The fan of plane waves inside every beam

Every front is a source began with Huygens’ rule: treat each point of a wavefront as a small source, and the wave further on is what those sources add up to. The rule needed repairs. The spiral that says how much light arrives added the phases, the wave that comes from the rim found the edge acting as a source of its own, and the backward wave Huygens had to remove found the obliquity factor and the quarter-period phase that the equation insists on and the geometry could not supply. In every version, propagation was an integral over the front.

There is a second description with exactly the same content, and it changes the kind of calculation. Instead of breaking a wavefront into points, break it into plane waves.

A field as a sum of plane waves #

Any field across a plane can be written as a sum of sinusoids across that plane, one for each spatial frequency — the Fourier decomposition that sharpness has to be paid for applied to time. In two dimensions, with $x$ across the beam and $z$ along it, a sinusoid $e^{ik_x x}$ across the plane is the trace of a plane wave crossing it at an angle whose sine is $k_x/k$, where $k$ is the wave’s own wavenumber. The decomposition of a field into spatial frequencies is therefore a decomposition into a fan of plane waves travelling in different directions: the field’s angular spectrum.

A plane wave’s future is trivial. Moving a distance $z$ along the axis, it gains a phase $k_z z$, with $k_z$ fixed by the requirement that the wavenumber’s components add to $k$:

$$k_z = \sqrt{k^2 - k_x^2}.$$

So propagating the whole field a distance $z$ is three steps: transform the field across the starting plane into its plane waves, multiply each by $e^{ik_z z}$, and transform back. No integral over the wavefront, no obliquity factor to be argued for, no approximation about small angles. The obliquity factor and the quarter-period phase that Huygens’ construction had to be given both come out of this square root automatically when the multiplication is turned back into an integral over the front.

Where the obliquity factor went #

The repairs that Huygens’ construction needed do not have to be supplied to the angular spectrum, and seeing why is the clearest way to see that the two descriptions are the same. Turn the multiplication back into an integral over the starting plane, and the kernel of that integral — the wave that a single point of the front sends out — is the transform of $e^{ik_z z}$ over all transverse wavenumbers. Carrying out that transform gives a wavelet whose amplitude falls with distance, whose phase is advanced by a quarter of a period relative to the field at its source, whose strength carries a factor of the wavelength, and whose amplitude varies with direction as the cosine of the angle from the axis. Those are the Rayleigh–Sommerfeld wavelet’s properties, and the cosine is the obliquity factor.

The backward wave disappears for a reason that is equally simple. The square root has two signs, and each describes a wave going one way along $z$. Propagating forward means choosing the positive root for travelling components and the decaying one for evanescent components — which is a statement about where the sources are, all of them behind the starting plane. Huygens’ spherical wavelets had no way to say that, so they radiated backwards as well as forwards; a sum of plane waves each committed to one direction cannot.

The waves that cannot travel #

The square root has a condition in it. For $|k_x| < k$ it is real and the component travels. For $|k_x| > k$ it is imaginary, and $e^{ik_z z}$ becomes $e^{-z\sqrt{k_x^2 - k^2}}$: the component does not travel at all, but decays away from the plane. A sinusoid across the plane finer than a wavelength cannot be the trace of any plane wave travelling in a real direction; it is an evanescent wave, the same kind of field that a complex angle of refraction describes at a surface.

A slit’s field, uniform across the opening and zero elsewhere, has a spectrum spread over a range of spatial frequencies inversely proportional to its width. A slit five wavelengths wide puts almost all of its power inside the travelling band; one a quarter of a wavelength wide puts more than half of it outside. That half is not scattered or absorbed — it simply never leaves the slit, decaying within a fraction of a wavelength into a field that carries no power away. Whatever it encoded about the slit’s edges is available only to something close enough to touch it.

Propagation is a multiplication #

Half a wavelength behind the slit, the intensity is the slit’s own shape with ripples at its edges — the edge waves of Fresnel’s construction, appearing without having been put in. At five wavelengths the ripples have spread inwards. At twenty-five the beam has developed a bright centre, and at a hundred it has spread into the pattern whose angular width is set by the slit’s width, the far-field pattern of Fraunhofer. All four come from one spectrum and one multiplication, differing only in the value of $z$.

