In the classical electromagnetic theory the wave-vector k = (2π/λ)σ underlies the Fourier space of propagating (or radiative) fields. The k-vector combines into a single entity the wavelength λ and the unit vector σ that signifies the beam's propagation direction. The Fourier transform relation between the three-dimensional space of everyday experience and the space of the wave-vectors (the so-called k-space) gives rise to relationships between the two domains analogous to Heisenberg's uncertainty relations.

Considering that in quantum theory the electromagnetic k-vector is proportional to the photon's momentum (p = ħk, where ħ = h/2π, h being the Planck constant), one should not be surprised to find relationships between dimensions of a beam in the X Y Z-space and its momentum spread in the k-space. Such relationships impose fundamental limits on the ability of measurement systems to determine the various properties of electromagnetic fields.

In this chapter we address two problems that have widespread applications in optical metrology, spectroscopy, telecommunications, etc., and discuss the constraints imposed by the uncertainty principle on these problems. The first topic of discussion is the separation of two overlapping beams of identical wavelength having slightly different propagation directions. This will be followed by an analysis of the limits of separating co-propagating beams having slightly different wavelengths.

Angular separation and the limit of resolvability

Figure 19.1 shows an aperture of diameter D, which transmits two plane waves of the same wavelength λ propagating in slightly different directions.