Mutual Coherence

Interference, the hallmark of wave phenomena, is responsible for some of the most striking effects in optics, from the iridescent colors of a soap bubble to the operation of a laser. However, the ability of waves to produce stable, observable interference patterns depends entirely on a crucial property: coherence. While the idealized cases of perfect coherence and complete incoherence are simple to conceptualize, most light sources in the universe—from distant stars to a common LED—exist in a complex middle ground of partial coherence. This article addresses the fundamental challenge of understanding and quantifying this "in-between" state, revealing it not as a defect but as a rich physical property that can be measured and engineered.

In the chapters that follow, we will embark on a two-part journey. First, under ​​Principles and Mechanisms​​, we delve into the core physics, defining the complex degree of mutual coherence and its relationship to interference fringe visibility. We will uncover the surprising origins of coherence with the van Cittert-Zernike and Wiener-Khinchin theorems. Following this, the ​​Applications and Interdisciplinary Connections​​ chapter will demonstrate the power of these concepts, showing how coherence is used to measure stars, optimize microscopes, and understand atmospheric effects, even providing a gateway to the quantum nature of light. We begin by building an intuition for what coherence truly means.

Imagine you are by a calm lake. If you dip two fingers into the water, side-by-side and in perfect rhythm, you will create a beautiful, stable pattern of ripples. Where crest meets crest, the water is high; where crest meets trough, the water is calm. This orderly pattern is ​​interference​​, and it arises because the two sources of ripples—your fingers—are acting in unison. They are ​​coherent​​.

Now, what if instead of your fingers, two random raindrops hit the water? Each creates its own ripples, but because they strike without any correlation, the resulting pattern on the water's surface is a chaotic, jumbled mess. You would not see a clear, stationary interference pattern. The sources are ​​incoherent​​.

This simple analogy captures the essence of coherence in light. It is the property of waves that enables them to produce stable, observable interference patterns. But the world is rarely so black and white. Most light is neither perfectly coherent nor perfectly incoherent. It exists on a spectrum, a state we call ​​partial coherence​​. Our mission here is to understand this "in-between" state, to quantify it, and to see how it arises from the most fundamental properties of light sources.

Let’s return to the classic double-slit experiment. When a wave passes through two slits, it creates two new wave sources. The intensity we see on a screen behind the slits is not simply the sum of the intensities from each slit alone. There is an additional "interference term." For waves that are not perfectly in sync, the observed intensity $I$ at any point on the screen is given by a wonderfully descriptive formula:

Here, $I_1$ and $I_2$ are the intensities we would measure if we covered up one slit at a time. The third term is where all the magic happens. The angle $\phi$ represents the phase difference between the two paths, which changes depending on where we look on the screen, creating the classic bright and dark fringes. But what is that new symbol, $\gamma_{12}$?

This is the ​​complex degree of mutual coherence​​. It is a measure of the correlation, the "in-sync-ness," of the waves from the two slits at the point of observation. Its magnitude, $|\gamma_{12}|$, is a number that ranges from 0 to 1.

• If $|\gamma_{12}| = 1$, the light is perfectly coherent. The waves interfere with maximum effect.
• If $|\gamma_{12}| = 0$, the light is completely incoherent. The interference term vanishes, and the total intensity is just $I_1 + I_2$, like with the random raindrops.
• If $0 \lt |\gamma_{12}| \lt 1$, the light is partially coherent. Interference still happens, but it's muted.

We quantify the "quality" of an interference pattern using a concept called ​​fringe visibility​​, $V$, defined as:

where $I_{max}$ and $I_{min}$ are the maximum and minimum intensities of the fringes. A high-contrast pattern where the dark fringes are truly dark has a visibility near 1. A washed-out, low-contrast pattern has a visibility near 0. Looking at our intensity equation, you can see that $|\gamma_{12}|$ directly controls the amplitude of the oscillatory part. When the individual intensities from each slit are equal ($I_1 = I_2$), the relationship is beautifully simple: the visibility is equal to the degree of mutual coherence, $V = |\gamma_{12}|$. If the intensities are unequal, or if there is background stray light, the visibility is reduced further, but it still remains proportional to $|\gamma_{12}|$. So, measuring fringe visibility is, in essence, measuring coherence.

You might have wondered why we call $\gamma_{12}$ the complex degree of coherence. We've seen that its magnitude, $|\gamma_{12}|$, determines the contrast of the fringes. But since it's a complex number, it also has a phase, or argument: $\gamma_{12} = |\gamma_{12}| e^{i\alpha}$. What does this phase $\alpha$ represent?