The calculation checks itself in three ways. The travelling part of the field keeps its power exactly, because multiplying by a phase changes no amplitude. Running the field back by the same distance recovers the starting field’s travelling part to a part in $10^{13}$, because multiplying by the opposite phase undoes the first. And at a hundred wavelengths, where the angles involved are small, the result agrees with Fresnel’s integral over the aperture computed directly. Near field and far field are not two theories; they are two values of one variable.

Why a Fresnel number of one is where the pattern changes #

The multiplication also explains where the near field turns into the far field, without calling on Fresnel zones. A slit of width $a$ spreads its spectrum over transverse wavenumbers of order $2\pi/a$. For small angles the phase each component gains is $kz - k_x^2 z/2k$, so components across the spectrum drift out of step with one another by about $(2\pi/a)^2 z/2k$. While that drift is small, the spectrum’s components still add up to the slit’s own shape; once it reaches a radian or so, they no longer do, and the shape dissolves into a diffraction pattern.

Setting the drift to one gives $z \approx a^2/\lambda$, give or take a factor of order one, and $a^2/\lambda z$ is the Fresnel number. The boundary between near and far field is the distance at which the spread of the spectrum’s phases across its width becomes a radian — which is why a wider slit keeps its shape for a distance growing as the square of its width, and why the propagation figure’s beam is still recognisably a slit at five wavelengths and a diffraction pattern by a hundred.

A beam that does not spread #

The same argument predicts something that looks impossible. A field whose plane-wave components all have the same $k_z$ gains the same phase everywhere as it propagates, so its intensity pattern never changes: it does not diffract at all. In two dimensions the simplest such field is two plane waves crossing at equal and opposite angles, whose interference fringes march down the axis unchanged for as far as the waves extend.

In three dimensions the components with one $k_z$ lie on a ring of transverse wavenumbers, and the field they make is a Bessel beam, a bright central line surrounded by rings, which Durnin demonstrated in 1987 keeping its central spot narrow over distances at which a Gaussian beam of the same width would have spread many times over. The trick has a price the angular spectrum states at once. Every ring of the pattern carries about as much power as the centre, so an ideal non-diffracting beam carries infinite power, and a real one made from a finite aperture stays non-diffracting only over the distance where the crossing waves still overlap. Nothing escapes the square root; a field can only choose which of its consequences to spend.

Detail that does not survive a wavelength #

The evanescent part of the spectrum has a consequence that no amount of engineering can change, and the angular spectrum states it more plainly than any other description.

The instrument in the figure is as good as an instrument can be: it records the whole field across a plane at some height, amplitude and phase, and computes the field back at the emitters by undoing each component’s travel, with the only limitation that a component which has decayed below a hundredth of its starting size is lost in noise. At a twentieth of a wavelength, it keeps spatial frequencies up to nearly fifteen times $k$, and two emitters 0.35 wavelengths apart are two sharp peaks. At a quarter of a wavelength it keeps three times $k$, and they are still two. From two wavelengths it keeps barely more than the travelling band, and they are one peak. Emitters 0.8 wavelengths apart remain separate at every height.

The loss is not in the instrument. The components that distinguished the close pair decayed exponentially with distance before any instrument could reach them, and what a wavelength or more away is left with is the travelling band, whose finest spatial period is a wavelength and which cannot separate emitters much closer than half of one. How far apart two things have to be found the Rayleigh limit from a lens’s aperture; this is the limit at an aperture of the whole half-space, and it is why a microscope’s resolution cannot beat about half a wavelength however large its lens. Near-field microscopes beat it by putting a probe inside the evanescent zone, and a negative-index slab was proposed as a way to amplify the evanescent components back — both are attempts to catch the part of the spectrum that does not travel.

Where Fresnel’s approximation earns its place #

The square root is what makes the method exact, and replacing it is what makes the older theory approximate. Expanding $k_z \approx k - k_x^2/2k$ for small angles gives the Fresnel approximation, which turns the propagation integral into one that can be done by hand with Fresnel’s integrals and Cornu’s spiral.

