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1 Lecture Notes Laser Systems and Applications

2 Syllabus NOE-043 LASER SYSTEMS AND APPLICATIONS L T P Unit Topic Lectures 1 & 2 Introduction: Review of elementary quantum physics, Schrodinger equation, concept of coherence, absorption, spontaneous emission and stimulated emission processes, relation between Einstein s A and B coefficients, population inversion, pumping, gain, optical cavities & 4 Lasers & Laser Systems: Main components of Laser, principle of Laser action, introduction to general lasers and their types, Three & four level Lasers, CW & Pulsed Lasers, atomic, ionic, molecular, excimer, liquid and solid state Lasers and systems, short pulse generation and Measurement Applications: Laser applications in medicine and surgery, materials processing, optical communication, metrology and LIDAR and holography. 7 Text/ Reference Books: 1. K.R. Nambiar, Laser Principles, Types and Application New Age International. 2. S. A. Ahmad, Laser concepts and Applications New Age International. Page i

3 Table of Contents UNIT S.No. Topic Page No. 1. Introduction of Laser: I & II III & IV V A S S I G N M E N T S 1.1 Review of elementary quantum physics Schrodinger Equation Concept of Coherence Absorption Spontaneous Emission Stimulated Emission Relation between Einstein s A and B coefficients Population Inversion Optical Pumping Optical Gain Optical cavities University question from unit-1 & II Lasers & Laser Systems: 2.1 Main components of Laser 2.2 Principle of Laser action 2.3 Introduction to general lasers and their types 2.4 Two level Lasers 2.5 Three level Lasers 2.6 Four level Lasers 2.7 CW & Pulsed Lasers 2.8 Atomic, ionic, molecular laser 2.9 Excimer 2.10 Liquid laser 2.11 Solid state laser 2.12 Short pulse generation and Measurement University question from unit-iii & IV 3. Applications of Laser: 3.1 Laser applications in medicine and surgery 3.2 Materials processing 3.3 Optical Communication 3.4 Metrology 3.5 LIDAR 3.6 Holography 3.7 University question from unit- V 4. Assignments & Tutorials 4.1 Assignment Assignment Assignment Tutorial Tutorial Tutorial Model university Paper Model university Paper Class Test Class Test-2 Page ii

4 Milestones in the Development of Lasers and Their Applications 1900: Planck's law describes the electromagnetic radiation emitted by a black body in thermal equilibrium at a definite temperature. Max Planck proposed that light is emitted in discrete quanta of energy. It was thoroughly challenged in Albert Einstein proposed that light is also propagated and absorbed in quanta. Light quanta are now called photons. 1905: Einstein postulated that light itself consists of localized particles quanta. Einstein's light quanta were nearly universally rejected by all physicists, including Max Planck and Niels Bohr. This idea only became universally accepted in 1919, with Robert Millikan's detailed experiments on the photoelectric effect, and with the measurement of Compton scattering. 1917: A Einstein postulated stimulated emission and laid the foundation for the invention of the laser by re-deriving Planck s law. 1922: Albert Einstein was awarded the 1921 Nobel Prize in Physics, 1924: R Tolman observed that molecules in the upper quantum state may return to the lower quantum state in such a way to reinforce the primary beam by negative absorption. De Broglie, in his 1924 PhD thesis, proposed that just as light has both wave-like and particle-like properties electrons also have wave-like properties. 1926: In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a physical system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. 1927: by the German physicist Werner Heisenberg, it states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa. 1928: R W Landenberg confirmed the existence of stimulated emission and negative absorption through experiments conducted on gases. 1940: V A Fabrikant suggests method for producing population inversion in his PhD thesis and observed that if the number of molecules in the excited state could be made larger than that of molecules in the fundamental state, radiation amplification could occur. 1947: W E Lamb and R C Retherford found apparent stimulated emission in hydrogen spectra. 1950: Alfred Kastler suggests a method of optical pumping for orientation of paramagnetic atoms or nuclei in the ground state. This was an important step on the way to the development of lasers for which Kastler received the 1966 Nobel Prize in Physics. 1951: E M Purcell and R V Pound: In an experiment using nuclear magnetic resonance, Purcell and Pound introduce the concept of negative temperature, to describe the inverted populations of states usually necessary for maser and laser action. 1954: J P Gordon, H J Zeiger and C H Townes demonstrate first MASER operating as a very high resolution microwave spectrometer, a microwave amplifier or a very stable oscillator. Page iii

