Interference and Coherence

1Introduction¶

Although the model of geometrical optics helps us to design optical systems and explains many phenomena, there are properties of light that require a more elaborate model. For example, interference fringes observed in Young’s double-slit experiment or the Arago spot (Figure 3) indicate that light is more accurately modeled as a wave.

In this chapter we will study the wave model of light. It will be shown that the extent to which light can show interference depends on a property called coherence. For most of the discussion we will assume that all light has the same polarization, so that we can treat the fields as scalar. In the last section we will look at how polarization affects interference, as described by the Fresnel-Arago laws.

It is important to note that the concepts of interference and coherence are not just restricted to optics. Since quantum mechanics dictates that particles have a wave-like nature, interference and coherence also play a role in e.g. solid state physics and quantum information.

2Interference of Monochromatic Fields of the Same Frequency¶

Let us first recall the basic concepts of interference. What causes interference is the fact that light is a wave, which means that it not only has an amplitude but also a phase. Suppose for example we evaluate a time-harmonic field at two points

Here $\varphi$ denotes the phase difference between the fields at the two points. If $\varphi=0$, or $\varphi$ is a multiple of $2\pi$, the fields are in phase, and when they are added they interfere constructively

However, when $\varphi=\pi$, or more generally $\varphi=\pi+ 2 m \pi$, for some integer $m$, then the waves are out of phase, and when they are superimposed, they interfere destructively.

We can sum the two fields for arbitrary $\varphi$ more conveniently using complex notation:

Adding gives

For $\varphi= 2 m \pi$ and $\varphi=\pi+2 m \pi$ we retrieve the results obtained before. It is important to realize that what we see or detect physically (the ‘brightness’ of light) does not correspond to the quantities $\mathcal{U}_1$, $\mathcal{U}_2$. After all, $\mathcal{U}_1$ and $\mathcal{U}_2$ can attain negative values, while there is no such thing as ‘negative brightness’. What $\mathcal{U}_1$ and $\mathcal{U}_2$ describe are the fields, which may be positive or negative. The ‘brightness’ or the irradiance or intensity is given by taking an average over a long time of $\mathcal{U}(t)^2$ (as discussed in Chapter 1). We shall omit the factor $\sqrt{\epsilon/\mu_0}$. As explained in Chapter 1, we see and measure only the long time-average of $\mathcal{U}(t)^2$, because at optical frequencies $\mathcal{U}(t)^2$ fluctuates very rapidly. We recall the definition of the time average over an interval of length $T$ at a specific time $t$ from Chapter 1:

where $T$ is a time interval that is the response time of a typical detector, i.e. $T\approx 10^{-6}\,\text{s}$ which is extremely long compared to the period of visible light which is of the order of $10^{-15}\, \text{s}$. For a time-harmonic function, the long-time average is equal to the average over one period of the field and hence it is independent of the time $t$ at which it is taken. Indeed for Eq. (5) we get

Using complex notation one can obtain this result more easily. Let

where

Then we find

hence

To see why this works, recall the time averaging formula and choose $A=B=U_1+U_2$.

Remark. To shorten the formulae, we will omit in this chapter the factor $1/2$ in front of the time-averaged intensity.

Hence we define $I_1=|U_1|^2$ and $I_2=|U_2|^2$, and we then find for the time-averaged intensity of the sum of $U_1$ and $U_2$:

where $\phi_1$ and $\phi_2$ are the arguments of $U_1$ and $U_2$ and $\phi_1-\phi_2$ is the phase difference. The term $2\text{Re}[U_1^* U_2]$ is known as the interference term. In the famous double-slit experiment (which we will discuss in a later section), we can interpret the terms as follows: let us say $U_1$ is the field that comes from slit 1, and $U_2$ comes from slit 2. If only slit 1 is open, we measure intensity $I_1$ on the screen, and if only slit 2 is open, we measure $I_2$. If both slits are open, we would not measure $I_1+I_2$, but we would observe fringes due to the interference term $2\text{Re}[U_1^* U_2]$.