The approximation is judged by where the light goes, not by the name of the regime. Each slit in the figure is viewed at the distance where its Fresnel number is one — conventionally squarely in Fresnel’s domain — and the approximation is off by 17.5 per cent for a slit two wavelengths wide and by 2.4 per cent for one sixteen wide. The error falls roughly as the inverse of the width, and the reason is the slit figure: the error lives in the light sent out at wide angles, where $k - k_x^2/2k$ is no longer close to the square root, and a sharp-edged slit’s share of power at wide angles falls as the wavelength over its width.

That is also the case for computing the exact version. On a computer, the exact multiplication and the approximate one cost exactly the same — both are one transform, one multiplication and one transform back — so the approximation’s advantage survives only where the calculation is to be done by hand.

Where it is used #

Nearly every computation of how a beam propagates in a uniform medium uses this method. Digital holography records a hologram on a camera and propagates the recorded field to any plane by multiplication, refocusing an image after it has been taken. Optical design software propagates beams between components with it, and the beam propagation method used for waveguides and fibres is the same idea applied in thin slices, alternating a multiplication in the plane-wave description with a correction for the medium’s variation in space.

The same mathematics works for any wave obeying the same equation. Near-field acoustic holography measures the sound pressure across a plane close to a vibrating panel, including the evanescent components that die before reaching a distant microphone, and propagates it back to image the panel’s vibration at detail finer than the sound’s wavelength. Seismic imaging propagates recorded waves back into the ground to locate the reflectors that produced them, and the image that is a diffraction pattern twice is Abbe’s theory of the microscope written in the same language of spatial frequencies passed and stopped.

Where the scalar picture stops #

The field was a scalar. Light is a vector field, and at large angles the polarisation components of a plane wave are not independent of its direction. For features of a few wavelengths and high numerical apertures the scalar result is wrong by amounts comparable to the paraxial errors in the last figure, and the vector angular spectrum is needed.

The medium was uniform. Multiplication by $e^{ik_z z}$ assumes the same $k$ everywhere. In a lens, a fibre or a turbulent atmosphere it is not, and the method has to be applied in slices with the medium’s effect added between them.

The slit was an ideal opening. The field just behind a real screen is not exactly one inside the opening and zero outside; the screen’s thickness and material change the field within a wavelength of its edges, which is precisely the region that decides the evanescent components.

And there was one transverse direction. A real aperture is two-dimensional, the transform is two-dimensional, and the travelling band is a disc rather than an interval. None of the conclusions change, but the numbers for a circular aperture differ from a slit’s.

What the figures leave out #

The resolution figure gives its ideal instrument a noise floor of one per cent in amplitude, and the heights at which the close pair is lost depend on that choice. A floor ten times lower would keep the pair resolved slightly further away — the decay is exponential, so a factor of ten in sensitivity buys a fixed extra distance of about a third of a wavelength divided by how far beyond $k$ the needed components lie — and no sensitivity keeps them a few wavelengths away.

The propagation figure scales each profile to its own peak, which hides how much dimmer the centre of the far pattern is than the field in the slit. The energy is conserved, as the figure’s check confirms, but it is spread over a width that grows in proportion to the distance.

Still open: how much band-limited light can be made to say #

A field made only of travelling components contains no spatial frequency above $k$, and it is tempting to conclude that it cannot vary faster than a wavelength anywhere. That conclusion is false. Band-limited functions can oscillate arbitrarily fast over a limited region — superoscillation — at the cost of enormous amplitude elsewhere, and focusing light into spots smaller than the diffraction limit has been demonstrated this way, with the spot surrounded by bright side lobes that carry almost all the energy.

How useful this can be is not settled. The sharper the superoscillating spot, the smaller the fraction of the power in it, and noise in the measurement grows correspondingly important. Whether superoscillation, or the computational methods that combine band-limited data with prior knowledge about what is being imaged, can recover information that is genuinely beyond the travelling band in realistic conditions — rather than detail that was in the band all along, disguised — is argued over with some heat.

The habit worth carrying away is to change the basis before doing the calculation. In a basis of plane waves, propagation through a uniform medium is a multiplication, and a problem that looks like an integral over a wavefront becomes, in that basis, a single square root whose real and imaginary parts divide what reaches the far field from what never leaves.