5 1956: N Bloembergen first proposed a three level solid state MASER 1958: A Schawlow and C H Townes, extend the concept of MASER to the infrared and optical region introducing the concept of the laser. 1959: Gordon Gould introduces the term LASER 1960: T H Maiman realizes the first working laser: Ruby laser 1960: P P Sorokin and M J Stevenson Four level solid state laser (uranium doped calcium fluoride) 1960: A Javan, W Bennet and D Herriott invent the He-Ne laser 1961: E Snitzer: First fiber laser. 1961: P Franken; observes optical second harmonic generation 1962: E Snitzer: First Nd:Glass laser 1962: R. Hall creates the first GaAs semiconductor laser 1962: R W Hellwarth invents Q-switching 1963: Mode locking achieved 1963: Z Alferov and H Kromer: Proposal of heterostructure diode lasers 1964: C K N Patel invents the CO 2 laser 1964: W Bridges: Realizes the first Argon ion laser 1964: Nobel Prize to C H Townes, N G Basov and A M Prochorov 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 1964: J E Geusic, H M Marcos, L G Van Uiteit, B Thomas and L Johnson: First working Nd:YAG laser 1965: CD player 1966: C K Kao and G Hockam proposed using optical fibers for communication. Kao was awarded the Nobel Prize in 2009 for this work. 1966: P Sorokin and J Lankard: First organic dye laser 1966: Nobel Prize to A Kastler for the discovery and development of optical methods for studying Hertzian resonances in atoms 1970: Z Alferov and I Hayashi and M Panish: CW room temperature semiconductor laser 1970: Corning Glass Work scientists prepare the first batch of optical fiber, hundreds of yards long and are able to communicate over it with crystal clear clarity Page iv

6 1971: Nobel Prize: D Gabor for his invention and development of the holographic method 1975: Barcode scanner 1975: Commercial CW semiconductor lasers 1976: Free electron laser 1977: Live fiber optic telephone traffic: General Telephone & Electronics send first live telephone traffic through fiber optics, 6 Mbit/s in Long Beach CA. 1979: Vertical cavity surface emitting laser VCSEL 1981: Nobel Prize to N Bloembergen and A L Schawlow for their contribution to the development of laser spectroscopy 1982: Ti:Sapphire laser 1983: Redefinition of the meter based on the speed of light 1985: Steven Chu, Claude Cohen Tannoudji, and William D. Phillips develop methods to cool and trap atoms with laser light. Their research is helps to study fundamental phenomena and measure important physical quantities with unprecedented precision. They are awarded the Nobel Prize in Physics in : Laser eye surgery 1987: R.J. Mears, L. Reekie, I.M. Jauncey and D.N. Payne: Demonstration of Erbium doped fiber amplifiers 1988: Transatlantic fiber cable 1988: Double clad fiber laser 1994: J Faist, F Capasso, D L. Sivco, C Sirtori, A L. Hutchinson, and A Y. Cho: Invention of quantum cascade lasers 1996: S Nakamura: First GaN laser 1997: Nobel Prize to S Chu, C Cohen Tannoudji and W D Philips for development of methods to cool and trap atoms with laser light 1997: W Ketterle: First demonstration of atom laser 1997: T Hansch proposes an octave-spanning self-referenced universal optical frequency comb synthesizer 1999: J Ranka, R Windeler and A Stentz demonstrate use of internally structured fiber for supercontinuum generation 2000: J Hall, S Cundiff J Ye and T Hansch: Demonstrate optical frequency comb and report first absolute optical frequency measurement Page v

7 2000: Nobel Prize to Z I Alferov and H Kroemer for developing semiconductor heterostructures used in high-speed- and opto-electronics 2001: Nobel Prize to E Cornell, W Ketterle and C E Wieman for the achievement of Bose- Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates 2005: H Rong, R Jones, A Liu, O Cohen, D Hak, A Fang and M Paniccia: First continuous wave Raman silicon laser 2005: Nobel Prize to R J Glauber for his contribution to the quantum theory of optical coherence and to J L Hall and T H Hansch for their contributions to the development of laserbased precision spectroscopy, including the optical frequency comb technique 2009: Nobel Prize to C K Kao for groundbreaking achievements concerning the transmission of light in fibers for optical communication Page vi

8 Chapter Review of elementary Quantum physics Light behaves like particles and in other respects like waves. Matter-particles such as electrons and atoms exhibits wavelike behavior too. Some light sources, including neon lights, give off only certain discrete frequencies of light. Quantum mechanics shows that light, along with all other forms of electromagnetic radiation, comes in discrete units, called photons, and predicts its energies, colors, and spectral intensities. Thermal radiation is electromagnetic radiation emitted from the surface of an object due to the object's temperature. If an object is heated sufficiently, it starts to emit light at the red end of the spectrum it is red hot. Heating it further causes the color to change from red to yellow, white, and blue, as light at shorter wavelengths (higher frequencies) begins to be emitted. It turns out that a perfect emitter is also a perfect absorber. When it is cold, such an object looks perfectly black, because it absorbs all the light that falls on it and emits none. Consequently, an ideal thermal emitter is known as a black body, and the radiation it emits is called black-body radiation. The first model that was able to explain the full spectrum of thermal radiation was put forward by Max Planck in He came up with a mathematical model in which the thermal radiation was in equilibrium with a set of harmonic oscillators. To reproduce the experimental results, he had to assume that each oscillator produced an integer number of units of energy at its single characteristic frequency, rather than being able to emit any arbitrary amount of energy. In other words, the energy of each oscillator was quantized. The quantum of energy for each oscillator, according to Planck, was proportional to the frequency of the oscillator; the constant of proportionality is now known as the Planck constant. The Planck constant, usually written as h, has the value of J s. So, the energy E of an oscillator of frequency f is given by where n= 0,1,2,3 To change the color of such a radiating body, it is necessary to change its temperature. Planck's law explains why: increasing the temperature of a body allows it to emit more energy overall, and means that a larger proportion of the energy is towards the violet end of the spectrum. Planck's law was the first quantum theory in physics, and Planck won the Nobel Prize in 1918 "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta" In 1905, Albert Einstein took an extra step. He suggested that quantization was not just a mathematical trick: the energy in a beam of light occurs in individual packets, which are now called photons. The energy of a single photon is given by its frequency multiplied by Planck's constant: For centuries, scientists had debated between two possible theories of light: was it a wave or did it instead comprise streams of tiny particles? By the 19th century, the debate was generally considered to have been settled in favor of the wave theory, as it was able to explain observed Page 1

