1. An expression for the Gain taking into consideration Doppler broadening : In the case of broadening due to thermal motion, the kinetic theory given the fraction of atoms whose component of velocity lies between v x and  v x as where m is the atomic mass,  is the Boltzman’s constant and  is the absolute temperature As previously explained due to the Doppler effect, these atoms will emit or absorb radiation propagating in the x direction of frequency

2. (1) where is the frequency of the line center. It follows that the fraction of atoms in a given level that can absorb or emit in the frequency range to is given by where from eqn. (1)) where

3. The rate of upward transition is The rate of stimulated or induced downward transitions The net time rate change of the spectral energy density in the interval is given by

4. where B 21 =B 12 =B where at

5. where is positive if N 2 <N 1 which is the condition for amplification. other wise if N 2 <N 1 which in the normal equilibrium condition ) then is negative, we have absorption.

6. Population Inversion In order to invert population of atomic levels the atoms must be excited by depositing energy in the medium using such method as to decrease the number of atoms at the lower level N L and to increase the number of atoms at the upper level N u. This process is called pumping since the atoms are redistributed as if pumped from the lower level to the upper level. The methods of pumping are i) optical pumping, where the atoms are excited by illumination of light ii) excitation by electric discharge in the case of gases iii) Injection of carriers by a forward current through a p-n junction in the case of semi-conductors iv) excitation by irradiation with electron beams v) excitation by chemical reaction. Historically, in 1954, Townes succeeded in realizing population inversion with a molecular beam of ammonia to make a maser at 1.25 cm wavelength.

7. As the ammonia molecular are distributed among energy levels in thermal equilibrium, the molecules at the upper level were collected and those in the lower level were eliminated by the action of an inhomogeneous electric field, so that population inversion can be achieved. However, such a method where population inversion is established by decreasing the number of atoms in the lower level cannot be applied successfully to optical transition. This is because the number of atoms N u and N L as related by Boltzmann’s formula namely KB Boltzmann const

8. yields N u  N L in the microwave case, since h > K B T at the optical frequency. Therefore, it is not sufficient by merely eliminating atoms at the lower level, but it is necessary to increase the number of atoms at the upper level by a process of pumping. For a two - level system, when its atoms are exited by irradiation or by electron collision, the number of atoms at the upper level will increase, but at the same time the probability of de-excitation that brings these excited atoms back to the lower level will increase with incident light or electrons.

9. Consequently no matter how strong the atoms may be excited, population inversion cannot be obtained. Therefore, three or four atomic systems must be used to achieve population inversion. It is not always necessary that the energy levels concerned should be discrete and sharp. Band levels may be used. Thus, dye lasers and semiconductor lasers can be considered as four -level lasers whose description follows.

10. Population Inversion in a Three- level Laser : There are many three – level lasers such as ruby laser and the optically pumped gas laser. Let the energies and populations of the relevant three levels of laser atomic system be denoted respectively by w 1, w 2, w 3 and N 1, N 2, N 3. If w 3 > w 2 > w 1 as shown in figure, then N 1 > N 2 > N 3 in the three – level system in thermal equilibrium. Here the lowest state 1 is not necessarily the ground state of the atom. Atoms in level 1 will be excited atoms of appropriate energy. We denote by  the probability of exciting the atoms from level 1 to level 3 by any such method of pumping.

11. Fig. (6( When the pumping is removed, the excited atoms will in general gradually return to the state of thermal equilibrium. This is termed relaxation. If we consider the atoms individually the relaxation process takes place at the same time as other atoms are excited.

12. Besides the radiative process, where the excited atoms make a transition to the lower state by emitting a photon, there are non-radiative processes such as collision of molecules in gases or the atom lattice interaction in solids, where the excited atom makes a transition to the lower state by releasing its energy in the form of molecular kinetic energy or vibrational energy of the lattice. Since relaxation is the results of such statistical processes, the relaxation rate or the relaxation constant is defined as a statistical average of the relaxation probabilities of the excited atoms per unit time. The reciprocal of the relaxation rate is the average life time of the excited atoms:

13. Now, the probability  Lu of an atom being thermally excited from the lower state w L to the upper state w u is related to the probability  uL of the reveres process from w u to w L by thermal relaxation. This relation in thermal equilibrium. is N u  uL = N L  Lu where Nu=Nu= Where T is the temperature of the medium. There for (1((1(

14. Fig.( 7 ) This last relation holds generally, even if N u and N L do not represent populations in thermal equilibrium. If these probabilities are constant under the conditions considered the rate equations expressing the rate of change at the number of atoms in each level of the three – level system under pumping are given as follows.

15. (2) (3) (4) Where N 1 + N 2 + N 3 = const. = N the total number of atoms in the three – level system.

