Bending and Bouncing Light Standing Waves, Reflection, and Refraction

What have we learned? Waves transmit information between two points without individual particles moving between those points Transverse Waves oscillate perpendicularly to the direction of motion Longitudinal Waves oscillate in the same direction as the motion Any traveling sinusoidal wave may be described by y = y m sin(kx   t +  )  is the phase constant that determines where the wave starts.

What else have we learned? The time dependence of periodic waves can be described by either the period T, the angular speed , or the frequency f, which are all related:  = 2  f = 2  /T The spatial dependence of periodic waves can be described by either the wavelength or the wave number k, which are related. k = 2  / The speed of a traveling wave depends on both spatial and time dependence: v = /T = f =  /k

Standing waves - graphically AnimationAnimation of Standing Wave Creation vv v vvv v=0

Standing waves - mathematically Take two identical waves traveling in opposite directions y 1 = y m sin (kx -  t) y 2 = y m sin (kx +  t) y T = y 1 + y 2 = 2y m cos  t sin kx This uses the identity sin a + sin b = 2cos½(a-b)sin½ (a+b) Positions for which kx = n  will ALWAYS have zero field. If kx = n  /2 (n odd), field strength will be maximum for particular time

Standing waves - interpretation y = 2y m cos  t sin kx Positions which always have zero field (kx = n  ) are called nodes. Positions which always have maximum (or minimum) field (kx = = n  /2 (n odd)) are called antinodes. The location of nodes and antinodes don’t travel in time, but the amplitude at the antinodes changes with time.

Standing waves - if ends are fixed If the amplitude must be zero at the ends of the medium through which it travels, then standing waves will only be created if nodes occur at the endpoints. –One example is a string with fixed ends, like a violin string Then the wavelength will be some fraction of 2L, where L is the length of the string/antenna/etc. L=n /2

Standing waves - if one end open If one end is open, the endpoint is an antinode –This is similar to waves in a cavity with an open end, like a wind instrument –Think about shaking a rope to set up a wave. Your end is free to move, and the wave amplitude cannot be greater than the amplitude of your motion Then the wavelength will be some odd fraction of 4L, where L is the length of the string/antenna L=n /4, n odd

Why care about Standing Waves? Electromagnetic signals are produced by standing waves on antenna, for example The length of the antenna can be no shorter than 1/4 the wavelength of the signal (since end of antenna is not fixed) This puts practical constraints on what wavelengths can be transmitted - need short wavelengths, or high frequencies They are similar in concept to Fourier spectra and modes in an optical fiber – both of which interest us

Summary of Reflection All angles determining the direction of light rays are measured with respect to a normal to the surface. Light always reflects off a surface with an angle of reflection equal to the angle of incidence. When light strikes a rough surface, each “ray” in the beam has a different angle of incidence and so a different angle of reflection – this is called Diffuse Reflection

Refraction When light travels into a denser medium from a rarer medium, it slows down and decreases in wavelength as the wave fronts pile up - animationanimation The amount light slows down in a medium is described by the index of refraction : n=c/v The wavelength in vacuum 0 is related to the wavelength in other media by the index of refraction too: n = 0 / The frequency of the light, and so the energy, remain unchanged.

Snell’s Law As light slows down and decreases in wavelength, it bends - animationanimation The relationship between angles of incidence and refraction (measured from the normal!) is given by Snell’s Law: n 1 sin  1 = n 2 sin  2 Do the “Before You Start” Questions in Today’s Activity

Total Internal Reflection Light traveling from a denser medium to a rarer medium bends away from the normal, so the angle in the rarer medium could become 90 degrees. When the angle of refraction is 90 degrees, the angle of incidence is equal to the critical angle: sin  c = n 2 /n 1, where n 1 is for the denser medium Any angles of incidence  1   c result in Total Internal Reflection, when the light cannot exit the denser material.

Do the Rest of the Activity

What have we learned today? Identical sinusoidal waves traveling in opposite directions combine to produce standing waves: y = y 1 + y 2 = 2y m cos  t sin kx Nodes, or locations for which kx = n , will not move but will always have zero displacement. If standing wave has both ends fixed (both nodes) a distance L apart, n = 2L, n any integer If standing wave has one end fixed (node) and one end open (antinode) a distance L apart, n = 4L, n odd integer

What else have we learned? The angle of incidence ALWAYS equals the angle of reflection Light reflecting off a smooth surface undergoes total reflection, while light reflecting off a rough surface can undergo diffuse reflection Light entering a denser medium will (a)slow down, v = c/n (b)decrease in wavelength, = 0 /n (c)and bend toward a normal to the interface of the media, n 1 sin  1 = n 2 sin  2

What else have we learned? Light entering a rarer medium can exhibit total internal reflection (TIR) if the angle of incidence is greater than the critical angle for the interface sin  c = n 2 /n 1 TIR is the phenomenon underlying fiber optics; the Numerical Aperture indicates the angles at which light can enter a fiber and remain trapped inside: NA = n 0 sin  m = (n n 2 2 ) 1/2.

Before the next class,... Read the Assignment on Fourier Analysis found on WebCT Read Chapter 3 from the handout from Grant’s book on Lightwave Transmission Do Reading Quiz 4 which will be posted on WebCT by Friday morning. Start Homework 5 (found on WebCT by Friday AM), due next Thursday on material from this and the previous class.