Bending and Bouncing Light

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 = ym sin(kx wt + f) • f 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 w, or the frequency f, which are all related: w = 2pf = 2p/T • The spatial dependence of periodic waves can be described by either the wavelength l or the wave number k, which are related. k = 2p/l • The speed of a traveling wave depends on both spatial and time dependence: v = l/T = lf = w/k

v v v v v v v=0 v=0 v=0 Standing waves - graphically Animation of Standing Wave Creation

Standing waves - mathematically • Take two identical waves traveling in opposite directions y1= ym sin (kx - wt) y2= ym sin (kx + wt) yT = y1 + y2 = 2ym cos wt sin kx This uses the identity sin a + sin b = 2cos½(a-b)sin½ (a+b) • Positions for which kx = np will ALWAYS have zero field. • If kx = np/2 (n odd), field strength will be maximum for particular time

Standing waves - interpretation y= 2ym cos wt sin kx • Positions which always have zero field (kx = np) are called nodes. • Positions which always have maximum (or minimum) field (kx = = np/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=nl/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=nl/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 - animation • The amount light slows down in a medium is described by the index of refraction : n=c/v • The wavelength in vacuum l0 is related to the wavelength l in other media by the index of refraction too: n = l0/l • The frequency of the light, and so the energy, remain unchanged.

Snell’s Law • As light slows down and decreases in wavelength, it bends - animation • The relationship between angles of incidence and refraction (measured from the normal!) is given by Snell’s Law: n1 sin q1 = n2 sin q2 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 qc = n2/n1, where n1 is for the denser medium • Any angles of incidence q1 qc 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 = y1 + y2 = 2ym cos wt sin kx • Nodes, or locations for which kx = np, will not move but will always have zero displacement. • If standing wave has both ends fixed (both nodes) a distance L apart, nl= 2L, n any integer • If standing wave has one end fixed (node) and one end open (antinode) a distance L apart, nl= 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 • slow down, v = c/n • decrease in wavelength, l = l0/n • and bend toward a normal to the interface of the media, n1 sinq1 = n2 sinq2

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 qc = n2/n1 • 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 = n0 sin qm= (n12 - n22)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 2 which will be posted on WebCT by Friday morning. • Start Homework 2 (found on WebCT by Friday AM), due next Thursday on material from the last two classes

Bending and Bouncing Light