5.5: Temporal Coherence and the Michelson Interferometer

To investigate the coherence of a field, the most general approach is to make the field in two different points \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) interfere for some time delay \(\tau\) and observe the fringe contrast. This means that one lets the fields \(U\left(\mathbf{r}_{1}, t\right)\) and \(U\left(\mathbf{r}_{2}, t-\tau\right)\) interfere. It is however customary to first look at the field in one point and let it interfere with itself but delayed in time, i.e. interfering \(U(\mathbf{r}, t)\) with \(U(\mathbf{r}, t-\tau)\). This special case is called temporal coherence. The other special case is spatial coherence in which the coherence of fields at two points is considered without time delay, by interfering \(U\left(\mathbf{r}_{1}, t\right)\) and \(U\left(\mathbf{r}_{2}, t\right)\). Spatial coherence will be treated later.

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 \(\PageIndex{1}\), 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 \(\left\langle|U(t)+U(t-\tau)|^{2}\right\rangle\). 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 \[\begin{aligned} I(\tau) &=\left\langle|U(t)+U(t-\tau)|^{2}\right\rangle \\ &=\left\langle|U(t)|^{2}\right\rangle+\left\langle|U(t-\tau)|^{2}\right\rangle+2 \operatorname{Re}\left\langle U(t) U(t-\tau)^{*}\right\rangle \\ &=2\left\langle|U(t)|^{2}\right\rangle+2 \operatorname{Re}\left\langle U(t) U(t-\tau)^{*}\right\rangle . \end{aligned} \nonumber \]

Figure \(\PageIndex{1}\): A Michelson interferometer to study the temporal coherence of a field. A beam is split in two by a beam splitter, and the two beams propagate over different distances which corresponds to a time difference \(\tau\) and then interfere at the detector.

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 \(i\). In studying time coherence we assume that these fields are propagating more or less parallel and that the light has a fixed polarisation, so that the fields can be added algebraically. The total complex field produced by a large number \(N\) of atoms is \[U(t)=U_{1}(t)+\ldots+U_{N}(t) . \nonumber \] During the integration time of the detector the fields \(U_{i}\) experience thousands of random phase jumps and therefore they do not interfere: the point sources of the extended source are mutually fully incoherent. The detected total intensity ( \(\PageIndex{1}\) ) is thus the sum of the intensities of the individual atoms: \[\begin{aligned} I(\tau) &=\left\langle|U(t)+U(t-\tau)|^{2}\right\rangle \\ &\left.\left.=\left\langle\sum_{i}\right| U_{i}(t)+U_{i}(t-\tau)\right]\left.\right|^{2}\right\rangle \\ &=\sum_{i} 2\left\{\left\langle\left. U_{i}(t)\right|^{2}\right\rangle+\operatorname{Re}\left\langle U_{i}(t) U_{i}(t-\tau)^{*}\right\rangle\right\}, \end{aligned} \nonumber \] The expression \(\left\langle|U(t)+U(t-\tau)|^{2}\right\rangle\) does not depend on position, so it cannot describe interference fringes in space. To better observe what happens when \(\tau\) is varied, we introduce interference fringes in space by tilting one beam so that the observed interference pattern is given by \[\begin{aligned} I(x, \tau) &=\left\langle\left|U(t) e^{i k_{x} x}+U(t-\tau)\right|^{2}\right\rangle \\ &=2\left\langle|U(t)|^{2}\right\rangle+2 \operatorname{Re}\left\langle U(t) U(t-\tau)^{*} e^{i k_{x} x}\right\rangle . \end{aligned} \nonumber \]

If \(\tau\) is changed, the maxima of the interference pattern translate as function of \(x\), which is easy to observe. How interference fringes for tilted collimated beams are observed in a Michelson interferometer is demonstrated in. It is possible to obtain different fringe patterns using diverging beams instead of collimated beams, as is demonstrated in.

