Based on transitions between bands electrons delocalized rather than bound to particular atom

Transcription

1 EE31 Lasers I 1/01/04 #6 slie 1 Review: Semiconuctor Lasers Base on transitions between bans electrons elocalize rather than boun to particular atom transitions between bans Direct electrical pumping high quantum efficiency (10s %) from Yariv Very high gains (10 s 100 s cm -1 ) very small structures (< 1 mm) Light confine to optical waveguie output not simple gaussian beam rapi ivergence outsie laser High irect moulation spees GHz Simple monolithic evice from Vereyen from Colren from Colren from Colren

2 EE31 Lasers I 1/01/04 #6 slie Review: Transparency an Gain from Yariv With no pumping, semiconuctor absorbs As pump rate increases, E fc increases an E fv ecreases When Efc Efv = Eg the semiconuctor just becomes transparent at ω = E g this ensity of carriers calle n trans For further increase in pumping: peak gain increases rapily gain banwith increases transparency E E = E fc fv g f c ( E ) u E ( k ) E ( k ) ω E Fc E f ( E ) v l Fv gain E E > E fc fv g k For GaAs at room temperature: n trans cm 18 1 Finite pumping rate require just to reach transparency (like three level lasers) transparency f c f ( E ) v u ( E ) l ω E Fc E u E l E Fv k Gain increases rapily for n > n trans

3 EE31 Lasers I 1/01/04 #6 slie 3 Toay Further iscussion of absorption an gain in semiconuctors 15.3 Yariv in aition to from #5 Note net problem set ue on We 3 December

4 EE31 Lasers I 1/01/04 #6 slie 4 Optical Pumping Simplest (though not usually most practical) way to pump is optically absorbe photons generate carriers steay state number from balance of generation an recombination Rate equation: I pump E( k) I0e α z ~1µ m call probability of recombination per unit time R rec call rate carriers generate G n = G R t = G β pn = G n/ τ steay state: n= G/ τ what rate require to reach transparency? G = n / τ trans trans ~ cm /5 10 s 6 3 ~10 carriers / cm s simplest moel: bimolecular recombination rate β~ cm 3 /s τ= 1/β p ~ 5 p = cm 3 ω p G I G = α ω pump p R rec what intensity to get transparency? G α I = trans trans ω p what is G? I trans ~ 5kW/ cm k

5 EE31 Lasers I 1/01/04 #6 slie 5 Homojunction Laser 1 Most all practical semiconuctor lasers are electrically pumpe Simplest possible structure is a simple p-n junction no longer use, but illustrates basic points Must have a region with electrons an holes simultaneously present oes not happen in homogeneous semiconuctor (unless optically pumpe) epletion region forms at unbiase p-n junction in thermal equilibrium, so still not goo enough: forwar bias injects minority carriers region near junction has both electrons an holes: EFc Efv > Eg forwar biase unbiase

6 EE31 Lasers I 1/01/04 #6 slie 6 Homojunction Laser What etermines epth of inversion? minority carriers iffuse a length ~ Dτ iffusivity D, lifetime τ for heavily ope GaAs: D n ~ 15 cm /s D p ~ 0.8 cm /s recall bimolecular recombination: τ 1/ β p 10 3 β = 10 cm /s for electrons injecte into p-type material: for p = cm -3 : τ n = 0.5 ns, ~ 1 µm Gain eists over region ~ 1 µm aroun junction problem: material is absorbing outsie this region Crue estimate of current neee to maintain the inversion: balance carrier injection rate with recombination N J A= e A for N = N τ trans ~10 18 cm -3 : J trans ~0kA/cm n high current ensity a problem n J carriers recombining : unit time recombination rate per volume carriers injecte unit time volume of slab: A N A τ n area A volume : J A

