Lec1 Light: Generationlight.ece.illinois.edu/ECE460/PDF/Lec1.pdf · Lec1 Light: Generation. Prof....

Light Emission, Detection, and StatisticsProf. Gabriel Popescu

Lec1Light: Generation


• Luminescence • Comprised of fluorescence and phosphorescence

• The phenomenon that a given substance emits light

• Light in the visible spectrum is due to electronic transitions

• Vibrationally excited states can give rise to lower energy photons, i.e. infrared radiation

• Rotational modes produce even lower energy radiation, in the microwave region.

• Application• Fluorescence-based measurements

• Microscopy, flow cytometry, medical diagnosis, DNA sequencing, and many other areas.

• Biophysics and biomedicine• Fluorescence microscopy, image at cellular and molecular scale, even single molecule scale.

• Fluorescent probes, specifically bind to molecules of interest, revealing localization information and activity (i.e., function).

• Challenge in microscopy: Merge information from the molecular and cellular scales.

Fluorescence


• Fluorescence • The radiative transition between two electronic states of the same spin multiplicity.

• Spin multiplicity: Ms=2s+1, with s the spin of the state.

• The spin s• s is the sum of the spins from all the electrons on that electronic state.

• s=0, two paired (antiparallel spins) electrons on the electronic sate.

• s=1/2, a single, unpaired electron in the state.

• s=1, two electrons of parallel spins.

• Electronic states • Singlets (Ms=2s+1=1, s=0)

• Doublets (Ms=2, s=1/2)

• Triplets (Ms=3, s=1)

• The singlet-singlet transition is the most common in fluorescence.

Jablonski Diagram


• Jablonski diagram• The energy levels of a molecule.

• 𝑆0, 𝑆1, 𝑆2 denote the ground, first, and second electronic state.

• A typical scenario:

• Absorb a photon of energy hvA , 𝑆0→𝑆2, at the 1fs=10-15 s scale

• Internal conversion:𝑆2→ 𝑆1, within 1ps=10-12s or less.

• Emit a photon of energy hvF , 𝑆2→ 𝑆1, in an average time of 10-9—10-8 s.

• Assume: The internal conversion is complete prior to emission.

Jablonski Diagram

Figure 5.1. a) Jablonski diagram depicting a three level system: 𝑆0, 𝑆1, 𝑆2 are singlet electronic states, each containing vibrational levels. Absorption takes place at frequencies A .T1 denotes a triplet state, from which phosphorescence takes place. b) Music score illustrating the similarity with the Jablonski diagram.

a

Intersystem CrossingInternal Conversion

1S

0S

2S

210

Absorption Fluorescence

PhosphorescenceAhv

AhvFhv Phv

T1

b


• Born-Oppenheimer approximation• A molecule’s wave function can be factorized into an electronic and nuclear (vibrational

and rotational) components.

• Phosphorescence

• Intersystem crossing: The conversion from the S1 state to the first triplet state T1

• Phosphorescence: Triplet emission (emission from T1 ), which is generally shifted to lower energy (longer wavelengths)

• Phosphorescence has lower rate constants than fluorescence. • Phosphorescence with decay times of seconds or even minutes.

• Molecules of heavy atoms tend to be phosphorescent.

Jablonski Diagram

total e N =


• Symmetry property between the power spectrum of absorption (or excitation) and emission (or radiative decay) —mirror symmetry

• Excitation: Absorption, S0 → S1 (higher vibrational levels)→ nonradiative decay, S1

(lowest vibrational level).

• Emission: Radiatively decays, S1 → S0 (high vibrational level) → thermal equilibration, S0

(ground vibrational state).

• The peaks in both absorption and emission correspond to vibrational levels, which results in perfect symmetry.

Emission Spectra

Exti

nci

on

co

effi

cien

t

1

1)

Flu

ore

scen

ce in

ten

sity

(n

orm

aliz

ed)

Wavenumber ( 1)

EmissionAbsorption

350 370 390 410 450 490 530 570

28,800 26,800 24,800 22,800 20,800 18,800

1.0

0.5

0

32

24

16

8

0

a Wavelength ( )

Figure 5.2. a) Mirror-image rule (perylene in benzene).


• Note: The local maxima are due to the individual vibration levels.

• The emission spectrum remains constant irrespective of the excitation wavelength (exceptions exist).

• Exception to the mirror rule

• Absorption spectrum: The first peak is due to S0 — S2, the larger wavenumber peak is from S0 — S1.

• Emission: Occurs predominantly from S1, the second peak is absent.