The self coherence function \(\Gamma(\tau)\) is defined by \[\Gamma(\tau)=\left\langle U(t) U(t-\tau)^{*}\right\rangle \quad \text { self-coherence. } \nonumber \] The intensity of \(U(t)\) is \[I_{0}=\left\langle|U(t)|^{2}\right\rangle=\Gamma(0) . \nonumber \]

The complex degree of self-coherence is defined by: \[\gamma(\tau)=\frac{\Gamma(\tau)}{\Gamma(0)} . \quad \text { complex degree of self-coherence } \nonumber \]

This is a complex number with modulus between 0 and 1 : \[0 \leq|\gamma(\tau)| \leq 1, \nonumber \]

The observed intensity can then be written: \[I(x, \tau)=2 I_{0}\left\{1+\operatorname{Re}\left[\gamma(\tau) e^{i k_{x} x}\right]\right\}, \nonumber \] Recall that we vary \(\tau\) by varying the length of one of the arms in the Michelson interferometer.

We consider some special cases. Suppose \(U(t)\) is a monochromatic wave \[U(t)=e^{-i \omega t} . \nonumber \]

In that case we get for the self-coherence \[\Gamma(\tau)=\left\langle e^{-i \omega t} e^{i \omega(t-\tau)}\right\rangle=e^{-i \omega \tau}, \nonumber \] and \[\gamma(\tau)=e^{-i \omega \tau} . \nonumber \]

Hence the interference pattern is given by \[I(x, \tau)=2\left[1+\cos \left(\omega \tau-k_{x} x\right)\right] . \nonumber \]

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

Next we consider what happens when the light is a superposition of two frequencies: \[U(t)=\frac{e^{-i(\bar{\omega}+\Delta \omega / 2) t}+e^{-i(\bar{\omega}-\Delta \omega / 2) t}}{2}, \nonumber \] where \(\Delta \omega \ll \bar{\omega}\). Then: \[\begin{aligned} \Gamma(\tau) &=\frac{1}{4}\left\langle\left(e^{-i(\bar{\omega}+\Delta \omega / 2) t}+e^{-i(\bar{\omega}-\Delta \omega / 2) t}\right)\left(e^{i(\bar{\omega}+\Delta \omega / 2)(t-\tau)}+e^{i(\bar{\omega}-\Delta \omega / 2)(t-\tau)}\right)\right\rangle \\ & \approx \frac{e^{-i(\bar{\omega}+\Delta \omega / 2) \tau}+e^{-i(\bar{\omega}-\Delta \omega / 2) \tau}}{4} \\ &=\cos (\Delta \omega \tau / 2) \frac{e^{-i-\bar{\omega} \tau}}{2} \end{aligned} \nonumber \] 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 of duration \(T\) satisfying \(T \Delta \bar{\omega} \gg 1\). Hence, the complex degree of self-coherence is: \[\gamma(\tau)=\cos (\Delta \omega \tau / 2) e^{-i \bar{\omega} \tau} \nonumber \] and ( \(\PageIndex{8}\) ) becomes \[I(x, \tau)=\left\{1+\operatorname{Re}\left[\gamma(\tau) e^{i k_{x} x}\right]\right\}=\left[1+\cos (\Delta \omega \tau / 2) \cos \left(\bar{\omega} \tau-k_{x} x\right)\right] . \nonumber \]

The interference term is the product of the function \(\cos \left(\bar{\omega} \tau-k_{x} x\right)\) ), which is a rapidly oscillating function of \(\tau\), and a slowly varying envelope \(\cos (\Delta \omega \tau / 2)\). It is interesting to note that the envelope, and hence \(\gamma(\tau)\), vanishes for some periodically spaced \(\tau\), which means that for certain \(\tau\) the degree of self-coherence vanishes and no interference fringes form. Note that if \(\Delta \omega\) is increased, the intervals between the zeroes of \(\gamma(\tau)\) decrease.

If more frequencies are added, the envelope function is not a cosine function but on average decreases with \(\tau\). The typical value of \(\tau\) below which interferences are observed is roughly equal to half the first zero of the envelope function. This value is called the coherence time \(\Delta \tau_{c}\). We conclude with some further interpretations of the degree of self-coherence \(\gamma(\tau)\).

• In stochastic signal analysis \(\Gamma(\tau)=\left\langle U(t) U(t-\tau)^{*}\right\rangle\) 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 \(t-\tau\).
• The Wiener-Khinchine theorem says that the Fourier transform of the self coherence function is the spectral power density of \(U(t)\) :

\[\hat{\Gamma}(\omega)=|\hat{U}(\omega)|^{2} . \nonumber \] This result can be proved for stationary fields using Parseval’s indentity. Using the uncertainty principle, we can see 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.