1. Transverse modes The distribution of the radiation intensity beam across the cross sectional area perpendicular to the optical laser axis has different shape. These different shapes depend on position of the mirror. Any change on the mirror position change these shapes which are known as Transverse Mode. Using a single lens and a screen to magnify the Transverse Mode in a laser beam. TEM mn m, n, are integer numbers. Assuming the beam advance in z direction : m = Number of points of zero illumination along x axis. n = Number of points of zero illumination (between illuminated regions) along y axis.

2. Population Inversion When the population number of a higher energy level is bigger than the population number of a lower energy level, a condition of "population inversion" is established. If a population inversion exists between two energy levels, the probability is high that an incoming photon will stimulate an excited atom to return to a lower state, while emitting another photon of light. The population inversion is essential for the laser generation. The distribution of the electrons in this case is different from the case of thermodynamic equilibrium which is controlled by the Maxwill-Boltezman distribution.

3. Thermal Equilibrium From thermodynamics we know that a collection of atoms, at a temperature T [ 0 K], in thermodynamic equilibrium with its surrounding, is distributed so that at each energy level there is on the average a certain number of atoms. The Boltzmann equation determines the relation between the population number (N i ) at specific energy level (E i ) and the temperature T [ 0 K] :

4. N i = Population Number = number of atoms per unit volume at certain energy level E i. k = Boltzmann constant: k = 1.38*10 23 [Joule/ 0 K]. E i = Energy of level i. We assume that E i > E i-1. Const = proportionality constant. It is not important when we consider population of one level compared to the population of another level. T = Temperature in degrees Kelvin [ 0 K] (Absolute Temperature). The Boltzmann equation shows the dependence of the population number (N i ) on the energy level (E i ) at a temperature T. From this equation we see that: 1. the higher the temperature, the higher the population number. 2. the higher the energy level, the lower the population number.

5. Relative Population (N 2 /N 1 ) The relative population (N 2 /N 1 ) of two energy levels E 2 compared to E 1 is: N 2 /N 1 = const* exp (-E 2 /kT)/ const* exp (-E 1 /kT) The proportionality constant (const) is cancelled by division of the two population numbers. The Figure below shows the population of each energy level at thermal equilibrium.

6. Conclusions: The relation between two population numbers (N 2 /N 1 ) does not depend on the values of the energy levels E 1 and E 2, but only on the difference between them: E 2 - E 1. For a certain energy difference, the higher the temperature, the bigger the relative population. The relative population can be between 0 and 1.

7. Population Inversion We saw that in a thermal equilibrium Boltzmann equation shows us that: N 1 > N 2 > N 3 Thus, the population numbers of higher energy levels are smaller than the population numbers of lower ones. This situation is called "Normal Population". In a situation of normal population a photon impinging on the material will be absorbed, and raise an atom to a higher level

8. An example is described in the Figure below. In this situation there are more atoms (N 3 ) in a higher energy level (E 3 ), than the number of atoms (N 2 ) in a lower energy level (E 2 ). The process of raising the number of excited atoms is called "Pumping".

9. Three Level Laser A schematic energy level diagram of a laser with three energy levels is the figure below. The two energy levels between which lasing occur are: the lower laser energy level (E 1 ), and the upper laser energy level (E 2 ). Atoms are pumped from the ground state (E 1 ) to energy level E 3 They stay there for an average time of 10 -8 [sec], and decay (usually with a non-radiative transition) to the meta-stable energy level E 2. Since the lifetime of the meta-stable energy level (E 2 ) is relatively long (of the order of 10 -3 [sec], many atoms remain in this level. If the pumping is strong enough, then after pumping more than 50% of the atoms will be in energy level E 2, a population inversion exists, and lasing can occur.

10. Four Level Laser The extra energy level has a very short lifetime The pumping operation of a four level laser is similar to the pumping of a three level laser. This is done by a rapid population of the upper laser level (E 3 ), through the higher energy level (E 4 ).

11. The advantage of the four level lasers is the low population of the lower laser energy level (E 2 ). To create population inversion, there is no need to pump more than 50% of the atoms to the upper laser level. The population of the lower laser level (N 2 (t)) is decaying rapidly to the ground state, so practically it is empty. Thus, a continuous operation of the four level laser is possible even if 99% of the atoms remain in the ground state (!) Advantages of four level lasers Compared to three level lasers: The lasing threshold of a four level laser is lower. The efficiency is higher. Required pumping rate is lower. Continuous operation is possible.

