1. Waves, the Wave Equation, and Phase Velocity
Prof. Rick Trebino
Georgia Tech
Waves, the Wave Equation, and Phase Velocity
f(x)
f(x-3)
f(x-2)
f(x-1) x
What is a wave?
Forward [f(x-vt)] and backward [f(x+vt)] propagating waves
The one-dimensional wave equation
Harmonic waves
Wavelength, frequency, period, etc.
Phase velocity Complex numbers Plane waves and laser beams
Photons and photon statistics

2. What is a wave? f(x) f(x-3) f(x-2) f(x-1) x
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 represents a rightward, or forward, propagating wave.
Similarly, represents a leftward, or backward, propagating wave.
v will be the velocity of the wave.
f(x)
f(x-3)
f(x-2)
f(x-1) x
Picture courtesy of the Consortium for Optics and Imaging Education
f(x - v t)
f(x + v t)

3. The one-dimensional wave equation and its solution
We’ll derive the wave equation from Maxwell’s equations next class.
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.
The wave equation has the simple solution:
where f (u) can be any twice-differentiable function.

4. Proof that f (x ± vt) solves the wave equation
Write f (x ± vt) as f (u), where u = x ± vt. So and
Now, use the chain rule:
So Þ and Þ   
Substituting into the wave equation:

5. The 1D wave equation for light waves
where E is the light electric field
We’ll use cosine- and sine-wave solutions: or
where:
The speed of light in vacuum, usually called “c”, is 3 x 1010 cm/s.

6. A simpler equation for a harmonic wave:
E(x,t) = A cos[(kx – wt) – q]
Use the trigonometric identity:
cos(z–y) = cos(z) cos(y) + sin(z) sin(y)
where z = k x – w t and y = q to obtain:
E(x,t) = A cos(kx – wt) cos(q) + A sin(kx – wt) sin(q)
which is the same result as before,
as long as:
A cos(q) = B and A sin(q) = C
For simplicity, we’ll just use the forward-propagating wave.

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

8. Definitions
Spatial quantities:
Temporal quantities:

9. The Phase Velocity v = l v v = w / k How fast is the wave traveling?
Velocity is a reference distance
divided by a reference time.
The phase velocity is the wavelength / period: v = l / t
Since n = 1/t :
In terms of the k-vector, k = 2p / l, and
the angular frequency, w = 2p / t, this is:
v = l v
v = w / k

10. Human wave
Picture courtesy of the Consortium for Optics and Imaging Education
A typical human wave has a phase velocity of about 20 seats per second.

11. The Phase of a Wave The phase is everything inside the cosine.
E(x,t) = A cos(j), where j = k x – w t – q
j = j(x,y,z,t) and is not a constant, like 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.

12. Complex numbers Let the x-coordinate be the real part
Consider a point,
P = (x,y), on a 2D
Cartesian grid.
Let the x-coordinate be the real part
and the y-coordinate the imaginary part
of a complex number.
So, instead of using an ordered pair, (x,y), we write:
P = x + i y
= A cos(j) + i A sin(j)
where i = (-1)1/2

13. Euler's Formula exp(ij) = cos(j) + i sin(j)
so the point, P = A cos(j) + i A sin(j), can be written:
P = A exp(ij)
where
A = Amplitude
j = Phase

14. Proof of Euler's Formula
exp(ij) = cos(j) + i sin(j)
Use Taylor Series:
If we substitute x = ij
into exp(x), then:

15. Complex number theorems

16. More complex number theorems
Any complex number, z, can be written:
z = Re{ z } + i Im{ z } So
Re{ z } = 1/2 ( z + z* )
and
Im{ z } = 1/2i ( z – z* )
where z* is the complex conjugate of z ( i ® –i )
The "magnitude," | z |, of a complex number is:
| z |2 = z z* = Re{ z }2 + Im{ z }2
To convert z into polar form, A exp(ij):
A2 = Re{ z }2 + Im{ z }2
tan(j) = Im{ z } / Re{ z }

17. We can also differentiate exp(ikx) as if the argument were real.

18. 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.