Emission Spectra

Figure 5.2. b) Exception to the mirror rule (quinine sulfate in H2SO4). Adapted from Lakowicz, J. R. (2013). Principles of fluorescence spectroscopy, Springer Science & Business Media.

1.0

0.5

0,

Wavenumber ( 1)

Absorption Emission

280 320 360 400 440 480 520 600

10

8

6

4

2

0 35,500 31,500 27,500 23,500 19,500 15,500

Exti

nci

on

co

effi

cien

t

1

1)

Flu

ore

scen

ce in

ten

sity

(n

orm

aliz

ed)b

Wavenumber ( )


• The population of each level is determined by the rate of absorption, nonradiative decay, spontaneous emission, and stimulated emission.

• Stimulated emission• The radiative transition from an excited state to the ground

state triggered (stimulated) by the excitation field.

• By stimulated emission, one excitation photon results in two identical emitted photons (development of lasers).

• Rate equations for levels S0 and S1

Rate Equations

Figure 5.3. Jablonski diagram for a three-level system.

12Relaxation(10 )s−

hA hA

S0

S2S1

dt

NdNNANBNB

dt

Nd

NNANBNBdt

Nd

nr

nr

1

1110110001

0

1110110001

1

−=+++−=

−−−=

N0,1 —concentrations, m-3

B01 — absorption rate, s-1

B10 — stimulated emission rate

(B01 = B10 for nondegenerate levels)

A10 — spontaneous emission rate

γnr —rate of nonradiative decay0 1N N+ 0/

10=+ dtNNd ）（Assume: =const, or


• Note: The energy lost from nonradiative decay, γnr , eventually converts into heat.

• For thermal equilibrium

• N1 << N0, the stimulated emission term can be neglected (B10 N1 ≈0)

• Note: The rate of decay is much higher than that of absorption, A01 + γnr >> B10.

• For active medium

• N1 > N0, when there is population inversion.

• The solutions for the populations of the two levels

Rate Equations

01110001=−− NNANB

nr

NNN =+10

100

01 10

011

01 10

nr

nr

nr

AN N

B A

BN N

B A

+=

+ +

=+ +


• The fluorescence quantum yield • The ratio of the number of photons emitted to that of photons absorbed

• For thermal equilibrium

• The quantum yield can approach unity if the nonradiative decay, γnr , is much smaller than the rate of radiative decay, A10.

• Note: Even for 100% quantum yield, the energy conversion is always less than unity because the wavelength of the fluorescent light is longer (the frequency is higher, i.e.hωA>hωF ).

Quantum Yield

nrA

A

BN

ANQ

+==

10

10

010

101

1 0 01 10/ / ( )nrN N B A = +


• The lifetime of the excited state• The average time the molecule spends in the excited state before returning to the

ground state.

• For a simplified Jablonski diagram, the lifetime is the inverse of the decay rate.

• Natural lifetime τ0

• The lifetime of a fluorophore in the absence of nonradiative processes, γnr →0.

• Depends on the absorption spectrum, extinction coefficient, and emission spectrum of the fluorophore.

• Natural lifetime τ0 and measured lifetime τ (τ > τ0, nonradiative processes)

Fluorescence Lifetime

10

1

nrA

=

+

0

10

1

A =

0

10

10

0

Q

A

A

nr

=+

=Q

=

0


• Fluorescence quenching• The nonradiative decay to the ground state.

• Collisional quenching• Due to the contact between the fluorophore and quencher, in thermal diffusion.

• Quenchers include: halogens, oxygen, amines, etc.

• Other quenching• Static quenching: The fluorophore can form a nonfluorescent complex with the quencher

• The attenuation of the incident light by the fluorophore itself or other absorbing species.

• Photon absorption takes place at femtosecond scale, 10-15 s, while emission occurs over a larger period of time.

Quenching


• During its lifetime, a fluorophore can interact with quenchers from within a radius of 7 nm.

The diffusion coefficient of oxygen in water ,the lifetime of a fluorophore

The root mean squared distance that the oxygen molecule can travel during this time

The significant distance , the thickness of a cell membrane bilayer is 4-5 nm .

• It is possible for a molecule to absorb light outside the cell and fluoresce from the inside (and vice versa), one lifetime later.

Quenching

252.5 10 mD

s−=

2 2 .x Dt =

2 7x nm =

nst 10=


Black Body Radiation


• Black body• An idealized model of a physical object that

absorbs all incident electromagnetic radiation, and an ideal emitter of thermal radiation.

• Spectral distribution of radiation by bodies at thermal equilibrium led into the development of quantum mechanics.

• Thermal radiation• Reverse process to absorption: Internal

energy → Thermal radiation.