9 Introduction of laser effects such as refraction, diffraction and polarization. James Clerk Maxwell had shown that electricity, magnetism and light are all manifestations of the same phenomenon: the electromagnetic field. Maxwell's equations, which are the complete set of laws of classical electromagnetism, describe light as waves: a combination of oscillating electric and magnetic fields. Einstein explained the effect by postulating that a beam of light is a stream of particles ("photons") and that, if the beam is of frequency, and then each photon has energy equal to. An electron is likely to be struck only by a single photon, which imparts at most an energy to the electron. Therefore, the intensity of the beam has no effect and only its frequency determines the maximum energy that can be imparted to the electron. Matter wave All matter can exhibit wave-like behavior. For example a beam of electrons can be diffracted just like a beam of light or a water wave. Matter waves are a central part of the theory of quantum mechanics an example of wave particle duality. The concept that matter behaves like a wave is also referred to as the de Broglie hypothesis due to having been proposed by Louis de Broglie in S.No Matter Waves Matter waves define nature of moving particles (charged or uncharged). Matter waves are not EM waves. all particles have wave property, it is called matter wave and the wavelength of this wave is inversely proportional to the momentum of the particle. Matter wave are neither radiated into space not emitted by particle they are associated with particles. Matter waves have shorter wavelength than EM waves Matter wave travel with different velocity. Matter waves require medium for propagation means they cannot travel in vacuum. Electromagnetic Waves EM waves are produced only by accelerated charged particles. Electromagnetic waves are associated with electric and magnetic fields perpendicular to each other and to the direction of propagation of radiation These waves can be radiated in to space. EM waves have larger wavelength than matter waves. All EM wave travel with speed of light. They do not require any medium for propagation means they can propagate through vacuum. de Broglie waves wavelength Matter waves are often referred to as de Broglie waves. The de Broglie wavelength is the wavelength, λ, associated with a particle and is related to its momentum, p, through the Planck constant, h. Page 2

10 Albert Einstein proposed that light is also propagated and absorbed in quanta. Light quanta are now called photons. These quanta would have energy and a momentum where (lowercase Greek letter nu) and (lowercase Greek letter lambda) denote the frequency and wavelength of the light, c the speed of light, and h Planck s constant. Heisenberg s Uncertainty principle in 1927, Werner Heisenberg stated that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa If and represent the uncertainties in position x and momentum p of the particle respectively then the uncertainty principle state that Thus if we want to locate the position of the particle precisely by making, then nothing will be known about its momentum because becomes infinite. Similarly if and are uncertainty in the measurement of E and t, respectively, then If and is uncertainty in momentum and energy respectively then partially differentiate both side Page 3

11 Note that the uncertainty in the simultaneous measurement of a pair of canonically conjugates variables. 1.2 Schrödinger Wave Equation In 1926, Erwin Schrödinger published an equation describing how a matter wave should evolve the matter wave analogue of Maxwell s equations and used it to derive the energy spectrum of hydrogen. A wave function in quantum mechanics describes the quantum state of a system of one or more particles, and contains all the information about the system considered in isolation. The most common symbols for a wave function are the Greek letters (lower-case and capital psi). The Schrödinger equation determines how the wave function evolves over time, that is, the wave function is the solution of the Schrödinger equation. There are several ways to develop the wave equation by applying quantum concepts to various classical equations of mechanics. One of the simplest approaches is to consider a few basic postulates to develop the wave equation. Basic Postulates 1. Each particle in a physical system is described by a wave function. This function and its space derivative are continuous, finite, and single valued. 2. In dealing with classical quantities such as energy E and momentum p, we must relate these quantities with abstract quantum mechanical operators defined in the following way: Classical Variables Quantum Variables and similarly for the other two directions. 3. The probability of finding a particle with wave function in the volume is. The product is normalized The classical equation for the energy of a particle can be written: Kinetic energy + potential energy = total energy Page 4