12. Round trip gain with losses The total losses of the laser system are due to a number of different processes these are: Transmission at the mirrors Absorption and scattering by the mirrors Absorption in the laser medium Diffraction losses at the mirrors All these losses will contribute to reduce the effective gain coefficient to (γ o - k)

13. Round trip Gain (G): Figure below show the round trip path of the radiation through the laser cavity. The path is divided to sections numbered by 1-5, while point “5” is the same point as “1”. By definition, Round trip Gain is given by: G = Round trip Gain. I 1 = Intensity of radiation at the beginning of the loop. I 5 = Intensity of radiation at the end of the loop.

14. Gain (G) Without Losses From privies we found that the intensity after one round trip is given by the equation I 5 = R 1 * R 2 *G 2 *I 1 Gain (G) With Losses We assume that the losses occur uniformly along the length of the cavity (L). In analogy to the Lambert formula for losses, we define loss coefficient (α), and using it we can define absorption factor k: k = exp(-2αL) k = Loss factor, describe the relative part of the radiation that remain in the cavity after all the losses in a round trip loop inside the cavity. All the losses in a round trip loop inside the cavity are 1-k (always less than 1). α = Loss coefficient (in units of 1 over length). 2L = Path Length, which is twice the length of the cavity.

15. Adding the loss factor (k) to the equation of I 5 : I 5 = R 1 * R 2 *G A 2 *I 1 *k From this we can calculate the round trip gain: G = I 5 /I 1 = R 1 * R 2 *G A 2 *k As we assumed uniform distribution of the loss coefficient (α), we now define gain coefficient (γ), and assume active medium gain (G A ) as distributed uniformly along the length of the cavity. G A = exp(+γL) Substituting the last equation in the Loop Gain: G = R 1 * R 2 * exp(2(γ-α)L)

16. When the loop gain (G) is greater than 1 (G > 1), the beam intensity will increase after one return pass through the laser. Laser beam can be generated When the loop gain (G) is less than 1 (G < 1), the beam intensity will decrease after one return pass through the laser. Laser oscillation decay and no beam will be emitted. There is a threshold condition for amplification, in order to create oscillation inside the laser. This Threshold Gain is marked with index “th”. For continuous laser, the threshold condition is: G th = 1 = R 1 R 2 G A 2 k = R 1 * R 2 * exp(2(γ-α)L)

17. Example Active medium gain in a laser is 1.05. Reflection coefficients of the mirrors are: 0.999, and 0.95. Length of the laser is 30 cm. Loss coefficient is: a = 1.34*10 - 4 cm -1. Calculate: 1.The loss factor k. 2.The round trip gain G. 3.The gain coefficient (g). Solution 1. The loss factor k: k = exp(-2aL) = exp[-2(1.34*10 -4 )*30] = 0.992 2. The Loop gain G: G = R 1 R 2 G A 2 k = 0.999*0.95*1.052*0.992 = 1.038 Since G L > 1, this laser operates above threshold. 3. The gain coefficient (γ): G = exp(γL), Ln G = γL γ= Ln G/L = ln(1.05)/30 = 1.63*10 -3 [cm -1 ] The gain coefficient (γ) is greater than the loss coefficient (α), as expected

18. Example Helium Neon laser operates in threshold condition. Reflection coefficients of the mirrors are: 0.999, and 0.97. Length of the laser is 50 cm. Active medium gain is 1.02. Calculate: 1.The loss factor k. 2.The loss coefficient a. Solution Since the laser operates in threshold condition, G = 1. Using this value in the round trip gain: G = 1 = R 1 R 2 G A 2 k 1. The loss factor k: k = 1/( R 1 R 2 G A 2 ) = 1/(0.999*0.97*1.02 2 ) = 0.9919 As expected, k 1, this laser operates above threshold. 2. The loss coefficient (α) is calculated from the loss factor: k = exp(-2 α L) lnk = -2 α L α = lnk/(-2L) = ln(0.9919)/(-100) = 8.13*10 -5 [cm -1 ] If the loss factor was less than 0.9919, then G < 1, and the oscillation condition was not fulfilled.

19. Example Reflection coefficients of the mirrors are: 0.999, and 0.95. All the losses in round trip are 0.6%. Calculate the active medium gain. Solution For finding the active medium gain G, the loss factor (k) must be found. All the losses are 1-k. 1-k = 0.006 k = 0.994 Using this value in the threshold loop gain: G th = 1 = R 1 R 2 G A 2 k (G A ) th = 1/sqrt( R 1 R 2 k) = 1/sqrt(0.999*0.95*0.994) = 1.03 The active medium gain must be at least 1.03 for creating continuous output from this laser.

