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Elitzur-Vaidman bomb-testing problem
suppose you have a super-sensitive bomb that explodes upon
the slightest interaction
can you make sure that such a bomb is reliable without
detonating it?
the light source A emits a single photon
the photon (i) passes through the bs or (ii) is reflected
•
if the bomb is dud
•
the system reduces to the Mach-Zehnder interferometer
otherwise:
•
logically, the bomb is real
•
if the photon took the lower route  the photon triggers the bomb  explosion
otherwise:
• logically, the photon took the upper route
• the photon on the upper route passes through the bs and is reflected
• the C detector monitors the photon, while D not
otherwise:
• logically, the photon passed through
• the D detector monitors the photon, while C not
•
•
the bomb exploded  real
the bomb did not explode and C detected the photon  logically the bomb
must be real
• the bomb did not explode and D detected the photon  either dud or real
if the third observation is made the experiment has to be run many times 
sometimes good bombs explode but at other times good bombs can be
found
Saturation density
Suppose that the density of light is larger and larger
Atoms would follow this increase - absorbing more and
more, but would ultimately reach their maximum capacity
- all atoms become excited
At this point light would just continue to propagate through
the material without being absorbed by the atoms since
they are saturated
The saturation radiative density, WS, is defined so that the
rate of spontaneous and stimulated emissions are equal
similar to
• A two-level atom therefore either emits or absorbs light - total
energy is in this case conserved
• How about the momentum conservation?
in stimulated emission the light is emitted in the same direction
as the absorbed light - the total net momentum transfer is zero
for an initially stationary atom - since the momentum has to be
conserved - spontaneously emitted light cannot be in any
particular direction
for example a gas of atoms - suppose that spontaneous
emission was directed in a particular way - then you would
expect the gas to drift in a particular direction (i.e., the centre
of mass would be moving) - this never happens in reality spontaneous emission has to be random - uniformly distributed
over the 4p solid angle centered on the atom
important consequences in laser cooling of atoms
Optical Excitation of Two-Level Atoms
rate equation
the radiation density <W> has two independent components
the thermal black body density <WT> and the external density <WE> (the latter
did not exist in Einstein's treatment)
at room temperature, blackbody radiation will be much smaller than from an
external source  blackbody radiation is absent
suppose that all the population is initially in the ground state (i.e., state 1), then
by solving the rate equation
for short times
the number of excited atoms increases
linearly with time
for times long enough, the number of atoms approaches its steady state value
Steady State
from the rate equation
Steady state rates for emission and absorption
when the laser is initially turned on, the population in the excited state
starts to increase linearly, finally reaching its steady state value
in the steady state the populations do not change any more and the total
amount of energy stored in the atoms is given by
once the laser action stops, these atoms release this energy through
spontaneous emission
Lifetime and amplification
• three different processes involved in the interaction between
a two level atom and light
• we may get the wrong impression that stimulated processes
are continuous in time, while the spontaneous emission is an
abrupt
• this is not correct – spontaneous emission is also continuous
the rate equation that we had previously
without external field, <W> = 0
The lifetime is
i.e., inversely proportional to the rate of spontaneous
emission. The population decreases exponentially
Amplification criterion
• for amplification the rate of stimulated emission should be
much bigger than for spontaneous emission
• Let's look at two different regimes: the microwave and the
visible light at the room temperature (T = 300K)
for = 0.1 m, the right hand side is close to 0
for = 500 nm it is huge, e100
Masers are possible
Lasers are impossible!
