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Lecture 2. Basic Lasers Principles The phenomenon of stimulated emission Einstein coefficients relationship with the construction of the particle emitting The shape of the emission lines Broadening of the emission lines Population inversion as a precondition for amplifying of radiation In 1915, Neils Bohr proposed a model of the atom that explained a wide variety of phenomena that were puzzling scientists in the late 19th century. This simple model became the basis for the field of quantum mechanics and, although not fully accurate by today’s understanding, still is useful for demonstrating laser principles. Under the right circumstances an electron can go from its ground state (lowest-energy orbit) to a higher (excited) state, or it can decay from a higher state to a lower state, but it cannot remain between these states. The allowed energy states are called “quantum” states and are referred to by the principal “quantum numbers” 1, 2, 3, etc. The quantum states are represented by an energy-level diagram. For an electron to jump to a higher quantum state, the atom must receive energy from the outside world. This can happen through a variety of mechanisms such as inelastic or semielastic collisions with other atoms and absorption of energy in the form of electromagnetic radiation (e.g., light). When an electron drops from a higher state to a lower state, the atom must give off energy, either as kinetic activity (nonradiative transitions) or as electromagnetic radiation (radiative transitions). Each photon has an intrinsic energy determined by the equation: E=hc/λ where λ is the wavelength of the light, c is the speed of light in a vacuum and h is Planck’s constant Lets consider two energy levels of an atom or molecule and interacting with the system of such particles beam of photons with energies corresponding to the difference between these two levels. Photons are absorbed and transfer particles into an excited state Probability dW of particle transitions between states 1 and 2 is described by the coefficients introduced by Einstein in the following way : dW12 = B12 ρν dt where ρν means the density of radiation in the system In general, when an electron is in an excited energy state, it must eventually decay to a lower level, giving off a photon of radiation. This event is called “spontaneous emission,” and the photon is emitted in a random direction and a random phase. The average time it takes for the electron to decay is called the time constant for spontaneous emission, and is represented by τ. When an electron is in energy state E2, and its decay path is to E1, but, before it has a chance to spontaneously decay, a photon happens to pass by whose energy is approximately E2 - E1, there is a probability that the passing photon will cause the electron to decay in such a manner that a photon is emitted at exactly the same wavelength, in exactly the same direction, and with exactly the same phase as the passing photon. This process is called “stimulated emission.” Excited particle may return to a base level spontaneously emitting a photon dWs21= A21 dt The beam of photons may also interact with the excited particles and the particle returns to the ground state by emitting a photon identical in all respects with that which forced emission dW21 = B21 ρν dt Balance of transitions increasing and decreasing the cast on level 2 is as follows : n1 dW12 = n2 dWs21 + n2 dW21 n1 B12 ρν = n2 A21 + n2 B21 ρν Accordingly to Boltzman distribution the cast per one level is given by: n2 = (g1/g2) n1 exp{-(E2-E1) / kT} where g1 and g2 means the degree of degeneracy of levels Thus: n1 B12 ρν = (g1/g2) n1 exp{-(E2-E1) / kT} (A21 + B21 ρν) After transformations we obtain the expression for the energy density : A 21 1 ρν B21 B12g1 exp E E /kT 2 1 B21g 2 Comparing to the Planck’s formula: ρ 8ππ /c hνν 21 2 3 we get the following equations: B12g1 = B21g2 A21 = (8πν2 / c3) hνB21 Now consider the group of atoms in exactly the same excited state, and most are effectively within the stimulation range of a passing photon. We also will assume that t is very long, and that the probability for stimulated emission is 100 percent. The incoming (stimulating) photon interacts with the first atom, causing stimulated emission of a coherent photon; these two photons then interact with the next two atoms in line, and the result is four coherent photons. At the end of the process, we will have eleven coherent photons, all with identical phases and all traveling in the same direction. Note that the energy to put these atoms in excited states is provided externally by some energy source which is usually referred to as the “pump” source. Of course, in any real population of atoms, the probability for stimulated emission is quite small. Furthermore, not all of the atoms are usually in an excited state; in fact, the opposite is true. Pumping medium (electrical, optical etc) for population inversion Define Ni , the population of level i Ni(t)= the # of electrons, per unit of volume, occupying energy level i at time t Boltzmann’s principle, a fundamental law of thermodynamics, states that, when a collection of atoms is at thermal equilibrium, the relative population of any two energy levels is given by: where N2 and N1 are the populations of the