Wave Optics

In 1637 Descartes gave the corpuscular model of light and derived Snell’s law. It explained the laws of reflection and refraction of light at an interface. The corpuscular model predicted that if the ray of light (on refraction) bends towards the normal then the speed of light would be greater in the second medium. This corpuscular model of light was further developed by Isaac Newton in his famous book entitled OPTICKS and because of the tremendous popularity of this book, the corpuscular model is very often attributed to Newton. In 1678, the Dutch physicist Christiaan Huygens put forward the wave theory of light – it is this wave model of light that we will discuss in this chapter. As we will see, the wave model could satisfactorily explain the phenomena of reflection and refraction; however, it predicted that on refraction if the wave bends towards the normal then the speed of light would be less in the second medium. This is in contradiction to the prediction made by using the corpuscular model of light. It was much later confirmed by experiments where it was shown that the speed of light in water is less than the speed in air confirming the prediction of the wave model; Foucault carried out this experiment in 1850. The wave theory was not readily accepted primarily because of Newton’s authority and also because light could travel through vacuum

and it was felt that a wave would always require a medium to propagate from one point to the other. However, when Thomas Young performed his famous interference experiment in 1801, it was firmly established that light is indeed a wave phenomenon. The wavelength of visible light was measured and found to be extremely small; for example, the wavelength of yellow light is about $0.6\mu \mathrm{m}$. Because of the smallness of the wavelength of visible light (in comparison to the dimensions of typical mirrors and lenses), light can be assumed to approximately travel in straight lines. This is the field of geometrical optics, which we had discussed in the previous chapter. Indeed, the branch of optics in which one completely neglects the finiteness of the wavelength is called geometrical optics and a ray is defined as the path of energy propagation in the limit of wavelength tending to zero. After the interference experiment of Young in 1801, for the next 40 years or so, many experiments were carried out involving the interference and diffraction of lightwaves; these experiments could only be satisfactorily explained by assuming a wave model of light. Thus, around the middle of the nineteenth century, the wave theory seemed to be very well established. The only major difficulty was that since it was thought that a wave required a medium for its propagation, how could light waves propagate through vacuum. This was explained when Maxwell put forward his famous electromagnetic theory of light. Maxwell had developed a set of equations describing the laws of electricity and magnetism and using these equations he derived what is known as the wave equation from which he predicted the existence of electromagnetic waves*. From the wave equation, Maxwell could calculate the speed of electromagnetic waves in free space and he found that the theoretical value was very close to the measured value of speed of light. From this, he propounded that light must be an electromagnetic wave. Thus, according to Maxwell, light waves are associated with changing electric and magnetic fields; changing electric field produces a time and space varying magnetic field and a changing magnetic field produces a time and space varying electric field. The changing electric and magnetic fields result in the propagation of electromagnetic waves (or light waves) even in vacuum. In this chapter we will first discuss the original formulation of the Huygens principle and derive the laws of reflection and refraction. In Sections 10.4 and 10.5, we will discuss the phenomenon of interference which is based on the principle of superposition. In Section 10.6 we will discuss the phenomenon of diffraction which is based on Huygens-Fresnel principle. Finally in Section 10.7 we will discuss the phenomenon of polarisation which is based on the fact that the light waves are transverse electromagnetic waves.

Maxwell had predicted the existence of electromagnetic waves around 1855; it was much later (around 1890) that Heinrich Hertz produced radiowaves in the laboratory. J.C. Bose and G. Marconi made practical applications of the Hertzian waves