• Cavity mode: The spatial frequency content of the electromagnetic field within the cavity, which satisfies the condition of vanishing electric field at the wall.

Black Body Radiation

Figure 6.1. Cavity modes: the sinusoids represent the real part of the electric field. All surviving field modes have zero values at the boundary. The modes are indexed by n, the number of zeros along an axis (orange dots).

11n =

0n =

...n =


• Planck’s Radiation Formula• Planck’s formula predicts the spectral density of the radiation emitted by a black body, at

thermal equilibrium, as a function of temperature.

• The radiated energy per mode, per unit volume is:

• f is the probability of occupancy associated with a given mode, obtained from the Bose-Einstein statistics.

• The average energy per mode:

Planck’s Radiation Formula

( ) 2hv

du v f dNV

=

1

1

−

=Tk

hv

Be

f 231.38 10BJk

K−= (Boltzmann’s constant)

V — Volume dN — The number of modes

v — Frequency

1−

==Tk

hv

Be

hvfhvE


• Calculate the number of modes per frequency interval dN

• The spherical shell in the first octant of the k-space is:

• Volume in k-space formed by the smallest spatial frequencies

• The volume of the cavity

• Dispersion relation


dk

zk

yk

xk

a

xk

yk

zk

/ zL

/ yL/ xL

b

min

Figure 6.2. a) Mode distribution in a cavity. b) The cube at the origin in a) is the smallest volume in k-space, defined by the inverse dimensions of the cavity.

dkkdkkVdk

22

24

8

1 ==

3min

k

x y z

VL L L

=

dvvc

Vdv

cc

vVdkk

V

V

VddN

k

k 2

32

22

3

2

3min

424

22

====

x y zL L L V=

2v

kc

=

• The number of modes per frequency interval


• The radiated energy per unit frequency range and unit volume, p(v)

• It is fundamental to calculating any quantity related to blackbody radiation.

• Calculate the power flowing from the cavity• Through a surface A and element of solid angle dΩ, per unit of

frequency v.


Figure 6.3. Power flow out of a black body surface.

1

8)()(

3

3

−

==Tk

hv

Be

v

c

h

dv

vduvp

A

d

2

cos)(

22

cos)(

2)(

=

=

dAdvvp

cdAvdu

cvPd 2

2 sind d =

=

0

sincos)(2

)( dAvduc

vdP

231.38 10BJk

K−=


• The spectral exitance• The power per surface area per frequency.

• The spectral exitance in term of the wavelength

• The Jacobian factor dv/dλ ensures that the Mλ is a distribution.


Figure 6.4. Spectral exitance of blackbodies at different temperatures.

1

2)(

4

1 3

2

−

==kT

hvv

e

v

c

hvcpM

2

v

d PM

dAd=

M d M d =

1

12/

5

2

−

===kT

hc

e

hc

d

dvcvMM

）（


• A function of temperature• Both p(v) and Mλ exhibit a maximum at a

particular frequency vmax.

• Wien’s displacement law• The relationship between the temperature of the source

and the peak wavelength of its emission.

• Note: λmax is obtained by finding the maximum of function Mλ with respect to λ. λmax ≠ c / vmax

• Color temperature: • Expressing the “color” of thermal light by the

temperature of the source.

Wien’s Displacement Law

( )

maxmax

0.v

v vv v

d v dM

dv dv

==

= =

maxv Tmax

a

T

a —constant

a=2,900 μmK

Figure 6.5. Color temperature for various thermal sources.

Temperature Source

1850 K Candle flame, sunset/sunrise

2400 K Standard incandescent lamps

2700 K"Soft white" compact fluorescent

and LED lamps

3000 K“Warm white” compact

fluorescent and LED lamps

3200 K Studio lamps, photofloods, etc.

5000 K Compact fluorescent lamps (CFL)

6200 K Xenon short-arc lamp

6500 K Daylight, overcast

6500 – 9500 K LCD or CRT screen

15,000 –

27,000 KClear blue sky


• Wien’s displacement formula in daily activities • Humans evolved to gain maximum sensitivity of their visual system at the most dominant

wavelength emitted by the Sun. • The Sun’s effective temperature is 5800K, which places its peak emission at ~500 nm (green), near the

maximum sensitivity of our eye.

• Dimming the light on an incandescent light bulb will result in shifting the color toward red (longer wavelengths).

• Heating a piece of metal will eventually produce radiation, of red color at first and blue-white when the temperature increases further. One can say that “white-hot” is hotter than “red-hot”.