12 Let be represented by the product. Using this product in above equation Now the variables can be separated to obtain the time-dependent equation in one dimension. Where E is total energy of the particle. This equation gives time dependent function and the time-independent equation We can show that the separation constant E corresponds to the total energy of the particle when particular solutions are obtained, such that is time independent part of wave function and time dependent part of wave function. Finally wave function can be written as Where A is a constant of integration. It is often more convenient to solve the time dependent equation first and the time dependence, when needed. 1.3 Optical Coherence Coherence is a measure of the correlation between the phases measured at different points on a wave. There are two types of optical coherence first one is Spatial coherence correlates the Page 5

13 phases at different points in space at a single moment in time, whereas temporal coherence correlates the phases at a single point in space over a period of time. Temporal coherence Temporal coherence is a measure of the correlation of light wave s phase at different points along the direction, it tell us how monochromatic source is. A light beam in which phase differences between points on the wave-front remain constant in time is said to be temporally coherent. Given the Fourier-transform relationship between time and frequency, this also implies that a temporally coherent beam has a high degree of spectral purity. Fig. 1 A Michelson interferometer setup In order to understand the concept of temporal coherence, we consider a Michelson interferometer arrangement as shown in Fig. 1. S represents an extended near monochromatic source, G represents a beam splitter, and M 1 and M 2 are two plane mirrors. The mirror M 2 is fixed while the mirror M 1 can be moved either toward or away from G. Light from the source S is incident on G and is divided into two equal portions; one part travels toward M 1 and is reflected back and the other part is reflected back from M 2. The two reflected waves interfere and produce interference fringes which are visible from E. When the mirrors M 1 and M 2 are nearly equidistant from G, then it is observed that the contrast of the interference fringes formed is good because two waves traversing the two different and paths take the same amount of time. If now the mirror M 1 is slowly moved away from G, then it is seen that for ordinary extended source of light (like a sodium lamp), the contrast in the fringes goes on decreasing and when the difference between the distances from G to M 1 and M 2 is about a few millimeters to a few centimeters, the fringes are no longer visible. This decrease in contrast of the fringes can be explained as follows. The source S is emitting small wave trains of an average duration (say) and there is no phase relationship between different wave trains this is in contrast to an infinitely long pure sinusoidal wave train, which is also referred to as a monochromatic wave. When the difference in time taken by the wave trains to travel the paths G to M 1 and back and G to M 2 and back is much less than the average duration then the interference is produced between two wave trains each one being derived from the same wave train. Hence even though different wave trains Page 6

14 emanating from the source S do not have definite phase relationship, since one is superimposing two wave trains derived from the same wave train, fringes of good contrast will be seen. On the other hand, if the difference in the time taken to traverse the paths to M 1 and back and to M 2 and back is much more than then one is superimposing two wave trains which are derived from two different wave trains, and since there is no definite phase relationship between two wave trains emanating from S, interference fringes will not be observed. Hence as the mirror M 1 is moved, the contrast in the fringes becomes poorer and poorer and for large separations no fringes would be seen. The time is referred to as the coherence time ( ) and the length of the wave train is referred to as the longitudinal coherence length ( ). It may be mentioned that there is no definite distance at which the interference pattern disappears; as the distance increases, the contrast in the fringes becomes gradually poorer and eventually the fringes disappear. As an example, for the neon nm line from a discharge lamp, the interference fringes would vanish if the path difference between the two mirrors is about a few centimeters. Thus for this source, ~100 ps. On the other hand, for the red cadmium line at nm, the coherence length is about 30 cm, which gives ~ 1 ns. The decrease in contrast of the fringes can also be interpreted as being due to the fact that the source S is not emitting a single frequency but emits over a band of frequencies. When the path difference is zero or very small, the different wavelength components produce fringe patterns superimposed on one another and the fringe contrast is good. On the other hand, when the path difference is increased, different wavelength components produce fringe patterns which are slightly displaced with respect to one another and the fringe contract becomes poor. Thus the non-monochromaticity of the light source can equally well be interpreted as the reason for poor fringe visibility for large optical path differences. If a light source emits a beam with frequencies ranging from to, hence with a spectral bandwidth of, the extreme frequencies in the beam will lose temporal synchronization in a comparatively short time, called the coherence time, given by The coherence length associated with this coherence time is simply The coherence volume for a source with given coherence properties is then The physical process giving rise to light emission largely determines the coherence of a source. In a thermal source, such as a gas-discharge lamp or an incandescent bulb, light is produced by microscopic or even atomic sources emitting spontaneously, hence at random times relative to one another. Wave packets from different emission events are essentially uncorrelated, and the degree of temporal and spatial coherence is low, though not zero. In a laser source, on the other hand, the light is produced by stimulated emission and the degree of Page 7

