Light as Waves In the previous chapter we discussed light as rays These rays traveled in a straight line except when they were reflected off a mirror or were refracted at the boundary between two optical media Physics for Scientists Engineers 2 Now we will discuss the implications of the wave nature of light Spring Semester 2005 Lecture 41 We know that light is an electromagnetic wave We normally do not think of light as a wave because its wavelength is so short that we usually do not notice this wave behavior April 8 2005 Physics for Scientists Engineers 2 1 April 8 2005 Wave Optics Physics for Scientists Engineers 2 2 Huygen s Constructions One way to reconcile the wave nature of light with the geometric optical properties of light is to use Huygens Principle developed by Christian Huygens a Dutch physicist who proposed a wave theory of light in 1678 before Maxwell developed his theories of light Diagrams of phenomena based on Huygens Principle are called Huygens constructions A Huygens construction for a light wave traveling in a straight line is shown below This principle states that every point on a propagating wave front serves as a source of spherical secondary wavelets At a later time the envelope of these secondary waves becomes a wave front If the original wave has frequency f and speed v the secondary wavelets have the same f and v April 8 2005 Physics for Scientists Engineers 2 3 April 8 2005 Physics for Scientists Engineers 2 4 Huygen s Constructions 2 Derivation of Snell s Law We start with a wave front traveling at the speed of light c We assume point sources of spherical wavelets along the wave front These wavelets also travel at c so at a time t the wavelets have traveled a distance of c t If we assume many point sources along the wave front we can see that the envelope of these wavelets forms a new wave front parallel to the original wave front Thus the wave continues to travel in a straight line with the original frequency and speed April 8 2005 Physics for Scientists Engineers 2 5 Now let s use a Huygens construction to derive Snell s Law for refraction between two optical media with different indices of refraction Assume that we have a wave with wave fronts separated by a wavelength 1 traveling with speed v1 in an optically clear medium incident on the boundary with a second optically clear medium as shown April 8 2005 Derivation of Snell s Law 2 Physics for Scientists Engineers 2 Derivation of Snell s Law 3 The angle of the incident wave front with respect to the boundary is 1 which is also the angle the direction of the wave makes with a normal to the boundary Thus the time interval between wave fronts for the first medium is 1 v1 and the time interval for the second medium is 2 v2 When the wave enters the second medium it travels with speed v2 This time interval is the same at boundary so we can write 1 2 v1 v2 According to Huygens Principle the wave fronts are the result of wavelet propagation at the speed of the original wave so we can write the separation of the wave fronts in the second medium in terms of the wavelength in the second medium 2 April 8 2005 Physics for Scientists Engineers 2 6 which we can rewrite as 1 v1 2 v2 The wavelengths of the light in the two media are proportional to the speed of light in those media 7 April 8 2005 Physics for Scientists Engineers 2 8 Derivation of Snell s Law 4 Derivation of Snell s Law 5 We can get a relation between the angle of the incident wave fronts 1 with the boundary and the angle of the transmitted wave fronts 2 with the boundary by analyzing an expanded region of the Huygens drawing We can see that sin 1 1 and sin 2 2 x x Solving for x we get sin 1 1 v1 sin 2 2 v2 Remembering that n c v we get sin 1 v1 c n1 n2 sin 2 v2 c n2 n1 or n1 sin 1 n2 sin 2 Which is Snell s Law April 8 2005 Physics for Scientists Engineers 2 9 Light Traveling in an Optical Medium April 8 2005 Physics for Scientists Engineers 2 Light Traveling in an Optical Medium 2 We have seen that the wavelength of light changes when traveling in an optical medium with index of refraction greater than one So the frequency of light traveling in an optical medium with n 1 is the same as the frequency of that light traveling in a vacuum Taking with case one as a vacuum and case two as a medium with index of refraction n we can write We perceive color by frequency rather than wavelength v n c n April 8 2005 Water has index of refraction n 1 33 The object appears to have the same color under water as in air v c n c f n n Physics for Scientists Engineers 2 Thus placing an object under water does not change our perception of the color of the object We can demonstrate that fact by taking a colored object and putting it in a jar of water Remembering that v f we can write the frequency fn of light traveling in a medium as fn 10 11 April 8 2005 Physics for Scientists Engineers 2 12 Interference Constructive Interference Sunlight is composed of light containing a broad range of frequencies and corresponding wavelengths The geometric optics of the previous chapter cannot be used to explain interference We often see different colors separated out of sunlight by refraction in rainbows To understand these interference phenomena we must take into account the wave nature of light We also sometimes see various colors from sunlight due to constructive and destructive interference phenomena on the surface of DVD s or CDs or in thin layers of oil or water April 8 2005 Physics for Scientists Engineers 2 Interference takes place when light waves of the same wavelength are superimposed If the light waves are in phase they interfere constructively as shown to the right 13 April 8 2005 Constructive Interference 2 A phase difference of 2 radians 360 or one wavelength will also produce two waves that are in phase If the light waves are traveling from some point then the phase difference can be related to the path difference between the two waves If the two light waves are out of phase as shown to the right the amplitudes of the waves will sum to zero everywhere and the two waves will destructively interfere Here the phase difference between the two waves is radians 180 or 2 The criterion for constructive interference is given by a path difference x given by April 8 2005 14 Destructive Interference The statement that the two waves interfere constructively is the same as saying that the phase difference between the two waves is zero x m Physics for Scientists …
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