• Warm-blooded animals at, say, T=310K (37 ℃), emit peak radiation at ~ 10 μm, in the infrared region of the spectrum, outside our eye sensitivity.

• Some reptiles and specialized cameras can sense these wavelengths and, thus, detect the presence of such animals.

• Most of the radiation is in the infrared spectrum, which we sense as heat, but only a small portion of the spectrum is visible.

• Wood fire can have temperatures of 1500-2000K, with peak radiation at ~2-2.5 μm.

Wien’s Displacement Law


• The Stefan-Boltzmann law• The frequency-integrated spectral exitance, meaning, the exitance (in W/m2), is proportional

to the 4th power of temperature. Stefan-Boltzmann constant σ = 5.67×10-8Wm-2K-4.

• The total power emitted by a black body is MA, where A is the area of the source.• Estimate the size of other stars: The total power emitted by the Sun (S) and the star of interest (X)

• Gray body• A body that does not absorb all the incident radiation and emits less total energy than a

black body, with an emissivity ε < 1.

Stefan-Boltzmann Law

4

0

TdvMMv

==

2 44S S SP R T =

2 44x x xP R T =

2

S xx S

x S

T PR R

T P

=

4M T=


• Wien approximation• The asymptotic behavior of Planck’s formula

(blue curve) for low temperature (high-frequency) hv>>KBT

• Rayleigh-Jeans law• At high temperatures (low-frequency) hv<<KBT

• Rayleigh-Jeans law is known as classic limit formula.

• “Ultraviolet catastrophe”: The amount of energy radiated over the entire spectral range diverges.

Asymptotic Behaviors of Planck’s Formula

Figure 6.6. The asymptotic behavior of Planck’s formula (blue curve) for high temperature (low-frequency, Rayleigh-Jeans, red curve) and low-temperature (high-frequency, Wien, green curve).

Tk

hv

Bevc

hvp

—3

3

8)(

Tk

hv

Bef—

Tkc

vvp

B3

28)(

hv

Tk

Tk

hvf B

B

=

−+

11

1

3

3

8( ) B

h

k Thp v e

c

−

( )2

3

8B

vv k T

c

Rayleigh-Jeans

Wien

Tk

hv

Bevc

hvp

—3

3

8)(

Tkc

vvp

B3

28)(


• The quantum of energy is assumed to be the difference between two atomic energy levels.

• The atomic system exchange energy with the environment by 3 fundamental processes .

• Absorption: An atom in state 1 absorbs an incident photon and is excited to energy level 2.

Einstein’s Derivation of Planck’s Formula

Figure 6.7. Radiative processes in a two-level atomic system:a) absorption, b) spontaneous emission, c) stimulated emission. Note how the stimulated rather than spontaneous emission is the reversed process to absorption.

c)

b)

a)

hv

2hv1

2

12B

21B

21A

Absorption

Spontaneous emission

Stimulated emission

1

2

1

2

1

2

1

2

1

2

hv

hv

2 1E E hv− =

( )212 1

dNB N v

dt=

N1 , N2 —the number density of each level

ρ — the incident energy density

B12 — the absorption rate, [B12]=s-1


• Spontaneous emission: An atom from level 2 decays radiatively (with emission of a photon) to the lower state.

• Note: The inverse of A12 can be interpreted as the decay time constant, or natural lifetime τ=1/ A12.

• Stimulated emission: Can be regarded as the exact reverse of absorption


Figure 6.7. Radiative processes in a two-level atomic system:a) absorption, b) spontaneous emission, c) stimulated emission. Note how the stimulated rather than spontaneous emission is the reversed process to absorption.

c)

b)

a)

hv

2hv1

2

12B

21B

21A

Absorption

Spontaneous emission

Stimulated emission

1

2

1

2

1

2

1

2

1

2

hv

hv

( )212 1

dNB N v

dt=

A12 — the spontaneous emission rate constant.

221 2

dNA N

dt= −

B12 — the absorption rate, [B12]=s-1


• Einstein combined all these processes to express the rate equations.

• At thermal equilibrium, the excitation and decay mechanisms must balance each other completely. Using the classic Boltzmann statistics to get the ratio of the two populations.

• p(v) and Planck’s formula are identical, provided two conditions are met


dt

dNNBNBNA

dt

dN1

221112221

2 )()( −=−+−=

2 1 0dN dN

dt dt= =

Tk

hv

Beg

g

vBA

vB

N

N —

=+

=1

2

2121

12

1

2

)(

)(

( ) 21

21 12 1

21 2

1.

