Lecture 5- ECE 240a Back to Basics Einstein Coefficients Thermal Equilibrium Mode and Energy Lecture 5- ECE 240a Density A and B coefficients Ver Chap. 7&8 Cross Section and Lineshape Broadening Homogeneous Broadening Inhomogeneous Broadening 1 ECE 240a Lasers - Fall 2019 Lecture 5
E-M Field Matter Interactions Lecture 5- ECE 240a Back to hν 1 Basics hν 2 hν 1 Einstein hν 1 Coefficients Thermal Equilibrium Mode and An incident photon is absorbed. Energy Density A and B coefficients An incident photon stimulates the creation of an identical, second Cross photon. This is stimulated emission. Section and Lineshape Fundamental quantum-level amplification process. Broadening Homogeneous Must have N 2 is greater than N 1 . Broadening Inhomogeneous Broadening Need external energy source that is not in thermal equilibrium with the system - pump. A photon is emitted spontaneously. This is spontaneous emission. No classical counterpart. Emitted photons are regarded as noise. Noise can be amplified by subsequent stimulated emission - amplified spontaneous emission (ASE) . 2 ECE 240a Lasers - Fall 2019 Lecture 5
Thermal Equilibrium Lecture 5- ECE 240a Back to Probability distribution, p ( E m ) , that an atom is in state E m is the Basics Boltzmann distribution Einstein Coefficients � � − E m Thermal p ( E m ) ∝ exp Equilibrium k B T Mode and Energy Density where k B is the Boltzmann constant. ( k = 1.38 × 10 − 23 J/K) and T is A and B coefficients the temp. in Kelvin. Cross Section and Lineshape The ratio N 2 / N 1 then Broadening Homogeneous � � Broadening − E 2 − E 1 N 2 = g 2 (1) Inhomogeneous exp , Broadening N 1 g 1 k B T where g is the degeneracy factor which is number of states at E 2 and E 1 . Probability that the energy state E n is occupied is � hf � �� p ( n ) = ( 1 − exp [ − hf / kT 0 ]) exp − n kT 0 3 ECE 240a Lasers - Fall 2019 Lecture 5
Energy per E-M Mode Lecture 5- ECE 240a Back to Basics Energy in a mode is the energy in a quantized harmonic oscillator Einstein Coefficients discussed in Lecture 2 and Problem Set 2 Thermal Equilibrium E ( n ) = hνn Mode and Energy Density A and B where zero-point energy is not included. coefficients Cross Section and Mean energy � E � is then Lineshape Broadening ∞ � Homogeneous � E � = E ( n ) p ( n ) Broadening Inhomogeneous n = 0 Broadening � hf ∞ � �� � = hν ( 1 − exp [ − hf / kT 0 ]) − n n exp kT 0 n = 0 hf = exp [ hf / ( kT 0 )] − 1 4 ECE 240a Lasers - Fall 2019 Lecture 5
Mode Density and Energy Density Lecture 5- ECE 240a Back to Basics Einstein The number of modes per frequency in 3-D is derived in Section 7.2 of Coefficients Thermal Verdeyen Equilibrium p ( ν ) dν = 8 πn 3 Mode and ν 2 dν Energy Density c 3 A and B coefficients where c is speed of light in vacuum and n is index of refraction. Cross Section and Lineshape Spectral energy density ρ ( ν ) is mode density × number photons/mode × Broadening energy per photon Homogeneous Broadening 8 πν 2 Inhomogeneous 1 Broadening ρ ( ν ) = × × hν e hν / k B T − 1 c 3 ���� � �� � � �� � energy per photon # modes/frequency # photons/mode = energy/frequency 5 ECE 240a Lasers - Fall 2019 Lecture 5
Einstein A and B coefficients Lecture 5- ECE 240a Now we use Verdeyen notation Back to Basics Einstein Write the rate for both stimulated emission and absorption as Coefficients Thermal R 21 = σ ( ν ) I ν Equilibrium = B 21 ρ ( ν ) Mode and hν Energy Density A and B where ρ ( ν ) is the energy density and B 21 is Einstein’s B coefficient. coefficients Cross Section and In thermal equilibrium for a closed system no external influence or pump Lineshape so that Γ 1 = Γ 2 = 0 Broadening dN 2 = − dN 1 Homogeneous Broadening dt dt Inhomogeneous Broadening Then dN 2 = R 2 ( t ) − N 2 − σ ( ν ) I ν ( N 2 − N 1 ) dt τ 2 hν becomes dN 2 = − dN 1 = − + B 21 N 1 ρ ( ν ) − B 21 N 2 ρ ( ν ) A 21 N 2 dt dt � �� � � �� � � �� � spon. emission absorption stim. emission 6 ECE 240a Lasers - Fall 2019 Lecture 5
Relationship Between A 21 and B 21 Lecture 5- ECE 240a Back to Basics Einstein Find ratio N 2 / N 1 in steady-state (equilibrium) Coefficients Thermal Equilibrium N 2 B 21 ρ ( ν ) = Mode and Energy N 1 A 21 + B 21 ρ ( ν ) Density A and B coefficients Equate with Boltzmann distribution Cross Section and � � Lineshape B 21 ρ ( ν ) N 2 = g 2 − hν = exp Broadening A 21 + B 21 ρ ( ν ) N 1 g 1 k B T Homogeneous Broadening Inhomogeneous Broadening Solve for ρ ( ν ) A 21 e − hν / k B T ρ ( ν ) = � � B 12 g 1 ( 1 − e − hν / k B T ) B 21 B 21 g 2 7 ECE 240a Lasers - Fall 2019 Lecture 5