15 coherence is high; however, since temporally and spatially independent transverse and longitudinal modes can coexist simultaneously, only single-mode lasers can achieve the highest degree of temporal coherence. In contrast to, for an ordinary source of light, for a well-controlled laser one can obtain, which gives ~ 2 ms. The corresponding coherence length is about 600 km. Such long coherence lengths imply that the laser could be used for performing interference experiments with very large path differences. Spatial coherence Spatial coherence is a measure of the correlation of light wave s phase at different points transverse to the direction of propagation, it tell us how uniform the phase of wavefront. A beam of light is said to be spatially coherent when the phase difference between points on the wavefront remains constant in time, even if the phase fluctuates randomly at any given point. Thus an extended source comprising an ensemble of randomly fluctuating point sources can produce spatially coherent light if the interference fringes from nearby point sources accidentally overlap. Fig: 2 Detection of spatially coherent and monochromatic wavelength from Non coherent light source In order to understand the concept of spatial coherence, we consider the Young s doublehole experiment as shown in Fig: 3. S represents a source placed in front of a screen with two holes S 1 and S 2 and the interference pattern between the waves emanating from S 1 and S 2 is observed on screen T. We restrict ourselves to the region near O for which the optical path lengths S 1 O and S 2 O are equal. If S represents a point source then it illuminates the pinholes S 1 and S 2 with spherical waves. Since the holes S 1 and S 2 are being illuminated coherently, the interference fringes formed near O will be of good contrast. Consider now another point source placed near S and assume that the waves from S and have no phase relationship. In such a case the interference pattern observed on the screen T will be a superposition of the intensity distributions of the interference patterns formed due to S and. If is moved slowly away from S, the contrast in the interference pattern on T will become poorer because of the fact that the Page 8

16 interference pattern produced by is slightly shifted in relation to that produced by S. For a particular separation, the interference maximum produced by S falls on the interference minimum produced by and the minimum produced by S falls on the maximum produced by. For such a position the interference fringe pattern on the screen T is washed away. Fig. 3 Young s double-slit experimental arrangement In order to obtain an approximate expression for the separation for disappearance of fringes, we assume that S and O are equidistant from S 1 and S 2. If the position of is such that the path difference between and is λ/2 (where λ is the wavelength of light used), then the source produces an interference minimum at O. If we assume, S 1 S 2 = d, and the distance between S and the plane of the pinholes is D, we obtain where we have assumed that Thus for disappearance of fringes, Or For an extended source made up of independent point sources, one may say that good interference fringes will be observed as long as Equivalently for a given source of width, interference fringes of good contrast will be formed by interference of light from two point sources S 1 and S 2 separated by a distance d as long as Page 9

17 Since is the angle (say θ) subtended by the source at the slits above equation can also be written as The distance is referred to as the lateral coherence width. It can be seen from above Equation that depends inversely on θ. 1.4 Absorption An atom in the lower energy level E 1 can absorb the incident radiation at a frequency ω = (E 2 E1) / and be excited to E 2 ; this excitation process requires the presence of radiation. The rate at which absorption takes place from level 1 to level 2 will be proportional to the number of atoms present in the level E1 and also to the energy density of the radiation at the frequency ω = (E 2 E 1 ) /. Thus if u(ω)dω represents the radiation energy per unit volume between ω and ω + dω then we may write the number of atoms undergoing absorptions per unit time per unit volume from level 1 to level 2 as where B 12 is a constant of proportionality and it is also referred as the Einstein B coefficient. u(ω) has the units of energy density per frequency interval. 1.5 Spontaneous Emission If a light source is in the excited state with energy E 2 it may spontaneously decay to a ground state with energy E 1, releasing the difference in energy between the two states as a photon. The photon will have angular frequency and energy (, where is the Planck constant and is the frequency). Light or luminescence from an atom is a fundamental process that plays an essential role in many phenomena in nature and forms the basis of many applications, such as fluorescent tubes, older television screens (cathode ray tubes), plasma display panels, lasers, and light emitting diodes. Where is the reduced Planck constant. The phase of the photon in spontaneous emission is random as is the direction in which the photon propagates. This is not true for stimulated. An energy level diagram illustrating the process of spontaneous emission is shown below: Rate at which electrons fall from an upper level of energy E 2 to a lower level of energy E 1 is at every instant proportional to the number of electrons remaining in E 2 (the population of E 2 ). Page 10

18 Where is proportionality factor for spontaneous Emission. The constant is referred to as the Einstein A coefficient and has units (. Fig: 4 Spontaneous Emission We shall now obtain the relationship between the Einstein A coefficient and the spontaneous lifetime of level 2. Let us assume that an atom in level 2 can make a spontaneous transition only to level 1. Then since the number of atoms making spontaneous transitions per unit time per unit volume is A 21 N 2, we may write the rate of change of population of level 2 with time due to spontaneous emission as the solution of which is Thus the population of level 2 reduces by 1/e in a time t sp = 1/A 21 which is called the spontaneous lifetime associated with the transition 2 1. If the number of light sources in the excited state at time t is given by N(t) the rate at which N decays is The above equation can be solved to give Where the initial number of light sources is in the excited state, t is the time and is the radiative decay rate of the transition. The number of excited states N thus decays exponentially with time, similar to radioactive decay. After one lifetime, the number of excited states decays to 36.8% ( ) of its original value. Page 11