1B

hv

k T

Av

B B ge

B g

=

−

212121BgBg =

3

3

21

218

c

hv

B

A =

g1, g2 — the degeneracy factors for the two states


LASER: Light Amplification by Stimulated Emission of Radiation


• Laser radiation can be highly monochromatic and collimated.

• Note: It is highly dissipative for thermal sources (e.g. incandescent filaments) to be filtered to approach the collimation and monochromaticity of a laser.

• Pumping• Delivering energy to the atomic system, by

electrically, optically, chemically, etc.

• A laser is an out-of-equilibrium system.

• Pumping need an active medium, which acts as an optical amplifier.

• Population inversion: An atomic system in which the population of the excited level (N1) is larger than that of the lower level (N2)

Figure 7.1. a) Pumping: empty circles represent atoms on ground energy levels and solid circles denote excited atoms. b) Active medium: population inversion is achieved. c) Active medium acts an optical amplifier of the incident radiation.

0I

0I I

Pump

Medium Active Medium

)a )b

)c

Optical amplifier

Population inversion, Optical Resonator, and Narrow Band Radiation


• Positive feedback mechanism• Amplifying the light characterized by narrow

spatial and temporal frequency bands.

• The optical resonator (cavity)• Converting the pump energy into a small

number of modes: both longitudinal (temporal frequencies) and transverse (spatial frequencies).

• Examples• Positive feedback phenomenon

• When a microphone is brought close to a speaker, a very narrow sound frequency band is amplified by the electronic circuit.

• Spatial, or transverse mode feedback• Vibrations of broad frequencies are excited at the

liquid surface, yet only those favored by the resonator (boundary conditions) become dominant.

Population inversion, Optical Resonator, and Narrow Band Radiation

Figure 7.1. d) Active medium in a resonator: A, spontaneous emission, B, stimulated emission. e) Spatial frequency feedback: the path in green is stable, while the one in red is not (exits the resonator). f) Temporal frequency feedback: only the modes with zeros at the boundary survive in the resonator: the mode in blue line survives, the black and red are suppressed.

Active Medium

Optical resonator

Pump)d

A B

)e

)f


• Power gain in spectral irradiance, Iυ

• Due to stimulated emission

• The amplification is proportional to the volume of active medium, thus, distance dz.

• Gain coefficient, γ• For small signals, or weak amplification, γ does not depend

on Iυ. The small signal gain is denoted by γ0.

• γ0 <0, denoting an exponential attenuation rather than gain.

Gain

Figure 7.2. Gain through an active medium.

I

Active medium

I dI +

dz

( )dI I dz =

( ) ( ) ( )00L

I L I e

=


• Taking the rate equation for the populations of the two levels into account.

• The line shape of the transition, where g1,2 denote the spectral distribution for each electronic level.

• Standard deviation of g(υ), ∆υ12.

• The emission spectrum is broader than either of the individual levels, as the variances add up.

Gain

Figure 7.3. a) Electronic states, 0, 1, 2, each containing vibrational level (v, blue lines) and respective rotational levels (r, red lines). The energy distribution of the upper levels 1 and 2 are g1, g2.b) The transition lineshape, g, is the cross-correlation function of g1 and g2, with the average and standard deviation as indicated.

( ) ( ) ( ) ( ) ( )221 2 21 2 12 1

0

dNA N g B N g B N g d

dt

= − − +

p(υ)—the spectral energy density of the pump

p(υ)dυ—the probability of emission of light in dυ

g(υ)—spectral line shape, probability density (emission &absorption)

2 22 2 2 2

12 12 12 1 2 1 2 1 2

2 22 2 2 2

1 1 2 2 1 2

= 2

= − + + − +

= − + − = +

( ) ( ) ( )1 2' ' 'g g g d = −

)a

2

1

0

v

2v

1v

1v

2v

1( )g v

2( )g v

vr

)b

v

( )g v

2 1v v−

2 2

1 2v v +


• The energy radiated per unit time and volume

• Assume that the pump spectral density, ρ(υ) , is much narrower than the absorption or emission linewidth, g(υ).

Gain

( )2 2/ ) / (d Q dtdV d Q dtAdz=

( ) ( ) ( ) ( )2 2

221 2 21 2 12 1

0

dI dNd Q d Qhv hv A N B N g B N g d

dz Adtdz dtdV dt

= = = = − − −

( ) ( )21 2 21 2

0 0

1,As g d then A N A N g d

= − = −

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

21 2 12 1

0

21 2 12 1

0

21 2 12 1

' ' ' ' '

( ') ' ( ') ' '

B N g B N g d

B N g B N g d

B N g B N g

−

= − − −

= −

A — the area normal to direction z

dV — the infinitesimal volume of interest

𝜌 𝜐′ ≃ 𝜌 𝜐 𝛿 𝜐 − 𝜐′)


• Invoke the relationships between Einstein’s coefficients

• For propagation along the z-direction

• σ quantifies how much stimulated emission power (in W) is emitted by each atom exposed to a certain irradiance (W/m2)

• Note that the spontaneous emission term, - A21N2 , represents noise in the laser system.