It turns out it represents an intrinsic, or "built-in," phase difference between the two wave sources. Let's look at the full interference term again, this time including the phase of $\gamma_{12}$:

where $\phi_{geom}$ is the phase difference due to the geometric path difference from the slits to the screen. Notice how the phase of the coherence, $\alpha$, simply adds to the geometric phase. The effect is profound yet simple: it causes a physical shift of the entire fringe pattern on the screen. The position of the central bright fringe, which would normally be at the point of zero path difference, is displaced. The visibility, which depends only on $|\gamma_{12}|$, remains unchanged.

So, the complex number $\gamma_{12}$ elegantly encodes two distinct physical properties: its ​​magnitude​​ tells us the degree of correlation (fringe contrast), and its ​​phase​​ tells us about any inherent phase lag between the sources (fringe position).

Where does spatial coherence come from? It's easy to imagine it from a perfect, laser-like source. But what about a more realistic source, like a star or a glowing filament? These are vast collections of countless atoms, each emitting light independently—the very definition of an incoherent source. It would seem that the light from such a source should be completely incoherent everywhere. But this is not true! In one of the most beautiful and surprising results in optics, we find that the very act of propagation can create coherence.

This is the essence of the ​​van Cittert-Zernike theorem​​. In plain English, it states that the ​​spatial coherence of light in the far field is related to the Fourier transform of the intensity distribution of the source​​.

Let's build our intuition. Imagine a "toy star" made of just two independent, monochromatic point sources, like two tiny, flickering light bulbs separated by a distance $d$. If we look at two points $P_1$ and $P_2$ on a very distant screen, what is the mutual coherence $\gamma_{12}$ between them? A calculation, which is really just careful bookkeeping of path lengths, reveals a stunning result. The magnitude of the coherence between two points separated by $\Delta x$ is:

where $R$ is the distance to the screen and $\lambda$ is the wavelength. The coherence is not zero! It varies across the screen in a periodic pattern. The light from our two incoherent sources has "gained" coherence simply by traveling through space. This is the principle behind stellar interferometry, where astronomers use widely separated telescopes to measure the tiny separation of double stars by analyzing the coherence of the starlight.

What happens if our source isn't two points, but a continuous disk, like a real star? The van Cittert-Zernike theorem gives us the answer. If the source has a certain size and shape, the theorem allows us to calculate the "coherence patch" on a distant screen. For example, for a circular or Gaussian-shaped star of a certain width, the coherence is high for two points close together but drops off as they move apart. A key insight from this Fourier relationship is an inverse scaling: the ​​larger the incoherent source, the smaller the area of coherence​​ in the far field. This is why it's impossible to do a double-slit experiment with a large household light bulb—the coherence area is microscopically small—but it is possible with a distant star, which, despite its immense physical size, has a tiny angular size, creating a broad area of coherence here on Earth.

So far, we have discussed ​​spatial coherence​​: the correlation between waves at two different points in space at the same instant. But there is another flavor: ​​temporal coherence​​. This refers to the correlation between a wave at one point in space at one time, and the wave at the same point at a later time. It asks: "How well can a wave interfere with a delayed copy of itself?"

This is precisely what is measured by an instrument like the Mach-Zehnder interferometer. Light is split, sent down two paths of different lengths, and then recombined. The path difference $\Delta L$ creates a time delay $\tau = \Delta L / c$. The visibility of the resulting interference fringes as we vary this delay tells us about the temporal coherence of the source.

And here, we find another moment of beautiful unity. The relationship between temporal coherence and the properties of the source is governed by the ​​Wiener-Khinchin theorem​​, which is the temporal twin of the van Cittert-Zernike theorem. It states that the ​​temporal coherence of light is the Fourier transform of its power spectral density​​ (i.e., its range of colors).

This leads to a similar inverse relationship:

• A purely monochromatic source (a single frequency, narrow spectrum) has an infinite ​​coherence time​​. It can interfere with a copy of itself no matter how long the delay.
• A source with a broad range of frequencies (like white light) has a very short coherence time. The visibility of interference fringes drops off rapidly as the path difference increases.

We can see this in action. For a source whose spectrum is a mix of two distinct frequencies, the visibility function itself shows a "beat" pattern, a direct consequence of the two frequencies interfering with each other in the Fourier domain. The coherence function carries a complete fingerprint of the source's spectrum.

The coherence can also be degraded by the medium it travels through. For instance, passing a perfectly coherent plane wave through a screen with random, fluctuating thickness (like looking through turbulent air) will scramble the wavefront and reduce its spatial coherence. The final coherence depends on the statistical properties of both the source and the path it has taken.

