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### Modern Optics I – wave properties of light

### Negative dispersion (vg = c0 / (n + wdn/dw) and dn/dw <0)

Special topics course in IAMS

Lecture speaker: Wang-Yau Cheng

2006/4

Outline

- Wave properties of light
- Polarization of light
- Coherence of light
- Special issues on quantum optics

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

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

where E is the light electric field

We’ll use cosine- and sine-wave solutions:

or

where:

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.

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.

- Propagation

Phase

Wave equation

Phase velocity

Group velocity

- Refraction of wave
- Interference of wave
- Electro-magnetic (EM) wave
- Spectrum of EM wave

Definitions: Amplitude and Absolute phase

E(x,t) = A cos[(k x – w t ) – q ]

A = Amplitude

q = Absolute phase (or initial phase)

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

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.

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

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 ºDw /Dk

In general, we define the group velocity as:

vgº dw /dk

Group velocity is not equal to phase velocity

if the medium is dispersive (i.e., n varies).

Calculating the Group velocity

vgº 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.

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.

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:

The group velocity can exceed c0 whendispersion 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.

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:

Naturally

Artificially

Grating pair

Optical fiber (or, some special designed waveguide)

Photonic crystal

Prisms in mode-locked lasers

EIT

- Propagation
- Phase
- Wave equation
- Phase velocity
- Group velocity

Refraction of wave

Interference of wave

- Electro-magnetic (EM) wave
- Spectrum of EM wave

An interesting question is what happensto 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.

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...

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."

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.

A Standing Wave: Experiment

Mirror

3.9 GHz microwaves

Input beam

The same effect occurs in lasers.

Note the node at the reflector at left.

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?

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.

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.

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:

Time

What do we hope to achieve with theFourier 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, 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.

- Propagation
- Phase
- Wave equation
- Phase velocity
- Group velocity
- Refraction of wave
- Interference of wave

Electro-magnetic (EM) wave

Spectrum of EM wave

“Light is just electromagnetic wave”

- Review of Maxwell equations
- The solutions which is convenient for optics
- EM wave spectrum

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.

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.

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

Note the vector symbol over the E.

which has the vector field solution:

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:

Note the arrows over the E’s!

The complex vector amplitude has six numbers that must be

specified to completely determine it!

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 m and e 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: r = 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, andEy = 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.

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:

An Electromagnetic Wave

The electric and magnetic fields arein phase.

The electric field, the magnetic field, and the k-vector are

all perpendicular:

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

Felectrical

Fmagnetic

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.

The Poynting Vector: S = c2eE 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

= c2e E B

And the direction is reasonable.

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

radio

gamma-ray

visible

microwave

infrared

UV

X-ray

106

105

wavelength (nm)

The transition wavelengths are a bit arbitrary…

- 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

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