The intensity Eq. (12) varies when the phase difference varies. These variations are called fringes. The fringe contrast is defined by

It is maximum and equal to 1 when the intensities of the interfering fields are the same. If these intensities are different the fringe contrast is less than 1.

More generally, the intensity of a sum of multiple time-harmonic fields $U_j$ all having the same frequency is given by the coherent sum

However, we will see in the next section that sometimes the fields are unable to interfere. In that case all the interference terms of the coherent sum vanish, and the intensity is given by the incoherent sum

3Coherence¶

In the discussion so far we have only considered monochromatic light, which means that the spectrum of the light consists of only one frequency. Although light from a laser often has a very narrow band of frequencies and therefore can be considered to be monochromatic, purely monochromatic light does not exist. One reason that light can not be perfectly monochromatic is that any source must have been switched on a finite time ago. Hence, all light consists of multiple frequencies and therefore is polychromatic. Classical light sources such as incandescent lamps and also LEDs have relatively broad frequency bands. The question then arises how polychromatic light behaves differently from idealized monochromatic light. To answer this question, we must study the topic of coherence. One distinguishes between two extremes: fully coherent and fully incoherent light, while the degree of coherence of practical light is somewhere in between. A broader frequency band generally gives a shorter temporal coherence time. Spatial coherence also depends on source size and geometry. It is a very important observation that no light is actually completely coherent or completely incoherent. All light is partially coherent, but some light is more coherent than others.

An intuitive way to think about these concepts is in terms of the ability to form interference fringes. For example, with laser light, which usually is almost monochromatic and hence coherent, one can form an interference pattern with clear maxima and minima in intensities using a double slit, while with sunlight (which is incoherent) this is much more difficult. Every frequency in the spectrum of sunlight gives its own interference pattern with its own frequency dependent fringe pattern. These fringe patterns wash out due to superposition and the total intensity therefore shows little fringe contrast, i.e. the coherence is less. However, it is not impossible to create interference fringes with natural lightYoung, 1804. The trick is to let the two slits be so close together (of the order of $0.02~\text{mm}$) that the difference in distances from the slits to the sun is small enough for the fields in the slits to be sufficiently coherent to interfere. For one point source, a path difference determines the time delay between fields at two observation points. For an extended source, distance matters as well: it changes the angular size of the source and thus the spatial coherence.

3.1Coherence of Light Sources¶

In a conventional light source such as a gas discharge lamp, photons are generated by spontaneous emission with energy equal to the energy difference between certain electronic states of the atoms of the gas. These transitions have a duration of the order of 10^-8 to $10^{-9} \, \text{s}$. Because the emitted wave trains are finite, the emitted light does not have a single frequency; instead, there is a band of frequencies around a center frequency with width roughly equal to the reciprocal of the duration of the wave train. This spread of frequencies is called the natural linewidth. Random thermal motions of the molecules cause further broadening due to the Doppler effect. In addition, the atoms undergo collisions that interrupt the wave trains and therefore further broaden the frequency spectrum.

We first consider a single emitting atom. When collisions are the dominant broadening effect and these collisions are sufficiently brief, so that any radiation emitted during the collision can be ignored, an accurate model for the emitted wave is a steady monochromatic wave train at frequency $\bar{\omega}$ at the center of the frequency band, interrupted by random phase jumps each time that a collision occurs. The discontinuities in the phase due to the collisions cause a spread of frequencies around the center frequency. An example is shown in Figure 4. The average time $\tau_0$ between the collisions is typically less than 10^-10 s which implies that on average between two collisions roughly 10⁵ harmonic oscillations occur and that during an atomic transition of the order of a hundred collisions may occur. A coherence time characterizes the delay over which a field remains strongly correlated with itself; its precise value depends on the chosen correlation threshold. In this collision-dominated model it is of the order of the mean time between collisions, $\tau_0$, which may be about 10^-10 s.

