Wavefronts and Huygens' Principle

Transcript

0:00Introduction

Hello everyone! Here is a famous argument about light. Newton's theory said light travels faster in water than in air; Huygens' theory said slower. Which one would you bet on? Hold on to your answer. Today, Huygens' idea of wavelets will tell us what each theory predicts, an experiment will settle the argument, and the same idea leads on to interference and diffraction, which come up often in JEE questions. So, let's begin.

0:29What is a wavefront?

So, what is a wavefront? Picture light spreading from a source, and join all the points oscillating in the same phase. That surface is a wavefront, okay? Rays are always perpendicular to the wavefronts, so a ray shows the direction in which the wavefront moves. Right? Now, for a tiny point source, all points at the same distance get the wave at the same moment. So what shape is the wavefront? Spherical, yes, exactly, with the rays going straight outwards! For a line source, like a long straight slit, the wavefronts are cylindrical. And for a very distant source, like a star, the small piece of the huge sphere reaching us is flat, a plane wavefront. Got it?

1:14Huygens' principle

Now, how does a wavefront move forward? Every point of a wavefront acts as a source of small secondary wavelets. Clear? These wavelets spread in the forward direction, with the speed of the wave itself. With me? So now we have a row of tiny wavelets. How do we get the new wavefront from them? Draw the surface that touches all of them in front, yes, exactly! This forward envelope is the new wavefront at the later time. That is Huygens' principle. Got it?

1:46Refraction by Huygens' construction

Now refraction, in the figure. The upper medium has index n one, the shaded lower one n two; the small numbers below are subscripts. The orange lines are rays, and the dark blue line, A B, is the incident wavefront. Its angle i with the surface equals the ray's angle with the normal, since the ray is perpendicular to the wavefront, and the normal is perpendicular to the surface. Remember this, okay? The wavefront first touches the surface at A, and a wavelet starts growing there. While B travels to C at speed v one, this wavelet, the dashed arc, grows at speed v two. So in time t, B C equals v one times t, and A D equals v two times t. The refracted wavefront is C D, the line from C touching the wavelet. Right? The two right triangles share the side A C, you see? So sine i is B C over A C, and sine r is A D over A C, with r the angle between C D and the surface. Divide, and sine i over sine r equals B C over A D, which is v one over v two, or n two over n one. That is Snell's law from wavelets! Got it? For reflection, the same construction gives i equals r, where r is now the angle of reflection. With me? Back to our argument. Say the second medium is denser, so n two is greater than n one; that pointed sign means greater than. Which is bigger, v one or v two? It is v one, yes, exactly, since v one over v two equals n two over n one, which is more than one. So v two is less than v one, the reverse sign: light is slower in the denser medium, as experiment confirms. Foucault measured the speed of light in water and found it less than in air. So Huygens wins the argument from our opening! Clear? Now let me ask you one more thing. When light goes from air into glass, does its frequency change? No, the frequency does not change. Each wavefront reaching the boundary sends exactly one wavefront into the glass, so the number per second stays the same. But the speed drops, so from vacuum into a medium of index n, the wavelength becomes lambda over n; lambda, the Greek letter, is the wavelength in vacuum. Okay?

4:05More results from wavefronts

Send a plane wavefront into a convex lens, or onto a concave mirror, and it comes out spherical, converging to the focus, right? Through a thin prism, the part of the wavefront crossing the thicker glass near the base is delayed more. So the wavefront turns towards the base, but stays plane. Got it? Now intensity, capital I, at a distance r. For a point source, light spreads over growing spheres, so capital I is proportional to one over r squared; the open loop sign means proportional to. Intensity goes as amplitude squared, so the amplitude goes as one over r. Okay? For a line source, light spreads over cylinders, so capital I goes as one over r, and the amplitude as one over root r, with the square root sign. Do not mix up intensity and amplitude. With me? The wave theory explains reflection, refraction, interference, diffraction and polarisation. Clear? But it does not explain the photoelectric effect or the Compton effect. This list comes up often in theory questions, okay?

5:12Example 1: refracted wavefront angle at a glass surface

Now example one. Here, a plane wavefront in air makes thirty degrees with a glass surface; the small raised circle means degrees. The glass has n equal to one point five. What angle does the refracted wavefront make with the surface? Pause the video and try it first. Done? Okay, let's check. Many students take ninety minus thirty, which is sixty degrees, as the angle of incidence. Is that right? No, no, wait, that is the trap! As in the figure, the wavefront's angle with the surface is the ray's angle with the normal, so i is just thirty degrees. With me? Now Snell's law, with n equal to one for air. Sine r equals sine thirty degrees over one point five, so zero point five over one point five, which is one third. Right? So r is sine inverse of one third, about nineteen point five degrees; the wavy equals sign means approximately. By the same rule, the refracted wavefront makes nineteen point five degrees with the surface. Clear?

6:18Example 2: width of a refracted beam entering glass

Now example two, a little harder. Here, a parallel beam two centimetres wide in air falls at sixty degrees on glass of index root three, the square root of three. Find the width of the refracted beam. Pause the video and try it first. Done? Okay, let's check. Sine r equals sine sixty degrees over root three. Sine sixty degrees is root three over two, so the root threes cancel, and sine r is one half, so r is thirty degrees. With me? Now the width. The beam was two centimetres wide in air, so it is still two centimetres wide in glass, right? Oh no, careful, it is not! The width is measured along the wavefront: w one in air, w two in glass. Both beams cover the same length of surface, capital L, okay? The wavefront makes the angle i with the surface, so w one equals capital L times cos i, and w two equals capital L times cos r. Got it? Divide, and capital L cancels. So w two over w one equals cos r over cos i, that is cos thirty degrees over cos sixty degrees. That is root three over two, over one half, which is root three. Okay? So w two equals two times root three centimetres, about three point four six centimetres. So the beam widens, as the rays bend towards the normal. Clear?

7:46Quick revision: wavefronts, Huygens and Snell's law

So, quick revision. The wavefront is a surface of the same phase, with rays perpendicular to it. Huygens: every point sends out forward wavelets, and their envelope is the new wavefront. And sine i over sine r equals v one over v two, which equals n two over n one; the frequency stays the same, and the wavelength becomes lambda over n. You have just learnt the idea that the whole of wave optics is built on, so well done. Now go and try the practice problems on this page. All the best!