ECE228B, Prof. D. J. Blumenthal Lecture 5, Slide 1Lecture 5: Single Mode LaserDesignsECE228B, Prof. D. J. Blumenthal Lecture 5, Slide 21.535 1.54 1.545 1.55 1.555 1.56 1.565x 10-6-50-45-40-35-30-25-20-15-10-50wavelength (µm)−gc(λΝ)Side Mode Suppression Ratio SMSR (1) The output optical spectrum of a laser can contain one or many frequencies For high performance communications (2.5Gbps and higher), it is important to use lasers that emitprimarily at one frequency (wavelength). The SMSR is a standard measure of how single frequency is a laser is Consider the following symmetrical model for a semiconductor gain medium embedded in an opticalresonator where the gain peak is aligned with one of the resonator modesFP cavity modesgeff(λ)SC optical gainλNλN+1λN-1αtotalΔλmΔλmλp−gc(λΝ+1)δgECE228B, Prof. D. J. Blumenthal Lecture 5, Slide 3Side Mode Suppression Ratio SMSR (2) Consider the time-averaged (Stationary) optical power for the dominant mode (N) and second mostdominant mode (N+1)dSNdt= 0 = !a" GN(N ) " (1 #$" SN) " SN#SN%p+!a&spN%ndSN +1dt= 0 = !a" GN +1(N ) " (1 #$" SN +1) " SN +1#SN +1%p+!a&spN%n The SMSR is defined asSMSR =SNSN +1 For a gain spectrum much larger than the cavity mode spacing, assume there is minimal wavelengthdependence to the last term in the rate equations (assuming non-linear gain is zero)SMSR =SNSN +1=!a" GN +1(N ) #1$p!a" GN(N ) #1$pECE228B, Prof. D. J. Blumenthal Lecture 5, Slide 4Mode Selectivity For single mode operation in a digitally modulated laser, numerical simulations of multi-mode rateequations show that the dominant mode gain must exceed gain of all other modes by order 5 cm-1.!gc= SMSRnsp2h"vg#m#i+#m( )1Poff Where nsp is the spontaneous emission factor, vg is the mode group velocity and Poff is the power in a“zero” bit Example: SMSR = 100; nsp = 3; hv = 0.8eV; vg = c/neff = 3x108/4; αm = αi = 30 cm-1; Poff=0.025mW Δgc = 10cm-1 Note: In practice it is very difficult to get (and keep) the gain peak aligned with a cavity resonance, so theSMSR not only decreases, but the laser can be unstable between two modes that are competing for thegain.ECE228B, Prof. D. J. Blumenthal Lecture 5, Slide 5Periodic Index Structures (1) Many of the SML lasers in use today rely on some form of periodic structure to create a wavelengthdependent loss designed to allow only one mode to dominate and a large resulting SMSR Examples include Distributed Bragg Reflector Laser (DBR) and the Distributed Feedback Laser (DFB) A periodic structure is defined as where the index of refraction varies periodically in the direction ofpropagation onlyn (z) = neff'+!n2cos(2"0z)Ex: neff = 4; Δn = 5% The Bragg period of the structure is defined as Λ = Mπ/β0, with M an integer. For M=1 (first orderstructure), the free space Bragg wavelength can be used to describe the Bragg period! ="B2neff'ECE228B, Prof. D. J. Blumenthal Lecture 5, Slide 6Periodic Index Structures (2) Defining the grating vector (related to the periodic structure) kg = 2π/Λ and the coupling coefficient κ, thewave equation for a field with free space propagation constant (k0 = 2π/λ ) propagating in the periodicmedium isd2Edz2+ n(z)k0[ ]2E =d2Edz2+ neff'+!n2cos 