Laser Physics Lecture Notes PDF

University of Technology Laser and Optoelectronics Engineering Department Laser Physics 2nd Year Lecture 2 Absorption of electromagnetic Radiation We saw that the proces...

University of Technology Laser and Optoelectronics Engineering Department Laser Physics 2nd Year Lecture 2 Absorption of electromagnetic Radiation We saw that the process of photon absorption by the atom is a process of raising the atom (electron) from a lower energy level into a higher energy level (excited state), by an amount of energy which is equivalent to the energy of the absorbed photon. When electromagnetic radiation passes through matter, part of it is transmitted, and part is absorbed by the atoms. The intensity (I) of the transmitted radiation through a thickness (x) of homogeneous material is described by the experimental equation of exponential absorption (Lambert Law): I=I0 exp (-x) I0 = Intensity of incoming radiation. = Absorption coefficient of the material. The thicker the material (bigger x), the lower the intensity after the material (the transmitted beam). It is common to use units of centimeter (10-2 m), to measure the width of the material (x), so the units of the absorption coefficient () are: cm-1 = 1/cm. Example 2.1: Absorption Coefficient (α) Calculate the absorption coefficient (α) of materials which transmit 50% of the intensity of the incident radiation on a 10 mm width, to the other side. Solution to example 2.1: Using the exponential absorption law: α = -1/x * ln (I/I0) = - 1/1 * ln (0.5/1) = 0.69 cm-1 1  University of Technology Laser and Optoelectronics Engineering Department Laser Physics 2nd Year Lecture 2 Spontaneous emission of electromagnetic Radiation One of the basic physical principles (which is the basis of a subject in physics called Thermodynamics) is that:  Every system in nature "prefers" to be in the lowest energy state.  This state is called the Ground state. As an example, we mentioned this principle in the Bohr model of the atom. When energy is applied to a system, the atoms in the material are excited, and raised to a higher energy level. (The terms "excited atoms", "excited states", and "excited electrons" are used here with no distinction).  These electrons will remain in the excited state for a certain period of time, and then will return to lower energy states while emitting energy in the exact amount of the difference between the energy levels (delta E).  If this package of energy is transmitted as electromagnetic energy, it is called photon. The emission of the individual photon is random, being done individually by each excited atom, with no relation to photons emitted by other atoms. When photons are randomly emitted from different atoms at different times, the process is called Spontaneous Emission. Boltzmann Distribution Equation From thermodynamics we know that a collection of atoms, at a temperature T [0K], in thermodynamic equilibrium with its surrounding, is distributed according to Boltzmann equation which determines the relation between the population number of a specific energy level and the temperature: Ni= Constant * exp (-Ei/kT) Ni = Population Number = number of atoms per unit volume at certain energy level Ei. k = Boltzmann constant: k = 1.38*1023 [Joule/0K]. Ei = Energy of level i. We assume that Ei> Ei-1 (ex: E2 > E1). Const = proportionality constant. 2  University of Technology Laser and Optoelectronics Engineering Department Laser Physics 2nd Year Lecture 2 The Boltzmann equation shows the dependence of the population number (Ni) on the energy level (Ei) 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. Relative Population (N2/N1): The relative population (N2/N1) of two energy levels E2 compared to E1 is: N2/N1 = const* exp (-E2/kT)/ const* exp(-E1/kT) N2/N1 = exp (-(E2-E1)/kT). The proportionality constant (const) is canceled by division of the two population numbers. Population at Thermodynamic Equilibrium Figure 2.1 shows the population of each energy level at thermodynamic equilibrium. Figure 2.1: Population Numbers at "Normal Population" 3  University of Technology Laser and Optoelectronics Engineering Department Laser Physics 2nd Year Lecture 2 Question 2.1: The difference between population numbers Prove that the difference in population numbers (N1, N2) between two energy levels E2 and E1 is given by: N1-N2 = N1*(1-exp (-h/kT). = 2 - 1 is the frequency which corresponds to the energy difference between the two levels E2 and E1. In the equation in question 2.1, the second term inside the parenthesis is always less than 1. So, the parenthesis is always less than 1. Thus the very important conclusions: 1. In a thermodynamic equilibrium, the population number of higher energy level is always less than the population number of a lower energy level. 