10 Feb 2014 No Comments

# What is the Difference between a Plane Wave and Collimated Field?

A Physics Stack Exchange User wonders whether Plane Waves and Collimated Optical Fields are indeed the same. The language around this point is often sloppy, so this is indeed a confusing point. It gets even more confusing when “collimated” is used in connexion with and optical fibre. Here’s how I think about it all.

In one sense you are right: the only free space “perfectly collimated” optical field is the plane wave in the sense that these are the only eigenfields of Maxwell’s equations, being fields which conserve their form under propagation and only undergo scaling by an eigenvalue in such propagation. Since Maxwell’s equations conserve energy in free space, propagation is represented by a unitary operator and the eigenvalue is therefore always the simple phase delay $e^{i\,\vec{k}\cdot \vec{r}}$. A wavefront of finite breadth must always undergo some “scrambling” by diffraction; Fourier analysis shows that it is always a superposition of plane waves with a spread of directions, so we can actually think of diffraction through a homogeneous medium as a three step process:

- Decompose the field at one infinite plane surface into a superposition of plane wave components through a Fourier transform;
- Impart the direction-dependent phase delay each of these constituent plane ways undergoes in crossing the homogeneous medium from infinite plane boundary to a parallel infinite plane boundary; and
- Reassemble the electromagnetic field from its constituent plane wave components at the output boundary. It is the fact that the phase delay is direction dependent (being $\delta\vec{z} \cdot \vec{k}$ radians where $\delta\vec{z}$ is parallel displacement between the input and output infinite planes) that scrambles the wavefront, thus begetting the phenomenon we call “diffraction”. So it is the propagation distance dependent interference between a field’s constituent plane waves.

So in this sense the only truly “collimated” field is indeed the plane wave. However the word “collimated” is to me a kind of “laboratory” word: one “collimates” a field, *i.e.* makes its phase front maximally flat, through a laboratory collimation procedure such as making adjustments to field-processing lenses and suchlike whilst the wavefront is observed through a suitable “collimation detector” such as a shear plate interferometer, wavefront sensor, wavefront camera or point diffraction interferometer. So a “collimated” field in this sense is one that has been flattened by such a procedure. Optically, a “perfectly” collimated field of this kind (i.e. with perfectly flat phasefronts but finite breadth) is actually the *focal plane field* of an electromagnetic field with an extremely small numerical aperture, with $NA$ of the order of $\lambda / w$ where $\lambda$ is the field’s wavelength and $w$ its transverse breadth. It diverges, its phasefronts take on curvature exactly as would any other focal plane field and the mathematics describing this process is precisely the same as for any focal plane field of course; it’s just that, owing to the extremely small numerical aperture, the divergence is very very slow with increasing axial distance from the focal plane.

For a Gaussian collimated field, the phase front at the focal plane is flat (as for any other collimated field), the phasefronts themselves are a family of ellipsoids and the integral curves of the rays (i.e. unit normals to the phasefronts) are the family of orthogonal hyperboloids (see the Wikipedia page for Gaussian beam).

An optical fibre is different again; here the matter of the fibre interacts with the electromagnetic field and there are many non-plane eigenfields (indeed all, even the radiation fields are non-plane). Again these fields are the ones that undergo only a simple scaling in propagation through the translationally invariant waveguide, and again, if the waveguide is made of lossless material, the scaling is a simple phase delay. The full tale is rather long winded: begin at chapter 11 of Snyder and Love, “Optical Waveguide Theory” (Chapman 1983) and keep reading! However you can get an intuitive feel for what’s going on by looking in more detail into the Beam Propagation Method. The finite difference finding of a solution of Maxwell’s equations is equivalent to solving a Master-like equation in the Lie group $U(N)$ for the square matrix $T \in U(N)$ which maps the input field to the output field with $N$ being of the order of the number of transverse grid points in the simulation (that’s an awfully big Lie group!) depending on whether a vector or scalar simulation is done:

$${\rm d}_z T(z) = K(z) T(z)$$

where $K(z) \in \mathfrak{u}(N)$ wanders around in the Lie algebra $\mathfrak{u}(N)$ of $N\times N$ skew-Hermitian matrices to represent the local action of diffraction and either focusing or de-focusing by the waveguide. A short, translationally invariant, length of waveguide has a transfer matrix of the form $\exp((D + L) z)$, where $D$ represents the diffraction through the “mean homogeneous” medium and $L$ the diffraction-free lensing of the waveguide’s local profile. The beam propagation method implements “operator splitting”, which is another way of saying splitting apart the effect of diffraction and lensing through the Trotter product formula of Lie theory:

$$\lim\limits_{m\to\infty}\left(\exp\left(D\,\frac{z}{m}\right)\,\exp\left(L\,\frac{z}{m}\right)\right)^m = \exp((D + L) z)$$

So the beam propagation method amounts to nothing more than a succession of diffractions through thin slivvers of homogeneous medium followed by the diffraction free lensing imparted by the equivalent local refractive index profile of each slivver. You can, with a bit of a stretch, imagine a step index refractive index profile to be a badly pixellated version of the smooth GRIN lens: the wave near the centre is delayed more by the higher refractive index of the core than the wave near the waveguide edges, so it tends to cancel out the divergence begotten of the pure diffraction of the foregoing slivver. It’s not too hard to imagine that for certain special field shapes this lensing, although arising from a “bad” lens, exactly cancels the diffraction and one has a waveguide mode whose phasefront stays perfectly flat and whose transverse profile perfectly conserved as it propagates through the waveguide.

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