16. In the steady state, the distribution of the number of atoms under constant pumping can be obtained by putting the left – hand side of equations 2,3&4 equal to zero. Although the solutions giving N 1, N 2 & N 3 can be readily calculated, yet we shall assume that the separations between the level are sufficiently greater than the thermal energy K B T, so that when applying equation (1) we find that, w u - w L >> KBT So that,

17. We can thus neglect  1 2,  1 3,  23 and equations 2,3&4 yield in the steady state (5) (6) (7) (8) Therefore (9) (10)

18. Therefore (11) From equations 5, 6, 7 and 11 we can write Thus we obtain the steady – state solution (12)

19. (13) From equation 12 (14)

20. from equations (12, 14)  (15) If the excitation is so strong such that we have N 2 > N 1 (15\)

21. This is the condition of population inversion. Thus to obtain population inversion with moderate pumping  2 1 should be small and  32 should be large compared with  3 1. This means that it is desirable that the relaxation from the upper laser level to the lower laser level should be slow, while the relaxation from the upper most level 3 to which the atoms was initially excited to the upper laser level 2 should be fast. Fig. (8)

22. The population inversion as defined by  N = N 2 - N 1 is calculated from 12 & 14 as a function of the excitation intensity  to be N put  o =

23. =  (16) Let us represent graphically the dependence of as a function of excitation intensity  expressed in terms of  o. Consider the two cases when )i)  32 =  21. Where  21 is the laser transition )ii(  32 = 9  21

24. (i) In the first case 151020 13/324/110 (ii) In the second case 24199410/90 0.820.810.710.520

25. Fig. (9(

26. Population inversion in a four- level laser Since the lower level of the laser transition is the lowest level in a there - level laser, the majority of atoms ( N 1  N ) are in this level at thermal equilibrium thus in order to invert the population, the number of atoms in the lowest level must be reduced to less than half by intense pumping. This demand is much reduced in a four - level system. Let us consider an atom, which has four energy levels as shown in fig (10). It is required to invert the population between levels 2 and 1. Since the lower level 1 lies at an energy higher than K B T above the ground level , the number of thermally excited atoms in the lower laser level 1 is so small that the population can be easily inverted by pumping a relatively small number of atoms into the upper level 2. The conditions for population inversion in this case are as follows.

27. Although separations between levels 1, 2 & 3 are assumed to be much greater than K B T as in the case of a three level laser, the number of thermally excited atoms go, No from the most population ground level O to level 1 are not neglected. The rate equations for atomic populations in the four-levels, than become.

28. Since N=No+N1+N2+N3 Laser Emission Fig. (11)Energy-level diagram of a four-level laser (17)

29. where  2 =  20 +  21 &  3 =  3o +  31 +  32 The steady –state solution is obtained as before  o1 N o -  1o N 1 +  21 N 2 +  31 N 3 = 0 -  2 N 2 +  32 N 3 = 0  N o -  3 N 3 = 0 Therefore, (18)

30. from equation 19, 20 N 2 is > N 1 when (19) (20)

31. (21) This is the condition for population inversion now  01 in the numerator of this equation is the probability of thermal excitation from level O to level 1, and is a small quantity as shown by the relation therefore, the excitation intensity necessary for population inversion is lowered. Since &

32. Then equation (21) can be approximated (22) Comparing equation (22) with equation (15\ )for population inversion in a three level laser, it is seen that they are similar except for the factor Since the four-level system has an extra level O, it is obvious that we have instead of instead of. Here it is the factor, which is important, because population inversion can be obtained even with very week pumping if the lower laser level 1 is above the ground level O by at least a few times K B T in energy.

33. Laser Operation (1) Essential Elements of Laser The laser device consists of basically of three elements; External source (pump), Amplifying medium and optical cavity (resonator(

34. The pump is an external energy source that produces a population inversion in the laser medium. Pumps can be optical, electrical, chemical or thermal in nature. For gas lasers (e.g. He-Ne laser), the used pump is an electrical discharge. The important parameters governing this type of pumping are the electron excitation cross-sections and the lifetimes of the energy levels.

35. In some lasers, the free electrons generated in the discharge process collide with and excite the laser atoms, ions, or molecules directly. In others, the excitation occurs by means of inelastic atom-atom (or molecule – molecule) collisions. In this case a mixture of two gasses is used such that the tow different species of atoms, say A and B, have excited states A* and B*. Energy may be transferred from one excited species to the other in a process as follows relation

36. A*+B A+B* e.g. He-Ne laser, where the laser – active neon atoms are excited by resonant transfer of energy from helium atoms in metastable state, where the He atoms receive their energy from free electrons via collisions.

37. 2- Laser medium The amplifying medium or laser medium is an important part of the laser device. Many laser are named after the type of laser medium used (e.g. He-Ne, CO 2 and Nd:YAG). This laser medium may be gas, liquid, or solid, determines the wavelength of the laser radiation. In some lasers the amplifying medium consists of two parts, the laser host medium and the laser atoms. For example, in Nd: YAG laser, the host medium is a crystal of yttrium Aluminum Garnet (or YAG), whereas the laser atoms are the Neodymium ions. The most important requirement of the amplifying medium is its ability to support a population inversion between two energy levels of the laser atoms.

38. 3-The Resonator The resonator is an optical “feed back device” that directs photons back and forth through the laser medium. Resonator or “optical activity” consists of a pair of carefully aligned plane or curved mirrors (see figure 2). One of them is chosen with a reflectivity 100% as possible. The other mirror is selected with a reflectivity somewhat less than 100% in order to allow part of the internally reflecting beam to escape and become the useful laser output beam. The geometry of the mirrors and their separation distance determine the structure of the electromagnetic field within the laser cavity and controlling the emerging laser beam.

39. Figure (2): four types of end mirrors in common use for lasers. (Mirror curvatures are exaggerated )