Project: LaserProject: Laser

By Mohamed SalamaBy Mohamed Salama

Jimmy NakatsuJimmy Nakatsu

IntroductionIntroduction

Lasers Lasers

1. Excited State of Atoms 1. Excited State of Atoms

2. Active Amplifying Material 2. Active Amplifying Material and Stimulated Emission and Stimulated Emission of photons of photons

3. Reflecting photons with 3. Reflecting photons with mirrors mirrors

Reflecting of photons Reflecting of photons allows for a chain allows for a chain reaction of reaction of amplification of energyamplification of energy

Laser animationLaser animation

PhotonPhoton

Statement Of The ProblemStatement Of The Problem

What are the steady states What are the steady states ofof our variables our variables in our modelin our modelss

Are the steady states stable or unstable?Are the steady states stable or unstable? What are the factors that determine stability What are the factors that determine stability

of the steady states ?of the steady states ? How do the steady states relate to the How do the steady states relate to the

operation of the laser?operation of the laser?

Description of the ModelsDescription of the ModelsWe have We have 3 main models3 main models to discuss : to discuss :• Simplified Simplified singlesingle differential Equation differential Equation

G: The gain coefficientG: The gain coefficient of photons of photons, G > 0, G > 0k: The inversion of the lifetime of a photon, k > 0k: The inversion of the lifetime of a photon, k > 0n: The number of photons in the system, n > 0n: The number of photons in the system, n > 0N: The number of excited atoms, N > 0N: The number of excited atoms, N > 0

knGnN

LossGaindt

dn

Assumptions for Model 1Assumptions for Model 1NN00: Number of excited atoms from the : Number of excited atoms from the

excitation of the pump, Nexcitation of the pump, N0 0 > 0> 0

: The rate at which atoms drop back to the : The rate at which atoms drop back to the ground states, ground states,

nNtN 0

20

0

)()(

)(

nGnkGN

knnNGndt

dn

Improved Model of a LaserImproved Model of a Laser A system of two differential equation, We are taking into A system of two differential equation, We are taking into

consideration the excitation of the atoms from the pump.consideration the excitation of the atoms from the pump.

f: The decay rate of spontaneous emission, f: The decay rate of spontaneous emission, f >0 f >0 p: the strength of the pump, p can take positive negative p: the strength of the pump, p can take positive negative

and zero and zero valuesvalues

knGnNdt

dn

pfNGnNdt

dN

Improved Model of a LaserImproved Model of a LaserA major assumption in model two is: A major assumption in model two is:

Finally, N(t) is expressed in terms of the Finally, N(t) is expressed in terms of the function of n(t)function of n(t)

0dt

dN

fGn

pN

pfNGnNdt

dN

0 0

Maxwell-Bloch ModelMaxwell-Bloch Model This model is much more sophisticated.This model is much more sophisticated. The system of three differential equationsThe system of three differential equations

E: The dynamics of the electric field, positive or negativeE: The dynamics of the electric field, positive or negative P: The mean polarization of the atoms, positive or P: The mean polarization of the atoms, positive or

negativenegative D: The population inversion:D: The population inversion: the the number of excited atoms number of excited atoms

divided by the number of ground state atomsdivided by the number of ground state atoms, D >= 0, D >= 0

)1(

)(

)(

2

1

EPDgdt

dD

PEDgdt

dP

EPkdt

dE

)1(

)(

)(

2

1

EPDgdt

dD

PEDgdt

dP

EPkdt

dE

k: The decay rate in the laser cavity due to beam k: The decay rate in the laser cavity due to beam transitions, k > 0transitions, k > 0

gg11: The decay rate of the atomic polarization, g: The decay rate of the atomic polarization, g11>0>0 gg22: The decay rate of the population inversion, g: The decay rate of the population inversion, g22>0>0 : The pumping energy parameter, can be negative, : The pumping energy parameter, can be negative,

positive, or zero positive, or zero

Maxwell-Bloch ModelMaxwell-Bloch Model Major assumption of this model:Major assumption of this model:

Eventually, the whole system is expressed in terms of E.Eventually, the whole system is expressed in terms of E.

g1,g2, mustg1,g2, must be be significantly greater than or you will get significantly greater than or you will get chaotic behaviors resulting.chaotic behaviors resulting.

0 and 0 dt

dD

dt

dP

)1(1

22

EE

Ek

dt

dE

k,gg 21k

Mathematical MethodsMathematical Methods

Continuous time models are solved by using Continuous time models are solved by using Euler’s Method.Euler’s Method.

Stability analysis (Graphically, mathematically)Stability analysis (Graphically, mathematically)

Jacobian Matrix and eigenvaluesJacobian Matrix and eigenvalues

Solving the system of equationsSolving the system of equations

Interpretation of the resultsInterpretation of the results

Model 1:Model 1:

Steady States:Steady States:

Stability: Stability:

G

kNnn 0

*2

*1

1or 0

stable is )(1

unstable is 0 , if

unstable is )(1

stable is 0 , if

0*

2

*1

0

0*

2

*1

0

G

kNn

n

G

kN

G

kNn

n

G

kN

Interpretation of Model 1Interpretation of Model 1

Having a steady state of nHaving a steady state of n11=0, there is no photons in the =0, there is no photons in the

cavity. The laser operation has seized and failed to cavity. The laser operation has seized and failed to function. There is more photons exiting the laser cavity function. There is more photons exiting the laser cavity than being emitted from the atoms. than being emitted from the atoms.

