Conditions for Producing Laser

1. Conditions for Producing Laser

2. Area A Io I L Absorption and Gain on a homogeneously Broadened Transition • Consider a beam of light having an intensity per unit frequency I(ν). • passing through a medium of thickness L and cross sectional area A as indicated in fig.

3. Let I denotes the intensity such that I = I(ν) ∆ ν • and Io the intensity before it enters the medium. • Let u and l represent the upper and lower energy levels respectively . • If El and Eu represent the energy of lower and upper energy levelsthen the energy difference ∆Eul = Eu – El = hνul • Let us determine the effect of the medium on the beam as it propagates through the medium. • Let Nu and Nl represent the total number of atoms per unit volume in the upper and lower levels of the transitions.

4. The Radiative transition probability for transitions occurring between energy levels u and l is given by Aul. • It can be easily proved that the transition probability per unit frequency range is given by • Aul(ν) = γulTAul /4π2/( ν - νo )2 + (γulT/4π)2 ………………….( 1 ) • Where Aul represents the total transition probability ; γulT = γu + • γl represents the total decay rate of upper and lower levels. • We consider the three possible radiative interactions between the two levels u and l.

5. Eu Nu NuAul NuBulu(ν) NlBluu(ν) Nl El • figure 2 • The first one on the left of the figure 2 represents the spontaneous emission from u to l at a spontaneous rate Aul

6. The other two processes are stimulated processes • (one corresponds to absorption and other corresponds to stimulated emission) • The number of stimulated transitions (upward or downward ) between two levels per unit volume per unit time will be given by • Nl Blu(ν ) ∆ν u(ν) = Nl Blu(ν ) I(ν) ∆ν η/c (absorption) ………………..(2) • And NuBul(ν) ∆ν u(ν) = NuBul(ν) I(ν) ∆ν η/c • (stimulated emission) ………..(3) • Where I is the intensity and η is the refractive index of the medium and c is the velocity of light in free space.

7. dA I (z) I (z+dz) dz • Now, we consider a small length dz of the medium as • shown in figure 3 • In this case we will estimate the energy that as a result of the • stimulated process given in equations (2) and (3) can be added • to or subtracted from the beam as it moves through a small • distance dz. • we are neglecting the spontaneous emission contribution since • the spontaneous emission is radiated in a solid angle 4π and • hence it contributes a very little in the direction of the incident • beam.

8. The amount of energy per unit time added when the beam passes through a region of length dz and cross sectional area dA within the medium can therefore be expressed as • [I(z+dz) – I(z)] dA = [NuBul(ν) - Nl Blu(ν )] I η hν dAdz/c ……….(4) • Energy is added to or subtracted from the beam in discrete amounts hν as a result of the two terms within the bracket of right hand side of equation (4). • Let dI = I(z + dz) – I(z) • Thus dI dA = [NuBul(ν) - Nl Blu(ν )] I η hν dAdz/c • dI / dz = [NuBul(ν) - Nl Blu(ν )] I η hν / c ……………(5) • dI / dz = g(ν ) I ………………………………(6)

9. Where g (ν) = [NuBul(ν) - Nl Blu(ν )] η hν / c ………………….(7) • The solution of equation (6) can be expressed as • I = Io egz …………………(8) • Thus g (ν) known as gain coefficient and has dimensions of reciprocal of length is directly proportional to the refractive index of the medium.

