1. Modern Optics I – wave properties of light
Special topics course in IAMS
Lecture speaker: Wang-Yau Cheng
2006/4

2. Outline Wave properties of light Polarization of light
Coherence of light
Special issues on quantum optics

3. Properties of wave Propagation Phase Wave equation Phase velocity
Group velocity
Refraction of wave
Interference of wave
Electro-magnetic (EM) wave
Spectrum of EM wave

4. Waves, the Wave Equation, and Phase Velocity
What is a wave?
Forward [f(x-vt)] vs. backward [f(x+vt)] propagating waves
The one-dimensional wave equation
Phase velocity
Complex numbers

5. What is a wave? A wave is anything that moves.
To displace any function f(x)
to the right, just change its
argument from x to x-a,
where a is a positive number.
If we let a = v t, where v is positive
and t is time, then the displacement
will increase with time.
So f(x-vt) represents a rightward, or forward,
propagating wave.
Similarly, f(x+vt) represents a leftward, or backward,
v will be the velocity of the wave.

6. The one-dimensional wave equation
We’ll derive the wave equation from Maxwell’s equations. Here it is in its one-dimensional form for scalar (i.e., non-vector) functions, f:
Light waves (actually the electric fields of light waves) will be a solution to this equation. And v will be the velocity of light.

7. Electromagnetism is linear: The principle of “Superposition” holds.
If f1(x,t) and f2(x,t) are solutions to the wave equation,
then f1(x,t) + f2(x,t) is also a solution.
Proof: and
This means that light beams can pass through each other.
It also means that waves can constructively or destructively interfere.

8. The solution to the one-dimensional wave equation
The wave equation has the simple solution:
where f (u) can be any twice-differentiable function.

9. The 1D wave equation for light waves
where E is the light electric field
We’ll use cosine- and sine-wave solutions: or
where:

10. Waves using complex numbers
The electric field of a light wave can be written:
E(x,t) = A cos(kx – wt – q)
Since exp(ij) = cos(j) + i sin(j), E(x,t) can also be written:
E(x,t) = Re { A exp[i(kx – wt – q)] } or
E(x,t) = 1/2 A exp[i(kx – wt – q)] + c.c.
where "+ c.c." means "plus the complex conjugate of everything before the plus sign."
We often write these expressions without the ½, Re, or +c.c.

11. Waves using complex amplitudes
We can let the amplitude be complex:
where we've separated the constant stuff from the rapidly changing stuff.
The resulting "complex amplitude" is:
So:
How do you know if E0 is real or complex?
Sometimes people use the "~", but not always.
So always assume it's complex.

12. Properties of wave Propagation  Phase  Wave equation
 Phase velocity
 Group velocity
Refraction of wave
Interference of wave
Electro-magnetic (EM) wave
Spectrum of EM wave

13. Definitions: Amplitude and Absolute phase
E(x,t) = A cos[(k x – w t ) – q ]
A = Amplitude
q = Absolute phase (or initial phase)

14. Definitions
Spatial quantities:
Temporal quantities:

15. The Phase Velocity How to measure the velocity of the moving wave?
First of all, measures the wavelength, secondly, count for how many wave peaks go through per second.
The phase velocity is the wavelength / period:
v = l / t
In terms of the k-vector, k = 2p / l, and
the angular frequency, w = 2p / t, this is: v = w / k

16. The Phase of a Wave The phase is everything inside the cosine.
E(t) = A cos(j), where j = kx – wt – q
In terms of the phase,
w = – ¶j/¶t
k = ¶j/¶x
and
– ¶j/¶t
v = –––––––
¶j/¶x
This formula is useful when the wave is really complicated.

17. When two waves of different frequency interfere, they produce "beats."
Indiv-
idual
waves
Sum
Envel-
ope
Irrad-
iance:

18. When two light waves of different frequency interfere, they produce beats.

19. Group velocity Light wave beats (continued):
Etot(x,t) = 2E0 cos(kavex–wavet) cos(Dkx–Dwt)
This is a rapidly oscillating wave [cos(kavex–wavet)]
with a slowly varying amplitude [2E0 cos(Dkx–Dwt)]
The phase velocity comes from the rapidly varying part: v = wave / kave
What about the other velocity?
Define the "group velocity:" vg º Dw /Dk
In general, we define the group velocity as:
vg º dw /dk

20. Group velocity is not equal to phase velocity
if the medium is dispersive (i.e., n varies).

22. Calculating the Group velocity
vg º dw /dk
Now, w is the same in or out of the medium, but k = k0n, where k0 is
the k-vector in vacuum, and n is what depends on the medium.
So it's easier to think of w as the independent variable:
Using k = w n(w) / c0, calculate: dk /dw = ( n + w dn/dw ) / c0
vg = c0 / ( n + w dn/dw ) = (c0/n) / (1 + w/n dn/dw )
Finally:
So the group velocity equals the phase velocity when dn/dw = 0,
such as in vacuum. Otherwise, since n increases with w, dn/dw > 0,
and:
vg < vphase.