• If you reach the steady stateIf you reach the steady state n n22, , then the laser is fully then the laser is fully

operational operational and the number of photons being emitted isand the number of photons being emitted is

equal to the number exitingequal to the number exiting photons photons from the laser from the laser cavity. cavity.

Improved model of a laserImproved model of a laser

Model 2:Model 2: Steady States:Steady States:

Stability: Stability:

G

kN

G

f

k

pn

f

pN

n

*2

*2

*1

*1

and 0

stable is ,-

unstable is ,0

0, - if

unstable is ,-

stable is ,0

0, - if

*2

*2

*1

*1

*2

*2

*1

*1

G

kN

G

f

k

pn

f

pNn

G

f

k

p

G

kN

G

f

k

pn

f

pNn

G

f

k

p

Interpretation of Model 2Interpretation of Model 2

If the n=0 and N= the laser is not If the n=0 and N= the laser is not operational, there are excited atoms but with no operational, there are excited atoms but with no photons to allow for stimulated emission. If the photons to allow for stimulated emission. If the pump is weak the laser will remain in this pump is weak the laser will remain in this situation.situation.

If n= and N= the laser is fully If n= and N= the laser is fully operational, the level of emission is dependent operational, the level of emission is dependent on the pump strength. The emission of photons on the pump strength. The emission of photons and number of excited atoms is constant.and number of excited atoms is constant.

f

p

G

f

k

p

G

k

Maxwell-Bloch ModelMaxwell-Bloch Model

Steady States:Steady States:

Stability: Depends on the values of Stability: Depends on the values of

1

and,

1

1

1

,

1

0

0

,

1

1

1

*4

*4

*4

*4

*4

*3

*3

*3

*2

*2

*2

*1

*1

*1

D

EP

PE

D

P

E

D

P

E

D

P

E

unstable

1

stable,

1

1

1

, unstable

1

0

0

, stable

1

1

1

,0

*4

*4

*4

*4

*4

*3

*3

*3

*2

*2

*2

*1

*1

*1

D

EP

PE

D

P

E

D

P

E

D

P

E

if

stable

1

unstable,

1

1

1

, unstable

1

0

0

, unstable

1

1

1

,0

*4

*4

*4

*4

*4

*3

*3

*3

*2

*2

*2

*1

*1

*1

D

EP

PE

D

P

E

D

P

E

D

P

E

if

unstable

1

unstable,

1

1

1

, stable

1

0

0

, unstable

1

1

1

,0

*4

*4

*4

*4

*4

*3

*3

*3

*2

*2

*2

*1

*1

*1

D

EP

PE

D

P

E

D

P

E

D

P

E

if

Maxwell Bloch ModelMaxwell Bloch Model If If 0 there are infinitely many steady 0 there are infinitely many steady

states( they will be on the line D=1, E=P in states( they will be on the line D=1, E=P in RR33))

If If there are two steady states, D=1, there are two steady states, D=1, E=1 P=1 and D=1, E=-1,P=-1. You will E=1 P=1 and D=1, E=-1,P=-1. You will converge to the steady state that is closest converge to the steady state that is closest to the initial conditions. However if you fall in to the initial conditions. However if you fall in the middle of the two steady states, you the middle of the two steady states, you converge to D=1,E=1 and P=1converge to D=1,E=1 and P=1

Maxwell Bloch ModelMaxwell Bloch Model

If If there is only one steady state. E=0, there is only one steady state. E=0, P=0, and D = 1+ P=0, and D = 1+ It will converge to the It will converge to the steady if E between -1 and 1.If the steady if E between -1 and 1.If the magnitude of E is greater than 1, E will magnitude of E is greater than 1, E will divergediverge away away from the steady state. from the steady state.

Interpretation of Maxwell BlochInterpretation of Maxwell Bloch

When When The population inversion is The population inversion is equal to equal to one for both one for both steady states. Thus the number ofsteady states. Thus the number of excited atoms is excited atoms is equal the number equal the number of of ground stateground state atoms. The laser is fully functioning, the excited atoms atoms. The laser is fully functioning, the excited atoms creates a polarized electric field either positive or negative.creates a polarized electric field either positive or negative.

When When = 0, there is no pumping, but the electric field must be equal = 0, there is no pumping, but the electric field must be equal to the polarization. to the polarization. The The population inversion population inversion isis 1. 1. The laser would not The laser would not operate because of loss of stimulated emission.operate because of loss of stimulated emission.

When When , The pump is absorbing energy from the laser, so the , The pump is absorbing energy from the laser, so the laser will be inactive. Your population inversion is less than 1, and so laser will be inactive. Your population inversion is less than 1, and so there isthere is less less excited atoms than excited atoms than ground state atomsground state atoms. Stimulated . Stimulated emission will not occur.emission will not occur.

CritiqueCritique

The delta t in Euhler’s methodThe delta t in Euhler’s method The assumption of the differentials of polarization The assumption of the differentials of polarization

and population inversion is equal to zero.and population inversion is equal to zero. Model 3 was the most sophisticated model, yet it Model 3 was the most sophisticated model, yet it

allowed for some odd results.allowed for some odd results. An improvement to the models would be an An improvement to the models would be an

inclusion of the reflectivity of the mirrors.inclusion of the reflectivity of the mirrors. Another variable that should be considered is heat Another variable that should be considered is heat

loss from the system.loss from the system.