We would first define a wavefront: when we drop a small stone on a calm pool of water, waves spread out from the point of impact. Every point on the surface starts oscillating with time. At any instant, a photograph of the surface would show circular rings on which the disturbance is maximum. Clearly, all points on such a circle are oscillating in phase because they are at the same distance from the source. Such a locus of points, which oscillate in phase is called a wavefront; thus a wavefront is defined as a surface of constant phase. The speed with which the wavefront moves outwards from the source is called the speed of the wave. The energy of the wave travels in a direction perpendicular to the wavefront. If we have a point source emitting waves uniformly in all directions, then the locus of points which have the same amplitude and vibrate in the same phase are spheres and we have what is known as a spherical wave as shown in Fig. 10.1(a). At a large distance from the source, a small portion of the sphere can be considered as a plane and we have what is known as a plane wave [Fig. 10.1(b)]. Now, if we know the shape of the wavefront at $t=0$, then Huygens principle allows us to determine the shape of the wavefront at a later time $t$. Thus, Huygens principle is essentially a geometrical construction, which given the shape of the wafefront at any time allows us to determine the shape of the wavefront at a later time. Let us consider a diverging wave and let $F_1{F}_2$ represent a portion of the spherical wavefront at $t=0$ (Fig. 10.2). Now, according to Huygens principle, each point of the wavefront is the source of a secondary disturbance and the wavelets emanating from these points spread out in all directions with the speed of the wave. These wavelets emanating from the wavefront are usually referred to as secondary wavelets and if we draw a common tangent to all these spheres, we obtain the new position of the wavefront at a later time.

Thus, if we wish to determine the shape of the wavefront at $t=\tau$, we draw spheres of radius $v\tau$ from each point on the spherical wavefront where $v$ represents the speed of the waves in the medium. If we now draw a common tangent to all these spheres, we obtain the new position of the wavefront at $t=\tau$. The new wavefront shown as $G_1{G}_2$ in Fig. 10.2 is again spherical with point $O$ as the centre. The above model has one shortcoming: we also have a backwave which is shown as $D_1{D}_2$ in Fig. 10.2. Huygens argued that the amplitude of the secondary wavelets is maximum in the forward direction and zero in the backward direction; by making this adhoc assumption, Huygens could explain the absence of the backwave. However, this adhoc assumption is not satisfactory and the absence of the backwave is really justified from more rigorous wave theory. In a similar manner, we can use Huygens principle to determine the shape of the wavefront for a plane wave propagating through a medium (Fig. 10.3).

We will now use Huygens principle to derive the laws of refraction. Let $P{P}^{\prime }$ represent the surface separating medium 1 and medium 2, as shown in Fig. 10.4. Let $v_1$ and $v_2$ represent the speed of light in medium 1 and medium 2, respectively. We assume a plane wavefront $AB$ propagating in the direction $A^{\prime }A$ incident on the interface at an angle $i$ as shown in the figure. Let $\tau$ be the time taken by the wavefront to travel the distance $BC$. Thus,

$$BC=v_1\tau$$

In order to determine the shape of the refracted wavefront, we draw a sphere of radius $v_2\tau$ from the point $A$ in the second medium (the speed of the wave in the second medium is $v_2$). Let $CE$ represent a tangent plane drawn from the point $C$ on to the sphere. Then, $AE=v_2\tau$ and $CE$ would represent the refracted wavefront. If we now consider the triangles $ABC$ and $AEC$, we readily obtain

$$\sin i=\frac{BC}{AC}=\frac{v_1\tau }{AC}(10.1)$$

and

$$\sin r=\frac{AE}{AC}=\frac{v_2\tau }{AC}(10.2)$$

where $i$ and $r$ are the angles of incidence and refraction, respectively. Thus we obtain

$$\frac{\sin i}{\sin r}=\frac{v_1}{v_2}(10.3)$$

From the above equation, we get the important result that if $r<i$ (i.e., if the ray bends toward the normal), the speed of the light wave in the second medium ($v_2$) will be less then the speed of the light wave in the first medium ($v_1$). This prediction is opposite to the prediction from the corpuscular model of light and as later experiments showed, the prediction of the wave theory is correct. Now, if $c$ represents the speed of light in vacuum, then,

$$n_1=\frac{c}{v_1}(10.4)$$

and

$$n_2=\frac{c}{v_2}(10.5)$$

are known as the refractive indices of medium 1 and medium 2, respectively. In terms of the refractive indices, Eq. (10.3) can be written as

