C 4646 Optics for ngineers Lecture 7 9 March, 00
Spontaneous mission Rate BFOR MISSION DURING MISSION AFTR MISSION electron hν hν The rate of spontaneous emission r sp can be written as: A f r sp A f[ ] f is the probability that the transition will occur. is the probability that the probability that the state contains an electron [ - f ] is the probability that the probability that the state is empty f and f are given by: f ( FP ) kt + e and f ( FN ) kt + e
Absorption Rate BFOR ABSORPTION DURING ABSORPTION AFTR ABSORPTION electron hν electron hν The rate of spontaneous emission r ab can be written as: r ab B f [ f] P( ) B ( ) P is the probability that the transition will occur is the density of photons of energy hν 3
Stimulated mission Rate BFOR MISSION DURING MISSION AFTR MISSION electron hν hν electron The rate of spontaneous emission r st can be written as: r st B f [ f] P( ) B P( ) is the probability that the transition will occur is the density of photons of energy 4
RadiativeTransition Rates A useful quantity for obtaining the relationships between the radiative transition rates is the number of photons per unit volume and unit energy To obtain it we need to determine: () Photon density of states (number of allowed solutions to the Maxwell s quations at a given energy) () Occupation probability (Bose-instein Statistics) At low temperatures (low energies) infinite number of particles can occupy a single quantum level (Bose-instein Condensation) lectromagnetic Radiation Quantum Particles (Photons) with energies r hν hc k We need to look at how individual levels are filled. 5
Derivation of Photon Density of States () Solving the wave equation (Maxwell s quations) with Born-Von Karman boundary conditions The BCs dictate that the allowed values of k x, k y, and k z, are discrete and are given by: The allowed states can be visualized as equally spaced points forming a 3D grid in k-space Note that there is only one grid point per every unit volume in the grid given by: Photon State 6
Derivation of Photonic Density of States () How many states N() do we have with energy less than? r The photon energy is related to k by: hν hc k Solving for k we get: k max hc where k max is the radius of the sphere in the k-space We can rewrite the total number of states per unit volume as: Where ρ ph ( ) in the integral is the density of states. Volume of a sphere in the k-space N( ) 4 4 3 πkmax π 3 V 3 ρ Polarization Factor 3 ( π ) 3 hc ( π ) ( ' ) d' Differentiating both sides of the equation with respect to we get the density of photon states: 3 4π 3 ρ ph( ) 3 π 3 ( hc) π ( hc) 3 3 0 ph 7
Photon State Occupancy Probability Occupancy of a particular photon state depends on temperature. Photons are bosons (not Fermions like electrons) and they obey different counting rules (Bose-instein statistics rather than Fermi-Dirac statistics) Average number of photons in the state near energy is given by: n Now we can write the spectral density at energy in the following way: e hv kt hv P( ) ρ ( ) n ( hc) (# of photons per unit volume per unit energy) ph π 3 e hv / kt Spectral nergy Density 8
Transition Rates: instein s Approach In thermal equilibrium r sp + r st r ab, which can be written as: A f [ f] + B f[ f] P( ) B f[ f] P( ) Solving for the photon density and simplifying we get: P( ) B f A [ f f [ ] B f f ] [ f ] B e A / kt B This must be identical to the solution of the black-body problem at all temperatures : P( ) ( hc) ) B Comparing the two terms we get: A 3 and ( hc) π Probability of spontaneous emission is related to absorption and stimulated emission probabilities. Once we know the spontaneous emission rate, we can determine the remaining two rates. 3 ( e π / kt B B 9
Transition Probabilities Transition probabilities can be computed from the fundamental quantummechanical principles using the time-dependent perturbation theory (Fermi Golden rule). (See e.g. Coldren and Corzine, Diode Lasers and Photonic Integrated Circuits Casey and Panish, Heterostructure Lasers They can be expressed in terms of the fundamental and material constants and momentum matrix elements: and similarly: B πq h ε nm 0 M hωm Oscillator Strength Stimulated mission Probability A 0 3 4πnq M ε m h c τ Spontaneous mission Probability is the electron-hole recombination lifetimeτ (e.g. τ 0.5ns for GaAs). 0