To understand coherence and incoherence, it is helpful to use this model for the emission by a single atom as harmonic wave trains of many thousands of periods interrupted by roughly a hundred random phase jumps. Due to the random phase jumps, the interference term of the sum of harmonic wave trains emitted by two atoms, when integrated over the relatively long integration time of a detector, becomes a sum over integrals over time intervals of average length $\tau_0$:

where the sum is over roughly one hundred random phase jumps during the total duration of the wave trains. The random phase jumps lead to cancellation of the integrals and hence the interference term vanishes. We conclude that over the integration time of typical detectors

For a narrow, smooth spectral line, the coherence time is of order the reciprocal bandwidth. If linewidth is quoted as an ordinary frequency width $\Delta f=\Delta\omega/(2\pi)$, an estimate is

The corresponding coherence length is estimated by

For a narrow band in vacuum, $\lambda\omega=2\pi c$ gives, to first order in bandwidth,

where $\bar{\lambda}$ and $\bar{\omega}$ are the wavelength and the frequency at the center of the line. Hence,

Order-of-magnitude coherence lengths and times of several sources are listed in Table 1. For a laser, the linewidth is extremely small and the coherence time very long. This is because the photons in a laser are not generated predominantly by spontaneous emission as they are in classical sources, but instead by stimulated emission. Lasers are discussed in Lasers.

3.2Polychromatic Light¶

When dealing with coherence one has to consider fields that consist of a range of different frequencies. Let ${\cal U}(\mathbf{r},t)$ be the real-valued field component. It is always possible to write ${\cal U}(\mathbf{r},t)$ as an integral over time-harmonic components:

where $A_\omega(\mathbf{r})$ is the complex amplitude of the time-harmonic field with frequency $\omega$. When there is only a certain frequency band that contributes, then $A_\omega=0$ for $\omega$ outside this band. We define the complex time-dependent field $U(\mathbf{r},t)$ by

Then

Remark: The complex field $U(\mathbf{r},t)$ now contains the time dependence in contrast to the notation used for a time-harmonic (i.e. single frequency) field introduced in Chapter 2, where the time-dependent $e^{-i\omega t}$ was a separate factor.

We now compute the intensity of polychromatic light. The instantaneous energy flux is (as for monochromatic light) proportional to the square of the instantaneous real field: $\mathcal{U}(\mathbf{r},t)^2$. We average the instantaneous intensity over the integration time $T$ of common detectors. As stated before, this integration time is very long compared to the period at the center frequency $2\pi/\bar{\omega}$ of the field. Using Eq. (6) and

we get

where the averages of $U(\mathbf{r},t)^2$ and $(U(\mathbf{r},t)^*)^2$ are zero because they are fast-oscillating and go through many cycles during the integration time of the detector. In contrast, $|U(\mathbf{r},t)|^2=U(\mathbf{r},t)^*U(\mathbf{r},t)$ has a DC-component which does not average to zero.

Remark: In contrast to the time-harmonic case, the long time average of polychromatic light depends on the time $t$ at which the average is taken. However, we assume in this chapter that the fields are emitted by sources that are stationary. The property of stationarity implies that the average over the time interval of long length $T$ does not depend on the time at which the average is taken. Many light sources, in particular conventional lasers, are stationary. (However, a laser source which emits short high-power pulses cannot be considered as a stationary source). We furthermore assume that the fields are ergodic, which means that taking the time-average over a long time interval amounts to the same as taking the average over the ensemble of possible fields. It can be shown that this property implies that the limit $T\rightarrow \infty$ in Eq. (6) indeed existsMandel & Wolf, 1995.

We use for the intensity again the expression without the factor $1/2$ in front, i.e.

The time-averaged intensity has hereby been expressed in terms of the time-average of the squared modulus of the complex field.

Quasi-monochromatic field. If the width $\Delta \omega$ of the frequency band is very narrow compared to the center frequency $\bar{\omega}$, we speak of a quasi-monochromatic field. In the propagation of quasi-monochromatic fields, we use the formula for time-harmonic fields at $\bar{\omega}$. The quasi-monochromatic assumption simplifies the computations considerably and will therefore be used frequently.