$$n_1\sin i=n_2\sin r(10.6)$$ This is the Snell's law of refraction. Further, if ${\lambda }_1$ and ${\lambda }_2$ denote the wavelengths of light in medium 1 and medium 2, respectively and if the distance $BC$ is equal to ${\lambda }_1$ then the distance $AE$ will be equal to ${\lambda }_2$ (because if the crest from $B$ has reached $C$ in time $\tau$, then the crest from $A$ should have also reached $E$ in time $\tau$); thus,

$$\frac{{\lambda }_1}{{\lambda }_2}=\frac{BC}{AE}=\frac{v_1}{v_2}$$

or

$$\frac{v_1}{{\lambda }_1}=\frac{v_2}{{\lambda }_2}(10.7)$$

Christiaan Huygens (1629 – 1695) Dutch physicist, astronomer, mathematician and the founder of the wave theory of light. His book, Treatise on light, makes fascinating reading even today. He brilliantly explained the double refraction shown by the mineral calcite in this work in addition to reflection and refraction. He was the first to analyse circular and simple harmonic motion and designed and built improved clocks and telescopes. He discovered the true geometry of Saturn's rings.

The above equation implies that when a wave gets refracted into a denser medium ($v_1>v_2$) the wavelength and the speed of propagation decrease but the frequency $\nu (=v/\lambda )$ remains the same.

We now consider refraction of a plane wave at a rarer medium, i.e., $v_2>v_1$. Proceeding in an exactly similar manner we can construct a refracted wavefront as shown in Fig. 10.5. The angle of refraction will now be greater than angle of incidence; however, we will still have $n_1\sin i=n_2\sin r$. We define an angle $i_c$ by the following equation

Demonstration of interference, diffraction, refraction, resonance and Doppler effect

$$\sin i_c=\frac{n_2}{n_1}(10.8)$$

Thus, if $i=i_c$ then $\sin r=1$ and $r=90^{\circ }$. Obviously, for $i>i_c$, there can not be any refracted wave. The angle $i_c$ is known as the critical angle and for all angles of incidence greater than the critical angle, we will not have any refracted wave and the wave will undergo what is known as total internal reflection. The phenomenon of total internal reflection and its applications was discussed in Section 9.4.

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We next consider a plane wave AB incident at an angle $i$ on a reflecting surface MN. If $v$ represents the speed of the wave in the medium and if $\tau$ represents the time taken by the wavefront to advance from the point B to C then the distance

$$BC=v\tau$$

In order to construct the reflected wavefront we draw a sphere of radius $v\tau$ from the point A as shown in Fig. 10.6. Let CE represent the tangent plane drawn from the point C to this sphere. Obviously

$$AE=BC=v\tau$$

If we now consider the triangles EAC and BAC we will find that they are congruent and therefore, the angles $i$ and $r$ (as shown in Fig. 10.6) would be equal. This is the law of reflection. Once we have the laws of reflection and refraction, the behaviour of prisms, lenses, and mirrors can be understood. These phenomena were discussed in detail in Chapter 9 on the basis of rectilinear propagation of light. Here we just describe the behaviour of the wavefronts as they undergo reflection or refraction. In Fig. 10.7(a) we consider a plane wave passing through a thin prism. Clearly, since the speed of light waves is less in glass, the lower portion of the incoming wavefront (which travels through the greatest thickness of glass) will get delayed resulting in a tilt in the emerging wavefront as shown in the figure. In Fig. 10.7(b) we consider a plane wave incident on a thin convex lens; the central part of the incident plane wave traverses the thickest portion of the lens and is delayed the most. The emerging wavefront has a depression at the centre and therefore the wavefront becomes spherical and converges to the point F which is known as the focus. In Fig. 10.7(c) a plane wave is incident on a concave mirror and on reflection we have a spherical wave converging to the focal point F. In a similar manner, we can understand refraction and reflection by concave lenses and convex mirrors. From the above discussion it follows that the total time taken from a point on the object to the corresponding point on the image is the same measured along any ray. For example, when a convex lens focusses light to form a real image, although the ray going through the centre traverses a shorter path, but because of the slower speed in glass, the time taken is the same as for rays travelling near the edge of the lens.

