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1 Stimulated Emission of Radiation - stimulated emission is referring to the emission of radiation (a photon) from one quantum system at its transition frequency induced by the presence of other photons at that frequency - introduced by Einstein in 1917 to derive Planck's blackbody radiation law - is the basis for the operation of a LASER (Light Amplification by Stimulated Emission of Radiation) Consider an assembly of N atoms with energy levels E i and E j in thermal equilibrium with a thermal radiation field with energy density u(ν) at temperature T - the transition frequency ν ij in the individual atom is - the probability of an atom in state i to absorb a photon is proportional to u(ν ij ) (related to the number of photons at that frequency) and to the probability B ij (Einstein B coefficient) of the atom to absorb a photon at that frequency phys4.15 Page 1 - thus, the total number of atoms per second that absorb a photon is where N i is the number of atoms in state i - an atom in state j has a probability A ji (Einstein A coefficient) to spontaneously decay into state i by emitting a photon at frequency ν in a process called spontaneous emission - also, an atom in state j has a probability B ji to decay into state i by emitting a photon at frequency ν ij stimulated by the presence of other photons at that frequency the number of which is proportional to u(ν ij ) - this process is called stimulated emission - thus the total number N ji of atoms per second that make a transition into the lower state is given by - in thermal equilibrium the number of atoms that make a transition to lower energy must be identical to the number of atoms making a transition to higher energy phys4.15 Page 2

2 Transitions in a collection of atoms interacting with a radiation field - in thermal equilibrium - solve for the energy density of the radiation field u(ν) - with the number of atoms in the ground N i or excited state N j is given by classical M.-B. statistics - thus phys4.15 Page 3 - hence the energy density of a radiation field in thermal equilibrium is given by - this is Planck's formula for blackbody radiation for - for identical upwards and downwards transition rates - and with the ratio of spontaneous emission to induced emission rate given by note: - stimulated emission does occur and is consistent with the form of the blackbody spectrum - the probability for stimulated absorption is equal to the probability stimulated emission - the likelihood of spontaneous emission relative to stimulated emission increases rapidly with frequency - knowing one of the parameters A ji, B ji, B ij we can determine the other ones - spontaneous emission is 'stimulated' by vacuum fluctuations (QED) phys4.15 Page 4

3 Light Amplification by Stimulated Emission of Radiation (LASER) LASER Light Amplification by Stimulated Emission of Radiation properties: monochromatic coherent directed bright phys4.15 Page 5 Properties of LASER light monochromatic - relative line width coherence - temporal coherence - spatial coherence limited only by optics brightness - intensities - large photon flux in pulsed operation - short pulses phys4.15 Page 6

4 The LASER 1. active LASER medium (gas, dye, solid) 2. pumping (light, electrical energy) % reflecting mirror 4. coupling mirror (< 1% transmission) 5. LASER beam phys4.15 Page 7 The LASER Principle - generation of a large number of excitations in a medium by (stimulated) absorption - a long lived metastable state to achieve population inversion - stimulated emission of radiation - storage of radiation field in a resonant cavity with partially transmitting mirror phys4.15 Page 8

5 Generation of Population Inversion - excitation by optical pumping in a three level system - transition (possibly non-radiative) to a metastable state - stimulated emission - at least a three level system is required to achieve population inversion as the symmetry between the probability for absorption (B ij ) and emission (B ji ) results at best in equal population of the two states such that emission cannot be the dominant process (i.e. there would be no amplification in number of photons) phys4.15 Page 9 Examples of Lasers Ruby Laser: Cr + ions in Al 2 O 3 crystal (sapphire) - optical pumping using flash light - Helium-Neon Laser: gas mixture (90 % He, 10 % Ne) where He is excited electrically in gas discharge and energy is transferred in collision to Ne phys4.15 Page 10

6 Nobel Prize in Physics 1964 for fundamental work in the field of quantum electronics, which has led to the construction of oscillators and amplifiers based on the maser-laser principle Charles H. Townes Nicolay G. Basov Aleksandr M. Prokhorov phys4.15 Page 11 Laser-Pointer housing electronics LASER and optics phys4.15 Page 12

7 CD Player compact disc the mechanism portable CD player (older modell) phys4.15 Page 13 Laser-Printer resolution up to 20 µm or 50 points per mm (= 1270 dpi) colors with multiple stages phys4.15 Page 14

8 Material Processing CO 2 gas LASER up to 20 kw power phys4.15 Page 15 Lasers in Medicine eye surgery dermatology etc. phys4.15 Page 16

9 Specific Heat of Solids - dependence of thermal energy stored in a solid on temperature - molar specific heat at constant volume c V : energy that needs to be added to 1 mol of a solid to increase its temperature by 1 Kelvin - thermal energy is stored in solids in the vibrations of its constituents (atoms, ions or molecules) - in a simple model the atoms can vibrate along 3 independent (x, y, z) directions that each can be considered as an independent harmonic oscillator - the equipartition theorem tells us the average energy stored in each degree of freedom, thus the total energy of a solid containing 1 mol = N 0 = of atoms is given by - R = 8.31 J/mol is the ideal gas constant (pv = nrt) - specific heat (Dulong Petit law) phys4.15 Page 17 Dulong petit law fails for light elements (e.g. carbon) and at low temperatures, see plot - problem can be solved using the Bose-Einstein distribution function for the probability of an harmonic oscillator to have energy hν - average energy for a harmonic oscillator of frequency ν - total energy per mol - thus Einstein's specific heat formula is phys4.15 Page 18

10 - the high temperature limit of Einstein's formula - therefore - at high temperatures the thermal energy is much larger than the typical level separation of the harmonic oscillators in the solid and the specific heat is close to the classical prediction - at low temperatures the thermal energy is comparable to the level separation and quantum statistics start to play a role - for lighter elements with smaller masses the oscillator frequencies are higher and thus quantum effects are more important even at higher temperatures - the zero point motion does not play a role for the specific heat as it is only a constant offset in energy phys4.15 Page 19 Debeye theory - Debeye regarded a solid as a continuous elastic medium with elastic standing waves instead of considering the atoms as individual oscillators the way Einstein did - these standing waves can have frequencies ν up to a maximum value ν max - they are like transverse and longitudinal elastic waves that propagate at the speed of sound in the solid - their energies are quantized as in a harmonic oscillator - each quantum of the elastic wave is called a phonon - Debeye assumed that 3N different modes exist per mol in a solid - the resulting formula describes the specific heat of the solid better than Einstein's formula - to be discussed in detail in solid state physiscs phys4.15 Page 20

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