# Optics basics: Coherence

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:

.

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, , coherence length
• filtered thermal light: coherence time, , coherence length

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

.

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: ,
• Filtered starlight: .

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 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.

• stuwat says:

Having a good grasp of coherence is key to understanding the principles behind many powerful tools which use optical interference. Optical interferometry allows us to make highly accurate measurements of distance, temperature, pressure, strain and velocity, to name but a few applications. Your blog is fast becoming a great resource for learning these fundamentals and I recommend it to anyone wanting to learn more about optics.

• meichenl says:

thanks for another great post!

i've heard it's possible to make two different lasers coherent for short periods. how difficult is that and how long can they be coherent? is it the same order of magnitude as the temporal coherence time for a single laser?

• meichenl: That's a great question, and the answer is yes - it can be done! I have to go track down the original paper again for the details, which I think will turn into another 'classic science papers' post.

In short, the work was done by Leonard Mandel in the 1960s, and the results stirred a significant amount of debate. In a great example of quantum weirdness, he and a collaborator later showed that one could get the two beams to interfere even if the light intensity was so low that only one photon was typically present in the interference pattern!

More to come...

• P.S. My intuition says that the fields will only interfere, as you've suggested, over times on the order of the coherence time.

Thanks for another excellent post! I do have a minor question about the discussion of coherence length in the Michelson interferometer, though.

I thought if you use a laser as your light source, then as you increase d, the interference fringes would disappear when 2d = 0.5 lambda * n, for n >= 1, due to destructive interference when the two beams are half a wavelength out of phase at the detector.

The explanation of the coherence length made it sound like the fringes just get gradually worse, without any sort of periodicity. Is this effect not visible under normal experimental conditions?

Thanks for any clarification you might offer -- I'm probably just confused about something.

• Wade: Thanks for the comment! I think you're mixing up the conditions for destructive interference (2d = 0.5 lambda + 2pi n) with the conditions for any sort of interference (2d < coherence time). My picture of the rings is produced in the Michelson interferometer is a little misleading: as d is increased, one will see the rings 'move': the light and dark rings will move outward, and new ones will appear in the center. This is due to the change in the specific conditions for constructive/destructive interference. However, one will gradually see this pattern vanish, i.e. the visibility of the fringes will decrease: this is the coherence effect. How 'quickly' this pattern vanishes is dependent on the bandwidth of the light, while the shape of the interference pattern is dependent on the center frequency.

In other words, "coherence" refers to the overall ability of the light field to produce constructive or destructive interference. When I talk about coherence, I'm talking about the presence or absence of an entire fringe pattern, rather than the presence or absence of an individual bright or dark fringe.

I think I understand my problem now -- I was thinking of what would happen at a single point on the screen, instead of to the entire interference pattern.

A single point on the screen should see a periodic variation in light intensity as d is adjusted, due to the rings in the interference pattern moving across it as they spread out from the center.

Then as d is made larger, the pattern as a whole will gradually fade out as the coherence length is approached.

Does this sound right to you?

• RWS says:

The paper you're looking for is by Magyar and Mandel from Nature in 1963. And it's worth noting that if you're working with sufficiently fast detectors, the coherence properties are rather meaningless (you trade the speed for time-averaging).

• Wade: Yep, that sounds about right. At a single point on the screen, as d is increased, one would see the intensity oscillate but the oscillations would gradually 'flatline' to a constant value of intensity.

RWS: Thanks for the comment! I actually got the Magyar paper via my precious ILL yesterday, and will hopefully discuss it in a future post, as it is quite interesting. One of the lessons of the paper, I would say, is an illustration that coherence properties necessarily involve averaging: fields always interfere over short time periods, but that interference is only visible over long times if the fields are coherent.

• juana says:

I have one question about the partially coherent beam
I read some stuff about partially coherent beam and geting rid that partially coherent beams are less effected by air turbulent but my problem is that how can I show MATHEMATICALL RELATIOSHIP BETWEEN PHASE, AMPLITUDE AND REFRACTIVE INDEX OR THE DECAY OF INTENSITY that will give me evedence that there is less effect on atmospheric turbulence

• juana wrote: "how can I show MATHEMATICALL RELATIOSHIP BETWEEN PHASE, AMPLITUDE AND REFRACTIVE INDEX OR THE DECAY OF INTENSITY that will give me evedence that there is less effect on atmospheric turbulence"

That's a tricky question to answer; mathematically, one looks at the scintillation index (fluctuations of intensity) of the wavefield on propagation. Conceptually, I tend to explain the 'resistance' of partially coherent beams as follows: A partially coherent beam consists of a large number of mutually incoherent 'modes' which propagate independently through the turbulence. At the detector, one sees the contributions from many modes at once. Because these modes are incoherent, they don't interfere with one another and produce a relatively uniform intensity pattern. Relatively uniform intensity low scintillation index.