Ultimately, the concept of mutual coherence is a powerful tool. It allows us to connect the fuzzy, macroscopic phenomenon of interference fringe contrast to the microscopic details of a light source—its size, shape, and spectral composition. It can even be used in clever ways to measure the correlation between different polarizations of light, turning a property of wave correlation into a directly measurable intensity change. It is a testament to the fact that even in the seemingly random flicker of an incoherent source, there is an underlying order waiting to be revealed by the elegant physics of wave propagation.

In our last discussion, we journeyed into the heart of what it means for light to be coherent. We saw that the concept of mutual coherence is not just an abstract mathematical tool, but the very essence of light's ability to create the exquisite and delicate patterns of interference and diffraction. It's the invisible thread that links different parts of a wavefront, dictating how they will conspire to produce brightness and darkness. Now, having grasped the principles, we are ready for the real fun. We are going to see what this idea is good for. Where does it show up in the world? How does it affect the starlight we see, the images in our microscopes, or the signals in our fiber-optic cables?

You will find, as is so often the case in physics, that a deep principle, once understood, illuminates an astonishingly broad landscape of phenomena, from the cosmic to the microscopic. Let us begin our tour.

Let's return to our old friend, Young's double-slit experiment. In an idealized world, with a perfect point source of light, we get a beautiful, crisp set of interference fringes. The visibility—the contrast between the brightest brights and the darkest darks—is perfect. But in the real world, sources are not infinitesimal points. A lightbulb filament has a size, the sun has a size, even a distant star, as we will see, is not a perfect point.

What happens if we illuminate our double slits with a more realistic source, say, a glowing, narrow line of hot gas? According to the van Cittert-Zernike theorem, which we have met before, each point on this extended source tries to produce its own interference pattern. Since the points on a hot, incandescent source are all jiggling independently, their patterns don't quite line up. The result? The final pattern on the screen becomes a washed-out, blurry version of the ideal.

The further apart we place our two slits, the more pronounced this blurring becomes. The "threads of correlation" from an extended source simply don't stretch that far. If the slits are close together, they are illuminated by light that is still highly correlated, and we see good fringes. As we move the slits apart, the correlation between the light they receive drops, and the fringe visibility fades. For a line source of width $W$ at a distance $L$, this fading follows a particularly elegant mathematical form, the sinc function, $\left| \frac{\sin(\alpha d)}{\alpha d} \right|$, where $d$ is the slit separation and $\alpha$ is a factor depending on the source geometry and wavelength. If the source has a "softer" edge, perhaps with a Gaussian intensity profile, the coherence it produces also falls off in a Gaussian manner, leading to fringe visibility that decays exponentially as the slits are separated.

This is the first and most direct application of coherence: it explains the imperfect patterns we see in the real world. The degree of spatial coherence, quantified by the mutual coherence function, is a direct measure of the "quality" of our light for the purpose of creating interference.

Now for a wonderfully clever twist. If the size and shape of the light source determine the coherence of the light, could we perhaps work backward? Could we measure the coherence of the light to figure out the size and shape of the source? The answer is a resounding yes! This idea revolutionized astronomy.

Imagine we are looking at what appears to be a single star, but we suspect it might actually be a binary system—two stars orbiting each other closely. How could we tell? We can't resolve them with a normal telescope. But we can measure the coherence of their combined light. We set up an interferometer, which is essentially a glorified version of Young's apparatus with movable mirrors instead of fixed slits. We start with the mirrors close together and see clear interference fringes. Then, we slowly increase the separation between our mirrors, say a distance $s$.

As the separation $s$ increases, the visibility of the fringes will change. For a specific separation, the light from the first star might be telling our two mirrors to make a bright fringe, while the light from the second star tries to make a dark fringe at the same spot. If the intensities are equal, they cancel out perfectly, and the fringes vanish entirely! By measuring the separation $s$ at which this first happens, we can directly calculate the angular separation of the two stars in the sky. This is the principle behind the Michelson stellar interferometer, which in 1920 was used to measure the angular diameter of the star Betelgeuse—the first time the size of a star (other than our sun) was ever directly measured. We used the "blurriness" of the light's coherence as a ruler to measure the cosmos.

Coherence is not just a passive property of light from natural sources; it is a parameter we can engineer and control in modern technology. Two domains where this is particularly powerful are fiber optics and microscopy.

Consider this remarkable idea: can we create coherence where there is none? Suppose you have a "messy" light source, like the face of a light-emitting diode (LED), which is spatially incoherent. Can you turn this into a clean, spatially coherent beam? It turns out you can, using the right kind of optical fiber. If you send this incoherent light into a special type of fiber called a graded-index (GRIN) fiber, something magical happens. The fiber's refractive index is highest at the center and decreases toward the edge. This structure guides light rays in undulating paths. After propagating a specific distance—a "quarter-pitch" length—the fiber effectively performs a Fourier transform on the light field at its input. Due to the deep connection between coherence and Fourier transforms embodied in the van Cittert-Zernike theorem, this process converts the spatial intensity distribution of the incoherent source at the input into a spatial coherence distribution at the output. An initially jumbled, incoherent field emerges with a beautifully ordered spatial coherence. We can literally manufacture coherence.