Example 10.1 (a) When monochromatic light is incident on a surface separating two media, the reflected and refracted light both have the same frequency as the incident frequency. Explain why? (b) When light travels from a rarer to a denser medium, the speed decreases. Does the reduction in speed imply a reduction in the energy carried by the light wave? (c) In the wave picture of light, intensity of light is determined by the square of the amplitude of the wave. What determines the intensity of light in the photon picture of light.

Solution (a) Reflection and refraction arise through interaction of incident light with the atomic constituents of matter. Atoms may be viewed as oscillators, which take up the frequency of the external agency (light) causing forced oscillations. The frequency of light emitted by a charged oscillator equals its frequency of oscillation. Thus, the frequency of scattered light equals the frequency of incident light.

(b) No. Energy carried by a wave depends on the amplitude of the wave, not on the speed of wave propagation.

(c) For a given frequency, intensity of light in the photon picture is determined by the number of photons crossing an unit area per unit time.

In this section we will discuss the interference pattern produced by the superposition of two waves. You may recall that we had discussed the superposition principle in Chapter 14 of your Class XI textbook. Indeed the entire field of interference is based on the superposition principle according to which at a particular point in the medium, the resultant displacement produced by a number of waves is the vector sum of the displacements produced by each of the waves. Consider two needles $S_1$ and $S_2$ moving periodically up and down in an identical fashion in a trough of water [Fig. 10.8(a)]. They produce two water waves, and at a particular point, the phase difference between the displacements produced by each of the waves does not change with time; when this happens the two sources are said to be coherent. Figure 10.8(b) shows the position of crests (solid circles) and troughs (dashed circles) at a given instant of time. Consider a point P for which

$$S_{1}P=S_{2}P$$

Since the distances $S_{1}P$ and $S_{2}P$ are equal, waves from $S_1$ and $S_2$ will take the same time to travel to the point $P$ and waves that emanate from $S_1$ and $S_2$ in phase will also arrive, at the point $P$, in phase. Thus, if the displacement produced by the source $S_1$ at the point $P$ is given by

$$y_1=a\cos \omega t$$

then, the displacement produced by the source $S_2$ (at the point $P$) will also be given by

$$y_2=a\cos \omega t$$

Thus, the resultant of displacement at $P$ would be given by

$$y=y_1+y_2=2a\cos \omega t$$

Since the intensity is proportional to the square of the amplitude, the resultant intensity will be given by

$I=4I_0$ where $I_0$ represents the intensity produced by each one of the individual sources; $I_0$ is proportional to $a^2$. In fact at any point on the perpendicular bisector of $S_1{S}_2$, the intensity will be $4I_0$. The two sources are said to interfere constructively and we have what is referred to as constructive interference. We next consider a point $Q$ [Fig. 10.9(a)] for which

$$S_{2}Q-S_{1}Q=2\lambda$$

The waves emanating from $S_1$ will arrive exactly two cycles earlier than the waves from $S_2$ and will again be in phase [Fig. 10.9(a)]. Thus, if the displacement produced by $S_1$ is given by

$$y_1=a\cos \omega t$$

then the displacement produced by $S_2$ will be given by

$$y_2=a\cos (\omega t-4\pi )=a\cos \omega t$$

where we have used the fact that a path difference of $2\lambda$ corresponds to a phase difference of $4\pi$. The two displacements are once again in phase and the intensity will again be $4I_0$ giving rise to constructive interference. In the above analysis we have assumed that the distances $S_{1}Q$ and $S_{2}Q$ are much greater than $d$ (which represents the distance between $S_1$ and $S_2$) so that although $S_{1}Q$ and $S_{2}Q$ are not equal, the amplitudes of the displacement produced by each wave are very nearly the same. We next consider a point $R$ [Fig. 10.9(b)] for which

