Photon Absorption & Emission

• Photon absorption raises an atom's electron to a higher energy level equivalent to the absorbed photon's energy.

Electromagnetic Radiation

• When electromagnetic radiation passes through matter, it is partially transmitted and partially absorbed by the atoms.
• The intensity (I) of transmitted radiation through a homogeneous material of thickness (x) is given by the Lambert Law: I = I₀ exp(-αx).
• I₀ is the intensity of incoming radiation.
• α is the absorption coefficient of the material.
• Thicker materials result in lower intensity after the material.
• Centimeters (10⁻² m) are commonly used to measure material width (x); the absorption coefficient (α) is measured in cm⁻¹.
• Materials transmitting 50% of incident radiation over 10 mm have an absorption coefficient (α) of 0.69 cm⁻¹.

Spontaneous Emission

• Systems naturally exist in the lowest energy state, known as the Ground state.
• Applying energy excites atoms to higher energy levels.
• Excited electrons return to lower energy states, emitting energy equal to the difference between energy levels (delta E).
• When transmitted as electromagnetic energy, this package of energy is a photon.
• Spontaneous Emission: Random photon emission by excited atoms at different times.

Boltzmann Distribution

• A collection of atoms at temperature T [K] in thermodynamic equilibrium distributes according to the Boltzmann equation.
• Boltzmann Equation: N₁ = Constant * exp(-Ei/kT).
• N₁ represents the population number, or the number of atoms per unit volume.
• k is Boltzmann constant: 1.38*10⁻²³ [Joule/°K].
• E₁ is the energy of level i, assuming E₁ > E(i-1).

Boltzmann Equation

• Boltzmann equation illustrates the dependence of population number (N₁) on energy level (E₁) at temperature T.
• Higher Temperatures: Correspond to higher population numbers.
• Higher Energy Levels: Correspond to lower population numbers.

Relative Population

• Relative population (N₂/N₁) of two energy levels E₂ compared to E₁ is expressed as N₂/N₁ = exp (-(E₂-E₁)/kT).
• This cancels the proportionality constant.

Population at Thermodynamic Equilibrium

• Population decreases as energy levels increase.
• The difference in population numbers (N₁, N₂) between two energy levels E₂ and E₁ is N₁-N₂ = N₁*(1-exp (-hv/kT).
• v = v₂ - v₁ is the frequency.
• In thermodynamic equilibrium, higher energy levels have lower population numbers.
• The lower the energy difference between levels, the smaller the population number difference.
• Electrons prefer to occupy the lowest energy level possible.

Room Temperature - Energy Level

• At room temperature (300°K) with a 0.5 eV energy level difference, the ratio of population inversion (N₂ / N₁) is approximately 4 * 10⁻⁹.
• For every 1,000,000,000 atoms at the ground level (E₁), there are only 4 atoms in the excited state (E₂).
• The wavelength of a photon emitted in transition from E₂ to E₁ is 2.48 µm, which falls in the Near Infra-Red (NIR) spectrum.

Thermodynamic Equilibrium

• Material in thermodynamic equilibrium at room temperature (3000K) emits a photon (0.5 µm wavelength, visible radiation).

Population Inversion

• When a photon of the same energy interacts with two levels, it is either absorbed by an atom in the lower state, raising it, or stimulates an atom in the upper state, causing it to move downward.
• In thermodynamic equilibrium (Boltzmann equation), N₁ > N₂ > N₃.
• Higher energy levels have smaller population numbers, resulting in Normal Population.
• Population Inversion: Achieved by inputting energy to achieve more atoms in higher levels

Population Inversion Condition

• In population inversion, at least one higher energy level has more atoms than a lower energy level.
• Pumping raises the number of excited atoms.
• Atoms remain in an excited level briefly (approximately 10⁻⁸ seconds), and return to a lower energy level by spontaneous emission.
• The high probability of an incoming photon stimulating an excited atom to return to a lower state and emit another photon of light exists when population inversion between two energy levels is present.

Photon and Atom Interaction

• Three possible processes exist between photons and atoms: absorption, spontaneous emission, and stimulated emission.
• Photon Absorption: A photon with frequency v12 excites an atom to a higher energy level (E2).
• Spontaneous Emission: A photon with frequency v12 is emitted by an atom in an excited state.
• Stimulated Emission: A photon with frequency v12 causes the emission of two photons with frequency v12.
• The ratio of N2/N1 is very small because E₂- -E₁ = hv₂₁ >>> kT,.

Laser Action and Emission

• To achieve laser action, POPULATION INVERSION is needed.
• In a two–level system, the number of atoms is constant (Ntotal = N₁ + N₂).
• Absorption and Emission processes occur between E₁ and E₂.
• The rate of depletion from ground (E₁) is proportional to radiative density p(v) and N1.
• dN1/dt = - B12 ρ(υ) N1 , with B12 the Einstein coefficient.
• Β12 ρ(υ ) is the probability per unit frequency that the transition occurs.
• Spontaneous emission decays the upper level population
• dN2/dt = A21 N2 where A21 is the probability of spontaneous emission; it can be expressed as N2(t) = N 2(0) exp (-t/t21).
• E.M radiation stimulates the atom, which creates a radiative field: dN2/dt = - B21 ρ(υ) N2 .

Einstein Coefficient

• B21 is required for lasing action while A21 represents losses.
• Einstein coefficients for stimulated emission and absorption are EQUAL ie The probability of absorption is equivalent to the probability of stimulated emission.
• The ratio of spontaneous to stimulated emission probabilities for a two-level system: R = A21 N2 / B21 ρ(υ) N2
• Substituting, R = exp ( hv/kT) – 1.

Tungsten Lamps

• For Tungsten lamps: spontaneous emission is PREDOMINENT because the emission frequency is 5x10¹⁴ Hz,

Thermal Equilibrium

• All thermal equilibrium systems cause stimulated emission through population inversion.

Explanation of Photon Absorption that raises an atom's electron to a higher energy level. Discussion of electromagnetic radiation, Lambert's Law, and intensity of transmitted radiation. Overview of Spontaneous Emission and excited electrons returning to lower energy levels.