7 EE31 Lasers I 1/01/04 #6 slie 7 Heterojunction Laser Several problems with the homojunction laser epth of inversion is not reaily controlle inepenently of other parameters of the system Ntrans high current ensity to reach transparency Jtrans = e τ proportional to, reuce to reuce threshol but uninverte region is highly absorbing increase to reuce losses no goo compromise Clever solution: heterojunction laser narrow bangap active region sanwiche between wier bangap barriers carriers confine by bangap iscontinuity control over by choice of active layer thickness unpumpe regions have wie bangap transparent at laser wavelength refractive ine step confines optical waveguie moe in a controllable fashion confinement of light in waveguie important part of ioe laser esign fortunately, high refractive ine narrow bangap from Yariv

8 EE31 Lasers I 1/01/04 #6 slie 8 Ternary an Quaternary Materials Important Allow continuous tuning of material properties especially ban gap, refractive ine important for carrier an photonfinement Eample: Al Ga 1- As ~lattice matche over 0<<1

9 EE31 Lasers I 1/01/04 #6 slie 9 Dielectric Waveguies Much interesting physics associate with ielectric waveguies see S. Fan EE 35 for etails here get general iea Basic waveguie: Ray picture region of high refractive ine (, core) surroune by lower ine material (, claing) light confine to high ine region by total internal reflections eact solution: solve Mawell s eqns with appropriate bounary conitions picture rays reflecting off ielectric interfaces maimum allowe angle from TIR conition: 1 θma = cos ( ncl / nco) θma < ( 1 ncl / nco ) Fiel picture λ fiel confine to with h iffracts: θd ~ π n h Reasonably confine moe requires θ D < θma λ λ < 1 ncl / nco h> π ncoh π n ( n n ) co co co cl Ray picture Fiel picture h θ θ D

10 EE31 Lasers I 1/01/04 #6 slie 10 Dielectric Waveguies Thickness to which moe can be confine ecreases with increasing refractive ine step n = n λ = 0.85µm GaAs 3.5 h > 0.1µm eact solutions show similar scaling: larger ine step allows tighter confinement Confinement factor GaAs nga Al As = only portion of gain that overlaps with waveguie moe is useful confinement factor quantifies this overlap small h leas to low transparency current one price to pay: very rapi iffraction when light eits en of laser ifficult to collimate, couple to fibers, θ out ~ λ/ πh 0º -- 40º common from Casey an Panish Heterostructure Lasers h > π : Al mole fraction λ n ( n n ) h co co cl

11 EE31 Lasers I 1/01/04 #6 slie 11 Threshol Current an Confinement Factor Alloy compositions like Al Ga 1- As allow ajustment of n an E g Traeoff in simple heterostructure small for low J trans too small leas to loose moe confinement poor overlap with gain leas to optimum for given n - note ecrease in opt with increasing n Leas to use of more complicate structures to allow separate confinement of carriers () an optical fiel (w): ouble heterostructure w E g from Yariv from Vereyen from Vereyen

12 EE31 Lasers I 1/01/04 #6 slie 1 Waveguie Moes: Formal Version How to solve Mawell s equations in meia with spatially varying refractive ine? Mawell s equations become: E+ ε () r k E 0 1 neglects RHS: ( Eiε ε ) OK for weak guiance see aenum i z For no z epenence, can take Eyz (,, ) = ψ ( ye, ) β waveguie moal fiel obeys t / / y ψ + [ ε( yk, ) β ] ψ = 0 t efineβ = kn = + eff k ω c free-space k-vector plane wave in z transverse envelope moal fiel z ε(r) = ε(,y) no z epenence n eff ψ + [ n (, y) n ] k ψ = 0 t eff eigenvalue ψ ( )

13 EE31 Lasers I 1/01/04 #6 slie 13 Comparison to D Schröinger Equation Close analogy between ielectric waveguie an D S eqn in attractive potential intuition from one can inform unerstaning of the other ψ + ε( yk, ) ψ = β ψ t n eff m t ψ (, ) V yψ = ψ V( ) ψ ( ) ψ ( )