Population inversion
• the ultimate condition for amplification is population inversion
between two levels, i.e., N2 > N1
• in thermal inversion equilibrium population inversion is
impossible as the weights of states go as
so that
N2 is always less populated than N1
• However, the steady state rate for N2 is given by
always less then N/2 (it approaches this limit for high intensities as
the population inversion is impossible not only in thermal equilibrium, but also under
the presence of an external coherent source - independent of the frequency of
radiation - wrong conclusion
Basic optical processes
Diffraction
Interference
Coherence
Light pressure
Optical absorption
Amplification
Atom-light interaction classical
Spectral lines
Mode locking
Diffraction
• diffraction of plane parallel light of wavelength λ from a
single slit of width d
far-field (Fraunhofer) limit - L ≥ d2/λ
near-field (Fresnel) regime - L ≤ d2/λ
In the Fraunhofer limit, the pattern on the screen observed at angle θ is
obtained by summing the field contributions over the slit:
where
principal maximum at θ = 0, and there are minima whenever β = mπ, m being an
integer. Subsidiary maxima - below β = (2m+1)π/2, for m ≥ 1. The intensity at the
first subsidiary maximum is less than 5% of that of the principal maximum
The angle at which the first minimum occurs is
or
diffraction from a slit causes an angular spread of ~λ/d
for a circular hole of diameter D. The intensity pattern has circular symmetry
about the axis, with a principal maximum at θ = 0 and the first minimum at θmin
calculates the resolving power of optical telescopes and microscopes
Interference
• Interference patterns - a light wave is divided and then
recombined with a phase difference between the two paths
Michelson interferometer – input parallel rays from a linearly polarized
monochromatic source of wavelength λ and amplitude E0
where ΔL = L2−L1 and k = 2π/λ, Δ - phase shifts
between the two paths even when L1 = L2
The field maxima and minima are
As L2 is scanned, bright and dark fringes appear at
with a period equal to λ/2
Interference
interference - the effect from a number of individual
sources (electromagnetic waves), is larger or smaller
than the sum of individual effects (constructive or
destructive)
suppose N atoms each emitting a light wave
ak is the (time-independent) amplitude of the k-th wave
Let's add up all the contributions to obtain the total amplitude
and then take the square to obtain the total effect (intensity)
when atomic emission is not coordinated - phases vary randomly
and in the end average to zero
this is the same as if there was no interference and the total
effect (intensity) is the same as the sum of individual effects
(intensities)
if all the phases are the same (laser)
this is N times more intense than when the phases are random
the sources are coherent
Coherence
light is a 3D spatial wave
evolving in time - two
different coherences:
• temporal coherence
concerns the same "beam of
light" at the same spatial
point, but two different times
• spatial coherence concerns
two or more different points
of the wave, but at the same
time
a monochromatic wave with infinite
coherence time
a wave whose phase drifts –
short coherence time
coherence - very important for obtaining interference
temporal coherence
Michelson interferometer
monochromatic source –
the spread in frequency
of light involved, dn < n
if the time delay between the
beams is dt than fringes are
formed only if
dtdn < 1
the same beam interferes with itself -at two different times
for a lamp dn ~ 108 s-1 so that dt ~ 108 s-1
the spatial coherence length is dl = c x 10-8 s = 3 m
for a laser, dn ~ 104 s-1 so that dl = 30 x 103 m
Spatial coherence – Young’s double slit
experiment
the same light illuminates two slits “close” to each other
the light is of size ds and the angle between the source and
the slits dq
interference fringes will form only if dqds < l
the source is composed of many point sources – different
fringe patterns
since all the sources are out of phase with each other we have
to add all the point intensities at the end to obtain the total
fringe pattern
if the distance between the slits increases – the patterns
become more and more out of step – fringe disappearance
Light pressure
a charge q interacts with an electric field E through the force
and with the magnetic field B through the Lorentz force
v is the charge velocity
the combined effect of the two forces leads to radiation pressure
(P) - the force on an area A per that area force is also the rate of
change of momentum (Dp)
the momentum density is given by the Poynting vector, S,
divided by c2. The volume is AcDt
considered just normal incidence, for isotropic radiation in a cavity
radiation pressure using Einstein's rate equations
the photons that make up an EM wave of wavevector k each carry a momentum
in a medium of refractive index the photon momentum is given
Suppose now that a photon is absorbed by an atom of mass M. It then gains the
velocity of
the atom then decays via
stimulated emission - the emitted photon carries away the momentum in the same
direction as the original photon
spontaneous emission - direction of the momentum of the emitted photon
anywhere within a 4p solid angle. The atom therefore recoils in some random
direction. On average – there is no cancellation of the momentum previously
gained as the net spontaneous emission transfer averages to zero
so in a cycle of absorption and emission we have the net
transfer of
from the photons to the atoms, in
the direction of the incident beam
every absorption = one momentum kick in laser direction
• net effect: laser pushes atom
• the momentum transfer to atoms gives rise to radiation
pressure
•
• the number of atoms N is large enough to produce small
time dependencies in the atomic populations and that P is
the total atom momentum
• The rate of change of P in the presence of radiative energy
density <W> at a frequency w (resonant with the ground and
the excited state of the atoms) is proportional to the
difference between the absorbed and the stimulated
emission rates
the rate of change of momentum is negative for N2 > N1
the number of atoms in steady state
for strong fields we have that WS << <W>
this is the saturation value for transfer rate for very strong beams
once saturation has been reached, any increase of beam strength produces very
little change in the momentum transfer rate
the steady state momentum transfer rate is equal to the force acting on each atom
how can this force be measured?
