The Fabry-Perot Cavity Reflecting Pf Pi Reflecting surface E surface f Ei 2 Steady state EM oscillations 1 Cavity axis x R R 2 L 1 Optical cavity resonator © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) Derivation of the Laser Threshold Condition In the steady-state, the light (plane wave assumption) should remain unchanged after one round trip (2L). In other words, the gain = loss at threshold. An energy pumped in above threshold is converted into photons. R1 and R2 are the power reflectivities of mirrors 1 and 2, respectively. g is the (intensity) gain per unit length, αint is the internal scattering loss in the waveguide per unit length, L is the cavity length of the laser, k is the wave number of the plane wave, E0 is the electric field amplitude. E0 exp()gL R1R2 exp(−α int L)exp(2ikL)= E0 By equating magnitude and phase on the two sides of this equation, one obtains: 1 ⎛ 1 ⎞ g = α + ln = α + α = α int ⎜ ⎟ int mir cav αmir is the mirror loss 2L ⎝ R1R2 ⎠ 2kL = 2mπ or ν m = mc /2ng L m is an integer Some typical device parameters for LD’s L = 200 - 400 µm, R1 = R2 = 0.3 (cleaved mirrors) -1 αmir = (1/2L)ln(1/R1R2) = 30 - 60 cm Γ ≤ 0.4 (the optical confinement factor) -1 αint = 2-4 cm -1 g = Γgm = 32 - 64 cm (gm is the material gain) ∆ν = 100-200 GHz (mode spacing in GHz) Look at fig. (d) for illustration of Γ (a) A double n p p heterostructure diode has two junctions which are (a) AlGaAs GaAs AlGaAs between two different bandgap semiconductors (~0.1 µm) (GaAs and AlGaAs). Electrons in CB Ec ∆Ec (b) Simplified energy Ec 2 eV band diagram under a 1.4 eV 2 eV large forward bias. Lasing recombination (b) Ev takes place in the p- Ev GaAs layer, the active layer Holes in VB Refractive (c) Higher bandgap index materials have a ∆n ~ 5% (c ) Active lower refractive region index Photon density (d) AlGaAs layers provide lateral optical (d) confinement. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) Equations for the Light-Current Curve Optical Power Laser Optical Power Optical Power LED Stimulated emission λ Optical Power Laser Spontaneous λ emission I 0 I th λ Typical output optical power vs. diode current (I) characteristics and the corresponding output spectrum of a laser diode. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall) Equations for Laser Diode Characteristics I take a slightly different approach here compared to Agrawal to motivate the power output per mirror facet and the threshold current. The “2” in the denominator takes into hω ηintα mir qnthV Pe = ()I − Ith account that the power is split between Ith = 2q α mir + α int symmetric mirrors τ c dP h ω η α e int mir η is close to 100% for modern = ηd and ηd = int dI 2q α mir + α int laser diodes dP 1.24 e = η dI 2λ(µm) d dPe/dI is called the slope efficiency (units of W/A) ηd is called the differential quantum efficiency (units of %) The Laser Rate Equations dP P = GP + Rsp − dt τ p Any term that subtracts from dN I N N creates damping in the = − − GP oscillator dt q τ c G =Γvg g = GN ()N − N0 τp = cavity photon lifetime τc = carrier (recombination) lifetime P = photon number N = electron number, N0 = the transparency electron number GN = differential gain G = the threshold gain, net rate of stimulated emission g = material gain vg = group velocity = c/ng Rsp = rate of spontaneous emission coupled into the lasing mode I = injected current The results of small-signal analysis on the laser rate equations 2 2 ΩR + ΓR H()ωm = A damped resonator ()ΩR + ωm − iΓR ()ΩR + ωm + iΓR 1/2 Ω = GG P −Γ−Γ 2 /4 R []N b ()P N ΓR =Γ()P +ΓN /2 Normally, Γp is ΓP = Rsp /Pb + εNLGPb assumed to be −1 zero ΓN = τ c + GN Pb ΩR is called the resonance frequency ΓR is called the damping rate ωm is the modulation frequency Typical modulation response of the LD Note the resonance damping at higher current levels f3db = 1.55fr for the Ib/Ith ≤ 4.6 The 3-dB modulation bandwidth and its dependence on certain device parameters 1 1/2 1/2 f = Ω2 +Γ2 + 2 Ω4 +Ω2 Γ2 +Γ4 3dB 2π [ R R ()R R R R ] at low power ΓR << ΩR (not much damping) 12 3Ω ⎛ 3G P ⎞ f R N b 3dB ≈ ≈ ⎜ 2 ⎟ 2π ⎝ 4π τ p ⎠ To get a high bandwidth semiconductor laser, you want to run it at high power, short cavity length (small τp) and large differential gain. High-speed laser cavities Temperature Performance of a 1.3 µm InGaAsP/InP LD used for uncooled applications Ith = I0exp(T/T0) T0 is called the characteristic temperature. Typical values are from 50 to 100 K..