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CHAPTER 1 PHYSICAL OPTICS: INTERFERENCE • Introduction CHAPTER 1 PHYSICAL OPTICS: INTERFERENCE What is “physical optics”? • Introduction The methods of physical optics are used when the • Waves wavelength of light and dimensions of the system are of • Principle of superposition a comparable order of magnitude, when the simple ray • Wave packets approximation of geometric optics is not valid. So, it is • Phasors • Interference intermediate between geometric optics, which ignores • Reflection of waves wave effects, and full wave electromagnetism, which is a precise theory. • Young’s double-slit experiment • Interference in thin films and air gaps In General Physics II you studied some aspects of geometrical optics. Geometrical optics rests on the assumption that light propagates along straight lines and is reflected and refracted according to definite laws, such Or the use of a convex lens as a magnifying lens: as Fermat’s principle and Snell’s Law. As a result the positions of images in mirrors and through lenses, etc. can be determined by scaled drawings. For example, the s ! s production of an image in a concave mirror. s Object object y Image • F • C f f y ! image 2 s ! 1 But many optical phenomena cannot be adequately The colors you see in a soap bubble are also due to an explained by geometrical optics. For example, the interference effect between light rays reflected from the iridescence that makes the colors of a hummingbird so front and back surfaces of the thin film of soap making brilliant are not due to pigment but to an interference the bubble. The color depends on the thickness of film, effect caused by structures in the feathers. ranging from black, where the film is thinnest, to magenta, where the film is thickest. Likewise, the colors you see in a thin film of oil floating on water are due to an interference effect between light rays reflected from the front and back surfaces of the oil Another example is the “spectrum of colors” you see film. when you reflect light from the active side of a CD. The effect that produces these colors is closely related to interference; it is called diffraction. Here is an example of diffraction caused by the edges of the razor blade when viewed from behind with Waves monochromatic blue light. In physics,we come across three main types of waves: Mechanical waves: water waves, sound waves, seismic waves, waves on a string ... Electromagnetic waves: visible and ultraviolet light, radio and television waves, microwaves, x-rays and radar waves. All e-m waves travel in vacuum with a speed of c = 2.99792458 !108 m s. The pattern of “fringes” is easily observable with monochromatic light. With white light, fringes due to Matter waves: the waves associated with electrons, the different wavelengths overlap making them more protons and other fundamental particles, atoms and difficult to observe. molecules. Diffraction and interference cannot be expained by However, all waves have features in common. geometrical optics; instead, light has to be treated as waves. Take the ripples (waves) in a pond caused by a small y $ stone being dropped into the water ... A x x " A ! Shown is a snapshot of the wave at some time t. It is described by the general equation: y (t,x) = A cos(!t " kx) At a position, x say, the disturbance y( t) varies x ! sinusoidally with time (i.e., simple harmonic motion). Similarly, at a time t, as shown here, the disturbance varies sinusoidally with position, x. The parameter y ! A k 2# is called the wavevector, where is the = $ $ x wavelength. The product kx is called the phase angle " A (= %). A negative (positive) sign indicates the wave is traveling to the right (left). Vertical and horizontal illustrations of the wave Also, ! is the angular frequency of the wave given by To describe the “disturbance” of such a wave we need ! = 2#f , and T = 1 is the periodic time of the wave. two variables, t and x. f y ! x y ! A x x wave at t wave at t + !t $ A Note that the disturbance at some fixed time t !, To find the speed of a wave, we take two snapshots at a time interval t apart. If the wave (i.e., the red dot) y (t!,x) = y(t!,x ± n!), ! i.e., the wave is reproduced at displacements of n !, travels a distance ! x in that time, then the