14 EE31 Lasers I 1/01/04 #6 slie 14 Step Profile Waveguie ψ + [ n (, y) n ] k ψ = 0 t eff core : ψ + ( n ) 0 co neff k ψ = u u < 0if neff < nco oscillatory solution ire ψ ( ) e ± iu cos( u) n eff claing: ψ + ( n ) 0 cl neff k ψ = w w 0if neff ncl > > ψ ( ) e eponential ecay iaing ± w w e e w Can have a moe that ecays for ± if < n eff < What etermines n eff? must meet bounary conition at core-claing interface leas to istinct eigenvalues associate with eigenmoes continuity of ψ an ψ/ at = ± equivalent to continuity of tangential electric an magnetic fiels suitable for TE moes (polarize along y) See etails in aenum

15 EE31 Lasers I 1/01/04 #6 slie 15 TM vs TE Moes So far have ignore polarization effects implicitly assume a TE (Transverse Electric) fiel TE z electric fiel polarize parallel to interface magnetic fiel fiel mostly normal to interface Transverse Magnetic fiels (TM) magnetic fiel polarize parallel to interface electric fiel mostly normal to surface TM z TM moal fiels quite similar to TE essentially ientical for small - for larger - begin to see ifferences small longituinal E-fiel component appears Differences ue to ifferent bounary conitions TE: tangential E continuous ψ continuous TM: normal D = ε E continuous εψ continuous

16 EE31 Lasers I 1/01/04 #6 slie 16 Aenum: Waveguie Moes

17 EE31 Lasers I 1/01/04 #6 slie 17 Waveguie Moes How to solve Mawell s equations in meia with spatially varying refractive ine? Recall back to PWE erivation in #.4: assume E(r)ep(iωt) time epenence, σ = 0, p a = 0 ω E+ ε () r E= ( E) can t take = 0 if c ε = ε(r) passive meium no loss what to o with RHS? id= 0 i( εe) = 0 ε ie+ Ei ε = 0 ω 1 E+ ε () r E= ( Eiε ε) c k Take refractive ine inep. of z: ε(r) = ε(,y) i z solution can then be taken in the form Eyz (,, ) = ψ ( ye, ) β ψ + [ ε( yk, ) β ] ψ = 0 t efine eigenvalue contains all polarization info usually small for weakly guiing waveguies can rop in step-profile waveguies ( ε = 0) β = kn eff t = / + / y Snyer an Love, Optical Waveguie Theory for etaile iscussion plane wave in z transverse envelope moal fiel z

18 EE31 Lasers I 1/01/04 #6 slie 18 Waveguie Moes How to solve Mawell s equations in meia with spatially varying refractive ine? Mawell s equations become: ε(r) = ε(,y) no z epenence E+ ε r k E () 0 1 neglects RHS: ( Eiε ε ) OK for weak guiance k ω c free-space k-vector z i z For no z epenence, can take Eyz (,, ) = ψ ( ye, ) β waveguie moal fiel obeys t / / y ψ + [ ε( yk, ) β ] ψ = 0 t efineβ = kn = + eff plane wave in z transverse envelope moal fiel n eff ψ + [ n (, y) n ] k ψ = 0 t eff eigenvalue ψ ( )

19 EE31 Lasers I 1/01/04 #6 slie 19 Comparison to D Schröinger Equation Close analogy between ielectric waveguie an D S eqn in attractive potential intuition from one can inform unerstaning of the other ψ + ε( yk, ) ψ = β ψ t n eff m t ψ (, ) V yψ = ψ V( ) ψ ( ) ψ ( )

