Lecture 14- ECE 240a Transient Response Ver Chap. 9.3 Linearized - - PowerPoint PPT Presentation

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Lecture 14- ECE 240a

Ver Chap. 9.3

ECE 240a Lasers - Fall 2019 Lecture 14 1

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Laser Dynamics

This lecture

AC response Step Response Gain switching

Next Lecture

Q-switching Mode locking

ECE 240a Lasers - Fall 2019 Lecture 14 2

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

System to Analyze

ℓg

ECE 240a Lasers - Fall 2019 Lecture 14 3

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Write Down Rate Equation Including Pump

dN2 dt′ = RN0 − N2 τ2 − σLIL hνL N2 = σP IP hνP N0 − N2 τ2 − σLIL hνL N2 = σP IP hνP N0 − N2 τ2

• 1 + IL

Is

• where

Is = hνL σLτ2 is the saturation intensity..

ECE 240a Lasers - Fall 2019 Lecture 14 4

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Write Down Equation for Energy in Cavity Including Pump

Start w/photon lifetime τp = 1 αtc = 1 αt2d/tRT = tRT L = tRT 1 − S where L = αt2d is the loss in the cavity per pass and S = 1 − L is the survival in the cavity. Rate of change in energy w per pass per time dw dt = − L tRT + β

• fraction in mode

hν N2ℓg τ2 total spon. emission = Sper pass − 1 tRT + βhν N2ℓg τ2 = SeN2σLℓg − 1 tRT + βhν N2ℓg τ2 where SeN2σLℓg is the survival including the gain in the medium

ECE 240a Lasers - Fall 2019 Lecture 14 5

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Define terms to Normalize Equations

t = t′/tRT Time normalized to the round-trip time tRT in the cavity a Round trip time to upper state lifetime tRT /τ2 g = N2σ2Ig Integrated gain where ℓg is the length of the gain medium ( not necessary equal to cavity length) β fraction of total spontaneous emission that couples into the lasing mode (key parameter) P = IL/Is relative number of photons with respect to saturation intensity R = σP N0ℓg × (λP /λL) × (IP /Is) pump rate normalized to the saturation intensity

Term (λP /λL) is the quantum efficiency of the pumping scheme Term σP N0ℓg 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

ECE 240a Lasers - Fall 2019 Lecture 14 6

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Coupled Normalized Equations

dP dt = (Seg − 1) P + βg dg dt = a [R − g(1 + P)] Study for two Cases: 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 tRT /τ2

ECE 240a Lasers - Fall 2019 Lecture 14 7

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Below Threshold

If the pump R is a step R(t) = aRau(t) but does not exceed the threshold, then can set P ≈ 0 and dg dt + g = aRa This has a solution g(t) = Ra(1 − e−at) and a steady-state value g = Ra that is not above the lasing threshold g = gth.

ECE 240a Lasers - Fall 2019 Lecture 14 8

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Linearized Solution to Equations

Assume that we have a zero-frequency terms and a time-varying term for each of the variables R(t), g(t) and P(t) R(t) = Rc + ∆r(t) g(t) = gth + ∆g(t) P(t) = Pc + ∆p(t) Steps for solution

1

Plug in forms for each variable

2

Collect terms in order: Order 1 terms are the zero-frequency terms, order δ-terms are (DC×δ) and order δ2 terms (product of δ and δ)

3

Solve for each group of terms separately

1

Solve for order 1 term - this gives the DC solution or the bias point

2

AC response are terms of order δ

3

Neglect the terms of δ2

ECE 240a Lasers - Fall 2019 Lecture 14 9

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Order δ Term for Power

Take the two equation and eliminate ∆g(t) to produce one equation of

• rder δ

d2∆p(t) dt2 + Ad∆p(t) dt + B∆p(t)

• 2nd order ODE

= C∆r(t) Driving term is pump where A = a(1 + Pc) B = gthC C = a(Pc + β) CW limit ∆p(t)/dt = 0, we obtain ∆p(t) = C B ∆r(t) = ∆r(t) gth

ECE 240a Lasers - Fall 2019 Lecture 14 10

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Sinusoidal Variation

Now assume we pump (above threshold) with a sinusoidal signal such that ∆r(t) = rmejωmt Order δ-equation is linear so output power has same form ∆p(t) = pmejωmt Substitute forms into equation to produce small-signal linearized AC response (transfer function) gthpm rm = B (B − ω2

m) + jωmC

ECE 240a Lasers - Fall 2019 Lecture 14 11

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Plot

Neglecting effect of spontaneous emission, the resonant frequency (called relaxation oscillation) is ω2

r = B ≈ aPcgth

Note that because we started with nonlinear equations this oscillation frequency depend on the DC power Pc

ECE 240a Lasers - Fall 2019 Lecture 14 12

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Measured Data

mode VCSEL. Fig. 2 LIV characteristics of duo-mod

850 nm 25 °C

5 10 15 20 25 30

• 40
• 30
• 20
• 10

10

Response (dB) Frequency (GHz) 2.5 mA ,1.9GHz 4.5 mA ,13GHz 6.5 mA ,14.5GHz 8.5 mA ,15.4GHz

• Fig. 4 E-O responses at 25 °C.

ECE 240a Lasers - Fall 2019 Lecture 14 13

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Step Response

Just like second order system

Increasing power

ECE 240a Lasers - Fall 2019 Lecture 14 14

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Gain Switching

Cold start from zero - do not assume that the laser is biased above threshold. Fundamental assumption is that when the gain is turned on it is so fast that power does not have time to build up Short times can set P = 0 and we have below threshold conditions dg dt + g = bRe where bRe = aR with b = τp/τ2 This has a solution g(t) = Re(1 − e−at) ≈ Re where Re 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.

ECE 240a Lasers - Fall 2019 Lecture 14 15

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Expression for the Power

Power is then P = δP0

• initial power in mode

exp[(Seg − 1)t]

• exponential small-signal gain

Some numbers - assume onset of saturation is P = 0.1 and δP = 10−7 - then argument to exponential function need to be about 14 before lasing starts

ECE 240a Lasers - Fall 2019 Lecture 14 16

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Complete Nonlinear Transient Solution

Define new set of normalized parameters

T = t′/tp time normalized to the photon lifetime in the cold cavity b = τp/τ2 ratio of lifetime in the cavity to the upper lasing state lifetime S = e−gth

Coupled equations become dP(t) dt =

• Seg(t) − 1
• P(t)

=

• eg(t)−gth − 1
• P(t)

and dg(t) dt = b [Re − g(t)(1 + P(t)]

ECE 240a Lasers - Fall 2019 Lecture 14 17

Lecture 14- ECE 240a Laser Dynamics

Hole Burning

Below Threshold Above Threshold Transient Response

Linearized Solution Sinusoidal Variation Step Response

Gain Switching

Plot

ECE 240a Lasers - Fall 2019 Lecture 14 18