10. Gain Coefficient and Stimulated Emission Cross Section for Homogeneously Broadening • In order to have closer look on gain coefficient on the basis of equation (7), we would like to make use of Einstein coefficients. • we know that • Aul(ν)/Bul(ν) = 8πhν3 η3/c3 Bul(ν) = c3Aul(ν)/ 8πhν3 η3 …………………………(9) Also glBlu / guBul = 1 Thus Blu = gu Bul/gl …………………(10)

11. Putting the values of Bul and Blu from (9) and (10) in equation (8), we get gH(ν) = [Nu – guNl/gl] c2 Aul(ν)/ 8π η2 ν2 ………..(11) Substituting the value of Aul(ν) from (1) in equation (11), we get: gH(ν) = [Nu – guNl/gl] c2 /8π η2 ν2 [γulTAul /4π2/( ν - νo )2 + (γulT/4π)2 ] ………..(12) Introducing ∆Nul = [Nu – guNl/gl] known as Population difference And stimulated emission cross section σHul (ν)= c2 /8π η2 ν2 [γulTAul /4π2/( ν - νo )2 + (γulT/4π)2 ] ……………… (13) Thus gH(ν) = ∆Nul σHul (ν) ………………………………………………(14)

12. Thus the intensity I at a specific distance z will be given by • equation (7) as • I = Ioexp[gH(ν).z] = Ioexp[σHul(ν) ∆Nulz] ………….(16) • Concluding Remarks • Since σHul (ν) and z are positive quantities so • if ∆Nul is positive then the beam intensity will go on increasing exponentially with distance z inside the material and will provide the amplification. • The materials in which this condition is satisfied are known as active materials. • if ∆Nul is negative then the beam will decrease in intensity with distance z exponentially and leads to absorption of the beam. • The first condition is required for laser action.

13. Statistical Weights and the gain equation • The statistical weights are important because they enter into the gain equation where their ratio gu/gl is a factor preceding the population density Nl as indicated in population difference ∆Nul. • The statistical weights arise in Boltzmann statistical theory, where they represent the number of levels at any specific energy.

14. Statistical weight for n = 2 in hydrogen is g = 2 + 6 = 8 for this energy level. • For solid state lasers, there are so many levels smeared together that the statistical weight for each level are approximately equal. • Therefore, we can ignore the statistical weights in the gain equation for solid state, dye, and semiconductor lasers (since gu/gl = 1). • But they can be very important for gas lasers e.g., if gu = 2 and gl = 4 for a specific laser transition then gu/gl = ½.

15. Hence the population difference would be ∆Nul = (Nu – guNl/gl) = (Nu – 0.5Nl). • This indicates that only half of the population in the lower laser level is applicable in the gain equation. • Thus we can face a situation in which Nu/Nl < 1 and yet still have gain! • This actually occurs in many gas lasers and has been shown to be important in those situations.

16. Population Inversion (Necessary condition for a Laser) • The intensity I at any point situated at a distance z from the surface is given by I = Ioexp[gH(ν).z] = Ioexp[σul(ν)(Nu – guNl/gl)z] Where Io is the intensity of the beam as it enters the medium • Amplification will occur only if Nu > guNl/gl • The case of upper level being more populated than the lower level taking into account the statistical weights is referred to as population inversion. • Population inversion is a necessary condition for amplification or laser action to occur but it is not a sufficient condition

17. Saturation Intensity (Sufficient Condition for a Laser) • It is well known that the intensity of the beam at some distance z from the surface is given by I = Ioexp[σul(ν)(Nu – guNl/gl)z] = Ioexp[σul(ν)ΔNulz] • Thus it is clear from the above equation that there is strong possibility of the growth of the beam when exponential coefficient is positive and of sufficient magnitude.

18. Let us assume a medium in which a population inversion exists. • Also assume that the value of the gain – the exponent of above equation is large enough to provide significant amplification of the beam. • Whether the beam will go on amplifying indefinitely while traversing through the medium? • The beam could eventually reach an intensity (at some specific length z) such that the energy stored in the upper laser level is not sufficient to satisfy the exponential growth demands of the beam.

19. Hence there must be a limiting expression to estimate the intensity at which this saturation occurs. • The length L at which the beam stops growing exponentially (i.e. saturation effect occurs) is known as saturation length Lsat. • The intensity I at z = Lsat is known as saturation intensity Isat. • The gain in this stage is known as saturation gain.

