**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben
***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.
Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**
Consortium leader
PETER PAZMANY CATHOLIC UNIVERSITY
Consortium members
SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER
The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***
PHYSICS FOR NANOBIO-TECHNOLOGY
Principles of Physics for Bionic Engineering
Chapter 11.
Interaction of Matter and Radiation - I
(A nanobio-technológia fizikai alapjai )
(A tér és az anyag kölcsönhatása)
Árpád I. CSURGAY,
Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ
Table of Contents
11. Interaction of Matter and Radiation – I 1. Experimental Foundation
2. On the Physics of Vacuum
3. Interactions in Thermal Equilibrium
4. LASER - The Ruby-laser and the VCSEL
1. Experimental foundation
At the end of the 19th century the discrete atomic spectra and the continuous spectrum of blackbody radiation were already
experimentally explored and well known.
All substances at finite temperatures radiate electro-magnetic waves. The spectrum emitted by a solid body depends on its temperature and composition. Objects that emit a spectrum of universal character, are called „blackbodies‟. Blackbodies
absorb all the radiation incident on them. E.g. a metallic cavity with a small hole is “black”. (H. Weyl proved that the
blackbody properties do not depend on the shape of the cavity.)
After multiple reflection by the inner walls of the cavity, the radiation is absorbed by the atoms of the wall. These atoms will reradiate electromagnetic waves into the cavity, and some of them will leak out through the hole.
4 8 2 4
, 5, 67 10 W/m K . RT T
At T in thermal equilibrium the energy absorbed by the walls is equal to the energy emitted.
The radiated power by a blackbody is given by the Stefan- Boltzmann Law
The free space wavelength (or the frequency) at which the intensity of radiation is maximum is given by Wien‟s Law
Lord J. W. S. Rayleigh and James Jeans assumed that the electromagnetic field in the cavity is determined by Maxwell‟s equations, and each waveguide mode is in thermal equilibrium, thus its energy is (1/2)kBT,
because of the equipartition principle. These assumptions led them to a radiation formula for the power radiated by a
blackbody in a frequency interval
maxnm T K 2,896 106
nm K .
( , d )
which means that the total radiation is infinite, and it is growing with square of the frequency („Ultraviolet catastrophe‟).
Max Planck in 1900 introduced the quantized energy and explained blackbody radiation.
The Planck‟s radiation formula:
turned to be in agreement with observations.
d 8π3 2 h /k de 1
T v T
v hv v v
c
d 8π k3B 2d ,T
v v T v
c
–34 2
h , where h 6, 626·10 Ws ,
E v
The discrete spectrum of the atomic absorption and emission is related to the fact that atoms do not radiate in stationary states, but absorb or emit a photons in case the stationary state
changes
Generation of light
by incandescence (heating) explores blackbody radiation,
by luminescence (fluorescence, phosphor-escence) explores quantum decay of excited electrons.
photon 2 1
E h E E
The excitation can be caused by
electrical current (electroluminescence),
electron bombardment (cathode luminescence),
photon bombardment (photoluminescence).
Fluorescence decay time is fast (electron-hole recombination happens in nanoseconds).
Phosphorescence: decay time is slow (electron-hole recombination happens in seconds, minutes, hours).
Luminescence is applied e.g. in Light Emitting Diodes (LED), Light Amplification by Stimulated Emission of Radiation
(LASER),
Flat Panel Display, etc, Fluorescence is applied e.g. in microscopy, bio- markers, etc.
The radiation–matter interaction happens in vacuum fluctuation, and the elementary events are photon absorption, spontaneous emission and stimulated emission of photons by matter.
Before we start to study the interactions, we look at the physics of quantized electromagnetic field in vacuum.
According to present world view, there is no vacuum in the ordinary sense of tranquil nothingness. There is instead a fluctuating quantum vacuum.
On the nanoscale the vacuum holds the key to full understanding of the forces of nature. Casimir effects, van der Waals forces, spontaneous emission, etc. play important roles in nanobio- technology.
Applied non-relativistic QED is becoming a badly needed discipline in nanobio engineering.
2. On the Physics of Vacuum
In free space there are no electrons and nuclei, but there can be electromagnetic energy. Classically the laws of the field are the Maxwell equations
2
, ) 1 , )
, ) , )
, ) 0 , ) 0
t t
c t
t t
t t
t
B(r E(r
E(r B(r
E(r B(r
Magnetic induction is source-less, thus a vector potential can be introduced
If we replace B in the Faraday equation by A
we get a potential vector field (its curl is zero), thus we can introduce a scalar potential
which is zero, because there are no charges present.
( , )t ( , )t (r, )t 0
t
E r A r
( , )t ( , )t , )t , )t 0
t t
E r B r E(r A(r ( , )t 0 ( , )t rot ( , )t ( , ).t
B r B r A r A r
From the vacuum Ampere‟s equation
using the identity on rot rot A, we get
Only rot A has been used, thus we can choose div A freely, e.g. we can apply the so called Coulomb gauge
2
( , )t 1 ( , )t c t
B r E r
2
2 2 2
( , ) ( , ) ( , )
1 1
( , ) ( , ).
t t t
t t
c t c t
A r A r A r
A r r
( , )t 0.