2"0z( )#$%&'(k0)*+,-.2E = 0d2Edz2+ k02neff2+ neff'!n cos 2"0z( )( ))*,-E =d2Edz2+"2+ 4"/cos 2"0z( ))*,-E = 0/=0!n21 Consider wavelengths λ close to the Bragg wavelength λΒ such that β = β0 + Δβ and Δβ << β0 Using the picture below, we describe the forward and backward propagating waves byE(z) = R(z)exp(! j"0z) + S(z)exp( j"0z)ΛR(z)exp(! j"0z)S(z)exp( j!0z)β0−β0ECE228B, Prof. D. J. Blumenthal Lecture 5, Slide 7Periodic Index Structures (3)d2R(z)exp(! j"0z) + S(z)exp( j"0z)[ ]dz2+"2+ 4"#cos 2"0z( )$%&'R(z)exp(! j"0z) + S(z)exp( j"0z)[ ]= 0d ! j"0R(z)exp(! j"0z) + j"0S(z)exp( j"0z) +(R (z)exp(! j"0z) +(S (z)exp( j"0z)[ ]dz+"2+ 4"#cos 2"0z( )$%&'R(z)exp(! j"0z) + S(z)exp( j"0z)[ ]= 0!"02R(z)exp(! j"0z) !"02S(z)exp( j"0z) ! j"0(R (z)exp(! j"0z) + j"0(S (z)exp( j"0z) ! j"0(R (z)exp(! j"0z) + j"0(S (z)exp( j"0z) +"2R(z)exp(! j"0z) +"2S(z)exp( j"0z) + 4"#cos 2"0z( )R(z)exp(! j"0z) + 4"#cos 2"0z( )S(z)exp( j"0z) = 0"2!"02( )R(z)exp(! j"0z) +"2!"02( )S(z)exp( j"0z) ! 2 j"0(R (z)exp(! j"0z) + 2 j"0(S (z)exp( j"0z) +4"#cos 2"0z( )R(z)exp(! j"0z) + 4"#cos 2"0z( )S(z)exp( j"0z) = 0exp(! j"0z)"2!"02( )! 2 j"0(R (z) + 4"#cos 2"0z( )R(z)$%&'+ exp( j"0z)"2!"02( )+ 2 j"0(S (z) + 4"#cos 2"0z( )S(z)$%&'= 0"2="02+ )"2+ 2"0)"*"02+ 2"0)"exp(! j"0z) 2"0)"! 2 j"0(R (z) + 4"#cos 2"0z( )R(z)$%&'+ exp( j"0z) 2"0)"+ 2 j"0(S (z) + 4"#cos 2"0z( )S(z)$%&'= 0 Inserting the backward and forward propagating field into the wave equation with periodically varyingindex of refraction Which can be described by the coupled-mode equations!R (z) + j"#R(z) = $ j%S(z)!S (z) $ j"#S(z) = j%R(z)ECE228B, Prof. D. J. Blumenthal Lecture 5, Slide 8Solution to Coupled Mode Equations (1) The coupled mode equations and wave equation describe the field in the periodic index structure. Assuming there are boundary conditions (e.g. R(0) and S(0) are known), we can write the fields asR(z) = cosh!z( )"j#$!sinh!z( )%&'()*R(0) "j+!sinh!z( )S(0)S(z) =j+!sinh!z( )R(0) + cosh!z( )+j#$!sinh!z( )%&'()*S(0)where at z = LR(L) = cosh!L( )"j#$!sinh!L( )%&'()*R(0) "j+!sinh!L( )S(0)S(L) =j+!sinh!L( )R(0) + cosh!L( )+j#$!sinh!L( )%&'()*S(0)R(L)S(L),-./01= Fper(L)R(0)S(0),-./01 Where we have defined the matrix Fper(L) and γ2=κ2-Δβ2 Note that Fper relates the right and left propagating waves at the left side (z=0) of the periodic indexstructure to the right and left propagating waves a the right side (z=L) of the structure.ECE228B, Prof. D. J. Blumenthal Lecture 5, Slide 9Solution to Coupled Mode Equations (2) We can define the field reflection coefficient rper and the power reflection coefficient Rper at z=0 as rper=S(0)R(0)=!j"#sinh#L( )cosh#L( )+j$%#sinh#L( )&'()*+,! j"L1 + j$%L[ ]Rper= rper2,"L( )2sin2-L( )2!"L( )2-L( )2!"L( )2, for -L ?"Ltanh2-L( )2!"L( )2, for -L ="L./00100 Where the final simplification for rper is for κL very close to ΔβL Note that | rper| increases with increasing κL which means a higher coupling coefficient leads to a strongerreflection. | rper| decreases with increasing ΔβL which means the reflection becomes smaller when thewavelength moves away from the Bragg reflection peak. The reflection experiences a π/2 phase shift when Δβ = 0.ECE228B, Prof. D. J.