Download Light-matter interaction Hydrogen atom Ground state – spherical

Light‐matter interaction Hydrogen atom
Ground state – spherical electron cloud
Excited state : 4 quantum numbers
n – principal (energy)
L – angular momentum , 2, 3.... L  n  1
Lz – projection of angular momentum LZ   L, ( L  1),....0
Sz projection of spin (S=1/2) S Z   1 .
2
Electron transition between levels – emission or absorption of a photon
Absorption Spontaneous
emission Selection rules
L  1, L Z  0,  
Photon carries angular momentum
Stimulated emission Relation between angular momentum of a photon and its polarization
Atomic configurations Atoms have hydrogen‐like orbitals.
Electrons are fermions, they follow Pauli exclusion principle: No more than one fermion can be placed in the same quantum state. Energy
Energy
Atoms come together, forms molecules and solids Band
Metals
Empty states
Occupied states Energy gap (no states allowed)
Filled band
Response to EM wave ‐ Attenuation 
 
B 
C Ed   A t dS

 

E  
C Bd   o A  J   0 t dS


We cannot neglect free current J   E
BUT
In UV range many metals becomes transparent Plasma frequency ne 2
 
 0m
2
P
Insulators Completely empty band
Excited electron
Energy gap
Completely filled band
Insulators are transparent for photons that have energy smaller than the band gap (long wavelength photon)
hole
Completely filled band – no current
States in the gap (impurity, vacancy)
Behave very much like isolated atoms;
Often gives color to crystals
Example: States of Cr in ruby Al2 O3 :Cr
Geometrical optics
Ideal optical system : all rays emitted from a point of an object arrive in the same point of the image at the same time. (We proved it for a spherical interface for small angles!) v  c / n Speed of light in media with index of refraction n
n depends on wavelength of the light; be aware.
Snell’s law:
Total internal reflection
Refraction, Spherical surface
n1 n2 n2  n1 1
 

s0 si
R
f
Thin lens
 1
nm nm
1  1

  nl  nm     
s0 si
 R1 R2  f
Thin lens’ combinations
From left to right: the image of lens#1 is the object for lens #2 and so on. This works always if you use formulas; the ray tracing can give wrong result.
Two thin lenses in contact 1 1 1
 
f
f1 f 2
Mirrors
1 1
2 1
  
s0 si
R f
Spherical mirrors and lens give sharp image only in paraxial (small angle) approximation.
Aspherical systems (ellipsoidal, hyperbolic and parabolic are free from these deficiency
We can say that they do not have spherical aberrations. We have proved this for parabolic mirror. Optical aberration Chromatic aberration Index of refraction depends on wavelength
Correction with negative lens
We notice that aberration correction requires addition of extra lenses. Recall that only 96% of light transmits via air/glass interface. In multi‐lens systems lenses must be covered by anti‐
reflection interference coating. I  I 0T N

1 

x 2 v 2 t 2
2
2
 1 ( x  vt )
Wave equation General solution   C1 1 ( x  vt )  C2 2 ( x  vt ) Superposition   A cos(kx  t   )  Re Aei ei ( kx t ) Harmonic wave
f  2
k  2 /    vk
v f
 2  2  2
1  2
 2  2  2 2
2
v t
x
y
z
1  2
  2 2
v t
2
Plane wave


 (r , t )  C1 (kr / k  vt )


i ( kr t  )
 (r , t )  C1e
Plane wave: wave front is perpendicular to the wave vector
Spherical and cylindrical waves
Maxwell equations and Electromagnetic waves
  1
EdS
 
0
A
 qi 
 
BdS
 0
1
 0 V
 dV Gauss law (from Coulomb law)
A

 
B 
C Ed    A t dS (Faraday law of electromagnetic induction)

 

E  
C Bd   o A  J   0 t dS (Ampere law with Maxwell displacement term)
In vacuum
 
 EdS  0
A
 
BdS
 0
A

 
B 
C Ed    A t dS

 
 E  
C Bd   o A   0 t dS
EM wave is transverse wave (in vacuum)
Poynting vector Energy flux of EM wave ‐ energy crossing unit area per unit time  1  
 
2
S
E  B  c 0E  B
0
Irradiance ‐ Poynting vector averaged over time

I S
T

c 0 2
Eo
2
E  cB
EM waves in matter   1
EdS
 
A
0
 
BdS
 0
q
i

1
  dV Gauss law (from Coulomb law)
0 V
A

 
B 
C Ed    A t dS (Faraday law of electromagnetic induction)

 

E  
C Bd   o A  J   0 t dS (Ampere law with Maxwell displacement term)
These equations are correct both in vacuum and in matter, but
  
E , B, J
must include all charges, free and bound
Bound charges and bound currents 
d  dipole moment

 d
P
polarization
V
D   0 E  P electic displacement
Linear media
P  o  E
D   E   R 0 E
D represent the effect of free charges