speed of the where n is an integer. wave is v !x . y T = !t A Since the disturbances (y) are equal t y (t,x) = y(t + !t, x + !x), T = 1 = 2" so "t # kx = "(t + !t) # k(x + !x), $ A f # i.e., "! t = k!x, Similarly, the disturbance at some fixed position x !, v " . $ = k y(t,x ) = y(t ± T, x ) = y(t ± m2" ,x ), ! ! # ! Note also v " 2%f f . = k = 2% & = & i.e., the wave is reproduced after time intervals of The velocity of a fixed point on a wave (such as the red m2" , where m is an integer. # dot) is called the phase velocity. Principle of superposition “If two or more waves are traveling through a medium, y 1 = 6sin x the resultant disturbance at any point is the algebraic sum of the individual disturbances.” y 2 = 5sin 2x Waves that obey this principle are called linear waves. y 3 = 4sin 3x One consequence is that two waves can “pass” through each other! y 4 = 3sin 4x y 2 y 5 = 2sin 5x y 1 y 2 y 1 y 6 = sin 6x y 1 + y2 y 1 + y2 y 1 + y2 y 2 y y 2 1 y 1 (Waves that do not obey this principle are called non- y = y1 + y2 + y3 + y4 + y5 + y6 linear waves.) Consider the superposition of two waves of equal So, the superposition of several waves of differing amplitude but slightly different frequencies and wavelengths and amplitudes produces complex wavelengths. Then, the resultant is waveforms. For example, to produce a square wave ... y (x,t) = Asin(!1t " k1x) + Asin(!2t " k2x) $ #! #k ' = 2Acos& t " x) sin(! t " k x), % 2 2 ( ! + ! where #! = ! " ! , #k = k " k , ! = ( 1 2) 1 2 1 2 2 (k1 + k2) and k = 2 . When plotted, at some time t, we WBX06VD1.MOV get A square wave can be expressed as a so-called Fourier y (x) series: A x ! 1 $ n# ' f(x) = " sin& x) , n=1,3,5… n % L ( y (x) where L = * , i.e., one-half of the wavelength. 2A 2 x A Fourier series decomposes a periodic function into a sum of simple oscillating functions, i.e., sines and/or cosines. i.e., the waves are separated into “groups”. y (x) y (x) 2A 2A x x x 1 x 2 x 1 x 2 ! x ! x $ !" !k ' So, successive minima (at time t) occur at x 2 and x 1 where y = (x,t) = 2Acos& t # x) sin(" t # k x) % 2 2 ( $ !" !k ' $ !" !k ' & t # x ) # & t # x ) = *, % 2( % 1( The first term is the envelope, i.e., the green curve. The 2 2 2 2 2* second term is the wave within the envelope. Both the i.e, x # x = !x = . 2 1 k envelope and wave within the envelope are traveling ! #1 waves. Thus, the spatial extent of the group is ! x + !k . If we plot the resultant as a function of t at a fixed point, we get The envelope moves with velocity v = !" , called y (t) g !k 2A the group velocity. The wave inside the envelope moves with velocity v = " , called the phase velocity. p k t The amplitude of the envelope is zero when $ !" !k ' * & t # x ) = (2n +1) , In the case of sound waves, this waveform produces the % 2 2 n( 2 phenomenon of “beats”. where n = 0,±1,±2 !. y (t) The group velocity (the velocity of the envelope) is 2A " d kv % d! ( p) " dvp % vg = = $ ' = $ vp + k ' . [ dk]k $ dk' # dk& t # & k k t 1 t 2 If the phase velocity is the same at all frequencies and ! t wavelengths, i.e., there is no dispersion then Successive minima (at point x) occur at t and t where dvp 2 1 dk = 0 i.e., v g = vp. $ !" !k ' $ !" !k ' & t2 # x) # & t1 # x) = *, % 2 2 ( % 2 2 ( 2* i.e, t2 # t1 = !t = . !" Hence, the temporal extent ! t + !"#1. So, we find that ! x.!k , constant ( 2*) and ! t.!" , constant ( 2*). The phase velocity of the individual harmonic waves is " v p = k , " = vpk. Case A.mpg Wave Packets dvp A medium in which dk = 0 is said to be non- dispersive. (An example is an electromagnetic waves in A “wave packet” can be created by superposing many vacuum.) Glass, for instance, is a dispersive medium. waves spanning a wavevector range k ± !k . ! 2 Shown here is a plot of the phase velocity in flint glass. a (kn) v !108 m/s y p( ) 1 .98 k " !k k + !k x Red ! 2 ! 2 dv p 2 k n 1 .97 # "39.3 m s k ! dk ! k 1 .96 To generate the wave packet shown above, we put 25 Violet y = #a(kn )cos(knx) 1 .95 n="25 # k !107 m"1 ( ) where k = k + n !k and the amplitudes a(k ) are n ( ! 50) n 0 1 .0 1 .2 1 .4 1 .6 a Gaussian distribution, i.e., dv "n2 $2 In this case, p 0, so v v . a(k ) = e . dk < p > g n dv We set k ! = 5, ! k = 3 and $ = 10. Thus, there are 51 If p 0, then v v .
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