Modern Optics I – wave properties of light
Special topics course in IAMS
Lecture speaker: Wang-Yau Cheng
2006/4
Outline
🡪Propagation
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
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,
propagating wave.
v will be the velocity of the wave.
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.
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.
The solution to the one-dimensional wave equation
where f (u) can be any twice-differentiable function.
The wave equation has the simple solution:
The 1D wave equation for light waves
We’ll use cosine- and sine-wave solutions:
or
where:
where E is the light electric field
Waves using complex numbers
The electric field of a light wave can be written:
E(x,t) = A cos(kx – ωt – θ)
Since exp(iϕ) = cos(ϕ) + i sin(ϕ), E(x,t) can also be written:
E(x,t) = Re { A exp[i(kx – ωt – θ)] }
or
E(x,t) = 1/2 A exp[i(kx – ωt – θ)] + 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.
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.
🡪 Phase
🡪 Wave equation
🡪 Phase velocity
🡪 Group velocity
Definitions: Amplitude and Absolute phase
E(x,t) = A cos[(k x – ω t ) – θ ]
A = Amplitude
θ = Absolute phase (or initial phase)
Definitions
Spatial quantities:
Temporal quantities:
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 = λ / τ
In terms of the k-vector, k = 2π / λ, and
the angular frequency, ω = 2π / τ, this is: v = ω / k
The Phase of a Wave
The phase is everything inside the cosine.
E(t) = A cos(ϕ), where ϕ = kx – ωt – θ
In terms of the phase,
ω = – ∂ϕ/∂t
k = ∂ϕ/∂x
and
– ∂ϕ/∂t
v = –––––––
∂ϕ/∂x
This formula is useful when the wave is really complicated.
When two waves of different frequency �interfere, they produce "beats."
Indiv-
idual
waves
Sum
Envel-
ope
Irrad-
iance:
When two light waves of different �frequency interfere, they produce beats.
Group velocity
vg ≡ dω /dk
Light wave beats (continued):
Etot(x,t) = 2E0 cos(kavex–ωavet) cos(Δkx–Δωt)
This is a rapidly oscillating wave [cos(kavex–ωavet)]
with a slowly varying amplitude [2E0 cos(Δkx–Δωt)]
The phase velocity comes from the rapidly varying part: v = ωave / kave
What about the other velocity?
Define the "group velocity:" vg ≡ Δω /Δk
In general, we define the group velocity as:
Group velocity is not equal to phase velocity
if the medium is dispersive (i.e., n varies).
Calculating the Group velocity
vg ≡ dω /dk
Now, ω 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 ω as the independent variable:
Using k = ω n(ω) / c0, calculate: dk /dω = ( n + ω dn/dω ) / c0
vg = c0 / ( n + ω dn/dω ) = (c0/n) / (1 + ω/n dn/dω )
Finally:
So the group velocity equals the phase velocity when dn/dω = 0,
such as in vacuum. Otherwise, since n increases with ω, dn/dω > 0,
and:
vg < vphase.
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 λ0.
The group velocity is the velocity of the envelope or irradiance: the math.
And the envelope propagates at the group velocity:
Or, equivalently, the irradiance propagates at the group velocity:
The carrier wave propagates at the phase velocity.
The group velocity can exceed c0 when�dispersion is anomalous.
vg = c0 / (n + ω dn/dω )
dn/dω 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/dω 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.
Beating the speed of light
To exceed c, we need a region of negative dn/dω 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:
Negative dispersion (vg = c0 / (n + ω dn/dω) and dn/dω <0)
🡪 Refraction of wave
🡪 Interference of wave
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.
Group velocity: the speed of information
Going faster than light...
Superposition again
Standing waves: the
sum of two oppositely
traveling waves
Beats: the sum of two different frequencies
Superposition allows waves to pass �through each other.
Otherwise they'd get screwed up while overlapping
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.
Adding waves of the same frequency, but opposite direction, yields a "standing wave."
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.
Waves propagating in opposite directions:
A Standing Wave
A Standing Wave�Again…
A Standing Wave: Experiment
3.9 GHz microwaves
Note the node at the reflector at left.
Mirror
Input beam
The same effect occurs in lasers.
Interfering spherical waves also �yield a standing wave
Antinodes
Two Point Sources
Different separations. Note the different node patterns.
When two waves of different frequency �interfere, they produce beats.
Take E0 to be real.
Young’s Two-Slit Experiment
What happens when light passes through two slits?
The idea is central to many laser techniques, such as holography, ultrafast photography, and acousto-optic modulators.
Tests of quantum mechanics also use it.
Light pattern
that emerges
“fringes”
Diffraction
Light bends around corners. This is called diffraction.
The diffraction pattern far away is the Fourier transform of the slit transmission vs. position.
Light patterns after passing through rectangular slit(s):
One slit:
Two slits:
Fourier decomposing functions plays a big role in optics.
Here, we write a square wave as a sum of sine waves of different frequency.
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:
Often, they do so by themselves.
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.”
Plane waves have only one frequency, ω.
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.
Light electric field
Time
🡪 Electro-magnetic (EM) wave
🡪 Spectrum of EM wave
The equations of optics are �Maxwell’s equations.
where is the electric field, is the magnetic field, ρ is the charge density, ε is the permittivity, and μ is the permeability of the medium.
Longitudinal vs. Transverse waves
Motion is along
the direction of
Propagation
Motion is transverse
to the direction of
Propagation
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.
Transverse:
Longitudinal:
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.
The 3D vector wave equation for the electric field
which has the vector field solution:
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.
Waves using complex vector amplitudes
We must now allow the complex field and its amplitude to be vectors:
The complex vector amplitude has six numbers that must be
specified to completely determine it!
Note the arrows over the E’s!
Derivation of the Wave Equation �from Maxwell’s Equations
Take of:
Change the order of differentiation on the RHS:
Derivation of the Wave Equation �from Maxwell’s Equations (cont’d)
But:
Substituting for , we have:
Or:
assuming that μ and ε are constant in time.
Derivation of the Wave Equation �from Maxwell’s Equations (cont’d)
Using the lemma,
becomes:
If we now assume zero charge density: ρ = 0, then
and we’re left with the Wave Equation!
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:
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.
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?
The magnetic-field strength in a light wave
Take Bz(x,0) = 0
Differentiating Ey with respect to x yields an ik, and integrating with respect to t yields a 1/-iω.
and
So:
But ω / k = c:
An Electromagnetic Wave
The electric field, the magnetic field, and the k-vector are
all perpendicular:
The electric and magnetic fields are in phase.
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.
Why we neglect the magnetic field
The force on a charge, q, is:
so:
Since B = E/c:
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.
Felectrical
Fmagnetic
where is the charge velocity
The Poynting Vector: S = c2 ε E x B
The power per unit area in a beam.
Justification (but not a proof):
Energy passing through area A in time Δt:
= U V = U A c Δt
So the energy per unit time per unit area:
= U V / ( A Δt ) = U A c Δt / ( A Δt ) = U c = c ε E2
= c2 ε E B
And the direction is reasonable.
V = A c Δt
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
The Irradiance (continued)
Since the electric and magnetic fields are perpendicular and B0 = E0 / c,
becomes:
where
The Electromagnetic Spectrum
infrared
X-ray
UV
visible
wavelength (nm)
microwave
radio
105
106
gamma-ray
The transition wavelengths are a bit arbitrary…
The Electromagnetic Spectrum
The Long-Wavelength Electro-magnetic Spectrum
Radio & microwave regions (3 kHz – 300 GHz)