Gain

( )3

221 2 21 2 13

1

1( ) .

8

dI gcA N A g N N

h dz h g

= − − −

( )I c =( )

21 2

1 IdIA N N

h dz h

= − −

( ) ( ) ( )2 2

21 212

22 1

1

8 8

cA g A g

gN N

st

N Ng

imulated emission cross section

population inversion

= =

= −

—

—


• The small signal gain, γ0

• The gain coefficient is positive only if there is a population inversion.

• For positive gain

• G0 denotes power gain, which depends on the length of the medium and acts as a band-pass filter on the initial irradiance Iυ.

• During normal laser operation, the spontaneous emission term becomes subdominant and can be neglected.

• The gain reduces or saturates if Iυ continues to increase via amplification.

Gain

( ) ( )v

dII I N

dz

= =

( ) ( ) ( )2

221 2 1

18

gA g N N N

g

= − =

( ) ( ) ( )00 ,I L I G L = ( ) ( )0

0 ,L

G v L e

=

v

0 0,G

0 0,G

0 ( )v

0 ( )v

𝐺0, 0

0 ( )

0 ( )G

0

Figure 7.4. The (small signal) power gain, G0, is narrower in frequency than the gain coefficient, 0.


• Uncertainty principle

• The line width of any transition has a finite bandwidth, simply because the light transit in the cavity is finite in time.

• Broadening the spectral line

• Homogeneous broadening, when the mechanism is the same for all atoms

• Inhomogeneous broadening, when different subgroups of atoms broaden the line differently.

Spectral Line Broadening

Homogeneous Broadening

• Natural (lifetime) broadening

• This broadening is due to the natural lifetime, i.e. without nonradiative decay like collisions.

• The natural lifetime is the largest possible, the natural line shape is the narrowest.



• The natural line shape is a Lorentzian distribution, whose standard deviation diverges and, thus, cannot be used as a measure of bandwidth.

• The temporal behavior of spontaneous emission (inverse Fourier transform of g(υ)).

2

1

v

2v

1v

1( )g v

2( )g v

v

21

a)

0 2 1 = −

b)

( )g

Figure 7.5. a) Transition between two levels, with the lifetime as indicated. b) Spectral line of the emission. c) Temporal autocorrelation function associated with spontaneous emission, i.e., the Fourier transform of the power spectrum in b).

( )( )

( )

2

0

2

1 / 2

1

2

g

=

−+

02 2( )i t tt e e − − =

e-|t|/τ21

0cos( )t

c)

t

(t)

( )0

1g d

=



• Collision broadening

• The process of collisions between atoms is essential in gas lasers. In solids, pressure broadening is absent.

• Assuming each atom experiences collisions at frequency υcol. Collisions interrupt the emission process, essentially shortening the lifetime of both levels, 1 and 2, by the same amount.

• Typically, collision broadening is expressed as MHz of broadening per unit of pressure, and is homogeneous.

21 21

10 10

col

col

col

col

A A

A A

= +

= + 21 10

12

2col colA A

= + +

𝑂𝑓𝑡𝑒 , 𝜐𝑐𝑜𝑙 ≫ 21, 10, 𝑡ℎ𝑒 𝑤𝑒 𝑔𝑒𝑡 Δ𝜐𝑐𝑜𝑙 ≃ 𝜐𝑐𝑜𝑙/𝜋


Inhomogeneous Broadening

• Inhomogeneous broadening

• Doppler broadening (velocity related)

• Isotope broadening (mass related)

• Zeeman splitting (nuclear spins in magnetic fields)

• Stokes splitting (nuclear spins in electric fields)

• Doppler broadening

• At thermal equilibrium, the atoms in a gas have a Gaussian distribution of velocities along one axis (say, z), vz.

M is the atomic mass, T is the absolute temperature, kB is the Boltzmann constant and ∆v is the standard deviation of the Gaussian distribution.

( ) ( )

22

2221 1

2 2

zz

B

vMv

vk T

zp v e ev v

−−

= =

Bk Tv

M = ( ) 1z zp v dv

−

=



• Doppler frequency, υD

• Natural line under Doppler effect

• Inhomogeneous Doppler line broadening, which is the convolution between the Lorentzian associated with the homogeneous line, and the Gaussian velocity distribution.