4Temporal Coherence and the Michelson Interferometer¶

To investigate the time coherence of a field at a certain point $\mathbf{r}$, we let the field at that point interfere with a delayed copy of itself, i.e. we let $U(\mathbf{r},t)$ interfere with $U(\mathbf{r}, t-\tau)$. Because, when studying temporal coherence, the point $\mathbf{r}$ is always the same, we omit it from the formula. Furthermore, for easier understanding of the phenomena, we assume for the time being that the field considered is emitted by a single atom (i.e. a point source).

Temporal coherence is closely related to the spectral content of the light: if the light consists of fewer frequencies (think of monochromatic light), then it is more temporally coherent. To study the interference of $U(t)$ with $U(t-\tau)$, a Michelson interferometer, shown in Figure 5, is a suitable setup. The light that goes through one arm takes time $t$ to reach the detector, while the light that goes through the other (longer) arm takes time $t+\tau$, which means that it was radiated earlier. Therefore, the detector observes the time-averaged intensity $\langle |U(t)+U(t-\tau)|^2 \rangle$. As remarked before, this averaged intensity does not depend on the time the average is taken; it only depends on the time difference $\tau$ between the two beams.

We have

The detected intensity varies with the difference in arm length.

So far we have considered a field that originates from a single atom. The total field emitted by an extended source is the sum of fields $U_i(t)$ corresponding to all atoms $i$. As has been explained already, fields from independent atoms have cross terms that vanish after averaging. But the field emitted by an atom can interfere with the delayed field of that same atom and for every atom the interference is given by the same expression Eq. (27). If the atoms contribute equally and share the same spectral shape, the total intensity is the single-atom result multiplied by their number. More generally, their intensities add, each with its own self-coherence term.

The self coherence function $\Gamma(\tau)$ is defined by

The intensity of $U(t)$ is

The complex degree of self-coherence is defined by:

Using the Cauchy–Schwarz inequality it can be shown that this is a complex number with modulus between 0 and 1:

The observed intensity can then be written:

We consider two special cases.

1. Suppose $U(t)$ is a monochromatic wave

In that case we get for the self-coherence

and

Hence the interference pattern is given by

So for monochromatic light we expect to detect a cosine interference pattern, which shifts as we change the arm length of the interferometer (i.e. change $\tau$). No matter how large the time delay $\tau$, a clear interference pattern should be observed.

2. Next we consider what happens when the light is a superposition of two frequencies:

where $\left(2\pi/T\right) \ll \Delta \omega \ll \bar{\omega}$, and $T$ is the integration time of the detector. Then:

where in the second line the time average of terms that oscillate with time is set to zero because the averaging is done over a time interval $T$ satisfying $T\Delta\omega \gg 1$. Hence, the complex degree of self-coherence is:

and Eq. (32) becomes

The interference term is the product of the function $\cos(\bar{\omega}\tau)$, which is a rapidly oscillating function of $\tau$, and a slowly varying envelope $\cos \left(\Delta\omega\,\tau/2\right)$. The visibility envelope $|\cos(\Delta\omega\tau/2)|$ vanishes at some delays and then revives periodically. Thus this ideal two-line source has no single coherence decay time. Increasing line separation brings the first zero closer to zero delay^[3][4]. For a smooth, broad spectrum the correlation envelope commonly narrows as bandwidth increases. A practical coherence time needs a stated visibility threshold or linewidth convention. We conclude with some further interpretations of the degree of self-coherence $\gamma(\tau)$.

Remarks.

1. In stochastic signal analysis $\Gamma(\tau)=\langle U(t)U(t-\tau)^* \rangle$ is called the autocorrelation of $U(t)$. Informally, one can interpret the autocorrelation function as the ability to predict the field $U$ at time $t$ given the field at time $t-\tau$.

2. For a stationary field, the Wiener–Khinchin theorem relates the self-coherence function to the power spectral density. With the Fourier convention used here,

The Fourier relation shows that the larger the spread of the frequencies of $U(t)$ (i.e. the larger the bandwidth), the more sharply peaked $\Gamma(\tau)$ is. Thus, the light gets temporally less coherent when it consists of a broader range of frequencies. Measuring the spectral power density with a spectroscope and applying an inverse Fourier transform is an alternative method to obtain the complex self-coherence function.

