Lecture 14- ECE 240a Laser Dynamics Hole Burning Below Threshold Above Threshold Lecture 14- ECE 240a Transient Response Ver Chap. 9.3 Linearized Solution Sinusoidal Variation Step Response Gain Switching 1 ECE 240a Lasers - Fall 2019 Lecture 14
Laser Dynamics Lecture 14- ECE 240a Laser Dynamics Hole Burning Below Threshold Above Threshold Transient This lecture Response AC response Linearized Solution Step Response Sinusoidal Gain switching Variation Step Response Next Lecture Gain Switching Q-switching Mode locking 2 ECE 240a Lasers - Fall 2019 Lecture 14
System to Analyze Lecture 14- ECE 240a Laser Dynamics Hole Burning Below Threshold Above Threshold Transient Response Linearized Solution Sinusoidal Variation Step Response Gain Switching ℓ g 3 ECE 240a Lasers - Fall 2019 Lecture 14
Write Down Rate Equation Including Pump Lecture 14- ECE 240a Laser Dynamics Hole Burning Below Threshold Above dN 2 RN 0 − N 2 − σ L I L Threshold = hν L N 2 Transient dt ′ τ 2 Response Linearized σ P I P hν P N 0 − N 2 − σ L I L Solution = hν L N 2 Sinusoidal τ 2 Variation � � Step Response σ P I P hν P N 0 − N 2 1 + I L = Gain τ 2 I s Switching where I s = hν L σ L τ 2 is the saturation intensity.. 4 ECE 240a Lasers - Fall 2019 Lecture 14
Write Down Equation for Energy in Cavity Including Pump Lecture 14- ECE 240a Start w/photon lifetime Laser 1 α t 2 d / t RT = t RT 1 t RT Dynamics τ p = α t c = = Hole Burning L 1 − S Below Threshold where L = α t 2 d is the loss in the cavity per pass and S = 1 − L is the Above survival in the cavity. Threshold Transient Response Rate of change in energy w per pass per time Linearized Solution hν N 2 ℓ g dw − L Sinusoidal = t RT + β Variation dt τ 2 ���� Step Response � �� � fraction in mode Gain total spon. emission Switching S per pass − 1 + βhν N 2 ℓ g = t RT τ 2 Se N 2 σ L ℓ g − 1 + βhν N 2 ℓ g = t RT τ 2 where Se N 2 σ L ℓ g is the survival including the gain in the medium 5 ECE 240a Lasers - Fall 2019 Lecture 14
Define terms to Normalize Equations Lecture 14- ECE 240a Laser t = t ′ / t RT Time normalized to the round-trip time t RT in the cavity Dynamics Hole Burning a Round trip time to upper state lifetime t RT / τ 2 Below Threshold Above g = N 2 σ 2 I g Integrated gain where ℓ g is the length of the gain medium ( Threshold Transient not necessary equal to cavity length) Response Linearized Solution β fraction of total spontaneous emission that couples into the lasing Sinusoidal Variation mode (key parameter) Step Response Gain P = I L / I s relative number of photons with respect to saturation Switching intensity R = σ P N 0 ℓ g × ( λ P / λ L ) × ( I P / I s ) pump rate normalized to the saturation intensity Term ( λ P / λ L ) is the quantum efficiency of the pumping scheme Term σ P N 0 ℓ g is the absorption efficiency of the pumping scheme S = 1 − L probability of survival of a photon in the cavity where L is the total loss 6 ECE 240a Lasers - Fall 2019 Lecture 14
Coupled Normalized Equations Lecture 14- ECE 240a Laser Dynamics Hole Burning Below Threshold dP dt = ( Se g − 1 ) P + βg Above Threshold Transient Response dg Linearized dt = a [ R − g ( 1 + P )] Solution Sinusoidal Variation Step Response Study for two Cases: Gain Switching Case 1 - bias laser above threshold Case 1a - AC response Case 1b - Step response Case 2 - bias below threshold - gain switching a Round trip time to upper state lifetime t RT / τ 2 7 ECE 240a Lasers - Fall 2019 Lecture 14
Below Threshold Lecture 14- ECE 240a Laser Dynamics Hole Burning Below Threshold Above If the pump R is a step R ( t ) = aR a u ( t ) but does not exceed the Threshold Transient threshold, then can set P ≈ 0 and Response Linearized dg Solution dt + g = aR a Sinusoidal Variation Step Response This has a solution Gain g ( t ) = R a ( 1 − e − at ) Switching and a steady-state value g = R a that is not above the lasing threshold g = g th . 8 ECE 240a Lasers - Fall 2019 Lecture 14
Linearized Solution to Equations Lecture 14- ECE 240a Laser Dynamics Hole Burning Assume that we have a zero-frequency terms and a time-varying term for Below Threshold each of the variables R ( t ) , g ( t ) and P ( t ) Above Threshold R ( t ) = R c + ∆ r ( t ) Transient Response g ( t ) = g th + ∆ g ( t ) Linearized Solution P ( t ) = P c + ∆ p ( t ) Sinusoidal Variation Step Response Gain Steps for solution Switching Plug in forms for each variable 1 Collect terms in order: Order 1 terms are the zero-frequency terms, order 2 δ -terms are (DC × δ ) and order δ 2 terms (product of δ and δ ) Solve for each group of terms separately 3 Solve for order 1 term - this gives the DC solution or the bias point 1 AC response are terms of order δ 2 Neglect the terms of δ 2 3 9 ECE 240a Lasers - Fall 2019 Lecture 14