• suppose that an atom beam passes
perpendicularly to a strong laser beam
• the atoms in the beam interact with light and absorb and
emit radiation
• deflection occurs when atoms absorb light followed by
spontaneous emission, in which case they gain
momentum perpendicular to their direction of motion
• the resulting deflection is about 10-5 rad
• the two isotopes have different transition frequencies
• tuning the laser to a resonant transition of one isotope
• only on-resonant atoms will be deflected, and therefore
the two isotopes would separate into two different beams
Optical absorption
light propagating through a medium is absorbed and reemitted
by atoms - its intensity will decrease
the intensity of radiation is I = uv
u - energy density of the radiation field
v - field velocity
for small distances traveled by light, the change in intensity has
to be proportional to the traveled medium length and to the
intensity itself
Kn - absorption coefficient
(frequency and medium dependent)
Beer’s law
the intensity changes exponentially as it propagates
Using the fact that I = vu we can write the equation for dI
from Einstein’s relationship
(
instead of n)
Fuchbauer- Ladendurg formula
for N1  N2 we obtain dI = 0
no absorption - saturation
for N2 >> N1 amplification – the basis of laser operation
this condition, cannot be achieved by exciting a two level
atom - when we have saturation then the No. of ground state
atoms > No. of excited state atoms
can be obtained in a three level atom
Amplification: three level system
the atom population can be stored in the third level - decay to the
second level and lase to the first level
S – transition probability due to both radiative and non-radiative processes
total No. of atoms conserved
use levels 1 and 2 for lasing, so the equations are solved for the steady state
case (i.e. , dNi /dt = 0)
where N = N1 + N2 + N3 is the total number of atoms in all
three levels
The number of atoms arriving at the level 2 per unit time is:
Let’s express the numerator of N2 - N1 (the denominator is positive at all times). It
is N( 􀀀 - A21)
The condition for lasing
Population inversion
3-level systems
Ruby-laser
Maiman (1960):
cavity
L =n l
Ruby: Al2O3 + Cr
Xe
t=0.003 s
coherent
monochromatic
collimated
four level laser
• atoms are pumped from ground
state to level 4, rapid decay to
level 3, creating population
inversion with respect to level 2
• the pumping to level 4 can be
optical (from a flashlamp or
another laser) or electrical
• the decay rate from level 2 to
ground state (level 1) must be fast
to prevent atoms accumulating in
that level and destroying the
population inversion
Nd:YAG (Nd:Y3AI5O12) laser
HeNe laser
•pump He to metastable state (20.61 eV)
•transfer excitation to Ne metastable state (20.66 eV)
•laser transition
•spontaneous emission (2 times) to deplete lower level ( low pumping)
•not very efficient! (20.6 eV vs 2 eV)
Classical treatment of atom-light interaction
the atom - a mass (electron) on a spring (attached to the nucleus)
this spring is then contracted and extended as it interacts with
light - an EM wave
as the spring extends the energy from the EM field gets stored absorption of radiation and is then released when the spring
contracts - radiation emission
F is the force on the electron due to the field F = qE0 cos(wt)
the atom oscillates at the frequency of the driving field
the highest amplitude of oscillation is when the field is on resonance - the driving
frequency is the same as the natural oscillator
Radiation damping
the solution decays exponentially to zero - all oscillations must
eventually die away as energy dissipated into the environment
the solution is not a monochromatic wave - more than one
frequency component is present in its expansion
We look at the Fourier spectrum
by taking a Fourier Transform
there are many frequencies in the spectrum and not just that of
the driving field
the intensities of various frequencies is given by
Lorentzian broadening
Spectral lines
• Atomic states have in principle well defined energies these energy levels, when analyzed spectroscopically,
appear to be broadened
• The shape of the emission line is is described by the
spectral lineshape function gw(w), which peaks at the line
center defined by
hw0 = E2 - E1
• Where do these come from?
atomic collisions
Doppler broadening
lifetime (natural) broadening
lifetime broadening
light is emitted when an electron in an excited state drops to a lower level by
spontaneous emission
The rate of decay is determined by the Einstein A coefficient  determines the
lifetime t
the finite lifetime of the excited state leads to broadening of the spectral line
according to uncertainty principle
DEDt > /2p
Dw =AE/  > 1/t
this broadening is intrinsic to the transition – natural broadening and the spectrum
corresponds to Lorentzian lineshape
collisional (pressure) broadening
The atoms in a gas frequently collide with each other and with walls of
the containing vessel, interrupting the light emission and shortening
the effective lifetime of the excited state
If the mean time between collisions, tcol , is shorter than the radiative
lifetime than we need to replace t by tcol in Dwlifetime = 1/t – resulting
in additional broadening
Based on the kinetic theory of gases tcol is given by
ss is the collision cross section and P the pressure
1/tcol and Dw are proportional to P
collisional broadening  pressure broadening
Doppler broadening
Emission of
all the atoms
atoms moving toward
the observer
originates from the random motion of the atoms in the gas Doppler shifts in
the observed frequencies
the Maxwell-Boltzmann velocity dis.