5Spatial Coherence and Young’s Experiment¶

Temporal coherence concerns the coherence of the field in one point. The absolute value of the degree of self coherence Eq. (30) quantifies how strong the interference is of the field in the point of interest with the field in that same point at a later time. In contrast, spatial coherence is concerned with determining how coherent the fields in two different points are. This is done by letting the fields interfere using a mask with two small holes at the positions of the points of interest and observing the fringe contrast at a distant screen (Young’s experiment).

While for temporal coherence we used a Michelson interferometer, the natural choice to characterize spatial coherence is Young’s experiment, because it allows the fields in two points $P_1$, $P_2$ which are separated in space to interfere with each other.

Let $\mathbf{r}_1$ and $\mathbf{r}_2$ be the position vectors of the points $P_1$ and $P_2$, respectively. We write the complex field in $P_1$ as a superposition of monochromatic fields as in Eq. (22):

The reason for doing this is that for a monochromatic field in the pinhole, i.e. a field with a well defined frequency, we can derive the disturbance in any point $\mathbf{r}$ behind the mask. In fact, according to the Huygens-Fresnel Principle, a monochromatic disturbance with frequency $\omega$ in the pinhole at $\mathbf{r}_1$ generates a radiating spherical wave with the same frequency $\omega$. Apart from a common diffraction and obliquity factor, its field at $\mathbf{r}$ is proportional to

Here $C$ is a common complex coupling factor that includes pinhole area, diffraction normalization, and the obliquity factor. We treat it as frequency independent over a narrow band and approximately equal for the two identical pinholes in the far field. The frequency remains in the propagation phase because even a small frequency change can produce a substantial phase change over a long path. The total field $U_1(\mathbf{r},t)$ in $\mathbf{r}$ due to the pinhole at $P_1$ is obtained by integrating the monochromatic components over frequency:

In words:

For the field in $\mathbf{r}$ due to pinhole 2 we have similarly

The total field in $\mathbf{r}$ is the sum $U_1(\mathbf{r},t)+U_2(\mathbf{r},t)$. Because of the difference in propagation distance $\Delta R=|\mathbf{r}-\mathbf{r}_2|-|\mathbf{r}-\mathbf{r}_1|$, there is a time difference $\tau$ between the times when the two fields were emitted by the two pinholes so that they arrive at a given time $T$ at point $\mathbf{r}$ on the screen in Figure 8. This time difference is given by

Furthermore, the amplitudes are reduced by a factor proportional to the reciprocal distance which is different for the two fields. But if the distance of the screen to the mask is large enough, we may assume these factors to be the same and then omit them. Using Eq. (46), the interference pattern on the screen is then, apart from a constant factor, given by

The second line uses stationarity: delaying a field does not change its mean intensity. We define the mutual coherence function by:

With the intensities

the complex degree of mutual coherence is defined by

It can be proved using Bessel’s inequality that

We can now write Eq. (47) as

By varying the point $\mathbf{r}$ over the screen we can vary $\tau$ and by measuring the intensities we can determine the real part of $\gamma_{12}(\tau)$ and hence the fringe contrast observed on the screen.

As an example, consider what happens when $U(\mathbf{r},t)$ is a monochromatic field

In that case

and

So we get

where $\varphi=\arg A(\mathbf{r}_1)-\arg A(\mathbf{r}_2)$. In this case $\gamma_{12}$ has modulus 1, as expected for a monochromatic field. The intensity on the screen becomes

The fields are fully coherent because $|\gamma_{12}|=1$. Their fringe contrast is $2\sqrt{I_1I_2}/(I_1+I_2)$, reaching 1 only when the pinhole intensities are equal. If $\varphi=0$, then interference maxima occur for

Because $\omega=c\frac{2\pi}{\lambda}$, and $\Delta R=c\tau$, we find that maxima occur when

For a large distance between the screen and the mask (in the Fraunhofer limit), these path length differences correspond to directions of the maxima given by the angles $\theta_m$ (see Figure 8):

where $d$ is the distance between the slits and $m$ is an integer^[5].

