamics (Prentice Hall, Upper Saddle River, 1999). [ii] R. Zia, Guided Polariton Optics: A Combined Numerical, Analytical and Experimetnal Investigation of Surface Plasmon Waveguides, PhD dissertation, StanfordUniversity, 2006. [iii] Melles Griot Catalog, 2005. [iv] M. Born and E. Wolf, Principles of Optics (CambridgeUniversity Press, Cambridge, 2003). [v] G.B. Airy, Transactions of the Cambridge Philosophical Society, v. 5, p. 283 (1835). 