# Addition of Waves

A quick note: some useful reference texts are Hecht’s “Optics” and his wonderful chapter on interference, and for those who are brave of heart, Born and Wolf’s “Principles of Optics” is also arguably useful for the math.

DERIVATION OF INTERFERENCE TERMS

As we have learned before, the principle of superposition of electromagnetic waves allows us to write that at a point in space with two electromagnetic waves, the total electric field at that point is given as:

$$\vec{E_{total}} = \vec{E_1} + \vec{E_2}$$

where then further terms could be added for additional electric fields (recall that the principle of superposition for electric fields can be seen more clearly starting from Coulomb’s Law). Notice here that the principle of superposition is stated in terms of the electric field; if we were to actually try to measure the electric field at a point, we would have a very hard time doing this as the rate of the electric field variation is in the 100s of THz range (in other words, a detector would need to have a response time on the order of 10 femtoseconds in order to measure the rapidly varying phase). Because of this, we can formulate the rest of our derivations in terms of the intensity of the two waves.

The intensity of two electric fields at a point is given as the time average of the magnitude of the total electric field at that point. The magnitude of the total electric field for two waves at a point is given as:

\begin{align} |E|^2 & = (\vec{E_1^*}+\vec{E_2^*})\cdot(\vec{E_1}+\vec{E_2}) \\ & = |\vec{E_1}^2|+|\vec{E_2}^2|+2\vec{E_1}\cdot\vec{E_2} \label{int} \end{align}

where the * denotes the complex conjugate of the field. As the intensity that we measure in the lab is the time average, we can take the time average of Equation \eqref{int} and write out the electric fields explicitly in complex notation to find the intensity:

$$I = \langle|E_1(t)|^2\rangle + \langle|E_2(t)|^2\rangle + \langle\hat{\epsilon_1}\cdot\hat{\epsilon_2}E_{0,1}E_{0,2}e^{i(\omega_1-\omega_2)t}e^{-i(k_1-k_2)\cdot{r}+i(\phi_1-\phi_2)}\rangle + c.c.$$

where “$\hat{\epsilon_{1,2}}$” refer to the polarization of the individual interfering waves, “$\omega_{1,2}$” refers to the oscillation frequency of the individual interfering waves, “$\phi_{1,2}$” refers to the initial angle (i.e. normally at time t=0) of each wave, and “c.c.” is the complex conjugate of the interference term. Here, the term with the additions of both waves is the interference portion of the intensity. Thus, the effect of interference is, in effect, to change the total intensity at a point by either adding or subtracting from the time averaged intensities of each individual wave, ie $\langle|E_1(t)|^2\rangle$ and $\langle|E_2(t)|^2\rangle$.

We can take stock here in the effect of the dot product of the polarization of the two waves in the total amount of interference. As we can clearly see mathematically, if the two waves have orthogonal polarization, there is no interference since the dot product will equal zero. Conversely, if they match in polarization, the interference is maximized with respect to polarization. We will discuss this in further detail under the “Conditions for Interference” portion of the page, as we talk about the Fresnel-Arago Laws. For now, we will suffice it to say that it is generally only the parallel polarization components that contribute to the interference.

We can also evaluate here the effect of the beat frequency of the interference “$\omega_1-\omega_2$”. If the detector response time is short compared to the beat period such that

$$T < \frac{2\pi}{|\omega_1-\omega_2|}$$

then the time averaged intensity would show temporal beats at the beat frequency. However, in the more likely case that the integration time is longer than the beat frequency, then the time average of the interference term will equal to zero since then $\omega_1 \approx \omega_2$ such that $\omega_1 – \omega_2 = 0$. This is why interference is normally referred to with two monochromatic waves interfering. However, the case of non-monochromatic waves creating interference is physically possible and leads to interesting results when observing the interference patterns. More on this later, though!

If we thus make the assumption that the two waves are of the same frequency, then the problem reduces to its dependence on the other exponential term, i.e. $e^{-i(k_1-k_2)\cdot{r}+i(\phi_1-\phi_2)}$. If we denote “$\delta$” as:

$$\delta = (k_1-k_2)\cdot{r} + (\phi_1-\phi_2)$$

then we can take the real part of the intensity to find the total intensity as:

$$I = I_1 + I_2 + 2\sqrt{I_1I_2}cos\delta$$

Here, it is obvious then that we would have constructive interference whenever

$$\delta = 0,\pm{2\pi},\pm{4\pi},…$$

and destructive interference whenever

$$\delta = \pm\pi,\pm{3\pi},\pm{5\pi},…$$

Analyzing $\delta = (k_1\cdot{r} – k_2\cdot{r})+(\phi_1-\phi_2)$ further, we can see that the first term arises from a difference in path length between the two waves.

CONDITIONS FOR INTERFERENCE

As we stated before, typically it is said that only monochromatic waves can interfere. However, in actuality, almost no source is actually monochromatic; as such, it is interesting to consider the interference properties of both a quasi-monochromatic source as well as a white light source. In both cases, an important property of the light source(s) for interfering is the temporal coherence, which refers to the amount of time that the waveform of a wave of light oscillates in phase as a sinusoid. For quasi-monochromatic light, the period over which the light oscillates coherently can be quite long such that relatively stable interference can still occur with perhaps some shifting in the pattern around a certain point. If the two waves that are interfering arise from the same source or very closely spaced sources, the two waves will still show strong spatial coherence even if the temporal coherence falls off. For white-light sources, the coherence time would be short, as the temporal coherence is also inherently a measure of the spectral purity. However, the different spectral components (red with red, blue with blue) will still interfere with one another; the spots of their interference will just shift up or down on the viewing screen. The same properties of spatial coherence would apply to the white-light case as well.

It is also interesting to consider the case of having many light sources of monochromatic light emitting independently with slight overlap between the waves. If the phase remains constant throughout the interference, the waves can still show temporal and spatial interference. However, if the phase varies randomly and rapidly, the coherence time is greatly diminished and the spatial coherence would be almost zero depending on the size of the source. Before the advent of lasers, it was taken for granted that two lightbulbs would not produce an interference pattern. Given the extremely long coherence time of lasers, however, it is now possible to have two different sources of laser light that interfere with relative stability. More on coherence and its properties in the last section of this “Classical Optics” portion of the website!

Fresnel-Arago Laws

The Fresnel-Arago Laws give the conditions under which interference of polarization states can occur. They found the following:

1. Two orthogonal, coherent polarization states cannot interfere.
2. Two parallel, coherent polarization states will interfere in the same manner as unpolarized light.
3. The constituent polarization states of natural light cannot be made to interfere, even when brought into alignment.