# Optics Series Lecture, Lecture – VIII.

“Harmonic Plane Waves” This lecture was delivered on 13th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics honors students.

In our last lecture, lecture-VII we began by discussing what are electromagnetic waves. We also discussed in good detail what are harmonic waves. Harmonic waves are those waves whose wave-profile is either sine, cosine or in general both sine and cosine combined with each other. Shortly (after within a few lectures) we will discuss what is wave profile and how to transform a wave profile into a traveling wave. A wave profile, wave form or wave shape is simply a time instant view of a more general moving wave. We also discussed what is a plane wave. We applied our harmonic plane waves to the interesting phenomena of interference between two plane waves that are in addition monochromatic that is have same wavelength. Such waves traveling in a homogeneous media do so at a  fixed frequency and as long as they are in free-space their speed remains unaltered at the sped of light value c = 3 × 108 m/s.

A plane wave is one traveling wave where the wave fronts are planar points with equal phases all over the plane. In that order a spherical wave front is the locus of uniform phase over spherical configuration and a cylindrical wave front would be a traveling wave where the locus of uniform phase is nothing but a cylindrical surface. In one of the future lecture, shortly, we will discuss in much detail what are spherical waves. Cylindrical waves are left to the advanced and willing students to work out by themselves. If time permits sometime in the future we can fall back and make a case for cylindrical wave fronts as well. But no promise at this point. Note that waves are simply motion of phase points as a function of space or location and time instant. A phase is nothing but the angular argument of the wave described in terms of harmonic functions.

## Plane Waves.

Let us begin studying Plane waves in detail. Here are some of its features.

1. A plane wave is the simplest example of a 3-dimensional wave.
2. These are so called, because plane wave wave fronts are planar in shape.
A wave-front is a locus of points on which the phase of the wave is same. Its a surface of wave-disturbances which move together, at the same speed.
3. Optical devices are often tuned to produce plane waves.
This necessitates the study of plane waves as base examples, where more complicated features can be assigned when they become pertinent.
4. Wave fronts are always perpendicular to the direction of wave propagation.

We need to find the mathematical expression for a plane which is perpendicular to a given vector k and passes through some point given by the constant position vector r0 = (x0, y0, z0). Let us remember that k is the wave propagation vector, hence the wave disturbance moves along the vector k. We have the position vector r which varies along from one space point to another. As a result we have the following. $\inline&space;\fn_jvn&space;\vec{r}\equiv&space;(&space;x_0,\,y_0,\,z_0)$$\inline&space;\fn_jvn&space;\vec{r}-\vec{r_0}\equiv&space;(x-x_0)\hat{i}+(y-y_0)\hat{j}+(z-z_0)\hat{k}$ $\inline&space;\fn_jvn&space;\boxed{(\vec{r}-\vec{r}_0)\cdot&space;\hat{k}=0}$ Let us draw a suitable diagram to see the above scenario. — See the diagram below. Let us rewrite the above equation by recognizing that: $\inline&space;\fn_jvn&space;\vec{k}=k_x&space;\hat{i}+k_y&space;\hat{j}+&space;k_z&space;\hat{k}$ which leads to: $\inline&space;\fn_cs&space;k_x&space;(x-x_0)+k_y&space;(y-y_0)+&space;k_z&space;(z-z_0)=0$ so that: $\inline&space;\fn_cs&space;k_x&space;x+k_y&space;y+&space;k_z&space;z=a;&space;\,\,&space;a&space;=&space;k_x&space;x_0+k_y&space;y_0+&space;k_z&space;z_0=constant$. We see that any plane perpendicular to wave propagation direction k passing through r0 = (x0, y0, z0) is given by: $\inline&space;\fn_jvn&space;\boxed{\vec{k}\cdot&space;\vec{r}=constant=a}$ and each point in this plane has the same projection on k; the wave propagation direction.

Let us now consider the harmonic or sinusoidal waves as we hinted at the beginning of the article. Our goal is to construct a set of planes over which ψ(r) varies in space in a harmonic or sinusoidal manner.