20. Nu u Pumping flux Ru NuAul NuBulIη/c Nl l • In order to estimate the saturation intensity Isat, we consider two energy levels l & u in case where steady state population density Nu exists in the upper level and Nl in the lower level as shown in figure below

21. Thus rate of change of population in level u can be expressed as dNu/dt = Ru – Nu [ Aul + BulIη/c] Or dNu/dt = Ru – Nu [ 1/τu + BulIη/c] • For steady state dNu/dt = 0 Thus Ru – Nu [1/τu + BulIη/c] = 0 Nu = Ru / [1/τu + BulIη/c] …………….(1) • When I = 0, the population density Nu is given as Nu = Ruτu …………………………….(2) • For a value of z = Lsat the intensity I and consequently the stimulated emission term in the denominator of equation (1), could eventually become as large as the term associated with the level lifetime 1/ τu.

22. Since the value of 1/τu can be quite large, a significant intensity ‘I’ would be required for that to happen. • When saturation occurs, the population of level u would decrease by a factor of 2 to a value of Ruτu /2 owing to stimulated emission. • The exponential growth factor σulΔNulz for z = Lsat would decrease by approximately by a factor of 2 because Nu would decrease by 2. • Further increase in ‘I’ would decrease the gain in regions where the beam propagates further into the medium as suggested by equation (1). • for our analysis we will arbitrarily define Isat as that intensity at which the stimulated emission rate becomes equal to the normal radiative decay rate.

23. Thus from equation (1) Isatη Bul /c = 1/τu Hence Isat = c / η Bul τu But Bul(ν) = c3 Aul(ν) / 8πh ν3 η3 Thus Isat = [c / η τu] [8πh ν3 η3 / c3 Aul(ν)] Or Isat = [8π η2ν2 / c2 Aul(ν)] [hν /τu] But 8π η2ν2 / c2 Aul(ν) = σulH Isat = hν / τu σulH • Thus larger the stimulated emission cross section and lifetime of upper laser level, smaller will be saturation intensity required.

24. Development and Growth of a Laser Beam for a Gain medium with homogeneous Broadening • It would be useful to obtain a value of the gain σulHΔNulz at which the beam would reach the saturation intensity. • consider a beam that starts by spontaneous emission and is subsequently amplified over a frequency width approximately equal to the homogeneous linewidth ΔνH centered at the center of the emission frequency νul.

25. l Area A= πd2/4 Gain medium L • Consider a cylindrical gain medium as shown in figure below that has a length L, a cross sectional area A and a diameter d.

26. Assume that population inversion exists. • Assume that upper laser level u is instantaneously populated by some pumping process to achieve a population density Nu and suppose that radiative decay rate from level u to l at frequency νul is Aul. • Also assume here that population inversion is very large so that we can neglect Nl i.e. ΔNul ≈ Nu. • For simplicity consider the beam starting at one end of the medium in a region of length l. • l is the gain length such that σulHΔNull ≈ σulHNul = 1 such that l < L. • Some of these photons are emitted in the elongated direction and would therefore be enhanced by stimulated emission as they transit through the length L of the medium.

27. The beam would evolve as the radiation propagates and the intensity grows exponentially. • We will calculate the conditions required for the beam to reach the saturation intensity as it arrives at the opposite end of the medium. • Consider a region (volume element) next to the region of length l. But the beam initiated by such a volume element would not reach the saturation intensity since it would not have traversed as much length as the beam originating from end region • The energy radiated per unit time into a 4π solid angle from within the volume A.l as Nu Al Aul hνul. • The saturation intensity will be given as (NuAl Aulhνul) (A / 4πL2) {exp [σulHNuL]}/A = Isat = hνul/ σulHτu …… (1)

28. [Nu(πd2/4) l Aul hνul] (A / 4πL2) {exp [σulHNuL]}/A = Isat = hνul Aul / σulH Or Nu(πd2/4)l(1/4πL2)exp [σulHNuL] = 1/σulH ……… (2) Where we have used area A = πd2/4 Also σulHNul = 1 Thus l = 1/ σulHNu Putting the value of l in equation (2), we get Nu (πd2/4) 1/ σulHNu (1/4πL2) exp [σulHNuL] = 1/ σulH

29. This implies that exp [σulHNuL] = 16(L / d)2 For saturation we replace L by Lsat Hence exp [σulHNuLsat] = 16(Lsat / d)2 ………… (3) • Thus from equation (3) it is clear that more the value of (Lsat / d)2, more will be the gain factor.

