Project: Laser By Mohamed Salama Jimmy Nakatsu

Introduction Lasers 1. Excited State of Atoms 2. Active Amplifying Material and Stimulated Emission of photons 3. Reflecting photons with mirrors

 Reflecting of photons allows for a chain reaction of amplification of energy Laser animation Photon

Statement Of The Problem  What are the steady states of our variables in our models  Are the steady states stable or unstable?  What are the factors that determine stability of the steady states ?  How do the steady states relate to the operation of the laser?

Description of the Models We have 3 main models to discuss : Simplified single differential EquationSimplified single differential Equation G: The gain coefficient of photons, G > 0 k: The inversion of the lifetime of a photon, k > 0 n: The number of photons in the system, n > 0 N: The number of excited atoms, N > 0

Assumptions for Model 1 N 0 : Number of excited atoms from the excitation of the pump, N 0 > 0  : The rate at which atoms drop back to the ground states, 

Improved Model of a Laser  A system of two differential equation, We are taking into consideration the excitation of the atoms from the pump.  f: The decay rate of spontaneous emission, f >0 f >0  p: the strength of the pump, p can take positive negative and zero values

Improved Model of a Laser A major assumption in model two is: Finally, N(t) is expressed in terms of the function of n(t)

Maxwell-Bloch Model  This model is much more sophisticated.  The system of three differential equations  E: The dynamics of the electric field, positive or negative  P: The mean polarization of the atoms, positive or negative  D: The population inversion: the number of excited atoms divided by the number of ground state atoms, D >= 0

 k: The decay rate in the laser cavity due to beam transitions, k > 0  g 1 : The decay rate of the atomic polarization, g 1 >0  g 2 : The decay rate of the population inversion, g 2 >0  : The pumping energy parameter, can be negative, positive, or zero

Maxwell-Bloch Model  Major assumption of this model:  Eventually, the whole system is expressed in terms of E.  g1,g2, must be significantly greater than or you will get chaotic behaviors resulting.

Mathematical Methods  Continuous time models are solved by using Euler’s Method.  Stability analysis (Graphically, mathematically)  Jacobian Matrix and eigenvalues  Solving the system of equations

Interpretation of the results  Model 1: Steady States: Steady States: Stability: Stability:

Interpretation of Model 1  Having a steady state of n 1 =0, there is no photons in the cavity. The laser operation has seized and failed to function. There is more photons exiting the laser cavity than being emitted from the atoms. If you reach the steady state n 2, then the laser is fully operational and the number of photons being emitted isIf you reach the steady state n 2, then the laser is fully operational and the number of photons being emitted is equal to the number exiting photons from the laser cavity. equal to the number exiting photons from the laser cavity.

Improved model of a laser  Model 2:  Steady States:  Stability:

Interpretation of Model 2  If the n=0 and N= the laser is not operational, there are excited atoms but with no photons to allow for stimulated emission. If the pump is weak the laser will remain in this situation.  If n= and N= the laser is fully operational, the level of emission is dependent on the pump strength. The emission of photons and number of excited atoms is constant.

Maxwell-Bloch Model Steady States: Stability: Depends on the values of Stability: Depends on the values of

Maxwell Bloch Model  If  0 there are infinitely many steady states( they will be on the line D=1, E=P in R 3 )  If  there are two steady states, D=1, E=1 P=1 and D=1, E=-1,P=-1. You will converge to the steady state that is closest to the initial conditions. However if you fall in the middle of the two steady states, you converge to D=1,E=1 and P=1

Maxwell Bloch Model  If  there is only one steady state. E=0, P=0, and D = 1+  It will converge to the steady if E between -1 and 1.If the magnitude of E is greater than 1, E will diverge away from the steady state.

Interpretation of Maxwell Bloch  When  The population inversion is equal to one for both steady states. Thus the number of excited atoms is equal the number of ground state atoms. The laser is fully functioning, the excited atoms creates a polarized electric field either positive or negative.  When  = 0, there is no pumping, but the electric field must be equal to the polarization. The population inversion is 1. The laser would not operate because of loss of stimulated emission.   When , The pump is absorbing energy from the laser, so the laser will be inactive. Your population inversion is less than 1, and so there is less excited atoms than ground state atoms. Stimulated emission will not occur.

Critique  The delta t in Euhler’s method  The assumption of the differentials of polarization and population inversion is equal to zero.  Model 3 was the most sophisticated model, yet it allowed for some odd results.  An improvement to the models would be an inclusion of the reflectivity of the mirrors.  Another variable that should be considered is heat loss from the system.