Remarks.

1. The mutual coherence $\Gamma_{12}(\tau)= \langle U(\mathbf{r}_1,t)U(\mathbf{r}_2,t-\tau)^* \rangle$ is the cross-correlation of the two signals $U(\mathbf{r}_1,t)$ and $U(\mathbf{r}_2,t)$.

2. As remarked above, by moving the point of observation $\mathbf{r}$ over the screen, one can obtain the real part of the complex degree of mutual coherence. To derive also the imaginary part, one can put a piece of glass behind one of the pinholes with thickness such that for the center frequency $\bar{\omega}$ an additional phase difference of $\pi/2$ is obtained between the fields in $\mathbf{r}_1$ and $\mathbf{r}_2$. If the frequency band $\Delta \omega$ is sufficiently narrow this phase difference applies to a good approximation to all frequencies in the band.

6More on Spatial Coherence¶

We first consider the case that the source is so small (e.g. a single emitting atom) that it can be considered to be a point source $S$. The correlation of the fields at two points $P_1$ and $P_2$ then depends on the travel-time difference from $S$. A path difference much smaller than the source coherence length $\ell_c$ generally gives high visibility; a much larger difference gives low visibility for a smooth broad spectrum.

An extended classical light source consists of a large set of emitting point sources that emit by spontaneous emission. As we have explained in Section 3.1, the wave trains emitted by different atoms (point sources) in the source suffer random phase jumps due to e.g. collisions and therefore their averaged cross terms vanish. Such a light source is called spatially incoherent. For a spatially incoherent light source, the spatial coherence in any two points $P_1$ and $P_2$ is determined by measuring the fringe contrast on a distant screen when a mask is used that is perpendicular to the mean direction of propagation of the light and which contains pinholes at $P_1$ and $P_2$. The fringe contrast and hence the mutual coherence at $P_1$ and $P_2$ are determined by two effects:

1. First of all it is determined by how coherent the contributions of the individual point sources $S$ in the extended source to the total fields at $P_1$ and $P_2$ are. This coherence is determined by the extent to which the difference between the distance of $S$ to $P_1$ and of $S$ to $P_2$ is smaller than the coherence length. If these differences in distances are for all point sources larger than the coherence length, the fringe contrast on the screen in Young’s experiment will be very low and hence the mutual coherence is very low.

2. The second effect is the size of the extended source. Even if for all point sources in the source the fields in $P_1$ and $P_2$ are coherent, the coherence of the total fields at $P_1$ and $P_2$ due to the entire source can be small. As we know, the cross terms between independent source points vanish after averaging. Hence the intensity observed in Young’s experiment is the sum of the intensities due to the individual point sources in the extended source. The total coherence can still be low because the fringe patterns from different source points shift relative to one another and cancel on averaging. The shift of the fringe patterns is due to the different positions of the point sources in the extended source, which cause the phase difference between the fields in $P_1$ and $P_2$ to vary with the point sources.

For a distant source, a small product of angular source size and pinhole separation keeps path delays small. This alone does not prevent fringes from different source points from canceling when their intensities are added.

To show this we consider two mutually incoherent point sources $S_1$ and $S_2$ in the $z=0$ plane. Their mutual coherence function satisfies:

where $\Gamma_0$ is the self-coherence which we assume to be the same for both point sources. $\Gamma_0(\tau)$ has a characteristic width set by the source spectrum. For a smooth broad spectrum its envelope generally decreases with delay, although it need not be monotonic. Eq. (61) expresses the fact that two point sources are mutually incoherent. Using the fact, based on the assumption that the source is stationary, that the long-time average does not depend on the origin of time, we find:

Furthermore, for $\tau=0$: $\Gamma_0(0)=I_0$, which is the intensity of either source.