19 1.6 Stimulated Emission Stimulated emission is the process by which an atomic electron (or an excited molecular state) interacting with an electromagnetic wave of a certain frequency may drop to a lower energy level, transferring its energy to that field. A new photon created in this manner has the same phase, frequency, polarization, and direction of travel as the photons of the incident wave. Fig: 5 Stimulated Emission If this process continues and other electrons are stimulated to emit photons in the same fashion, a large radiation field can build up. This radiation will be monochromatic (each photons have same energy), coherent (each photons have same phase). The rate of stimulated emission is proportional to the instantaneous number of electrons in the upper level ( ) and the energy density of the stimulating field. Where proportionality factor for stimulated Emission. The constant is referred to as the Einstein B coefficient. We notice that no energy density is required a transition from upper energy level to lower energy level. 1.7 Relation between Einstein s A and B coefficients Boltzmann distribution In statistical mechanics, a Boltzmann distribution is a probability distribution of particles in a system over various possible states. The distribution is expressed in the form Page 12

20 Where E is state energy (which varies from state to state), k B T (a constant of the distribution) is the product of Boltzmann's constant and thermodynamic temperature. The ratio of a Boltzmann distribution computed for two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference. Let N 1 and N 2 be the number of atoms per unit volume present in the energy levels E 1 and E 2, respectively, then we can write At thermal equilibrium between the atomic system and the radiation field, the number of upward transitions must be equal to the number of downward transitions. Hence, at thermal equilibrium or Using Boltzmann s law, the ratio of the equilibrium populations of levels 1 and 2 at temperature T is where (k B = J/K) is the Boltzmann s constant. Hence Now according to Planck s law, the radiation energy density per unit frequency interval is given by where c is the velocity of light in free space and n 0 is the refractive index of the medium. Comparing above both Equations, we obtain B 12 = B 21 = B and Thus the stimulated emission rate per atom (B 21 ) is the same as the absorption rate per atom (B 12 ) and the ratio of spontaneous ( ) to stimulated emission ( ) coefficients is given. The coefficients A and B are referred to as the Einstein A and B coefficients. Page 13

21 At thermal equilibrium, the ratio of the number of spontaneous to stimulated emissions is given by Thus at thermal equilibrium at a temperature T, for frequencies, of spontaneous emissions far exceeds the number of stimulated emissions., the number At equilibrium ratio of the stimulated to spontaneous emission rates is generally very small, and the contribution of stimulated emission is negligible. With a photon field present, ω ω Above equation indicates, the way to enhance the stimulated emission over spontaneous emission we have a very large photon energy density ω. In the laser, this is encouraged by providing an optical resonant cavity in which the photon density can build up to a large value through multiple internal reflections at certain frequencies ( ). Similarly, When we take ratio of the stimulated emission rate to absorption rate, we observe If stimulated emission is to dominate over absorption of photons from the radiation field, we must have more electrons populated in upper energy level than lower energy level. This condition is known as population inversion. It is also called negative temperature. Actually such phenomena do not occur in lesser. This is just a terminology emphasizes the nonequilibrium nature of population inversion. In the above equation ω ω ; ω are positive quantities, If we want to obtain, temperature must have negative value (from above equation) but in actual if we want dominance of stimulated emission over both spontaneous emission and absorption, two requirements must be met (1) To obtain population inversion condition ( ) (2) An optical resonant cavity to encourage the photon energy density. 1.8 Population Inversion Population inversion is a necessary condition for stimulated emission. Under normal circumstances, there is always a larger number of atoms in the lower energy state as compared Page 14

22 to the excited energy state, and an electromagnetic wave passing through such a collection of atoms would get attenuated rather than amplified. Thus, in order to have amplification, one must have population inversion. For a material in thermal equilibrium condition, the distribution of electrons in various energy states is given by the Boltzmann distribution law. Fig: 6a Fig: 6b Fig. 6a Relative populations in two energy levels as given by the Boltzmann relation for thermal equilibrium Fig. 6b Inverted population difference required for optical amplification According to the Boltzmann law, the higher energy states are least populated and population of electrons in higher energy states decreases exponentially with energy shown in figure 6a. Population inversion corresponds to non equilibrium distribution of electrons such that higher energy states have a larger numbers of electrons than the lower energy states shown in figure 6b. The process of achieving the population inversion by exciting the electrons to the higher energy states is known as pumping. Population inversion for the two level energy system is not possible only possible for the three and four level energy systems. For a three level energy system, electrons are first pumped from energy level E 1 to E 3 by the absorption of radiation of frequency from a pumping source. The life time of the electrons in the higher energy level E 3 is generally very short and electrons from the energy level E 3 rapidly decay into metastable energy level E 2 without any radiation. The net population inversion is achieved between the E 2 (metastable energy state) and E 1, which is responsible for emission of laser radiation. Page 15

