A function with a large curl
r f ( x, y, z ) = ( y, x, 0) r f (1, 0, 0) = (0,1, 0) r f (0,1, 0) = (1, 0, 0) r f (1, 0, 0) = (0, 1, 0) r f (0, 1, 0) = (1, 0, 0)
y
x
r r f z f y f y f x f x f z f = , , y z z x x y = ( 0 0, 0 0, 1 (1) ) =( 0, 0 , 2
)
$ So th
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 its made up of particles. We call them photons.
Waves using complex vector amplitudes
We must now allow the complex field E and its amplitude E0 to be % % vectors:
rr rr r E ( r , t ) = E0 exp i k r t % %
(
)
Note the arrows over the Es!
The complex vector amplitude has six numbers that must be specifi
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:
r r2 r E E 2 = 0 t
2
Note the arrow over the E.
which has the solution: where and and
r r k ( k x , k y , k z ) r (
Vector fields
Light is a 3D vector field.
rr A 3D vector field f (r )
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 fie
Longitudinal vs. Transverse waves
Motion is along the direction of propagation longitudinal polarization
Longitudinal:
Transverse:
Motion is transverse to the direction of propagation transverse polarization
Space has 3 dimensions, of which 2 are transver
exp(-x2)
Laser pulses
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.
E
t
t2 x2 + y 2 E ( x, y, z , t ) = E0 exp 2 exp exp[i (kz t )] 2 % % w
This is t
Laser beams vs. Plane waves
A plane wave has flat wave-fronts throughout all space. It also has infinite energy. It doesnt 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:
rr E0 exp[i (k r t )] is called a plane wave. %
A plane waves 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 wavefronts swe
Photons
The energy of a single photon is: h or h = (h/2 )
where h is Planck's constant, 6.626 x 10-34 Joule-sec. One photon of visible light contains about 10-19 Joules, not much!. is the photon flux, or the number of photons/sec in a beam. = P / h where
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 mo
Div, Grad, Curl, and still more all that
r The Curl of a vector function f :
rr f z f y f x f z f y f x f , , y dz z dx x dy
The curl can be treated as a matrix determinant :
x r r f = x fx
y y fy
z z fz
Functions that tend to curl around have large
Div, Grad, Curl, and more all that
TheLaplacianofascalarfunction: rr r f 2
f
f
f f = , , x y z
=
2 f 2 f 2 f + + 2 2 x y z 2
TheLaplacian of a vectorfunctionisthesame, butforeachcomponentoff:
r 2 fx 2 fx 2 fx 2 f y 2 f y 2 f y 2 fz 2 fz 2 fz 2 f = 2 + +
Div, Grad, Curl, and all that
The Divergence of a vector function:
rr f x f y f z f + + x y z
The Divergence is nonzero if there are sources or sinks.
A 2D source with a large divergence:
y x
Note that the x-component of this function changes rapidly in t
Div, Grad, Curl, and all that
Types of 3D vector derivatives:
The Del operator:
r , , x y z
The Gradient of a scalar function f :
r f f f f , , x y z
If you want to know more about vector calculus, read this book!
The gradient points in the direction of
Maxwell's Equations and Light Waves
Vector derivatives: Div, grad, curl, etc. Derivation of wave equation from Maxwell's Equations Why light waves are transverse waves Why we neglect the magnetic field
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.
PhotonsRadiation Pressure
Photons have no mass and always travel at the speed of light. The momentum of a single photon is: h/ , or hk Radiation pressure = Energy Density (Force/Area = Energy/Volume)
When radiation pressure cannot be neglected: Comet tail
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.
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
r2 2 E E 2 = 0 t
2 E 2 E 2 E 2 E + 2 + 2 2 = 0 2 x y z t
which has the solution: where a
Waves using complex amplitudes
We can let the amplitude be complex:
E ( x, t ) = A exp i ( kx t )
E ( x, t ) = cfw_ A exp(i ) exp i ( kx t )
cfw_
where we've separated the constant stuff from the rapidly changing stuff.
The resulting "complex amplitude
Waves using complex amplitudes
We can let the amplitude be complex:
E ( x, t ) = A exp i ( kx t )
E ( x, t ) = cfw_ A exp(i ) exp i ( kx t )
cfw_
where we've separated the constant stuff from the rapidly changing stuff.
The resulting "complex amplitude
A simpler equation for a harmonic wave:
E(x,t) = A cos[(kx t) ]
Use the trigonometric identity:
cos(zy) = cos(z) cos(y) + sin(z) sin(y)
where z = k x t and y = to obtain:
E(x,t) = A cos(kx t) cos( ) + A sin(kx t) sin( )
which is the same result as before,
The 1D wave equation for light waves
2 E 2 E 2 = 0 2 x t
Well use cosine- and sine-wave solutions: where E is the light electric field
E ( x, t ) = B cos[k ( x vt )] + C sin[k ( x vt )]
or
kx (kv)t
E ( x, t ) = B cos(kx t ) + C sin(kx t )
where:
k
=v=
1
Proof that f (x vt) solves the wave equation
u = 1 and Write f (x vt) as f (u), where u = x vt. So x
Now, use the chain rule:
f f = x u
f f u = x u x f f u = t u t
2 2 f 2 f =v 2 t u 2
u =v t
So
2 f 2 f x 2 = u 2
and
f f =v t u
Substituting into the wave
The one-dimensional wave equation and its solution
Well derive the wave equation from Maxwells equations next class. Here it is in its one-dimensional form for scalar (i.e., non-vector) functions, f: 2 2
f 1f 2 2 =0 2 x v t
Light waves (actually the elect
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
Waves, the Wave Equation, and Phase Velocity
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. 0 1 2 3
f(x) f(x-2) f(x-1) f(x-3)
x
Phase velocity