LINE-BROADENING MECHANISMS A line- broadening mechanism is referred to as homogeneous when it broadens the line of each atom in the same way. In this case the lineshape of the single-atom cross section and that of the overall absorption cross section are identical. A line-broadening mechanism is said to be inhomogeneous when it distributes the atomic resonance frequencies over some spectral range. Thus, in the discussion that follows, we consider the line- shape function in either absorption or emission, whichever is more convenient.

1- Homogeneous Broadening: The first homogeneous line-broadening mechanism we consider is due to collisions, and is known as collision broadening. In a gas it is due to the collision of an atom with other atoms, ions, free electrons, etc., or with the walls of the container. In a solid it is due to the interaction of the atom with the lattice phonons. A second homogeneous line-broadening mechanism originates from spontaneous emission. Since this emission is an inevitable feature of any transition, the corresponding broadening is called natural or intrinsic. In the case of natural broadening, it is easiest to consider the behaviour in terms of the spectrum of the emitted radiation.

The Laser light has a very important property which is very sharp or very narrow range of frequencies. This is in contradiction with the uncertainty principal which demand that the exited atom most stay for an infinite time in the exited state to define the state energy with zero uncertainty. Heisenberg uncertainty principle: ΔE* ∆ t > h ∆ E = h* ∆n ∆n > 1/ ∆t If ∆ t = 10-8 [s]        = = > ∆ n = 108 [Hz] If ∆ t = 10-4 [s]        = = > ∆ n = 104 [Hz]

The longer the specific energy level transition lifetime, the narrower is its linewidth Dn . When the atom become exited, it will stay exited only for the duration of the lifetime and return to the ground level emitting the excess photons To overcome this contradiction between uncertainty and lifetime of the exited atom it can be conceded that the exited levels has a range of energy and the atoms will be distributed over these levels and the most probability is at the frequency νo as in the fig

2- Inhomogeneous Broadening As a first case of inhomogeneous broadening we consider that which occurs for ions in ionic crystals or glasses. Such ions experience a local electric field produced by surrounding atoms of the material. Due to material inhomogeneities that are particularly significant in a glass medium, these fields differ from ion to ion. Via the Stark effect, local field variations then produce local variation of energy levels and thus of the ions' transition frequencies. (The term inhomogeneous broadening originates from this case.) For random local field variations, the corresponding distribution of transition frequencies g*(ν´o -νo) turns out to be given by a Gaussian function.

A second inhomogeneous broadening mechanism, typical of gas, arises from atomic motion and is called Doppler broadening. Assume that an incident em wave of frequency v is propagating in the positive z-direction and let vz be the component of atomic velocity along this axis. According to the Doppler Effect, when two sources are moving the relative motion affect one another. If a moving ambulance toward observe the frequency increases. If a moving ambulance away from observe the frequency decreases.

The atoms emit the electromagnetic spectrum in continues motion The atoms emit the electromagnetic spectrum in continues motion. The spectrometer will measure different frequencies depending in the direction of motion. The following equation gave the measured relative Doppler frequencies. The flowing fig. Gave a comparison between the line width broadening due to Doppler, Collision, and Natural Effect

each absorption removes a photon, SdF must equal the THE LASER IDEA Consider two arbitrary energy levels 1 and 2 of a given material, and let N1 and N2 be their respective populations. If a plane wave with a photon flux F is travelling in the z- direction in the material , the elemental change dF of this flux along the elemental length dz of the material is due to both stimulated absorption and emission processes occurring in the shaded region . Let S be the cross-sectional area of the beam. SdF = (W2lN2 – W12N1)(Sdz) each absorption removes a photon, SdF must equal the difference between stimulated emission and absorption events occurring in the shaded volume per unit time

With the help of Esq. , , nonradiative decay does not add new photons, while photons created by radiative decay are emitted in any direction and thus give negligible contribution to the incoming photon flux F. Equation (1.2.1) shows that: the material behaves as an amplifier (dF/dz>0) if N2 > g2N1 /g1 while it behaves as an absorber if N2 < g2N1 /g1. At thermal equilibrium populations are described by Boltzmann statistics

where k is Boltzmann's constant Then if N1 and N2 are the thermal equilibrium populations of the two levels: where k is Boltzmann's constant and T is the absolute temperature of the material. In thermal equilibrium we thus have the material then acts as an absorber at frequency ν0 if a nonequilibrium condition is achieved for which , then the material acts as an amplifier. In this case we say that there exists a population inversion in the material.

This means that the population difference is opposite in sign to what exists under thermodynarnic equilibrium the condition for thermodynarnic equilibrium where the material is an obsorber the condition for population inversion where the material is an active A material in which this population inversion is produced is referred to as an active medium.

Example: Calculate the population ratio N1,N2 for the two energy levels E1,E2 at temperature (300K) if the energy difference is 0.5ev, what is the wavelength of the photons produced by this transition? Solution: This ratio means that at temperature 300K for 109 atoms in the ground level E1 there are only 4 atoms in the exited energy level E2 The wavelength of the photons:

How to make an oscillator from an amplifier material: it is necessary to introduce suitable positive feedback by placing the active material in a resonant cavity having a resonance at frequency ν0. In the case of a laser, feedback is often obtained by placing the active material between two highly reflecting mirrors, such as the plane parallel mirrors in Fig. 1.3. In this case a plane em wave travelling in a direction perpen -dicular to the mirrors bounces back and forth between the two mirrors, and is amplified on each passage through the active material.

If one of the two mirrors (e. g If one of the two mirrors (e.g. mirror 2) is partially transparent, a useful output beam is obtained from that mirror. It is important to realize that, for masers and lasers, a certain threshold condition must be reached. In the laser case, oscillation begins when the gain of the active material compensates the losses in the laser (e.g., losses due to output coupling). According to Eq. (1.2.1)

the gain per pass in the active material the ratio between output and input photon flux) is exp{σ[N2 — (g2N1/g1)]Ɩ} where we denote for simplicity σ = σ21, and where Ɩ is the length of the active material. Let now R1and R2 is the power reflectivity of the two mirrors (Fig. 1.3), respectively, and let Li be the internal loss per pass in the laser cavity.

If, at a given time, F is the photon flux in the cavity leaving mirror 1 and travelling toward mirror 2, then the photon flux F´ leaving mirror 1 after one round trip is: At threshold we must have F' = F and therefore: This equation shows that threshold is reached when the population inversion is reaches a critical value, called the critical inversion, given by: 1.2.3

Equation (1.2.3) can be simplified if one defines

Example: A laser cavity consists of two mirrors with reflectivities R1 = 1 and R2 = 0.5, while the internal loss per pass is Li = 1%. Calculate total logarithmic losses per pass. If the length of the active material is Ɩ =7.5 cm and the transition cross section is σ = 2.8 x 10-19 cm2, calculate the threshold inversion.