Pulsed Lasers 14: Pulsed lasers: gain-switching and Q- switching CONTENTS: • Methods for LASER switching • Gain switching • Relaxation oscillations • Q-switching basics • Q-switching methods • Q-switching dynamics 1 Methods for laser switching So far, we have looked at the steady-state (CW) output from a laser: optical emitted gain γ power Pe γ = α +α th i m P =P threshold e s γ=α i transparency γ=0 inversion pump P =0 pump 0 rate R e rate R Rth 0 Rth 2Rth But how can we turn a laser on and off ? 1. Change the pump rate R to modify the gain (“gain switching”) 2. Change the cavity loss αi+αm to modify the lasing threshold (“Q-switching”) 3. Employ beating between multiple modes of the laser cavity (“mode-locking”) 2 531 PHYS- Dr. A. M. Azzeer 1 Pulsed Lasers Gain switching How fast can we switch a laser on and off, by modulating the pumping rate (i.e. what are the dynamics)? ⎧dN N ⎛ g ⎞ We’ve written all these rate 2 = R − 2 − ⎜ N − 2 N ⎟B ρ υ equations, so let’s use ‘em !! ⎪ 2 1 21 () ⎪ dt τ sp ⎝ g1 ⎠ ⎨ Remember the rate equations ⎪dφ ⎛ g ⎞ φ for an idealised 4-level system: = ⎜ N − 2 N ⎟B ρ υ − ⎪ 2 1 21 () ⎩ dt ⎝ g1 ⎠ τ c Re-writing the energy density in terms of photon number: c3 A c2 A ρ B 21 c 21 c 21ρ = 3 ρ = 2 = σ 0φ Assuming a further idealised 8πhν 8πν hν system in which N1≈0 , we end up with coupled rate equations ⎧dN N ⎪ = R − − cσ 0Nφ for the photon density φ and ⎪ dt τ sp ⎨ the population inversion N. dφ φ ⎪ = Ncσ φ − ⎛ g2 ⎞ 0 N = ⎜ N2 − N1 ⎟ ⎩⎪ dt τ c ⎝ g1 ⎠ 3 Gain switching 2 We can write these more simply in terms of the dN N N φ ⎧ = R − − population inversion at threshold Nth. ⎪ dt τ N τ We should recognise these as a pair of differential ⎪ sp th c ⎨ equations describing a system of electrons N dφ ⎛ N ⎞ φ ⎪ = ⎜ −1⎟ coupled to a system of photons φ by the stimulated ⎪ dt ⎜ N ⎟τ emission term. ⎩ ⎝ th ⎠ c We can obtain an analytical solution only in the ⎧N(t)()= N0 + N′ t small-signal limit N’(t)<<N : ⎨ 0 ⎩φ()t = φ0 +φ′ ()t Ignoring the N’(t)·φ’(t) terms: ⎧dN′()t ⎧ 1 φ 1 ⎫ N φ′ ⎪ = −N′()t ⎨ 0 + ⎬ − 0 ⎪ dt ⎩Nth τ c τ sp ⎭ Nth τ c ⎨ ⎪dφ′ N′()t φ0 ⎛ N0 ⎞φ′()t ⎪ = + + ⎜ −1⎟ ⎩ dt Nth τ c ⎝ Nth ⎠ τ c 4 531 PHYS- Dr. A. M. Azzeer 2 Pulsed Lasers Gain switching 3 Differentiating the equation for N’(t) , substituting for dφ’/dt and setting the equilibrium terms to zero, we obtain: d 2N′()t dN′()t ⎧ 1 φ 1 ⎫ 1 ⎧ 1 φ ⎫ + 0 + + 0 N′ t = 0 2 ⎨ ⎬ ⎨ ⎬ () dt dt ⎩Nth τ c τ sp ⎭ τ c ⎩Nth τ c ⎭ With an identical expression for φ’. This is the equation for a damped harmonic oscillator. It can be solved (an exercise for the student!) by solutions of the form ⎧N′()t = N(ω)exp(− t t0 )cos(ωt +θ ) ⎨ ⎩φ′()t = φ (ω )exp()()− t t0 sin ωt +θ The strength of the coupling relative to the excitation and decay rates determines whether the system is underdamped (exhibits relaxation oscillations) or overdamped (relaxes exponentially to steady-state values) 5 Gain switching: relaxation oscillations N • Takes a time (~t0) to re- establish steady-state after a perturbation • Limits the modulation rate / t switching speed