 - magnetic dipole

M 


magnetization
V
B  0 M  0 H
H auxiliary magnetic field, represents free charges
Linear media
M  R H
B  0  R H   H
Examples Spontaneous polarization Ferroelectic materials
Spontaneous magnetization
Ferromagnetic materials Piezoelectric materials Applied voltage shrinks/extends a crystal and shrinking the crystal generates voltage across. Quartz
Semiconductors: doping Free carriers –
electrons
Free carriers –
holes
The state of extra electron on P at low temperatures
  1
EdS
 
0
A
 qi 
 
DdS
   qFREE
A
After taking surface integral around point charge we have
D
q
4 R 2
E is the real field, which determines forces between charges
D   E   R 0 E
E
q
4 R 0 R
 R  11
Hydrogen atom Silicon P
In Silicon 2
E
q
4 0 R 2
Borh radius of orbit aB  10 nm
aB  0.05 nm
EM waves in matter

 
B 
C Ed    A t dS

 
B 
C Ed   A t dS
Real current created by bound charges 

 


P
E  
C Bd   o A  J FREE  J BOUND  t   0 t dS

 

E  
C Bd   o A  J   0 t dS
Wave equation in matter 
2

 E
 2 E   2
t
r  1
typical case
   r 0
v
1


   r 0
c
r

c
n
Fresnel equations
Polarization
Linear polarization Circular polarization
Right, clockwise
Left, counter‐clockwise
Polarizers
The only component that is passes through polarizer is the component of with electrical field along the axis of polarizer. (You need cos() factor. Birefringent Crystals


D   ij E
 Dx    xx
 D   
 y   yx
 D  
 z   zx
 xy  xz   Ex 

 yy  yz   E y 
 zy  zz   Ez 
E perpendicular to optical axis – ordinary ray
E along optical axis – extraordinary ray
Retarders
Interference Principle of superposition The first slit is needed ensure spatial coherence of the wave‐front E  E1 ( x, t )  E2 ( x, t )
Photons (electrons) sent one by one
Formation of interference pattern is a property of a single photon
Diffraction and Interference with Fullerenes:
Wave-particle duality of C60
Markus Arndt, Olaf Nairz, Julian Voss-Andreae, Claudia Keller,Gerbrand van der Zouw,
and Anton Zeilinger Nature 401, 680-682, 14.October 1999
Diffraction 1) Superposition principle 2) Huygens‐Frensel “wavelet” principle Phase of EM wave is taken into account Diffraction grating   (kb / 2)sin 
  (ka / 2)sin 
b width of the slit
a
distance between the slits
Fraunhofer (long distance) diffraction  (kb/2)sin
Single slit b width of the slit
Circular aperture
a radius
J1 – Bessel function
Diffraction limit imposed by a circular aperture
  1.22 / D
Angular spread of the first maxima
Angular resolution of optical instruments
Geometrical resolution Distance between photoreceptors d=6‐10 micron
Focal length of an eye f=2 cm
Angular resolution Teta=d/f=3x10‐4
Resolution set by diffraction   1.22 / D
Wavelength 600 nm
Diameter of pupil 4 mm
Teta=1.22L/D=1.8x10‐4
Eye is well optimized optical instument
Angular resolution of Hubble space telescope Geometrical resolution Distance between photo‐detectors
in CCD camera d=3 microns
Effective Focal length f=10 m
Angular resolution Teta=d/f=3x10‐7 rad
Resolution set by diffraction Wavelength 600 nm
Diameter of primary mirror 2 m
Teta=1.22L/D=3.5x10‐7 radians
The optics of the telescope is also optimazed
Component of the first ruby laser
Al2 O3 :Cr
Elementary processes
1) Excitation 2) Stimulated emission
Fabry‐Perot resonator
Elementary process
What is needed; physical principle Stimulated emission 1) We need to create so‐called inverted population of electrons
nE
 E 
 exp  

nG
 kT 
Excitation using three‐level process Equilibrium concentration 2) We need external excitation, but we also can not excite electrons directly to lasing level because we have stimulated excitation , which has the same rate as stimulated emission 3) For optical excitation at least three‐level system must be used
Photon are bosons and like to stay in the same quantum state
This is actually wrong way of thinking about lasing
Laser Fabry Perot cavity
Quantum state of a photon. From energy in a particular mode we can compute the number of photons in it
The resonance lines are extremely narrow, when reflections at the is high. Stimulated emission, a bit more details Possible emission of stimulated photon
Photons are bosons. They “like” to be together. P( N )
N
P (1)
Probability of emission in a particular quantum state is proportional to the number of photon occupying this state.
Semiconductor lasers .
Excitation is not optical