Spontaneous mission Rate The rate of spontaneous emission r sp (ν) is determined by: Conservation of nergy: Conservation of Momentum: (k-selection rule) () optical joint density of states () occupancy probabilities (3) transition probabilities k ~ k Conservation Laws and Photon Interaction: Considering the carrier dispersion relations we get: Which can be solved for k: where: m r h k m m c v + + m v g k hν p p p hν/c h/λ or k k k π/λ h k + m c hν mr ν h ( h ) g g hν k
Optical Joint Density of States The energy levels with which the photon with energy hν interacts are: m r c + ν mc ( h ) mr v + g m v ( hν ) hν We can write the expression for density of states ρ(ν) with which the photon interacts. The density of states in the conduction band ρ c ( ) is related to ρ(ν) by ρ c ( ) d ρ(ν) dν, from which we get: g d hmr ρ ( ν ) ρc ( ) ρc ( ) dν m c ρ(ν) We can thus write the joint DOS as: ( m ) 3 r ρ( ν ) hν πh g for hν > g g hν
Occupancy Probabilities Recall the general form of the Fermi-Dirac distribution function: f ( ) + e We can therefore write the emission and absorption probabilities as: f e f kt F [ f ] + exp[( FN ) / kt] + exp[( FN FP FP ) / kt ] at T > 0 k (MISSION) f c () f v () f a f [ f ] + exp[( ( ) mr where c + hν g and m r + ( hν ) hν c ) / kt] + exp[( and FC and FV are the quasi-fermi levels of the electrons in the conduction band and holes in the valence band, respectively. FP FN m m ) / kt] v g v (ABSORPTION) 3
Transition Probabilities Radiative transition probability density (per unit time) for the spontaneous emission of a photon into any of the available radiation modes in the frequency band ν and ν + dν is given by: P sp ( ν ) dν I( ν ) dν τ Where I(ν ) is the line-shape function centered at the transition frequency ν 0 ( )/h. The rate of spontaneous emission r sp (ν) is then given by: Which simplifies to r sp ( ν ) Iν 0 ( ν ) f ( f) ρ( ν 0 ) dν 0 τ r sp ( ν ) ~ f τ ( ν ) ρ( ν ) when the line-shape function is significantly narrower that the spontaneous I(ν ) emission spectra, i.e. can be treated as a delta function δ (ν ν 0 ) e 4
Spontaneous mission Rate under Weak Injection Consider a non-degenerate semiconductor in under weak injection. Find an expression for the spectral intensity of the direct recombination rate. Assuming c - FC >> kt and FV - v >> kt you will get: r sp ( ν ) ( m ) τ 3 r h exp π FC kt FV g hν g exp hν kt g Linewidths observed experimentally are usually symmetric and broader. r sp (ν) ~.8 kt ~0.5 kt λ.45λ max kt Area R sp Things we ignored: Subtle adjustments to DOS Role of impurities and doping Line-shape broadening due to energy uncertainty g hν 5
Materials for Semiconductor Light mitters Source: www-opto.e-technik.uni-ulm.de/lehre/cs/ 6
LD Applications 7
LDsfor Telecom Applications Consider an InGaAsP LD emitting at λ.55µm. What is it s linewidth? λ.45λ max kt Recall, λ.45x(.55µm) 0.06eV ~ 90nm The radiative lifetime of the LD material (and therefore the approximate rise time) is ~50ns. Haw fast can we transmit information with this device? ~ 0Mbps. Note that the current commercial optical networks can operate at 0Gbps. Would such a diode be suitable for broad-band, high-bit-rate fiber-optic communication? No, much narrower (spectrally) and faster light sources are needed for telecom applications. What we need is a LASR Light Amplification by Stimulated mission of Radiation 8
RadiativeTransition Rates Semiconductors Radiative transition rates in semiconductors can be obtained by considering the spontaneous emission rate r sp ( ν ) ~ f τ The absorption and stimulated emission rates can be written in the following form: r ( ) φ ab λ ν ν a 8πτ ρ( ν ) f ( ν ) and ( ν ) ρ( ν ) where φ ν is the internal flux of photons with energy hν We can now proceed to define the gain and the requirements for lasing in semiconductors. e r ( ) φ st λ ν ν e 8πτ ρ( ν ) f ( ν ) 9
Population Inversion In Semiconductors For light amplification (lasing) we need: r st > r ab B f f ] P( ) > B f [ f ] P( ) [ f [ f] > f[ ] f f > f This is the condition for population inversion which essentially says that the occupancy probability of the higher energy state must be larger than the occupation probability of the lower energy state. e FN )/ kt ( FP )/ kt e + + ( Finally we get the lasing condition: < FN FP ffectively this means that the quasi-fermi levels must be separated by more than the width of the bandgap. 0
Gain and Carrier Injection in Semiconductors Consider a cylinder of unit cross-sectional area and incremental lengths dz. Assume the spectral photon flux density (number of photons crossing the unit area per unit time interval) is directed along the axis of the cylinder. The incremental number of photons per unit area per unit time per unit energy dφ ν (z) is equal to the number of photons gained per unit time per unit volume per unit frequency ( rst ( ν ) rab ( ν ) ) multiplied by the thickness of the cylinder dz d φ ν ( z) [ rst ( ν ) rab ( ν )] dz Which can be rewritten as: Gain Coefficient dφν ( z) λ ρ( v)[ f f] φ ν ( z) dz 8πτ