Relationship -Cont. Lecture 5- ECE 240a Let g 1 / g 2 = 1 and B 12 = B 21 so that Back to Basics B 12 g 1 = 1 Einstein B 21 g 2 Coefficients Thermal Equilibrium Now divide through by e − hν / k B T Mode and Energy Density ρ ( ν ) = A 21 1 A and B coefficients ( e hν / k B T − 1 ) B 21 Cross Section and Lineshape Substitute the expression for the energy density Broadening Homogeneous ρ ( ν ) = 8 πν 2 Broadening 1 × e hν / k B T − 1 × hν Inhomogeneous c 3 Broadening Equate expressions and solve 8 πν 2 e hν / k B T − 1 × hν = A 21 1 1 × ( e hν / k B T − 1 ) c 3 B 21 or = 8 πhν 3 n 3 = 8 πhn 3 A 21 B 21 c 3 λ 3 8 ECE 240a Lasers - Fall 2019 Lecture 5
Cross-Section and Lineshape function Lecture 5- ECE 240a Assume for rest of class that g 1 = g 2 and n g = n (group index is same as index) Back to Basics Look at frequency dependence of stimulated emission (absorption) rate Einstein Coefficients R 21 = σ ( ν ) I ν Thermal = B 21 ρ ( ν ) Equilibrium hν Mode and Energy Density where A and B coefficients I ν I ν n g ≈ I ν n ρ ( ν ) = v g = Cross c c Section and Lineshape energy intensity velocity = power/area length/time = energy/time/area = energy = Broadening volume length/time volume Homogeneous Broadening Now use Inhomogeneous Broadening λ 3 B 21 = A 21 8 πh where A 21 = 1 / τ 21 is decay time from upper to lower state. Equate and solve for cross-section � � λ 3 λ 2 hν I ν n σ ( ν ) = A = A 21 × g ( ν ) I ν 8 πhn 3 c 8 πn 2 ���� � �� � lineshape function Cross-section 9 ECE 240a Lasers - Fall 2019 Lecture 5
Atomic Lineshape Function Lecture 5- ECE 240a Back to Now write rate in terms of density matrix formulation Basics Einstein p 2 E 2 β Coefficients 0 = R 21 Thermal h 2 β 2 + ( ω 21 − ω ) 2 6¯ Equilibrium Mode and � � πp 2 E 2 Energy β / π Density 0 = A and B h 2 β 2 + ( ν 21 − ν ) 2 6¯ coefficients � �� � Cross g ( ν ) Section and Lineshape λ 2 I ν Broadening = A 21 hν g ( ν ) Homogeneous 8 πn 2 Broadening � �� � Inhomogeneous cross-section Broadening Thus the Einstein A coefficient A 21 , which governs the rate of spontaneous emission can be derived (at least in principle) from quantum dipole moments p , which depend on the wavefunctions of the material the field interacts with Expression in M&E 10 ECE 240a Lasers - Fall 2019 Lecture 5
Broadening Lecture 5- ECE 240a Back to Basics Gain sites can interact in same manner with different frequencies or each Einstein Coefficients site may have slightly different gain characteristics. Thermal Equilibrium Mode and Gain sites that have the same characteristics are called homogeneous Energy Density A and B coefficients When the gain characteristics of each site are different are called Cross inhomogeneous Section and Lineshape All materials exhibit both types of gain characteristics on different time Broadening Homogeneous scales. Broadening Inhomogeneous Broadening Examples of materials that are mostly homogeneously broadened Semiconductors such as GaAs; isolated atoms or ions in crystalline hosts. Materials that are mostly inhomogeneously broadened Gases (Doppler) Atoms or ions in glass or other noncrystalline materials 11 ECE 240a Lasers - Fall 2019 Lecture 5
Atoms in Crystal vs. atom in glass Lecture 5- ECE 240a Back to Basics Einstein Coefficients Thermal Equilibrium Example: Atom (ion) glass: Mode and Energy Density Each Er ion experiences a different local field because of the A and B inhomogeneous nature of the glass. coefficients Within the subset of Er ions that experience the same local field, the gain Cross Section and is homogeneously broadened. Lineshape However there is an overall distribution of local fields and the total gain Broadening profile is inhomogeneously broadened producing a large gain bandwidth Homogeneous and wavelength dependence. Broadening Inhomogeneous Atom in Crystal Broadening Each atom experiences regular long range order. 12 ECE 240a Lasers - Fall 2019 Lecture 5
Picture Lecture 5- ECE 240a Back to Basics Einstein Coefficients Thermal Equilibrium Mode and Energy Density A and B coefficients Cross Section and Lineshape Broadening Homogeneous Broadening Inhomogeneous Broadening Glass Crystal (short-range order) (regular long-range order) (no long range order) 13 ECE 240a Lasers - Fall 2019 Lecture 5
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