• juana says:

Thank you for your insight .
you give me one step ahead to break the queestion a little further

• Alex says:

So I'm fuzzy on electron interference. The idea, quantum mechanically, is that a single electron when shot at two slits actually interferes with itself right? It takes all possible paths, including bouncing off the the divider containing the two slits. My question is, if detectors were set up in a 360 degree frame, shouldn't you, after shooting one electron, also get a electron behind the cannon, or above, along with the electron detected by the plate in front of the slits. Mainly my question is how can the electron exist simultaneously in two states, interfere with itself but not show up also as one of the infinite other possibilities at the same time in another place?

• Alex: It's a fair question. I think the resolution of your question, if I understood it correctly, is as follows:

A single electron, though it has wave properties, is still only detected at a single location effectively as a 'point' particle. The electron interference pattern only builds up on average as many, many electrons are sent through the system; I sketched a picture of this in my 'basics' post here.

The wave nature of any particular electron does include the possibilities of the electron being reflected/absorbed by the screen; if it does so, though, the electron is not detected in the interference pattern and does not contribute to it.

So, in essence, the electron acts like a wave up until the point that it 'chooses' to take a stand and be detected somewhere, which is the idea of the collapse of the wavefunction in quantum mechanics. You may justifiably ask: when does the electron 'choose' to take a stand? This is, in essence, one of the great unsolved problems in physics, so I can't really give an answer!

• Alex says:

So I understand it travels as a wave, but when shot one at a time, it creates an interference pattern, meaning somehow the particles are interfering with themselves correct?

• Alex wrote: "but when shot one at a time, it creates an interference pattern, meaning somehow the particles are interfering with themselves correct?"

Well, yes and no! 🙂 We never see the wave of an individual electron; as I said, the electron is detected as a point-like particle. If you were to do the experiment with a single electron, you wouldn't see any reason to believe that an electron has wave-like properties.

What the wave function represents, roughly, is the probability of finding the electron at a particular point in space. This `probability wave' is what creates an interference pattern, but we don't see this pattern directly -- the electron is finally detected only at one place. When we send many electrons through the same double slit interferometer, however, the screen acts as a histogram, counting how many electrons appear at a particular point on the screen. We indirectly see the probability wave by making the relationship: 'large # of electrons = high probability', 'small # of electrons = low probability'.

This is actually a very subtle point (and, as I pointed out, still not perfectly understood); I may come back and write a more detailed post on this in the near future.

• Andrew says:

Have you seen the remarkable paper by Lindner et al (quant-ph/0503165) that describes observations of quantum interference in time? Horwitz analyses the results in Phys Lett A v355 (2006) 1-6.

He write on p2 to the effect that e/m waves can interfere in time since they obey 2nd order d.e.'s, but Schrodinger particles cannot since they obey 1st order d.e.'s. (I do accept his earlier argument that the recombined electron state is a mixed state, thus no interference).

Well, the Schrodinger eq is a special case of the Pauli eq which is a low energy approx to the Dirac eq which looks 1st order but the dispersion relation is as for the Klein-Gordon eq, that is, second order. This seems quite a separate matter from choosing the retarded propagator.

Can you relate differential order and temporal coherence, please?

Thanks

• Tom says:

Thanks! This is one of the most concise explanations of coherence I have found. Kudos on avoiding equations! (I took Wolf's graduate modern coherence class).

• You're welcome! I still remember taking Wolf's first midterm -- and basically leaving the correlation function completely out of one of my solutions.

• tm says:

I liked the article but didnt find what I was looking for, the coherence time of sunlight. I found an article giving a solar radiation coherence time of 10^-12 sec, but I dont see how they got it. Also, does the solar radiation coherence time vary from the radio through the IR and visible ? thanks.

• Formally, coherence time is the inverse of the spectral bandwidth of the radiation -- if you can find a number for the spectral bandwidth of sunlight, you'll have the coherence time.

It's a little subtle, however, I imagine, because one could also talk about an "effective" coherence time. Because sunlight is so broadband, there are hardly any detectors that are sensitive to all frequencies! This means that the coherence time might seem effectively larger than it "physically" is.

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