23. 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 l0.

24. The group velocity is the velocity of the envelope or irradiance: the math.
The carrier wave propagates at the phase velocity.
And the envelope propagates at the group velocity:
Or, equivalently, the irradiance propagates at the group velocity:

25. The group velocity can exceed c0 when dispersion is anomalous.
vg = c0 / (n + w dn/dw )
dn/dw is negative in regions of anomalous dispersion, that is, near a
resonance. So vg can exceed c0 for these frequencies!
One problem is that absorption is strong in these regions. Also, dn/dw is
only steep when the resonance is narrow, so only a narrow range of
frequencies has vg > c0. Frequencies outside this range have vg < c0.
Pulses of light (which are broadband) therefore break up into a mess.

26. Beating the speed of light
To exceed c, we need a region of negative dn/dw over a fairly large
range of frequencies. And the slope should not vary much—to avoid
pulse break-up. And absorption should be minimal.
One trick is to excite the medium in advance with a laser pulse, which
creates gain (instead of absorption), which inverts the curve.
Then two nearby resonances have a region in between with minimal
absorption and near-linear negative slope:

27. Negative dispersion (vg = c0 / (n + w dn/dw) and dn/dw <0)
Naturally
Artificially
Grating pair
Optical fiber (or, some special designed waveguide)
Photonic crystal
Prisms in mode-locked lasers
EIT

28. Properties of wave Propagation Phase Wave equation Phase velocity
Group velocity
 Refraction of wave
 Interference of wave
Electro-magnetic (EM) wave
Spectrum of EM wave

29. An interesting question is what happens to wave when it encounters a surface.
At an oblique angle, light can be completely transmitted
or completely reflected.
"Total internal reflection" is the basis of optical fibers,
a billion dollar industry.

30. 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...

31. Superposition allows waves to pass through each other.
Otherwise they'd get screwed up while overlapping

32. 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:
where all initial phases are lumped into E1, E2, and E3.

33. Adding waves of the same frequency, but opposite direction, yields a "standing wave."
Waves propagating in opposite directions:
Since we must take the real part of the field, this becomes:
(taking E0 to be real)
Standing waves are important inside lasers, where beams are
constantly bouncing back and forth.

34. A Standing Wave

35. A Standing Wave Again…

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

37. Interfering spherical waves also yield a standing wave
Antinodes

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

39. When two waves of different frequency interfere, they produce beats.
Take E0 to be real.

40. Young’s Two-Slit Experiment
What happens when light passes through two slits?
Light pattern
that emerges
“fringes”
The idea is central to many laser techniques, such as holography, ultrafast photography, and acousto-optic modulators.
Tests of quantum mechanics also use it.

41. Diffraction Light bends around corners. This is called diffraction.
Light patterns after passing through rectangular slit(s):
One slit:
Two slits:
The diffraction pattern far away is the Fourier transform of the slit transmission vs. position.

42. Fourier decomposing functions plays a big role in optics.
Here, we write a square wave as a sum of sine waves of different frequency.

43. The Fourier transform is perhaps one of the most important equation in optics.
It converts a function of time to one of frequency:
and converting back uses almost the same formula:

44. Often, they do so by themselves.

45. What do we hope to achieve with the Fourier Transform?
We desire a measure of the frequencies present in a wave. This will
lead to a definition of the term, the “spectrum.”
Light electric field
Time
Plane waves have only one frequency, w.
This light wave has many
frequencies. And the
frequency increases in
time (from red to blue).
It will be nice if our measure also tells us when each frequency occurs.

46. Properties of wave Propagation Phase Wave equation Phase velocity
Group velocity
Refraction of wave
Interference of wave
 Electro-magnetic (EM) wave
 Spectrum of EM wave

47. “Light is just electromagnetic wave”
Review of Maxwell equations
The solutions which is convenient for optics
EM wave spectrum

48. The equations of optics are Maxwell’s equations.
where is the electric field, is the magnetic field, r is the charge density, e is the permittivity, and m is the permeability of the medium.

