18. Standing Waves, Beats, and Group Velocity

Transcription

1 18. Standing Waves, Beats, and Group Velocity Superposition again Standing waves: the sum of two oppositely traveling waves Beats: the sum of two different frequencies Group velocity: the speed of information Going faster than light... (but not really)

2 Sometimes, the refractive index is less than one. Example: a bunch of free electrons, above the plasma frequency n 2 1 P2 In the x-ray range (where >> P ), the refractive index of most materials is slightly less than unity: 1 n is small and positive So, how fast do waves propagate in these situations? What is the speed of light? To answer this question, we need to think more carefully about what we mean by speed.

3 Superposition allows waves to pass through each other. Recall: If E 1 (x,t) and E 2 (x,t) are both solutions to the wave equation, then so is their sum. Otherwise they'd get screwed up while overlapping, and wouldn t come out the same as they went in.

4 Adding waves of the same frequency, but different initial phase, yields a wave of the same frequency. This isn't so obvious using trigonometric functions, but it's easy with complex exponentials: Etot ( x, t) E1exp j( kxt) E2exp j( kxt) E3exp j( kxt) ( E1E2 E3)exp j( kxt) where all the phases (other than the kxt) are lumped into E 1, E 2, and E 3.

5 Adding waves of the same frequency, but opposite direction, yields a "standing wave." Waves propagating in opposite directions: Etot ( x, t) E0exp j( kx t) E0exp j( kx t) E0 exp( jkx)[exp( jt) exp( jt)] 2E exp( jkx)cos( t) 0 Since we must take the real part of the field, this becomes: E ( xt, ) 2E cos( kx)cos( t) tot 0 (taking E 0 to be real) Standing waves are important inside lasers, where beams are constantly bouncing back and forth.

6 A Standing Wave E ( x, t) 2E cos( kx)cos( t) tot 0

7 A Standing Wave You ve seen the previews. Now, the movie! Question: what is the speed of energy propagation here?

8 A Standing Wave: Experiment 3.9 GHz microwaves Mirror Input beam Note the node at the reflector at left. The same effect occurs in lasers.

9 Interfering spherical waves also yield a standing wave Antinodes

10 Two Point Sources Different separations. Note the different node patterns.

11 When two waves of different frequency interfere, they produce beats. Etot ( xt, ) E0exp( j1t) E0exp( j2t) Let ave and 2 2 So : Etot ( x, t) E0exp j( avett) E0exp j( avett) E0 exp( javet)[exp( jt) exp( jt)] 2E0 exp( javet)cos( t) Taking the real part yields the product of a rapidly varying cosine ( ave ) and a slowly varying cosine ().

12 When two waves of different frequency interfere, they produce "beats." Individual waves Sum Envelope Irradiance

13 When two light waves of different frequency interfere, they produce beats. Etot (,) x t E0exp( j k1x1t) E0exp( j k2x2t) Let Similiarly, So: k ave ave k1k2 k1k2 and k and 2 2 Etot (,) xt E0exp( jkavexkxavett) E0exp( jkavexkxavett) E0 exp j( kavex avet) exp j( kx t ) exp{ j( kx t )} 2E0 exp j( kavexavet)cos( kxt) Real part : 2E cos( k x t)cos( kxt) 0 ave ave

14 Group velocity Light-wave beats (continued): E tot (x,t) = 2E 0 cos(k ave x ave t) cos(kx t) This is a rapidly oscillating wave: [cos(k ave x ave t)] with a slowly varying amplitude: [2E 0 cos(kx t)] The phase velocity comes from the rapidly varying part: v = ave / k ave What about the other velocity the velocity of the amplitude? Define the "group velocity:" v g /k In general, we define the group velocity as: v g d dk

15 Usually, group velocity is not equal to phase velocity, except in empty space. For our example, v g k ck ck nk nk where the subscripts 1 and 2 refer to the values at 1 and at 2. k 1 and k 2 are the k-vectors in vacuum. If k , g nk1k2 n n n n v c k c phase velocity If n n, v 1 2 g phase velocity

16 Calculating the Group velocity v g d /dk Now, is the same in or out of the medium, but k = k 0 n, where k 0 is the k-vector in vacuum, and n is what depends on the medium. So it's easier to think of as the independent variable: vg dk/ d 1 Using k = n() / c 0, calculate: v c n d 0 g So dn or So, the group velocity equals the phase velocity when dn/d = 0, such as in vacuum. Otherwise, since n usually increases with (normal dispersion), dn/d > 0 and so usually v g < v. dk d n 1 dn n d d c0 c0 d v g v dn 1 nd

17 Why is this important? You cannot send information using a wave, unless you make it into some kind of pulse. You cannot make a pulse without superposing different frequencies. Pulses travel at the group velocity. sum of 2 different frequencies sum of 6 different frequencies sum of many different frequencies

18 Group velocity (v g ) vs. phase velocity (v ) vg v vg v vg v vg v v g 0 v 0 Source:

19 Calculating Group Velocity vs. Wavelength We more often think of the refractive index in terms of wavelength,so let's write the group velocity in terms of the vacuum wavelength 0. Use the chain rule : Now, dn dn d0 d d d c0 d0 2c0 2c0 0 0, so: 2 2 d (2 c0/ 0) 2c0 Recalling that : we have: v v g g c0 dn /1 n n d c 2c dn /1 n n0 d0 2c0 or: v g c / 1 dn 0 0 n n d0 c n 0 0 dn d 0

20 The group velocity is less than the phase velocity in regions of normal dispersion v g c0 dn n d In regions of normal dispersion, dn/d is positive. So v g < c 0 /n < c 0 these frequencies. for

21 The group velocity often depends on frequency We have seen that the phase velocity depends on, because n does. v n c 0 It should not be surprising that the group velocity also depends on. v g n c0 dn d When the group velocity depends on frequency, this is known as group velocity dispersion, or GVD. Just as essentially all solids and liquids exhibit dispersion, they also all exhibit GVD. This property is crucially important in the design of, e.g., optical data transfer systems that use fiber optics.

22 GVD distorts the shape of a pulse as it propagates in a medium v g (blue) < v g (red) GVD means that the group velocity will be different for different wavelengths in a pulse. GVD = 0 GVD = 0 Source:

23 The group velocity can exceed c 0 when dispersion is anomalous c0 vg dn n d dn/d is negative in regions of anomalous dispersion, that is, near a resonance. So v g exceeds v, and can even exceed c 0 in these regions! We note that absorption is strong in these regions. dn/d is only steep when the resonance is narrow, so only a narrow range of frequencies has v g > c 0. Frequencies outside this range have v g < c 0.

24 The group velocity can exceed c 0 when dispersion is anomalous There is a more fundamental reason why v g > c 0 doesn t necessarily bother us. The interpretation of the group velocity as the speed of energy propagation is only valid in the case of normal dispersion! In fact, mathematically we can superpose waves to make any group velocity we desire - even zero! For discussion, see:

25 In artificially designed materials, almost any behavior is possible Here s one recent example: Science, vol. 312, p. 892 (2006) A metal/dielectric composite structure In this material, a light pulse appears to exit the medium before entering it. Of course, relativity and causality are never violated. experiment theory