Skulls in the Stars

In previous optics basics posts, the interference of waves has played a major role.  When two or more monochromatic (single-color) waves are combined, they form a pattern of light and dark regions, in which the combined light fields have constructively or destructively interfered, respectively.  The simplest of these patterns can be created by the interference of two plane or spherical waves, and would appear as shown below:

One way of producing such a pattern is by Young’s two-pinhole (or double-slit) experiment, which we will have cause to discuss in more detail below.  The actual size of the interference pattern depends on the experimental setup, including the wavelength of the light, but can be easily made visible to the naked eye.

An astute observer of nature, however, will find something fishy about this whole discussion of interference: it does not seem to manifest itself in everyday experiences with light.  Sunlight streaming through a window, for instance, doesn’t interfere with the light emanating from a lamp inside the room.  Something is missing from our basic discussion of interference which explains why some light fields, such as those produced from a single laser source, produce interference patterns and others, such as sunlight, seemingly produce no interference.  The missing ingredient is what is known as optical coherence, and we discuss the basic principles of coherence theory and its relationship to interference below the fold.

Optical coherence refers to the ability of a light wave to produce interference patterns, such as the one illustrated above.  If two light waves are brought together and they produce no interference pattern (no regions of increased and decreased brightness), they are said to be incoherent; if they produce a ‘perfect’ interference pattern (‘perfect’ in the sense that regions of complete destructive interference exist), they are said to be fully coherent.  If the two light waves produce a ‘less-than-perfect’ interference pattern, they are said to be partially coherent.

Coherence is one of those topics which is best explained by starting with experimental observations.  We first describe two interference experiments, which highlight the two broad ‘types’ of coherence, and then explain in more detail the origins of the phenomena.

We first consider the Michelson interferometer, illustrated schematically below:

Light from a source S (which could be anything: sunlight, laser light, starlight) is directed upon a half-silvered mirror M₀, which sends 50% of the light towards mirror M₁ and 50% of the light towards mirror M₂.  The light is reflected from each of these mirrors, returns to the beamsplitter M₀, and equal portions of the light reflected from M₁ and M₂ is combined and projected on the screen B.  The interferometer is ‘tunable’, in that we can adjust the distance of the mirror M₁ from the beamsplitter.

The Michelson interferometer, in essence, interferes a light beam with a time-delayed version of itself.  The light which travels along the path to mirror M₁ must travel a distance 2d further than the light which travels along the path to mirror M₂.

What do we see on the screen?  When d = 0, we see a circular set of very clear interference fringes.  When d increases, however, the fringes becomes less distinct: the dark rings become brighter and the bright rings become dimmer.  Finally, for very large d, past some critical value D, the bright and dark rings have vanished completely, leaving only a diffuse spot of light:

Evidently, a light field cannot interfere with a time-delayed version of itself if the time delay is sufficiently large.  The distance 2D is referred to as the coherence length: interference effects are only noticeable when the path length difference is less than the coherence length.  This quantity can be converted into a coherence time T, by dividing by the speed of light c:

$$T = 2D/c$$.

The Michelson experiment measures the temporal coherence of a light wave: the ability of a light wave to interfere with a time-delayed version of itself.  Some typical values of coherence length:

• well-stabilized laser: coherence time, $$T sim 10^{-4} mbox{ seconds}$$, coherence length $$2D = 30mbox{ km}$$
• filtered thermal light: coherence time, $$T sim 10^{-8} mbox{ seconds}$$, coherence length $$2D = 3 mbox{ m}$$

We can use a different sort of interferometer — Young’s two-pinhole (double slit) interferometer — to measure another, distinct, type of coherence.  Young’s interferometer, discussed briefly in the basics post on interference, is illustrated below:

A light source is used to simultaneously illuminate a pair of small pinholes, separated by a distance d,  in an opaque screen.  The light emanating from the pinholes interferes on a second screen some distance away.  In a manner similar to that described for temporal coherence, we imagine increasing the separation d of the two pinholes.  For small values of d, one sees a distinct pattern of bright and dark interference fringes on the screen.  As d is increased, however, the fringes become less distinct: the dark lines become brighter and the bright lines become dimmer.  Finally, for sufficiently large d, past some critical value D’, no fringes are visible and only a diffuse spot of light remains on the screen.  This critical value D’ is referred to as the transverse coherence length of the light field.  More commonly, one refers to the coherence area of the light field as the area within which the pinholes must lie to observe interference.  This area is approximately

$$Asim pi D’^2$$.

Young’s interferometer measures the spatial coherence of a light field: the ability of a light field to interfere with a spatially-shifted version of itself.  It is clear that this cannot be just another manifestation of temporal coherence because the light arriving at the two pinholes has traveled the same distance from the source.  Typical values of the coherence area:

• Filtered sunlight: $$A sim 4 times 10^{-3} mbox{mm}^2$$,
• Filtered starlight: $$A sim 6 mbox{m}^2$$.