49. Longitudinal vs. Transverse waves
Motion is along
the direction of
Propagation
Longitudinal:
Motion is transverse
to the direction of
Propagation
Transverse:
Space has 3 dimensions, of which 2 directions are transverse to
the propagation direction, so there are 2 transverse waves in ad-
dition to the potential longitudinal one.

50. Vector fields Light is a 3D vector field.
A 3D vector field assigns a 3D vector (i.e., an arrow having both direction and length) to each point in 3D space.

51. The 3D vector wave equation for the electric field
Note the vector symbol over the E.
This is really just three independent wave equations, one each for the x-, y-, and z-components of E.
which has the vector field solution:

52. Waves using complex vector amplitudes
We must now allow the complex field and its amplitude to be vectors:
Note the arrows over the E’s!
The complex vector amplitude has six numbers that must be
specified to completely determine it!

53. Derivation of the Wave Equation from Maxwell’s Equations
Take of:
Change the order of differentiation on the RHS:

54. Derivation of the Wave Equation from Maxwell’s Equations (cont’d)
But:
Substituting for , we have:
Or:
assuming that m and e are constant in time.

55. Derivation of the Wave Equation from Maxwell’s Equations (cont’d)
Using the lemma,
becomes:
If we now assume zero charge density: r = 0, then
and we’re left with the Wave Equation!

56. Why light waves are transverse
Suppose a wave propagates in the x-direction. Then it’s a function of x and t (and not y or z), so all y- and z-derivatives are zero:   
Now, in a charge-free medium,
that is,
Substituting, we have:

57. The magnetic-field direction in a light wave
Suppose a wave propagates in the x-direction and has its electric field along the y-direction [so Ex = Ez= 0, and Ey = Ey(x,t)].
What is the direction of the magnetic field?
Use:
So:
In other words:
And the magnetic field points in the z-direction.

58. The magnetic-field strength in a light wave
Suppose a wave propagates in the x-direction and has its electric field in the y-direction. What is the strength of the magnetic field?
and
Take Bz(x,0) = 0
Differentiating Ey with respect to x yields an ik, and integrating with respect to t yields a 1/-iw.
So:
But w / k = c:

59. An Electromagnetic Wave
The electric and magnetic fields are in phase.
The electric field, the magnetic field, and the k-vector are
all perpendicular:

60. The Energy Density of a Light Wave
The energy density of an electric field is:
The energy density of a magnetic field is:
Using B = E/c, and , which together imply that
we have:
Total energy density:
So the electrical and magnetic energy densities in light are equal.

61. Why we neglect the magnetic field
Felectrical
Fmagnetic
The force on a charge, q, is:
so:
Since B = E/c:
where is the charge velocity
So as long as a charge’s velocity is much less than the speed of light, we can neglect the light’s magnetic force compared to its electric force.

62. The Poynting Vector: S = c2 e E x B
The power per unit area in a beam.
Justification (but not a proof):
Energy passing through area A in time Dt:
= U V = U A c Dt
So the energy per unit time per unit area:
= U V / ( A Dt ) = U A c Dt / ( A Dt ) = U c = c e E2
= c2 e E B
And the direction is reasonable.
V = A c Dt

63. The Irradiance (often called the Intensity)
A light wave’s average power per unit area is the “irradiance.”
Substituting a light wave into the expression for the Poynting vector,
, yields:
The average of cos2 is 1/2:
real amplitudes

64. The Irradiance (continued)
Since the electric and magnetic fields are perpendicular and B0 = E0 / c,
becomes:
where

65. The Electromagnetic Spectrum
radio
gamma-ray
visible
microwave
infrared UV
X-ray
106
105
wavelength (nm)
The transition wavelengths are a bit arbitrary…

66. The Electromagnetic Spectrum

67. The Long-Wavelength Electro-magnetic Spectrum

68. Radio & microwave regions (3 kHz – 300 GHz)

69. Key points Why light is just the electromagnetic wave?
Why light is a transverse EM wave?
Speed of light is by definition
How to use Maxwell eq. depends on your conditions
What’s the so-called “instantaneous frequency”?
What’s the three ways of the solutions of wave equation?
What’s the amplitude and phase of EM wave?
Wave equation