23 1.9 Optical Pumping In general, population inversion is achieved by optical pumping and electrical pumping. In optical pumping gas filled flash lamp are most popular. Flash lamps are essentially glass or quartz tube filled with gases such as xenon and krypton. Some wavelength of flash (emission spectrum of flash lamp) matches with the absorption characteristics of the active laser medium facilitating population inversion. This is used in solid state laser like Ruby Laser and Nd:YAG (Yttrium- Aluminum-Garnet). Typical circuit of flash lamp operation is shown in Figure- Electrical pumping is used in Gas laser, is achieved by passing a high voltage electric current directly through the mixture of active gas medium. The collision of discharge electrons of sufficient kinetic energy excites one of the gases to high energy levels, which subsequent transfer its excitation energy to second gas through collision, achieving the population inversion Optical Gain The optical gain: Where is the photon flux (the number of photons per cross section area unit in the unit of time) and z is the direction of the electromagnetic field propagation, and: The optical gain experienced by an incoming photon is very much dependent on the photon s energy. When, is positive and incoming light wave with photon energy will be amplified by the material. The requirement for gain at a photon energy is: The quasi-fermi level separation must be greater than the band-gap to achieve optical gain in the material. Under equilibrium conditions, and optical gain is impossible to achieve Optical Cavities or Optical Resonator An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric oscillators and some interferometers. Light confined in the cavity reflect multiple times producing standing waves for certain resonance frequencies. The standing wave patterns produced are called modes; longitudinal modes differ only in frequency while transverse modes differ for different frequencies and have different intensity patterns across the cross section of the beam. Page 16

24 Light confined in a resonator will reflect multiple times from the mirrors, and due to the effects of interference, only certain patterns and frequencies of radiation will be sustained by the resonator, with the others being suppressed by destructive interference. In general, radiation patterns which are reproduced on every round-trip of the light through the resonator are the most stable, and these are the Eigen modes, known as the modes, of the resonator. Resonator modes can be divided into two types: longitudinal modes, which differ in frequency from each other; and transverse modes, which may differ in both frequency and the intensity pattern of the light. The basic or fundamental transverse mode of a resonator is a Gaussian beam. Fig: 7 Types of two-mirror optical cavities, with mirrors of various curvatures, showing the radiation pattern inside each cavity. Optical Resonant Cavity An optical resonant cavity can be obtained using reflecting mirrors to reflect the photons back and forth, allowing the photon energy density to build up and get amplified by the amplifying medium and also suffers losses due to the finite reflectivity of the mirrors and other scattering and diffraction losses. One or both of the end mirrors are constructed to be partially transmitting so that a fraction of the light will "leak out" of the resonant system. This transmitted light is the output of the laser. The gain in photons per pass between the end plates must be larger than the transmission at the ends, scattering from impurities, absorption, and other losses. Page 17

25 If the oscillations have to be sustained in the cavity then the losses must be exactly compensated by the gain. Thus a minimum population inversion density is required to overcome the losses and this is called the threshold population inversion. Fig. 8 A typical optical resonator consisting of a pair of mirrors facing each other. The active medium is placed inside the cavity In order to obtain an expression for the threshold population inversion, let d represent the length of the resonator and let R1 and R2 represent the reflectivity s of the mirrors. Let 1 represent the average loss per unit length due to all loss mechanisms (other than the finite reflectivity) such as scattering loss and diffraction loss due to finite mirror sizes. Let us consider a radiation with intensity I 0 leaving mirror M1. As it propagates through the medium and reaches the second mirror, it is amplified by and also suffers a loss of ; for an amplifying medium is positive and. The intensity of the reflected beam at the second mirror will be. A second passage through the resonator and a reflection at the first mirror leads to an intensity for the radiation after one complete round trip of oscillation to begin. Hence for laser the equality sign giving the threshold value for (i.e., for population inversion). Indeed, when the laser is oscillating in a steady state with a continuous wave oscillation, then the equality sign in above equation must be satisfied. If the inversion is increased then the LHS becomes greater than unity; this implies that the round trip gain is greater than the round trip loss. Page 18