Order δ Term for Power Lecture 14- ECE 240a Laser Dynamics Hole Burning Take the two equation and eliminate ∆ g ( t ) to produce one equation of Below order δ Threshold d 2 ∆ p ( t ) Above + Ad ∆ p ( t ) Threshold + B ∆ p ( t ) = C ∆ r ( t ) Transient dt 2 dt � �� � Response � �� � Driving term is pump Linearized 2nd order ODE Solution Sinusoidal where Variation Step Response A = a ( 1 + P c ) Gain Switching B = g th C C = a ( P c + β ) CW limit ∆ p ( t ) / dt = 0 , we obtain B ∆ r ( t ) = ∆ r ( t ) ∆ p ( t ) = C g th 10 ECE 240a Lasers - Fall 2019 Lecture 14
Sinusoidal Variation Lecture 14- ECE 240a Laser Dynamics Hole Burning Now assume we pump (above threshold) with a sinusoidal signal such Below Threshold that Above ∆ r ( t ) = r m e jω m t Threshold Transient Response Linearized Solution Sinusoidal Order δ -equation is linear so output power has same form Variation Step Response ∆ p ( t ) = p m e jω m t Gain Switching Substitute forms into equation to produce small-signal linearized AC response ( transfer function) g th p m B = ( B − ω 2 r m m ) + jω m C 11 ECE 240a Lasers - Fall 2019 Lecture 14
Plot Lecture 14- ECE 240a Laser Dynamics Hole Burning Below Threshold Above Threshold Transient Response Linearized Solution Sinusoidal Variation Step Response Gain Switching Neglecting effect of spontaneous emission, the resonant frequency (called relaxation oscillation) is ω 2 r = B ≈ aP c g th Note that because we started with nonlinear equations this oscillation frequency depend on the DC power P c 12 ECE 240a Lasers - Fall 2019 Lecture 14
Measured Data Lecture 14- ECE 240a mode VCSEL. Fig. 2 LIV characteristics of duo-mod Laser Dynamics 10 Hole Burning Below Threshold 0 Above Threshold Transient Response Response (dB) Linearized -10 Solution Sinusoidal 850 nm 25 °C Variation Step Response -20 Gain 2.5 mA ,1.9GHz Switching 4.5 mA ,13GHz -30 6.5 mA ,14.5GHz 8.5 mA ,15.4GHz -40 0 5 10 15 20 25 30 Frequency (GHz) Fig. 4 E-O responses at 25 °C. 13 ECE 240a Lasers - Fall 2019 Lecture 14
Step Response Lecture 14- ECE 240a Just like second order system Laser Dynamics Hole Burning Below Threshold Above Threshold Transient Response Increasing Linearized Solution Sinusoidal power Variation Step Response Gain Switching 14 ECE 240a Lasers - Fall 2019 Lecture 14
Gain Switching Lecture 14- ECE 240a Laser Dynamics Cold start from zero - do not assume that the laser is biased above Hole Burning threshold. Below Threshold Fundamental assumption is that when the gain is turned on it is so fast Above Threshold that power does not have time to build up Transient Response Short times can set P = 0 and we have below threshold conditions Linearized Solution dg Sinusoidal dt + g = bR e Variation Step Response Gain where bR e = aR with b = τ p / τ 2 Switching This has a solution g ( t ) = R e ( 1 − e − at ) ≈ R e where R e is the pump rate. This is valid when t ≫ a − 1 and means that gain turns on rapidly and can be treated as a constant with respect to the build-up of the power. 15 ECE 240a Lasers - Fall 2019 Lecture 14
Expression for the Power Lecture 14- ECE 240a Laser Dynamics Hole Burning Below Threshold Above Threshold Power is then Transient Response exp [( Se g − 1 ) t ] Linearized P = δP 0 Solution ���� � �� � Sinusoidal initial power in mode Variation exponential small-signal gain Step Response Gain Some numbers - assume onset of saturation is P = 0.1 and δP = 10 − 7 - Switching then argument to exponential function need to be about 14 before lasing starts 16 ECE 240a Lasers - Fall 2019 Lecture 14
Complete Nonlinear Transient Solution Lecture 14- ECE 240a Laser Dynamics Hole Burning Below Define new set of normalized parameters Threshold T = t ′ / t p time normalized to the photon lifetime in the cold cavity Above Threshold b = τ p / τ 2 ratio of lifetime in the cavity to the upper lasing state lifetime Transient S = e − g th Response Linearized Coupled equations become Solution Sinusoidal Variation Step Response � � dP ( t ) Se g ( t ) − 1 = P ( t ) Gain dt Switching � � e g ( t ) − g th − 1 = P ( t ) and dg ( t ) = b [ R e − g ( t )( 1 + P ( t )] dt 17 ECE 240a Lasers - Fall 2019 Lecture 14
Recommend
More recommend