xample: Absorption/Gain Coefficient of GaAs Casey and Panish, Heterostructure Lasers
Inversion in pn-homojunction N-side P-side C N-side P-side C FN F g V qv f FP V UNDR QUILIBRIUM UNDR FORWARD BIAS Simple band-structure analysis suggests that population inversion required for lasing action cannot be achieved in a non-degenerately doped homojunction. FN - FP < g 3
Inversion in pn-homojunction Degenerate Doping N-side P-side N-side P-side Inversion Region F FN g UNDR QUILIBRIUM Inversion required for lasing action can be achieved in the inversion region where arly semiconductor lasers were based on degenerately doped pn-junctions g FN - FP > g UNDR FORWARD BIAS FP 4
Modern Lasers: pn-heterojunctions N-side P-side N-side P-side C Inversion Region g g V F FN FP UNDR QUILIBRIUM UNDR FORWARD BIAS Inversion can be readily achieved in the junction region where: FN - FP > g Current state-of-the-art semiconductor lasers are based on pn-heterojunctions (e.g. GaAs/AlGaAs and InP/InGaAsP heterostructures) Herbert Kroemer, 000 Nobel Laureate in Physics for developing semiconductor heterostructures used in high-speed- and opto-electronics. 5
Summary:Optical Processes in Gain Media Physical processes in gain media: -Photon density builds stimulating further recombination, which in turn causes additional buildup, etc. -Photons at peak energies dominate and therefore stimulate more transitions gradual spectral narrowing of emission -Superradiance: population inversion is achieved high output powers, narrower spectra ( <.8kT) photons of all phases are multiplied (incoherent output) -Lasing action requires: () Gain larger than losses (losses other than material absorption play a role) () Lasing radiation is coherent (optical feedback is needed) Feedback is provided by optical cavities 6
MIRROR FabryPérotOptical Cavities L MIRROR Allowed wavelengths supported by the cavity must satisfy: L λ q /(n), where q,, and n is the refractive index of the material filling the space between the mirrors Similarly, the allowed frequencies are: ν q qc/(l), where q,, Frequencies of the adjacent modes are separated by : ν F ν q+ - ν q (q+)c/(l) - qc/(l) c/(l) Where ν F is called free spectral range (FSR). ν F ν q-4 ν q-3 ν q- ν q- ν q ν q+ ν q+ ν q+3 ν q+4 ν 7
Optical Cavities with Losses Sources of cavity losses are: () Imperfect reflections of the mirrors. If the mirror reflactances are R and R respectively, the wave intensity attenuation is the product R R () Absorption and scattering in the regions between mirrors Roundtrip attenuation factor is proportional to exp(-α s L), where α s is the loss coefficient for scattering and absorption Combining the two effects we get the total attenuation factor given by: R R exp(-α s L) exp(-α r L) where α r is called the distributed loss coefficient. Solving for α r we get: ν F α α r s + L ln R R ν q-4 ν q-3 ν q- ν q- ν q ν q+ ν q+ ν q+3 ν q+4 ν 8
Optical Cavities for Semiconductor Lasers -Feedback is provided by a pair of mirrors (cleaved semiconductor facets). -Mirror reflectances are determined by the refractive index contrast (dielectric/air interface). -Recall the introductory &M result (Fresnel Relations) for the reflectance: n R n (e.g. if n n(gaas) 3.59 and n n(air) which yields: R~0.3) Useful website for material parameters: http://www.luxpop.com/ + n n α r, g R Losses and gain are present in the cavity Gain coefficient (g) is determined by the material properties and the injection level R L Losses are represented by: α r α s + L ln R R 9
Semiconductor Laser Output Characteristics BLOW THRSHOLD AT THRSHOLD ABOV THRSHOLD Cavity Modes hν F Gain Cavity Loss Gain Gain Light Intensity ~.8 kt0.05mev Light Intensity x000 Light Intensity ~0-9 ev 30
Light-Current and Spectral Characteristics Light Intensity Spontaneous mission Stimulated mission I TH Current Mode Competition 3
Materials for Semiconductor Light mitters () Available Substrates Source: Coldren and Corzine, Diode Lasers and Photonic Integrated Circuits 3
Semiconductor Laser Structures } MB-grown Heterostructure Mirrors Cleaved Facets 33
Cavity Design for Single Mode Operation Distributed Feedback (DFB) and Distributed Bragg Reflector (DBR) Lasers Vertical Cavity Surfase mitting Lasers (VCSLs) 34