2. The lower the energy difference between the energy levels, the less is the difference between the population numbers of these two levels. Physically, the electrons inside the atom prefer to be at the lowest energy level possible. Example 2.2: Calculate the ratio of the population inversion (N2 / N1) for the two energy levels E2 and E1 when the material is at room temperature (3000K), and the difference between the energy levels is 0.5 eV. What is the wavelength () of a photon which will be emitted in the transition from E2 to E1? Solution to example 2.2: When substituting the numbers in the equation, we get: = 4 * 10-9 This means that at room temperature, for every 1,000,000,000 atoms at the ground level (E1), there are 4 atoms in the excited state (E2). To calculate the wavelength: 4  University of Technology Laser and Optoelectronics Engineering Department Laser Physics 2nd Year Lecture 2 This wavelength is in the Near Infra-Red (NIR) spectrum. Question 2.2: A material is in thermodynamic equilibrium at room temperature (3000K).The wavelength of the photon emitted in the transition between two levels is 0.5 µm (visible radiation). Calculate the ratio of the population numbers for these energy levels. Population Inversion When a photon of the same energy between two levels is incident on the sample, TWO possibilities may realized: 1. It is absorbed by an atom in the lower state; moving upward. 2. Stimulating an atom in the upper state; moving downward. BOTH ARE POSSIBLE. If N2 > N1 2nd possibility stimulated emission). If N1 > N2 Absorption But We saw that in a thermodynamic equilibrium Boltzmann equation shows us that: N1 > N2 > N3 Thus, the population numbers of higher energy levels are smaller than the population numbers of lower ones. This situation is called "Normal Population" (as described in Figure 2.2a). By putting energy into a system of atoms, we can achieve a situation of "Population Inversion". 5  University of Technology Laser and Optoelectronics Engineering Department Laser Physics 2nd Year Lecture 2 In population inversion, at least one of the higher energy levels has more atoms than a lower energy level. An example is described in Figure 2.2b. In this situation there are more atoms (N3) in a higher energy level (E3), than the number of atoms (N2) in a lower energy level (E2). Figure 2.2: "Normal Population" compared to "Population Inversion".  This is one of the necessary conditions for lasing.  The process of raising the number of excited atoms is called "Pumping". Stimulated Emission Atoms stay in an excited level only for a short time (about 10-8 [sec]), and then they return to a lower energy level by spontaneous emission. 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. 6  University of Technology Laser and Optoelectronics Engineering Department Laser Physics 2nd Year Lecture 2 Possible Processes between Photons and Atoms Figure 2.4 summarizes the three possible processes between photons and atoms: absorption, spontaneous emission, and stimulated emission. Photon Absorption: A photon with frequency 12 hits an atom at rest (left), and excites it to higher energy level (E2) while the photon is absorbed. Spontaneous emission of a photon: An atom in an excited state (left) emits a photon with frequency 12 and goes to a lower energy level (E1). Stimulated emission of a photon: A photon with frequency 12 hit an excited atom (left), and cause emission of two photons with frequency 12 while the atom goes to a lower energy level (E1). Now, E2 –E1 = hυ21 >>> kT, Hence the ratio of N2 / N1 is very small, i.e only few atoms are in the upper level. So: To measure kT at room temperature 7  University of Technology Laser and Optoelectronics Engineering Department Laser Physics 2nd Year Lecture 2 (T=300 K), so at kT=hυ, Hence υ = 6x1012 Hz So λ = 50 μm this is I.R But for transition frequency in the visible and N.I.R (optical region): So: N2