upper and lower energy states, respectively, T is the equilibrium temperature, and k is Boltzmann’s constant. Substituting hν for the E2 – E1 we obtain: For a normal population of atoms, there will always be more atoms in the lower energy levels than in the upper ones. Since the probability for an individual atom to absorb a photon is the same as the probability for an excited atom to emit a photon via stimulated emission, the collection of real atoms will be a net absorber, not a net emitter, and amplification will not be possible. Achieving population inversion in a two-level atom is not very practical. Such a task would require a very strong pumping transition that would send any decaying atom back into its excited state. Consequently, to make a laser, we have to create a “population inversion.” For photon flux Φ The electron is pumped (excited) into an upper level E4 by some mechanism (for example, a collision with another atom or absorption of high-energy radiation). It then decays to E3, then to E2, and finally to the ground state E1. Let us assume that the time it takes to decay from E4 to E3 is much longer than the time it takes to decay from E2 to E1. In a large population of such atoms, at equilibrium and with a continuous pumping process, a population inversion will occur between the E3 and E2 energy states, and a photon entering the population will be amplified coherently. Although with a population inversion we have the ability to amplify a signal via stimulated emission, the overall single-pass gain is quite small, and most of the excited atoms in the population emit spontaneously and do not contribute to the overall output. We use semi-classic approach: electromagnetic radiation which a source of disturbance will be decribed by classics theory thus the atom with the aid of quantum mechanics Solving Schröding’s equation we can find expression for Einstein’s coefficients Within the spectral lines of light emitted by any body the light intensity changes you can therefore speak of the power spectral density. The primary cause of the spectral lines of finite thickness is the Heisenberg uncertainty principle: E Such broadening of the line is always present and is called the natural broadening Doppler Broadening also affects the width of the spectral line for a group of light-emitting particles Broadening of the spectral lines can also occur as a result of particle collisions Kryterium pozwalające stwierdzić czy dana linia emisyjna jest jednorodnie czy niejednorodnie poszerzona: jeśli ma miejsce zróżnicowanie warunków emisji dla różnych cząstek to dają one różne wkłady do całkowitego poszerzenia linii widmowej i linia emisyjna jest niejednorodna, w przeciwnym wypadku uważamy linię za jednorodną Skończona szerokość linii (nawet tylko naturalnie poszerzonej) zmniejsza prawdopodobieństwo emisji wymuszonej. To turn this system into a laser, we need a positive feedback mechanism that will cause the majority of the atoms in the population to contribute to the coherent output. This is the resonator, a system of mirrors that reflects undesirable (off-axis) photons out of the system and reflects the desirable (on-axis) photons back into the excited population where they can continue to be amplified. The lasing medium is pumped continuously to create a population inversion at the lasing wavelength. As the excited atoms start to decay, they emit photons spontaneously in all directions. Some of the photons travel along the axis of the lasing medium, but most of the photons are directed out the sides. The lasing medium is pumped continuously to create a population inversion at the lasing wavelength. As the excited atoms start to decay, they emit photons spontaneously in all directions. Some of the photons travel along the axis of the lasing medium, but most of the photons are directed out the sides. In the simplest case consists of two flat resonator, fully reflecting mirrors, which are arranged perfectly parallel to each other at a distance L = n λ/2 , where (n is an integer, λ is laser light wavelength), it allows the creation of standing waves inside resonator. Warunek rezonansu spełniać również mogą fale rozchodzące się pod określonymi kątami Θ w stosunku do osi rezonatora: 2L = n λ cosΘ, Takie zjawisko najłatwiej można zaobserwować w rezonatorach o wklęsłych zwierciadłach np. rezonatorze konfokalnym (współogniskowym). It is therefore possible formation of resonant vibration, which, due to misaligned propagation direction is called the transverse or non-axial mods Laser resonator: Provides positive feedback (amplifier becomes a generator) Force oscillations in the resonant frequencies - standing waves (modes) are formed Interferes with the emission line halfwidth (the better resonator-the narrower line and the better monochromaticity) Interferes with the geometry of the beam Resonator’s quality is defined as the the ratio of the energy contained in the resonator Wst to the energy P lost during one period Finesse determines the quality of the resonator The smaller the total loss of the resonator, the higher the finesse and the narrower is the resonance line 4R F 2 1 R The intensity of transmitted light K. Łukaszewski, Wstęp do elektroniki kwantowej i optyki nieliniowej, skrypt WFTIMS, Łódź 1999. F. Kaczmarek, Wstęp do fizyki laserów, PWN, Warszawa 1979