where N(v) is the number of atoms moving with v
The line shape is a Gaussian
the Doppler broadening gives a Gaussian profile rather than a Lorentzian
its half width at half maximum is
The dominant broadening in low pressure gases at room temperature
is usually Doppler broadening and the lineshape is closer to Gaussian
Line broadening in solids
• the spectra will be subject to lifetime broadening as in gases – a
fundamental property of radiative emission
• the atoms are locked in their positions – neither pressure nor Doppler
broadening are relevant
• The emission and absorption lines can be broadened by other
mechanisms
non-radiative transitions (phonons)
dN 2
N
1
= - A21 N 2 - 2 = -( A21 
)N2
dt
t NR
t NR
The non-radiative transitions shorten the lifetime of the excited
1
1
state according to
= A21 
t
t NR
the phonon emission times in solids are often very fast – substantial broadening
inhomogeneity of the host medium
More detailed principles of laser
laser elements:
• a cavity with two or more highly reflecting mirrors
• a gain medium - support inverted atomic population
• an energy source which can excite the atoms in the gain
medium to achieve the population inversion
•
a loss mechanism by which the stored energy is dissipated
• absorption and amplification in a certain medium, but now
• an electro-magnetic field propagates through a medium,
the medium responds to it – polarization
• the medium is composed of a bunch of classical oscillating
springs
• the microscopic dipole is then
p = -ex
P = Np = -Nex
the driven harmonic oscillator equation for polarization is
solutions
the proportionality constant between P and E a complex number
the measured quantity is the intensity which is
the mod square of E
the real part represents the intensity decay due
to the absorption in the medium, while the
imaginary part the oscillatory behavior of E
the response of the medium is linear, therefore
the dielectric
susceptibilty
E a plane wave
the wavevector is also
is a complex number, so is the refractive index
the real part n’ is the "normal" refractive index, the
imaginary component
absorption or gain
in vacuum
in medium
• the electric field is modified as it propagates. It has the
form
the last term is equal to
in terms of intensity
the absorption (gain) coefficient
which proves Beer’s law
the laser cavity has two laser mirrors
if intensity I impinges on a mirror of reflectivity r will return with the intensity
rI
the gain due to the medium
for combination of one round trip reflections and the gain coefficient
where
For oscillations to build up
The maximum value of the gain coefficient is
threshold population inversion
The size of population difference N2 -N1
amplification is achieved for rate of emission into a cavity mode
exceeding the rate of loss of photons from the cavity in that mode
The No. of photons into a mode is
where
The rate of loss of photons is
where
is
the lifetime of a photon within the laser cavity and r is the reflectivity of
the mirrors (the same for both mirrors)
The amplification condition is
very different from the “zero" coming out
from Einstein's analysis (but still small
compared to the number of atoms which
is of the order of 1023
Mode locking
• radiation in a cavity will be a mixture of many different modes
oscillating at different frequencies (rates)
• they will all oscillate with different phases as the atoms in the cavity
mirrors and the lasing medium radiate randomly at different times 
contributing to different initial phases
• why do not all these interfere destructively, canceling each other out
and leaving no signal?
• the state of the field
• the intensity of radiation to obtain is proportional to the number of
modes
arrange so that all of the fields in different modes have the same phase
mode locking condition
the frequency separation between different modes is
the mode-locked intensity is
the peak intensity is
N times stronger than without mode locking
the maximum happens
the time between the maxima
if modes can be locked, then high intensities and small pulse widths can be obtained
the width of the pulse
it is allowed to achieve shorter and shorter pulses (ultra-short pulses) by "locking in"
more and more modes (N)
problem
The organic dye Rhodamine 6G can lase at wavelengths
from 550 to 630 nm. Estimate the shortest possible pulse
duration that can be achieved by mode-locking this
material in a laser. How many optical cycles does this
represent? If the cavity of the laser has a length d = 1.5
m, and the average power output is 100 mW, what is the
peak power of a mode-locked pulse having the shortest
duration allowed by the gain medium.