We assume for convenience that the two points $P_1$, $P_2$ are at a large distance $z$ from the two point sources and that the line $P_1P_2$ is parallel to the line joining the two point sources as shown in Figure 9. We will compute the mutual coherence $\Gamma_{P_1P_2}(0)$ for zero time delay $\tau=0$ (we can also compute the mutual coherence for more general time delays $\tau>0$, i.e. $\Gamma_{P_1P_2}(\tau)$, but it will suffice for our purpose to take $\tau=0$). The fields in $P_1$ and $P_2$ are the sum of the fields emitted by $S_1$ and $S_2$. Since $S_1$ and $S_2$ are point sources they emit spherical waves. Therefore, similarly to Eq. (44) we find that the field in $P_1$ is proportional to

and

where we omitted the constant factors in front of Eq. (44).

For sufficiently large $z$, all distances $|S_iP_j|$ in the denominators may be replaced by $z$. We absorb their common amplitude factor, including the coupling coefficient, into the field normalization. By substituting Eq. (64) and Eq. (65) into Eq. (48) with $\tau=0$, we find for the mutual coherence of $P_1$ and $P_2$:

Now we use Eq. (61) and Eq. (62) to get

Similarly,

Because the width of the self-coherence function $\Gamma_0$ is set by the coherence time $\tau_c$, result Eq. (67) confirms that for the fields in $P_1$ and $P_2$ to be coherent, the path difference from each source point to $P_1$ and $P_2$ should be small compared with the coherence length $\ell_c=c\tau_c$ for that source contribution to retain appreciable visibility. To express the result in terms of the angle $\alpha$ subtended by the source at the midpoint of $P_1P_2$ we choose coordinates such that $P_j=(x_j,0,z)$ for $j=1,2$. If the distance to the source is so large that $S_1P_1$ and $S_1P_2$ are almost parallel, we see from Figure 9 that

Similarly,

Hence, with $\Gamma_0(-\tau)=\Gamma_0(\tau)^*$, Eq. (67) becomes

The argument of $\Gamma_0$ shows how finite bandwidth reduces each source contribution as its path difference grows. The sum of those contributions can also cancel because their phases differ, even when each is individually coherent.

At fixed pinhole separation, moving a source farther away reduces its angular size. Near zero angular size the contributions have nearly the same phase and spatial coherence approaches one. For a discrete source, visibility can have additional zeros and revivals at larger angular sizes.

As an example, consider quasi-monochromatic light for which (see Eq. (34)):

Here $\bar\omega$ is the center frequency. This idealization treats each source as monochromatic over the delays of interest, so its temporal coherence length is effectively much larger than all relevant path differences. Hence the only remaining criterion for coherence of the total fields in $P_1$ and $P_2$ is that the fringe patterns due to the different point sources in Young’s experiment sufficiently overlap. Indeed, in this case of very long coherence time $\tau_c$ we have

and hence the degree of mutual coherence is:

For these two equal point sources, with baseline $b=|x_1-x_2|$, the first visibility zero occurs at $b=\bar\lambda/(2\alpha)$. This is a first null, not a maximum allowed separation: $|\gamma_{P_1P_2}|$ revives at larger baselines. A continuous source has a different visibility function.

6.1Example: Solar coherence across a baseline¶

The Sun has an angular diameter of about $\alpha=2R_\odot/\mathrm{AU}\simeq0.0093$ radians, or $0.53^\circ$. The two-point-source cosine derived above does not describe a filled solar disk. For a uniformly bright circular disk observed through a narrow spectral band, the van Cittert–Zernike result gives the modulus of the spatial degree of coherence at baseline $b$:

where $J_1$ is a Bessel function. The first zero occurs at $b\simeq1.22\lambda/\alpha$. At $\lambda=550\,\mathrm{nm}$, this is about $72\,\mu\mathrm{m}$. A baseline of $20\,\mu\mathrm{m}$ still gives appreciable visibility; it is not a hard maximum coherence distance. Solar limb darkening and finite bandwidth alter the precise visibility curve.