The coherence area of sunlight gives us an idea of why interference effects are not observed in everyday experience: interference can only be created with sunlight which lies within a spot roughly a tenth of a millimeter across!

Why are some light fields coherent and others incoherent?  Interference requires a definite phase relationship between the interfering fields.  If we look at the interference of two monochromatic waves as a function of time, we see that the waves undergo complete constructive interference if they are both ‘up’ simultaneously and both ‘down’ simultaneously:

The dashed lines show that the two waves are both up/down at the same time.  They therefore enhance each other, producing a larger wave.  Similarly, two waves undergo complete destructive interference if they have opposing amplitudes:

Realistic waves, however, are never perfectly monochromatic.  Ordinary thermal sources of radiation, such as a lightbulb or the Sun, consist of a large number of atoms radiating light almost independently of one another.  The field emanating from such a source has random fluctuations of amplitude and phase.  An example of a wavefield which is ‘almost’, but not truly, monochromatic is pictured below:

This wave contains what seem to be regular up-and-down motions but clearly has large fluctuations in the amplitude (size) of the wave.  Less clear is that the phase of the wave (the spacing of the peaks) also changes.  We can see this by superimposing a monochromatic field on top of this one:

At the point labeled ‘in phase’, the monochromatic wave and the random wave both go up and down together at the same time.  At the point labeled ‘out of phase’, however, the two waves are oscillating in the opposite direction. The spacing of the wavefronts only gradually changes, and for a significant period of time the random wave looks almost perfectly monochromatic.

Here is our explanation of coherence time: the coherence time of a wave is the duration over which the field looks approximately monochromatic.  As long as we interfere a wave with itself within the coherence time, it is effectively the same as interfering two monochromatic waves together.

The less monochromatic a wave is, the broader its spectrum.  A wavefield containing many colors, such as sunlight, therefore has a very low coherence time.  A measurement of the coherence time can be used to estimate the spectral width of a light wave, and vice versa.

What about spatial coherence?  Spatial coherence is a measure of the interference-causing capabilities of a wavefield at two different points in space.  To see what factors influence spatial coherence, let’s first imagine a single point source, producing light which is not necessarily monochromatic.  A point source produces spherical waves, as illustrated schematically below:

The wavefronts (the peaks of the wave, illustrated in blue) are not equally spaced for a non-monochromatic wave.  However, at points P₁ and P₂ which are equally distant from the source, the fields are identical.  If the wavefields have a definite relationship to each other — in other words, they are correlated — at those two points in space for all time, they can interfere with one another; this is illustrated schematically below:

The two examples on the left show fields which are correlated with one another: if they are combined using a Young’s interferometer, they will produce an interference pattern.  When the two fields ‘do their own thing’, however, as in the example on the right, they produce no interference pattern.  As an example of a source which produces an uncorrelated field, we imagine the field produced by two independent point sources.

In the figure on the left, the wavefronts from the red and blue sources arrive at points P₁ and P₂ at the same instant of time.  In the figure on the right, however, the wavefronts from the red and blue sources arrive at points P₁ and P₂ at different times, resulting in different fields at the two points.  In short, the field at P₁ changes independent of the field at P₂.  There is no definite phase relationship between the fields at the two points, and they will not produce an interference pattern when added together.

As a consequence of this argument regarding point sources, one can show that a larger (or closer) thermal source of light produces light of smaller coherence area than a smaller (or more distant) thermal source of light.  Sunlight therefore has low spatial coherence, while starlight has significantly higher spatial coherence.  This argument holds only for thermal light sources, i.e. collections of atoms/molecules radiating independently.  In lasers, which emit light by a very different physical process, light is typically spatially fully coherent.

Coherence theory can be thought of as the merging of electromagnetic theory with statistics, just as statistical mechanics is the merging of ordinary mechanics with statistics.  It is used to quantify and characterize the effects of random fluctuations on the behavior of light fields.  We have roughly divided coherence effects into two classes — spatial and temporal coherence — which are summarized by the following picture:

It is worth noting that we cannot typically measure the fluctuations of the wavefield directly.  The individual ‘ups and downs’ of visible light is not something that we can detect directly with our eyes, or even with sophisticated sensors: the frequency of visible light is on the order of $$10^{15}$$ cycles/second (a ‘1′ with fifteen zeros after it).  We only measure the average properties of the light field.

As a field of study, coherence has a number of applications.  It has been shown that partially coherent fields are less affected by atmospheric turbulence, which makes such fields useful in laser communications.  Partially coherent fields have been applied in studies of laser-induced fusion: the reduction of interference effects results in a ‘smoother’ beam impinging on the fusion target.  Coherence effects have been used in astronomy for a number of applications, notably to determine the size of stars and the separation of binary star systems.

Coherence is important in the study of quantum-mechanical fields as well as classical fields.  In 2005, Roy Glauber won half of the physics Nobel Prize “for his contribution to the quantum theory of optical coherence.”

There’s a lot more I could say about coherence (it is one of my specialties, actually), but we’ll leave further discussion for future posts.