$$S_{2}R-S_{1}R=-2.5\lambda$$

The waves emanating from $S_1$ will arrive exactly two and a half cycles later than the waves from $S_2$ [Fig. 10.10(b)]. Thus if the displacement produced by $S_1$ is given by

$$y_1=a\cos \omega t$$

then the displacement produced by $S_2$ will be given by

$$y_2=a\cos (\omega t+5\pi )=-a\cos \omega t$$

where we have used the fact that a path difference of $2.5\lambda$ corresponds to a phase difference of $5\pi$. The two displacements are now out of phase and the two displacements will cancel out to give zero intensity. This is referred to as destructive interference. To summarise: If we have two coherent sources $S_1$ and $S_2$ vibrating in phase, then for an arbitrary point $P$ whenever the path difference,

$$S_{1}P\sim S_{2}P=n\lambda (n=0,1,2,3,\dots )(10.9)$$

we will have constructive interference and the resultant intensity will be $4I_0$; the sign $\sim$ between $S_{1}P$ and $S_{2}P$ represents the difference between $S_{1}P$ and $S_{2}P$. On the other hand, if the point $P$ is such that the path difference,

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$$S_{1}P\sim S_{2}P=\left(n+\frac{1}{2}\right)\lambda (n=0,1,2,3,\dots )(10.10)$$

we will have destructive interference and the resultant intensity will be zero. Now, for any other arbitrary point $G$ (Fig. 10.10) let the phase difference between the two displacements be $\varphi$. Thus, if the displacement produced by $S_1$ is given by

Ripple Tank experiments on wave interference

$y_1=a\cos \omega t$ then, the displacement produced by $S_2$ would be

$$y_2=a\cos (\omega t+\varphi )$$

and the resultant displacement will be given by $$y=y_1+y_2=a\left[\cos \omega t+\cos (\omega t+\varphi )\right]=2a\cos (\varphi /2)\cos (\omega t+\varphi /2)$$

The amplitude of the resultant displacement is $2a\cos (\varphi /2)$ and therefore the intensity at that point will be

$$I=2I_0(10.12)$$

at all points.

When the phase difference between the two vibrating sources changes rapidly with time, we say that the two sources are incoherent and when this happens the intensities just add up. This is indeed what happens when two separate light sources illuminate a wall.

We will now discuss interference using light waves. If we use two sodium lamps illuminating two pinholes (Fig. 10.11) we will not observe any interference fringes. This is because of the fact that the light wave emitted from an ordinary source (like a sodium lamp) undergoes abrupt phase changes in times of the order of $10^{-10}$

seconds. Thus the light waves coming out from two independent sources of light will not have any fixed phase relationship and would be incoherent, when this happens, as discussed in the previous section, the intensities on the screen will add up. The British physicist Thomas Young used an ingenious technique to “lock” the phases of the waves emanating from $S_1$ and $S_2$. He made two pinholes $S_1$ and $S_2$ (very close to each other) on an opaque screen [Fig. 10.12(a)]. These were illuminated by another pinholes that was in turn, lit by a bright source. Light waves spread out from $S$ and fall on both $S_1$ and $S_2$. $S_1$ and $S_2$ then behave like two coherent sources because light waves coming out from $S_1$ and $S_2$ are derived from the same original source and any abrupt phase change in $S$ will manifest in exactly similar phase changes in the light coming out from $S_1$ and $S_2$. Thus, the two sources $S_1$ and $S_2$ will be locked in phase; i.e., they will be coherent like the two vibrating needle in our water wave example [Fig. 10.8(a)]. The spherical waves emanating from $S_1$ and $S_2$ will produce interference fringes on the screen $G{G}^{\prime }$, as shown in Fig. 10.12(b). The positions of maximum and minimum intensities can be calculated by using the analysis given in Section 10.4.

We will have constructive interference resulting in a bright

region when $\frac{xd}{D}$

$$x=x_n=\frac{n\lambda D}{d};n=0,\pm 1,\pm 2,\dots (10.13)$$

On the other hand, we will have destructive