A r
We see that the state variable of the electromagnetic field in free space is the vector potential A(r, t), which satisfies the wave equation
and from the vector potential we can recalculate the electric and magnetic fields
2
0 0
2 2 2
1 1
( , )t ( , ),t ,
c t c
A r A r
( , )t ( , ),t t
E r A r
( , )t ( , ).t
B r A r
If we look for the solution of the time-dependent wave equation for the vector potential as a product of an only time-dependent function and an only space-dependent one
we get an eigenvalue problem for A(r), and an ordinary differential equation for f (t)
( , )t f t( ) ( ),
A r A r
2
,k k k
A r A r
2
2 2 2
2
d ( )
( ) ( ).
d
k
k k k
f t k c f t f t
t
General solution is the linear combination of the fields of eigen- modes. The fields of the eigen-modes
Eigenvalues Eigen-mode functions
Normalization General solution
...
, , ...
,
, 2
1 k ki
k k
1 , 2 , ... , i , ...
A r A r A r
, k ( ) k
k
t
f tA r A r
, ( )
, , ) ( ) ( ), ( ) d ( ).d
k
k k k k k k
t f t t f t f t f t
t
E r A r B (r A r
d 1
( )
d 2k k
V V
V V k
A r 2
A r 2The electric energy stored in a resonant mode of a cavity
The magnetic energy stored in a resonant mode of a cavity
The Lagrangian
2
E 0
2 2
0
2 0
1 ( , ) d
2
1 d
2
1 ( )
2
V
k k
V
k
E t V
f t V
f t
A r
A r
2 M
0
2 2
0
2 2
0
1 , d
2
1 ( ) d
2 2 ( )
V
k k
V
k
E t V
f t V
k f t
A r
A r
2
2 2
E M 0
1
2 2
k k k k k
L E E f k f
Parameters m, C State variable x(t)
Kinetic energy Potential energy
Lagrangian
„Quantization”
There is an analogy between the mechanical harmonic oscillator and an eigenmode of an electromagnetic cavity.
Mechanical, one-dimensional harmonic oscillator
1/ 2 m x 2
1/ 2 Cx2
1/ 2
( ) 2
1/ 2
( ) 2L m x t C x t
j 1 0,1,...
n 2
p mx E n n
x
Harmonic oscillator Cavity mode
Quantization Quantization
2 2
kin pot
1 1
2 2
L E E mx Cx E M 0 2 2 2
0
1
2 k 2 k
L E E f k f
2
1 2
2 2
/
1 2 0,1, 2,..., ,...
n
m Cx C m
E n
n n
H
2 2
2
0 0
2
0 0
1
2 2
/
1 2 0,1, 2,..., ,...
k
n
k f
k k c
E n
n n
H
We have illustrated that every orthonormal mode of an
electromagnetic resonant cavity mathematically shows an analogy with a one-dimensional mechanical harmonic
oscillator, thus formally it can be quantized.
The energy of a single cavity mode is
For every cavity mode with frequency (m, l, p are integers defining a mode), there belong infinite number of (quantized) energy
1 0,1, 2,..., ,...
n 2
E n n n
m p
1 , 0,1, 2,..., ,...
2
m p
n m p
E n n n
The energy of a photon of frequency is
where n is the number of photons (In Greek photon = light)
What does n = 0 mean? Zero point energy of the field in a cavity?
There are infinite number of modes, each of them has zero point energy
m p m p 1 ,
2
m p
n m p
E n
0
1 .
2
m p
E m p
0
1 2 .E
If there is no real photon in the cavity, i.e.
the expectation values of the field are zero
but the average of the square of the fields (energy) is not zero (vacuum fluctuation). This state is denoted by
(This vector is not the zero vector of the state space!)
The impact of vacuum fluctuation on the nanoscale can be significant.
See e.g.: V.A. Parsegian, Van der Waals Forces:
A Handbook for Biologists, Chemists, Engineers, and Physicists, Cambridge, 2006
0 for , , n m p ( , )t 0, ( , )t 0, ( , )t 0,
A r E r B r
vacuum 0 .
vac vacdV 0 0 1.
Casimir effect
If we assume that the vacuum possess a zero point fluctuation in each mode, it can be shown that between two metal plates a significant force appear. (H. Casimir, Philips, 1948)
Between two mirrors, the zero point energy density is much smaller than outside, because the boundary conditions at the mirrors allow only those modes which satisfy them.
The force between the plates is
where A is the surface, d the distance.
4
π c 480 F A
d
d F
Notice, that the Casimir force is inversely proportional to the fourth degree of the distance between the plates.
For d < 10 nm, it is the largest force in nature between neutral bodies.
For
For
2
7 3
1cm , 1micron
10 N or pressure 10 pascal
A d
F
5
10nm
pressure 10 pascal 1atm d
2. Radiation Matter Interaction in Thermal Equilibrium (Einstein’s Approach)
Interactions happen between photons and atoms.
In thermal equilibrium the number of absorptions is equal to the number of emissions.
Atoms in equilibrium follow the Maxwell – Boltzmann statistics (population distribution among energy levels).