The function ψ(r) is called the wave function and according to our choice it is sinusoidal — that is, sine or cosine or a combination thereof, hence has the following form: $\inline&space;\fn_jvn&space;\psi&space;(\vec{r})&space;=&space;\Big&space;\{&space;A\,&space;sin\,&space;(\vec{k}\cdot\vec{r})\,\,\,or&space;\,\,\,&space;A\,&space;cos\,&space;(\vec{k}\cdot\vec{r})&space;\Big\}$ or in general: $\inline&space;\fn_jvn&space;\psi&space;(\vec{r})&space;=&space;A\,&space;e^{\,\,i\,\vec{k}\cdot\vec{r}}$. As we saw above ψ(r) is  constant over any plane defined by: $\inline&space;\fn_jvn&space;\boxed{\vec{k}\cdot&space;\vec{r}=constant=a}$The wave-function ψ(r) repeats itself in space after a displacement of λ, in the direction of vector k. λ is called as the wavelength of the wave. Let us display the wave-fronts of harmonic plane waves — or plane harmonic waves as an equally valid appellation, by a diagram, shown below. According to this definition of wavelength λ, we have: $\inline&space;\fn_jvn&space;\psi&space;(\vec{r})&space;=\psi&space;(\vec{r}+\frac{\lambda&space;\vec{k}}{k})\\&space;Note;\,&space;\hat{k}\ne&space;\frac{\vec{k}}{k},&space;\,&space;we&space;\,\,can\,\,&space;use&space;\,\,&space;\hat{n}&space;\,\,&space;instead&space;\,&space;\,for&space;\,&space;\,&space;\frac{\vec{k}}{k}$. So k is the magnitude of the vector-k. The magnitude is known as wave number or propagation number and the vector k is known as the propagation vector or wave vector. Do not confuse between the unit vector along z-axis, denoted as k-cap and the unit vector along vector k, the propagation vector, which as a good measure, we can denote rather by n-cap. For a plane harmonic wave therefore the periodicity condition that we just introduced in terms of wavelength λ, is given by: $\inline&space;\fn_jvn&space;A\,e^{\,i\,\vec{k}\cdot\vec{r}}&space;=&space;A\,e^{\,i\,\vec{k}\cdot(\vec{r}\,+\lambda\frac{\vec{k}}{k})}=A\,e^{\,i\,\vec{k}\cdot\vec{r}}e^{\,i\lambda&space;k}$. This means: $\inline&space;\fn_cs&space;e^{\,i\lambda&space;k}=1&space;=&space;e^{\,i2\pi}$ so, $\inline&space;\fn_cs&space;\lambda&space;k=&space;2\pi&space;\,&space;or\,&space;\boxed{k&space;=&space;\frac{2\pi}{\lambda}}$.

Our constructed plane harmonic wave-fronts are such that at any fixed point in space r is constant and this means the phase is constant as is the wave function ψ(r). In other words planes defined by: $\inline&space;\fn_jvn&space;\boxed{\vec{k}\cdot&space;\vec{r}=constant=a}$ are not yet moving. The planes inherit motion when the wave function varies as a function of time, that is ψ(r) becomes a time dependent function. We introduce a time dependence into our harmonic wave function as we would in case of traveling waves, to be discussed a few lectures from today. $\inline&space;\fn_jvn&space;\psi&space;(\vec{r},\,t)&space;=&space;A&space;\,&space;e&space;^{\,i(\vec{k}&space;\cdot&space;\vec{r}&space;\,\mp&space;\,\omega&space;t)};\,&space;A,&space;\omega&space;\,&space;and\,&space;k\,&space;are\,&space;constant.$ We see that each (space, time) point is associated with a phase corresponding to the value of the wave-function at this point. We have seen that surfaces that join all points of equal phase are known as wave-fronts. Any wave front will have the same value of wave function at all points of the wave-front if amplitude A is independent of points on the wave front. That is amplitude A is uniform over the wave-front. Such waves are known as homogeneous waves. We would not be dealing with in-homogeneous waves at the moment. Our amplitude would be independent of various points on the wave-front.

### Wave velocity:

The phase of a plane harmonic wave moves at certain speed, it gives us the propagation velocity of the wave front and is known as phase velocity. Thus wave velocity is the phase velocity, when there is only a monochromatic wave passing through a homogeneous medium. This will change if the wave consists of many different wave-length or frequency. In such a case the speed of the wave depends on wavelength or frequency of the wave, which are now multiple in number, either discrete or continuous. Thus the frequency ω is a function of wave number k or equivalently wavelength λ. Such interdependence of speed, frequency, wavelength or wave number, expressed mathematically: eg ω = ω(k) are known as dispersion relations. Accordingly wave velocity can be of two types, for monochromatic components, its the phase velocity and for multi-chromatic waves, its the group velocity, where group velocity is the combined value of different phase velocities.

Let us find out the phase velocity of our monochromatic plane harmonic wave given as: $\inline&space;\fn_jvn&space;\psi&space;(\vec{r},\,t)&space;=&space;A&space;\,&space;e&space;^{\,i(\vec{k}&space;\cdot&space;\vec{r}&space;\,\mp&space;\,\omega&space;t)}$. When r has a component rk along vector k we can write: $\inline&space;\fn_jvn&space;\psi&space;(\vec{r},\,t)&space;=&space;\psi(r_k&space;+&space;dr_k,&space;\,&space;t+dt)=\psi(r_k,&space;\,t)$because disturbance is constant on the wave front which moved a distance drk in time dt. In terms of the harmonic wave function we had conjured above: $\inline&space;\fn_jvn&space;A&space;\,&space;e&space;^{\,i(\vec{k}&space;\cdot&space;\vec{r}&space;\,\mp&space;\,\omega&space;t)}=A&space;\,&space;e&space;^{\,i(kr_k&space;+&space;\,kdr_k&space;\,\mp\,&space;\omega&space;t&space;\,\mp\,&space;\omega&space;dt&space;)}=&space;A&space;\,&space;e&space;^{\,i(kr_k&space;\,\mp\,&space;\omega&space;t)}$. This gives us: k × drk = ± ω × dt. Since drk is the distance moved by the wave in a time dt we have the wave velocity v given by: $\inline&space;\fn_cs&space;\boxed{\frac{dr_k}{dt}=\pm\,\,&space;\frac{\omega}{k}=\,&space;\pm\,&space;v}$.