In the world of the very small, coherence is just as crucial. One might think that for the best microscope image, one should use the most coherent light possible, like a laser. But this often produces images riddled with distracting speckles and artifacts. Conversely, completely incoherent illumination can wash out fine details. The sweet spot is often somewhere in between: partially coherent illumination. By adjusting the aperture of the condenser lens that illuminates the sample, a microscopist is, in effect, tuning the coherence area of the light—the small patches within which the light waves are well-correlated. Tailoring this coherence allows for enhancing the contrast of specific features, making transparent biological cells or subtle material defects pop into view. Controlling coherence becomes a knob for optimizing what we can see.

Let's return to the stars. A star is so incredibly far away that the light arriving at Earth from it is, for all practical purposes, a perfect plane wave. A plane wave has perfect spatial coherence; the field is perfectly correlated across the entire wavefront. So, if we put our double-slit apparatus in this plane wave, we should see perfect fringes, no matter how far apart we place the slits.

But this isn't what happens. We all know that stars twinkle, and when astronomers take pictures with large telescopes, the image of a star is not a perfect point but a fuzzy, dancing blob. Why? The culprit is the Earth's atmosphere. Our air is not a placid, uniform medium. It is a turbulent soup of swirling pockets of air at different temperatures and densities, and therefore different refractive indices.

As the perfect, coherent plane wave of starlight passes through this turbulent layer, different parts of the wavefront are delayed by slightly different, random amounts. The perfectly flat wavefront becomes corrugated and bent. The "unseen threads" of coherence are stretched and snapped. This process severely degrades the spatial coherence of the light by the time it reaches the ground. The twinkling we see with our eyes is the result of these corrupted wavefronts sweeping past our pupils, causing rapid fluctuations in brightness. For a large telescope, this loss of coherence is what fundamentally limits its resolving power, a phenomenon known as "atmospheric seeing." This is the great challenge for ground-based astronomy, which has led to the incredible technology of adaptive optics—systems that measure the incoming wavefront's distortion in real-time and use deformable mirrors to correct it, effectively trying to restore the light's lost coherence.

So far, we have seen how coherence affects the images we see. But can we think of coherence itself as something that can be imaged and transformed? In modern statistical optics, we do exactly that. When a beam of partially coherent light passes through a lens system, the system doesn't just form an image of the intensity; it also transforms the mutual coherence function in a predictable way. For example, a special type of beam known as a Gaussian Schell-model beam, which has both a Gaussian intensity profile and a Gaussian coherence function, will emerge from an imaging system as another Gaussian Schell-model beam, but with its size and coherence properties magnified and scaled by the system. This means that coherence is a physical property of the beam that propagates and transforms right along with the light's intensity and phase.

Finally, our journey into coherence leads us to the doorstep of the quantum world. All the interference phenomena we’ve discussed—the visibility of fringes—are related to what we call first-order coherence, which measures correlations in the electric field. But there is another, deeper level: second-order coherence, which measures correlations in the intensity of the light itself.

If you point a fast detector at a chaotic thermal light source, like a star or an incandescent bulb, you’ll find that its intensity is not constant but fluctuates rapidly. In a famous experiment, Hanbury Brown and Twiss showed that these intensity fluctuations are themselves correlated. This "intensity interferometry" led to the definition of the normalized second-order coherence function, $g^{(2)}(0)$. For chaotic thermal light, an amazing and fundamental relationship, known as the Siegert relation, connects the two worlds: $g^{(2)}(0) = 1 + |g^{(1)}(0)|^2$. We know that the maximum possible fringe visibility we can get from a source is $V_{\text{max}} = |g^{(1)}(0)|$. For thermal light, one can achieve $V_{\text{max}}=1$, which implies $g^{(2)}(0) = 2$. This value, 2, means that photons in thermal light have a tendency to arrive in "bunches."

This is profoundly different from an ideal laser, for which $g^{(2)}(0) = 1$ (photons arrive randomly), or a true single-photon source used in quantum computing, for which $g^{(2)}(0) = 0$ (photons always arrive alone). Therefore, measuring not just the fringe visibility but also the intensity statistics gives us a powerful tool to probe the very nature of light and distinguish between different kinds of light sources. The concept of coherence, which began as a simple descriptor for interference, ultimately becomes a gateway to the fascinating statistical and quantum character of light itself.