Photons follow the Bose Einstein statistics (Planck‟s law of radiation).
Absorptions are proportional to the number of atoms at the lower energy level and to the photonic energy density
Spontaneous emissions are proportional to the number of atoms at the higher energy level
Stimulated emissions are proportional to the number of atoms at the higher energy level and to the energy density of photons
12 1
. B N 2. AN
21 2
. B N According to Maxwell Boltzmann statistics
In thermal equilibrium the number of absorptions are equal to the number of emissions
Where B12, A and B21 are constants. If we knew them, we could derive the Planck‟s Law of radiation:
T v T
E E
N N
N k
h k 1
1
2 e e
1
2
12 1 1 21 1
h h
kT kT
B N AN e B N e
h h
k k
12 21 h
k
12 21
e e .
e
v v
T T
v T
B B v A v A
B B
Comparing the two expressions for
we get that
These are the „Einstein relations‟.
Note, that the generated emission and absorption are symmetric The system either dissipate or amplify electromagnetic energy,
depending on the ratio
h
3 3 hk k
12 21
8π h 1
and
e e 1
v v
T T
A v
v v
B B c
3
12 21 3
; A 8π hv .
B B B
B c
12 21. B B
2 / 1.
N N
Energy
W4
W1
W3
W2
exp k
W T
Pumping
of energy LASER action
De-excitation
Population inversion between W1 and W2
The coherent radiated power can be calculated.
Radiating transitions (number per unit time):
Energy radiated during a single transition Absorbed power
Coherent radiated power:
The spontaneous radiation AN2 does not contribute to the output power, only to the noise. Condition for amplification is N2 > N1. The electron population of the higher energy level should exceed
the population of the lower one.
The natural population of thermal equilibrium should be inverted.
“Population inversion”, “Pumping”.
2 21. N w
1 12h . N w v
2 21h 1 12h 2 1 21h .
P N w v N w v N N w v
h .v
MASER: Microwave Amplification by Stimulated Emission of Radiation
LASER: Light Amplification by Stimulated Emission of Radiation Three level LASER Four level LASER
E
Pumping
Spontaneous
”get together”
Stimulated emission
E
Pumping
Spontaneous
”get together”
Stimulated emission Spontaneous
”get together”
4. LASER - The Ruby-laser and the VCSEL
Example 1: LASER: ‘ruby’ laser.
The ruby laser is a three level laser. Aluminum oxide (Al2O3) crystal is doped by 0,05% Cr3+. The highest energy is a band.
694 nm Energy levels 3
2
1 From this band the electrons are spontaneously
jumping down to level 2, which is a sharp energy level. The energies are absorbed by the crystal atoms, thus these transitions do not generate photon radiation.
Electrons are jumping from level 2 to level 1 through stimulated emission of radiation.
The wavelength of the radiation is 694 nm, thus the ruby laser generate red laser
radiation.
A ruby rod is surrounded by a spiral of flash-lamp. The flash-lamp provides a strong flash light of proper frequency. In order to “pump” the electrons from energy level 1 to band 3.
694 nm Energy levels 3
2
1 The two bottoms of the ruby rod are covered by
mirrors. One of them is a semi-transparent mirror, which admits the coherent light ray to leave the ruby rod. The two mirrors form a optical resonant cavity.
The spontaneous light radiation is always preset in this cavity. If the generated and amplified coherent radiation is reflected by the mirrors (and it does not leave the ruby crystal through the side walls of the crystal) an avalanche process emerges and a strong coherent light ray is generated.
The process continues as long as the population difference between energy levels 2 and 1 are maintained (as long as N2> N1).
The ruby laser is an impulse mode laser.
Example 2: Vertical Cavity Surface Emitting Laser VCSEL
A gain region is un-doped, and it forms a quantum well, between two Bragg reflector region, which are distributed periodic quarter wavelength dielectric layers of
GaAs
1 x
AlGa xAs
The index of refraction can be controlled by the composition of the layers:
The reflectivity of the lower mirror is better than 99%.
The reflectivity of the upper mirror is lower to couple some light out.
Gain region – Quantum well GaAs p-type Bragg reflector
n-type Bragg reflector Radiation out
AlGa1xAs .x n Bragg reflector
p Bragg reflector
Cathode ( - )
Anode(+) Anode(+)
Gain region
Example 3: Light Emitting Diode LED (Electroluminescence)
“Injection” electroluminescence: a photon is produced when a conduction electron falls into a valence hole. Line width ~ 10 nm (too broad)
2 1
h E E E
Cathode ( - )
Anode(+) Anode(+)
n
p 0
p
EC
p
EV
0
n
EC
n
EV
E
The color of LED depends on
as x is varied from 0 to 0.45 the band gap changes from ΔE =1.4 eV to 2 eV.
With Nitrogen impurity doping 2.2 eV, i.e. green light can be achieved.
GaN (gallium nitride) band gap can be increased up to 3.4 eV, blue, even UV LED can be made.
h E
GaAs1xPx
15 15
1.4 eV ~ 0.34 10 Hz 880 nm Infrared 2 eV ~ 0.48 10 Hz 625 nm Red