of lasers φ t 6 531 PHYS- Dr. A. M. Azzeer 3 Pulsed Lasers Q switching basics A way of obtaining short light pulses lamp with high peak intensity current THE BASIC IDEA: • Pump the laser hard BUT: cavity t • Prevent lasing by temporarily loss High-Q screwing up the laser cavity (low Q) αi+αm Low-Q Stimulated emission is suppressed, so a very large population inversion t inversion can build up Nmax THEN: • Restore the laser cavity Q The laser finds itself with a gain much higher than the steady state t threshold value. Stimulated photon φmax flux emission occurs, generating a ∆τ bright pulse, until the gain decreases to its steady state value. t 7 Q switching methods 1. Mechanical • e.g. rotating chopper, or vibrating mirror mount to mis-align cavity 2. Electro-optic switch (Pockels cell) • Fast electro-optic switching in materials such as KDP. • Requires voltage pulse in 1-10kV range 3. Acousto-optic modulator • Ultrasonic transducer generates standing compression wave in optical material, hence standing refractive index grating (via photoelastic effect). • Laser beam undergoes Bragg scattering from the grating. 4. Passive Q-switching • Saturable absorber α • Historically, laser dyes with nonlinear absorbtion α = 0 • Currently: semiconductor multilayers 1+ P P • New development: carbon nanotubes! s 8 531 PHYS- Dr. A. M. Azzeer 4 Pulsed Lasers Q switching dynamics The rate equation for photons can be written as: dφ ⎛ N ⎞⎛ φ ⎞ where N is the population inversion = ⎜ −1⎟⎜ ⎟ dt Nth τ c and Nth is its value at threshold. ⎝ ⎠⎝ ⎠ The rate equation for N (if there is no pumping when dN N ⎛ φ ⎞ the Q is restored and spontaneous emission is = −2 ⎜ ⎟ negligable compared to stimulated emission) is: dt Nth ⎝τ c ⎠ A factor of 2 because N2 decreases by 1 while N1 increases by 1, for each stimulated emission process. These equation can be solved, for an initial inversion N0 : see {AY6.9} or {CD8.16}. hν Maximum power is Pmax ≈ N0 2τ c N Max. number of photons in cavity is: φ ≈ 0 max 2 Decay rate of Q-switched pulse is: τ c 9 Gain switching 4 We can obtain the parameters in “gain switching 3” more simply if we write: ⎧N()t = N0 + N′exp(−αt)exp( jωt) ⎨ ⎩φ()t = φ0 +φ′exp (−αt )exp (jωt ) ⎧ 1 ⎧ 1 φ 1 ⎫ obtaining: ⎪α = ⎨ 0 + ⎬ ⎪ 2 ⎩Nth τ c τ sp ⎭ ⎨ ⎪ 2 1 ⎧ 1 φ0 ⎫ 2 ⎪ω = ⎨ ⎬ −α ⎩ τ c ⎩Nth τ c ⎭ If α<<ω (underdamping): 1 ⎧ 1 φ ⎫ ω ≈ ⎨ 0 ⎬ τ c ⎩Nth τ c ⎭ 10 531 PHYS- Dr. A. M. Azzeer 5 Pulsed Lasers What U need 2 know: LASER switching 14.1 Be aware of the existence of : Be aware of the existence of : coupled equations relaxation oscillations for electron and photons spontaneous N pumping emission net stimulated ⎧dN N N φ t ⎪ = R − − emission ⎪ dt τ sp Nth τ c ⎨ φ dφ N φ φ ⎪ = − ⎩⎪ dt Nth τ c τ c t net stimulated cavity emission loss 11 What U need 2 know: LASER switching 14.2 Describe briefly with the aid of these lamp current diagrams, the basic principles of: Q-switching cavity t loss αi+αm inversion t Nmax t photon φ flux max ∆τ t Note that : Q-switch pulse duration ~ cavity lifetime 12 531 PHYS- Dr. A. M. Azzeer 6.