26 1.12 University question from Unit-I & II Theory questions from Elementary quantum physics 1. What is the concept of matter waves? Derive an expression for de-broglie wavelength. [UPTU ] 2. State and explain Heisenberg s uncertainty principle. An electron has a speed of 10.5km/sec with an accuracy of 0.01%. Calculate the uncertainty in measurement of position of the electron. [UPTU ] 3. Describe the De-Broglie theory of matter/wave. How it is experimentally verified? [UPTU ] 4. Discuss in brief the basic idea of elementary quantum theory and some observed experimental phenomena which could not be understood on this basis. [UPTU ] Theory questions from Schrodinger equation 5. What is wave function? Write the conditions fulfilled by it. Derive the time independent Schrodinger s wave equation. [UPTU ] 6. Derive the time independent and time dependent Schrodinger equation for a non relativistic particle. [UPTU ] 7. Establish time dependent Schrödinger wave equation and then convert it into time independent wave equation. What happened if the particle is free? [UPTU ] 8. Explain the physical significance of the wave function. Derive Schrodinger s time dependent wave equation starting from Schrodinger s time independent wave equation. [UPTU ] Theory questions from concept of coherence 9. What is the concept of coherence in laser? Explain the temporal and spatial coherence. [UPTU ] 10. Illustrate about the following: (i) temporal coherence (ii) spatial coherence. [UPTU ] 11. Obtain the relationship between the size of the source and the coherence of the field. [UPTU ] 12. What do you mean by coherence? Explain temporal coherence. How is temporal coherence related with coherence length? [UPTU ] 13. What do you mean by coherence? Explain different types of coherence with the help of suitable diagrams. [UPTU ] Numerical questions from Concept of coherence 14. Calculate the coherence length of a laser beam for which the bandwidth equal to 3000Hz. The speed of light is m/s. [UPTU ] Coherence Time = Page 19

27 Introduction of laser Coherence Length = 15. The coherence length of a light source is m and its wavelength is 5500AO, calculate: (i) Frequency (ii) Coherence Time. [UPTU ] Coherence Length = Theory questions from absorption, spontaneous and stimulated emission 16. Explain spontaneous emission and stimulated emission of radiation. Obtain a relation between transition problems of spontaneous and stimulated emission. [UPTU ] 17. Explain the phenomenon of induced emission indicating the features which differentiate it from spontaneous emission. [UPTU ] Theory question from Einstein s constant 18. Establish relation between Einstein s coefficients. [UPTU ] 19. Write the significance of Einstein s coefficients and explain the relation between Einstein s A and B coefficients. [UPTU ] 20. What are Einstein Coefficients? [UPTU ] 21. Derive an expression for Einstein s coefficient of stimulated and spontaneous emission. [UPTU ] 22. What are the Einstein s coefficients? Discuss in details. [UPTU ] Theory question from Population Inversion, Pumping, Gain 23. What is population inversion? Describe two methods to achieve it. [UPTU ] 24. Discuss the different types of pumping used in laser. What is the advantage of using lasers at the place of flash lamp in optical pumping? What type of pumping is suitable for HF, HCl lasers? [UPTU ] 25. What do you understand by laser gain? Derive an expression for the loop gain. [UPTU ] 26. What is gain in Lasers? Obtain the condition for threshold gain. [UPTU ] 27. What is pumping? How can it help in achieving population inversion? Differentiate between optical pumping and electrical pumping scheme. [UPTU ] 28. What do you understand by the term light amplification and how is it achieved in the case of laser. [UPTU ] 29. How many methods do you know for optical pumping? Discuss various methods citing examples. [UPTU ] Page 20

28 Numerical question from Population Inversion 30. The ratio of population of two energy levels (N 2 /N 1 ) is half. Find the wavelength of light emitted at 300K. [UPTU ] 31. The ratio population of two energy levels out of which upper one corresponds to a metastable state is Find the wavelength of light emitted at a temperature T= 330K. [UPTU ] 32. Calculate the population ratio of two states in He-Ne Laser that produces light of wavelength 6000A O an 300K. [UPTU ] Hint: Theory question from Optical Resonant Cavity 33. What is an optical cavity? Describe various types of optical cavities. [UPTU ] 34. Define Q-factor of an optical resonator. Show that Q =, where V o resonant frequency and A V full width at half maximum. [UPTU ] 35. What are optical cavities? Discuss briefly the different configurations of optical cavities. [UPTU ] 36. Derive the expression for the resonance frequency in a resonator cavity with two mirrors of radius r 1 and r 2 separated by a distance d. [UPTU ] 37. What do you mean by modes in a resonator? Derive an expression for frequency difference between successive modes for a cavity of length L. [UPTU ] 38. What are optical cavities? How they are useful in laser action. [UPTU ] 39. Explain (i) Active mode locking (ii) Passive mode locking [UPTU ] 40. Discuss the working of resonators of laser systems. How many types of such resonators you know? Explain the working of one resonator in detail? [UPTU ] Page 21

29 1. Elementary quantum physics Study Materials Wikipedia Video lecture 2. Schrodinger equation Wikipedia Video lecture Animation 3. Concept of coherence Wikipedia Video lecture Animation 4. Absorption, Spontaneous and Stimulated emission Wikipedia Video lecture Animation Page 22

30 5. Einstein s constant Wikipedia Video lecture Population Inversion, Pumping, Gain Wikipedia Video lecture Animation 7. Optical Resonant Cavity Wikipedia Video lecture Reference Book: 1. K. Thyagarajan, Ajoy Ghatak Lasers- Fundamentals and Applications Second Edition Springer Publication 2. Ben g. Streetman & Sanjay Banerjee Solid State Electronic Devices Sixth edition PHI Page 23