0 0.z

D

v

c − =

0 2

0 0

1

2

1

2

z h

z

h

vg

c v

c

− =

− −

+



• Voigt distribution

• The Doppler broadening is dominant over the natural broadening when g approaches a δ-function.

• Doppler broadening is reduced in low-temperature gases and is absent in solid state lasers.

)a ( )g

D0

zD

v

c

)b ( )zp v

zv

2

2

zv

ve−

)c ( )Dg p g = ⓥ

0

Figure 7.6. a) Doppler frequency shift of the “natural” transition line, for a single velocity vz. b) Maxwell-Boltzmann distribution of velocities. c) Voigt distribution: the Doppler-broadened spectral line, averaged over all velocities.

( ) ( ) 0z

D z z

vg p v g dv p g

c

−

= − =

ⓥ ( )

20

a t bt i t

Dg t e e− −



• Isotopic Broadening

• The active medium contains isotopes of the same chemical element.

• The helium-neon (He-Ne) laser, as neon occurs naturally in two isotopes 20Ne and 22Ne.

• The spectral line is slightly shifted depending on which isotope emits light.

• Stokes and Zeeman Splitting

• Stokes and Zeeman effects describe the splitting of the line due to, respectively, electric and magnetic fields present.

• These phenomena are important in solid state lasers, where atoms in the lattice are exposed to inhomogeneous (local) fields.


Threshold for Laser Oscillation

• Population inversion is not sufficient for laser oscillation

• The resonator introduces losses which may overwhelm the gain

• The threshold condition

• Upon a round trip propagation in the resonator, the irradiance experiences a net gain.

• Threshold gain coefficient

• The threshold condition acts as a filter in the frequency domain.

1R2R

L

0 ( )v

Figure 7.7. Losses in the resonator due to mirror reflection.

( )02

1 2 1.L

e R R

( )1 2

1ln .

2th R R

L = −

• For calculating the threshold gain, the stimulated emission can be ignored.

• As the power increases, spontaneous emission can be ignored.


Laser Kinetics

• The time evolution of laser radiation is dictated by the kinetics of the population in each level.

• Three-level system

• The rate equations for the upper two levels of interest.

Figure 7.8. Three-level system: laser emission takes place due to radiative transition between levels 2 and 1.

0

1

2

12B 21B

1

1

2P

1P

20

1

21

1

h

( )( )

( ) ( ) ( )( ) ( )

( )( )

( ) ( ) ( )( ) ( )

2 2

2 2 1

2

1 211 2 1

1 21

( )

dN t N t v I vP t N t N t

dt hv

dN t N t v I vN tP t N t N t

dt hv

= − − −

= − + + −

( )2 20 21

2.1 1 1

is the inverse lifetime decay rate of level

= +

1/τ20— decay rate to ground state

1/τ21— rate of spontaneous emission

P1, P2 — the pump rates of levels 1 and 2

σ— spontaneous emission cross section

I(v) —spectral irradiance

N1－N2 — population inversion

τ1 — lifetime of level 1


Laser Kinetics

• Using Laplace transform to solve N1(t) and N2(t)

• The pumps P12 turn on at t=0 and staying constant for t>0, i.e. a step function, Г(t)

• Laplace Transform

( )

1,2

1,2 1,2 1,2

, 0

1( ) , 0

2

0,

P t

P t P t P t

rest

= = =

( )

( )( )

1 t

s

df tsf s

dt

( )( )

( ) ( )

( )( ) ( )

( ) ( )

222 2 1

2

1 211 2 1

1 21

N sP IsN s N s N s

s hv

N s N sP IsN s N s N s

s hv

= − − −

= − + + −


Laser Kinetics

• Various particular cases can be solved, depending on which terms can be safely neglected.

• P1 =0, no pumping of level 1.

• 1/τ2 =0, no spontaneous decay of level 2 (large signal gain).

• σ =0, no stimulated emission (weak signal).

• 1/τ21 =0, no spontaneous emission.

• dN1 /dt= dN2 /dt=0, steady state.

( )( )

( ) ( )

( )( ) ( )

( ) ( )

222 2 1

2

1 211 2 1

1 21

N sP IsN s N s N s

s hv


s hv

= − − −

= − + + −


Laser Kinetics

• Example 1: P1 =0, σ =0.