30. Gain medium Intense beam output d d L L = d No net effect for a sphere Effect of Shape or Geometry of Amplifying Medium on Growth Factor • consider two differently shaped gain media as shown in figure below

31. In case of a long cylinder having diameter d and length L while other is a sphere having same diameter d. • For the long cylinder, say Lsat / d = 100, spontaneous emission will originate at one end of the medium and will emerge at another end in an elongated shape. • Thus the value of gain factor exp [σulHNuLsat] = 16(100)2 = 1.6 ×105. This is an extremely large increase resulting in a very intense beam with an extremely low divergence. • In case of a sphere, where Lsat / d = 1 then the value of exponential growth factor exp [σulHNuLsat] = 16, which is significantly lower than the value 1.6 ×105 obtained for the cylindrical shape medium.

32. Thus the radiation originating from different locations within the sphere would cause the beam to diverge rapidly (rather than concentrated in a specific direction) in the same manner as radiation emitting from a spherically shaped incoherent source. • Hence the same amount of energy would emerge from the sphere under the presence of gain as would occur with no gain in the medium!

33. Exponential Growth Factor (gain) • The exponential growth factor σHul(ν) ∆NulL consists of mainly three parts • Stimulated Emission Cross Section • Population Difference • Gain Length • In most of the cases the stimulated emission cross section σul is of the order of 10-16 m2 on the basis of the expression for σul = c2 Aul/ 8πη2 ν2.

34. Exponential Growth Factor (gain) • There is not much scope of increasing the stimulated emission cross section. • Similar is the case with population difference. • Third component of the gain i.e. requirements of length of the gain medium has a lot of scope. • But long lasers are too cumbersome, too difficult to set up and operate and not very practical. • This problem was overcome by increasing the effective length by using the mirrors.

35. Threshold Requirements for a Laser • The threshold gain conditions are defined as the necessary requirements for the beam to grow to the point at which it reaches the saturation intensity Isat. • Achieved by generally using two mirrors. • Consider a round trip pass of the beam through the amplifier and assume that the gain is uniform over the amplifier length (not changing in time).In this case the beam will experience an exponential growth of exp[g(νo)2L] for a round trip

36. Threshold Requirements for a Laser • Loss at each mirror of (1-R); where R is the reflectivity of the mirror. • Hence the beam will be reduced by the factor R after it is reflected from each mirror. • The minimum round trip steady state requirement for the threshold of lasing is that the gain exactly equals the loss. • Any increase in gain beyond the threshold will cause the beam to grow.

37. Threshold Requirements for a Laser • For the simple situation in which both the mirrors have same reflectivity R, this threshold can be expressed as R2 exp [gth 2L] = 1 Where gth represents the threshold gain. • gth = (1 / 2L) ln(1/R2) ; hence larger the value of L lesser will be gth.

38. Threshold Requirements for a Laser • Consider a general case in which mirrors have different reflectivities R1 and R2 (say). • Consider fractional losses a1 and a2 at the Brewster windows or at any other region in the path of the beam other than inside the amplifier. • Also assume a possible distributed loss α within the gain medium;

39. Threshold Requirements for a Laser • For this general situation, the threshold equation for a round trip pass becomes R1R2 (1- a1) (1-a2) exp [gth – α] 2L = 1 Thus gth = (1 / 2L) ln [1/ R1R2 (1- a1) (1-a2)] + α • The threshold gain gth is the value at which the amount of gain just equals the loss • Hence the gain has to increase above that value in order for a beam to develop.