20 EE31 Lasers I 1/01/04 #6 slie 0 Step Profile Waveguie ψ + [ n (, y) n ] k ψ = 0 t eff core : ψ + ( n ) 0 co neff k ψ = u u < 0if neff < nco oscillatory solution ire ψ ( ) e ± iu cos( u) n eff claing: ψ + ( n ) 0 cl neff k ψ = w w 0if neff ncl > > ψ ( ) e eponential ecay iaing ± w u e e w Can have a moe that ecays for ± if < n eff < What etermines n eff? must meet bounary conition at core-claing interface leas to istinct eigenvalues associate with eigenmoes continuity of ψ an ψ/ at = ± equivalent to continuity of tangential electric an magnetic fiels suitable for TE moes (polarize along y)

21 EE31 Lasers I 1/01/04 #6 slie 1 Step Profile Waveguie Can show that two continuity conitions lea to: W = U tan( U) secon conition erive from efinitions of u an w: ientical to result for QM square well leas to one or more solutions U(V), W(V) U + W = k nco n V number of solutions increases with V given U(V), W(V) have n eff, moal fiels, etc. note that these are functions only of V V is an important quantity in waveguie theory normalize frequency, a measure of strength of waveguie larger V leas to more moes, tighter confinement all properties (appropriately normalize) epen only on V Some etails in Yariv ifferent approach, same results here post some notes from a previous course on waveguies more like approach in Snyer an Love, or Marcuse show some results without intervening algebra w u ( ) cl normalize frequency important quantity in waveguie theory

22 EE31 Lasers I 1/01/04 #6 slie Eigenvalue an Moal Fiel General solutions given waveguie structure, can calculate V = π ( nco ncl ) λ given V, eigenvalue equation yiels one or more solutions U an hence have W = V U given U(V), W(V), have moal fiels cos( U)ep[ W( / 1)] > ψ ( ) = cos( U / ) < < cos( U)ep[ W( / + 1)] < cos( u) n eff W = 0"cutoff" GaAs core Al Ga 1- As cla w e e w Casey an Panish

23 EE31 Lasers I 1/01/04 #6 slie 3 Effective Ine Given V an U(V), can fin n eff normalize form convenient: b n n n eff cl co ncl bv ( ) = 1 ( U/ V) For small V, n eff b 0 For large V, n eff b 1 n eff V = π n n λ ( co cl ) GaAs core Al Ga 1- As cla At cutoff, n eff b 0 Casey an Panish

24 EE31 Lasers I 1/01/04 #6 slie 4 Moe Size Moe size important in SC lasers overlap with gain region ivergence of output beam coupling to optical fibers Various measures of size useful in ifferent contets effective with simple an useful: D = ( + 1/w) 1/ normalize form: D= k n n D= ( V + b ) co cl D Casey an Panish

25 EE31 Lasers I 1/01/04 #6 slie 5 Confinement Factor Fraction of power ire is important for SC laser overlap of moal fiel with gain region in simple heterostructures Can show that in our normalize parameters: P V + b P + P V + b core Γ = core cla 1/ 1/ Casey an Panish

26 EE31 Lasers I 1/01/04 #6 slie 6 Moal Gain Gain g eperience by a moe is intensity-weighte average moal intensity ψ ( ) consier case of heterojunction α ( m ) ψ( ) ψ ( ) take gain ire to constant an loss iaing to be constant a mco, mcl, g = a αmcl, ψ( ) + αmco, ψ( ) αmcl, ψ( ) a a g = ψ ( ) αm, copco αm, clpcl = Γ Ptot = α Γ α (1 Γ) α mcl, < < a αm( ) = αm, co a< < a α mcl, a < < Recall: J th (an thus α m,co ) ~ Traeoff: Γ improves with J th improves with V n n co cl

27 EE31 Lasers I 1/01/04 #6 slie 7 Threshol Current an Confinement Factor Alloy compositions like Al Ga 1- As allow ajustment of n an E g Traeoff in simple heterostructure small for low J trans too small leas to loose moe confinement poor overlap with gain leas to optimum for given n - note ecrease in opt with increasing n Leas to use of more complicate structures to allow separate confinement of carriers an optical fiel: ouble heterostructure w E g from Yariv from Vereyen from Vereyen