19. 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:
As written, this entire field is complex!
How do you know if E0 is real or complex?
Sometimes people use the "~", but not always.
So always assume it's complex.

20. Complex numbers simplify optics!
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.

21. The 3D wave equation for the electric field and its solution!
A light wave can propagate in any direction in space. So we must allow the space derivative to be 3D: or
which has the solution:
where
and

22. Usually, we just draw lines; it’s easier.
is called a plane wave.
A plane wave’s contours of maximum field, called wave-fronts or phase-fronts, are planes. They extend over all space.
Wave-fronts are helpful for drawing pictures of interfering waves.
A wave's wave-fronts sweep along at the speed of light.
Usually, we just draw lines; it’s easier.
A plane wave's wave-fronts are equally spaced, a wavelength apart.
They're perpendicular to the propagation direction.

23. Laser beams vs. Plane waves
A plane wave has flat wave-fronts throughout all space. It also has infinite energy. It doesn’t exist in reality.
A laser beam is more localized. We can approximate a laser beam as a plane wave vs. z times a Gaussian in x and y:
Localized wave-fronts z
Laser beam spot on wall w x y x
exp(-x2)

24. Laser pulses E t exp(-x2) x
If we can localize the beam in space by multiplying by a Gaussian in x and y, we can also localize it in time by multiplying by a Gaussian in time. t E
This is the equation for a laser pulse.

25. Longitudinal vs. Transverse waves
Motion is along the direction of propagation—longitudinal polarization
Longitudinal:
Motion is transverse to the direction of propagation—transverse polarization
Transverse:
Space has 3 dimensions, of which 2 are transverse to the propagation direction, so there are 2 transverse waves in addition to the potential longitudinal one.
The direction of the wave’s variations is called its polarization.

26. Wind patterns: 2D vector field
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.
Wind patterns: 2D vector field
A light wave has both electric and magnetic 3D vector fields:

27. The 3D wave equation for the electric field is actually a vector equation!
A light-wave electric field can point in any direction in space:
Note the arrow over the E.
which has the solution:
where
and

28. 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!
x-component
y-component
z-component

29. Light is not only a wave, but also a particle.
Photographs taken in dimmer light look grainier.
Very very dim
Very dim
Dim
Bright
Very bright
Very very bright
When we detect very weak light, we find that it’s made up of particles.
We call them photons.

30. Photons The energy of a single photon is: hn or = (h/2p)w
where h is Planck's constant, x Joule-sec.
One photon of visible light contains about Joules, not much!.
F is the photon flux, or
the number of photons/sec
in a beam.
F = P / hn
where P is the beam power.

31. Counting photons tells us a lot about the light source.
Random (incoherent) light sources,
such as stars and light bulbs, emit
photons with random arrival times
and a Bose-Einstein distribution.
Laser (coherent) light sources, on
the other hand, have a more
uniform (but still random) distribution: Poisson.
Bose-Einstein
Poisson

32. Photons have momentum
If an atom emits a photon, it recoils in the opposite direction.
If the atoms are excited and then emit light, the atomic beam spreads much more than if the atoms are not excited and do not emit.

33. Photons—Radiation Pressure
Photons have no mass and always travel at the speed of light.
The momentum of a single photon is: h/l, or
Radiation pressure = Energy Density (Force/Area = Energy/Volume)
When radiation pressure cannot be neglected:
Comet tails (other forces are small)
Viking space craft (would've missed Mars by 15,000 km)
Stellar interiors (resists gravity)
PetaWatt laser (1015 Watts!)
Comet image from

34. Photons "What is known of [photons] comes from observing the
results of their being created or annihilated."
Eugene Hecht
What is known of nearly everything comes from observing the
results of photons being created or annihilated.