20. Summary G = round trip Gain, determines if the output power of the laser will increase, decrease, or remain constant. It includes all the losses and amplifications that the beam has in a complete round trip through the laser. G L = R 1 R 2 G A 2 k R 1, R 2 = Reflection coefficients of the laser mirrors. G A = Active medium gain as a result stimulated emission. G A = exp(+γL) γ = Gain coefficient. L = Active Medium length. k = Optical Loss Factor in a round trip path in the laser cavity. k = exp(-2αL) α = Loss coefficient. When G = 1, The laser operate in a steady state mode, meaning the output is at a constant power. This is the threshold condition for lasing, and the active medium gain is: (G A ) th = 1/sqrt( R 1 R 2 k) The round trip Gain is: G L = R 1 * R 2 * exp(2(γ-α)L) round trip Gain

21. Steady State Oscillation and Gain Saturation Population in version and pumping threshold condition: From the equation of small signal gain one can conclude that the population inversion required for reaching the lasing threshold: or At threshold the population inversion Note that the lasing threshold will be readily when g(v) is maximum at v = v o corresponding to the centre of the natural linewidth.

22.  Pumping power required to reach threshold condition  To find the power required for a 4-level laser system to reach the threshold we will use the rate equations. 1.First we assume that E 1 >>KT so the thermal population of the energy level 1 is negligible. 2.Second we assume that the population of the ground state does not change during lasing action

23. R 1 and R 2 are the rate of pumping then the rate equation for the population for the change in N 2 and N 1 In steady state condition dN 2 /dt = dN 1 /dt = 0 ( we assumed that g 1 =g 2 and R 1 =0) By solving the above two rate equations we get We get

24. For population inversion A 21 T 10 (The upper lasing level has a longer spontaneous emission life time than the lower level. In most laser T 21 >>T 10 and hence (1-A 21 /A 10 ) ≈ 0 At threshold At threshold the radiation density ρ v is very small and we can assume that (ρ v =0) In steady state In steady state situation the gain becomes equal to the losses then we can write (N 2 -N 1 ) ss = (N 2 -N 1 ) th

25. From the two equations and we get and hence the radiation density ρ v This means that the power output is directly proportional to the pumping power within the laser cavity

27. Modifying the laser output In this chapter we examine the cavity resonator in detail. There are a number of intracavity devices that may be used to enhance specific features of laser output, such as operating mode and spectral width LONGITUDINAL MODES We now know that the cavity itself is an interferometer and is resonant only at wavelengths spaced apart by the free spectral range FSR of the arrangement. Refining the model above, we can see that the actual output of the laser will not be a smooth curve as depicted in Figure 6.4.1, but rather, a series of closely spaced wavelengths which follow the general envelope of the curve and exist at points where the gain exceeds losses in the cavity.

28. The development of the output is seen in Figure 6.4.2, where the top diagram shows the gain curve of the laser medium along with the lasing threshold (the point at which gain equals losses). In a simple model one would expect an output spectrum resembling the shaded area on the top diagram. The response of the cavity, resonant Atfre quencies spaced apart by the FSR, is shown in the middle diagram. The result is depicted on the lower diagram, in which the output is seen to be several modes (11 in this case), with the strongest at the centre of the curve (where the gain is highest) and the weakest near the edges, where gain just slightly exceeds losses.

29. Since the laser cavity is an interferometer the resonant modes of the cavity are spaced apart by free spectral range (FSR), which is defined as where n is the refractive index of the medium inside the cavity. The FSR, in this case, is in terms of frequency (Hz). The number of modes that will oscillate simultaneously may be determined by dividing the spectral width of the laser by the FSR. Example: An argon laser with a 90-cm cavity has a spectral width of 5 GHz. The FSR of the cavity is The number of modes

30. Selection of the laser emission lines The addition of a prism between the plasma tube and the HR allows selection of a single line, since only one unique wavelength can make the path through the prism, reflecting off the HR and back through the prism in exactly the same path. By changing the angle of the prism and HR relative to the laser’s axis, this arrangement allows tuning through a large range.

31. Alternatively, the HR may be replaced by a diffraction grating which, in a similar manner, reflects only a single wavelength of light back into the plasma tube for amplification Diffraction gratings, on the other hand, feature higher angular dispersions of incident wavelengths and so are easier to tune when multiple wavelengths are close together or the medium has a large continuous wavelength range, such as a dye laser.