• Using partial fraction decomposition (or expansion)

• The shift property of the Laplace transform

( )( )

( ) ( )

( )( ) ( )

( ) ( )

222 2 1

2

1 211 2 1

1 21

N sP IsN s N s N s

s hv


s hv

= − − −

= − + + −

( )

( )( )

22

2

2

1

1 21

1

1

Ps N s

s

N ss N s

+ =

+ =

( ) 22 2 2

2

2

1 1

11

PN s P

s ss s

= = − +

+

( ) ( )atf s a e f t−+ → ( ) 2

2 2 2 1

t

N t P e

− = −


Laser Kinetics

• Expressing N1 (s) as a sum of simple fractions, of the form a/(s+b).

2 1 1 2

1 1 1 11A s s B s s C s s

+ + + + + + =

( ) 21

21

2 1

1

1 1

PN s

s s s

=

+ +

( ) 21

21

2 1

1 1

P A B CN s

ss s

= + + + +

2 1

1

21 2 1 2 1 2

1 1

2 21 12 1 1 20 0 0

1 1 1 1

1 1 1 11 1

s s s

A B C

s s s s s s

= + = + =

= = = = = =

− −+ + + +

，，


Laser Kinetics

• Expressing N1 (s) as a sum of simple fractions, of the form a/(s+b).

• If τ2 > τ1, the density N2 is greater than N1 and gain is obtained across all values of t. If τ1 > τ2, the excited level is depleted and gain quickly diminishes.

• Note: The two steady states are obtained as the limit of N1 (t), N2 (t), for t → ∞.

( ) 1 2

1

1 2 21 2

1 12

2 2

11

1 1

t t

N t P e e

− −

= + − − −

1 2 0.1 =

1 2 0.5 =

(a)

(b)

Figure 7.9. Population dynamics for the two levels expressed in Eqs. 7.35 (N2) and 7.39 (N1) for two ratios of the lifetimes, as indicated.( )

( )

1 21 2

21

2 2 2

N t P

N t P

→ =

→ =


Laser Kinetics

• Partial fraction decomposition

• g and h have no common factors and g of lower degree than h, we can expand f as

• an are the roots of h, bn are constants that result from the identity g(s)/h(s).

• A certain root, a1, has a multiplicity m.

• Combining shift theorem with the Laplace transform pair.

• Suitable for the polynomial in the denominator that has roots of multiplicity greater than 1.

( ) ( ) / ( )f s g s h s=

𝑓 𝑠) =

𝑏

𝑠 − 𝑎 = 𝑔 𝑠)/ℎ 𝑠) 𝑓 𝑡) =

𝑏 𝑒𝑎𝑛𝑡

( )1

1m

s a-𝑓 𝑠) =

≠1

𝑏

𝑠 − 𝑎 +

𝑘=1

𝑚𝑏1𝑘

𝑠 − 𝑎1𝑘

𝑠 ↔

𝑡 1

− )!

𝑏1𝑘

𝑠 − 𝑎1𝑘 ↔

𝑏1𝑘

𝑘 − !𝑡𝑘 1𝑒𝑎1𝑡


Gain Saturation

• The effect of stimulated emission (σ ≠0)

• Assume τ1 =0, level 1 decays infinitely fast, i.e., N1 =0

• The population dynamics of level 2 in the Laplace domain.

• New lifetime τ2’, adjusted for the effects of stimulated emission.

• Whenever I approaches Is , the gain saturates. (I=Is, τ2’= τ2 /2, faster depletion of level 2)

( ) 22

2

1.

I PN s s

h s

+ + =

( )( )

( ) ( )222 2 1

2

N sP IsN s N s N s

s hv

= − − −

2

2 2 2

1 1 11 1

' s

I I

h I

= + = +

2

s saturation irrad n ehI ia c

= ( ) 2 '

2 2 2 ' 1

t

N t P e

− = −

2N

2N

t

sI I𝑃2𝜏2/3

𝑃2𝜏2

Figure 7.10. Effects of the saturation intensity on the N2 kinetics.


Gain Saturation

• Saturation at steady state• The general equations at steady state.

• The gain is obtained as σ(N2－N1).

• Note: γ0 is small signal gain, i.e. I →0, γ → γ0.

• For τ2 >> τ1, or τ21 = τ2 , Is=hv/στ2, the gain drops to ½ γ0.

( )

( )

22 2 1

2

2 11 2 1

21 1

0

0

N IP N N

hv

N NIP N N

hv

− − − =

+ + − − =

12 2 1 1

21

2 1

1 21 2

21

1

1

P P

N NI

hv

− −

− =

+ + −

( )( )0

1s

vv

II

=

+( ) ( )

( )1

0 2 2 1 1

21 2 1 2

2 21

11

1 1

s

hvv v P P I

v

= − − =

+ −

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