Ver Chap Lecture 15- ECE 240a. Q-Switching. Mode Locking. ECE 240a Lasers - Fall 2017 Lecture Q-Switch Discussion

Transcription

1 ing Ver Chap. 9.3 Lasers - Fall 2017 Lecture 15 1

2 ing ing (Cavity Dumping) 1 Turn-off cavity - prevent lasing 2 Pump lots of energy into upper state - use pulsed pump 3 Turn cavity back on - all the energy in the upper state depleted in a few round trips 1 If single-(round-trip) pass gain is 5 then after 10 round-trips AC response Step Response Gain switching d l g Gain medium l s Shutter (Q-switch) Lasers - Fall 2017 Lecture 15 2

3 Geometry Lecture 15 ing Lasers - Fall 2017 Lecture 15 3

4 Analysis ing Total gain in one round trip Total loss G = Exp[2 N(t)σl g }{{} ] line integrated gain = e 2g(t) e 2g th. = Πj R j Π i T i }{{}} e 2αsls {{} mirrror and transmission losss scattering loss The loss term defines the threshold gain g th Photon lifetime (from before) τ p = = τ RT 1 S τ RT 1 e 2g th Lasers - Fall 2017 Lecture 15 4

5 Rate of Net Photon Gain ing Define N p as the net change in the photon number during one round trip. Rate of change is dn p(t) dt = = = ( ) L G 1 N p(t) τ RT ( e 2(g(t) g th ) ) 1 N p(t) τ RT ( e 2(g(t) g th ) ) 1 Np(t) 1 e 2g th τ p Assume net increase per round-trip is small and expand e x 1 + x where dn p(t) dt = Np(t) τ p ( ) n(t) 1 n th n(t). = (N 2 (t) N 1 (t)) Al g is the total number of excited states (atoms) in the cavity that interact with the mode cross-sectional area A. Lasers - Fall 2017 Lecture 15 5

6 ing If ratio n(t) n th If ratio n(t) n th > 1, then there is a net gain in the photon number N p(t) < 1, then the net photon number decreases. Now for every photon that is added, there is a decrease in the upper state and an increase in the lower state. Therefore, the rate of change of the total number of excited states n(t) that can interact is given by or n(t) dt = }{{} 2 n(t) N p(t) n th τ p lose on upper/gain on lower dn(t) dt = 2 n(t) n th N p(t) τ p Lasers - Fall 2017 Lecture 15 6

7 Governing Equations ing In order to solve, normalize the time by the photon lifetime so that T = t/τ p. The two governing equations become dn p(t ) dt = N p(t ) ( ) n(t ) 1 n th dn(t ) dt = 2 n(t ) n th N p(t) These are a set of nonlinear coupled first order equations. Lasers - Fall 2017 Lecture 15 7

8 ing Occurs when all of the modes add in phase. Look at spectrum Spectrum may be written as S(ω ω 0 ) = g 1 (ω ω 0 ) L(ω ω 0 l f) l= For simplicity assume g 1 (ω) is a gaussian g 1 (f) = e ω2 /2σ 2 and L(ω) is a Lorentzian of the form L(ω) = 1 ω 2 + B 2 Lasers - Fall 2017 Lecture 15 8

9 Plot of Spectrum ing S(f f0) f f 0 Lasers - Fall 2017 Lecture 15 9

10 Rewrite Spectrum ing Now re-write l= L(ω ω 0 l f) as a convolution of the Lorentzian and an impulse train so that l= L(ω ω 0 l f) = L(ω ω 0 ) S(ω ω 0 ) = g 1 (ω ω 0 ) L(ω ω 0 ) l= l= δ(ω l ω) δ(ω l ω) Now take the Fourier transform to determine the ( { }) F T 1 {L(ω ω 0 )} F T 1 δ(ω ω 0 i ω) Γ(τ) = F T 1 {g 1 (ω ω 0 )} i= Lasers - Fall 2017 Lecture 15 10

11 Time Response ing First term is the Fourier transform gaussian which is a gaussian F T 1 {g 1 (ω ω 0 )} = σe 2π2 σ 2 τ 2 The second term is the transform Lorentzian which is a double-sided exponential F T 1 {L(ω ω 0 )} = B exp [ τ ] Final term is the transform n impulse train which is another impulse train δ(ω i ω) 1 δ(t i ω ω ) i= i= Put it all together 2Bπ 3/2 [ Γ(τ) = Bπ exp 2π i ] [ ( exp 2π 2 σ 2 τ i ) 2 ] ν ν ν i= Lasers - Fall 2017 Lecture 15 11

12 Plot ing Lasers - Fall 2017 Lecture 15 12

13 Time Response as Sum of Phasors ing Write the field in time as a sum of complex amplitudes (phasors) e(t) = (N 1)/2 (N 1)/2 E n(t) exp [j(ω 0 + nω c)t + φ n(t)] where N is total number of modes mplitude E n and phase φ n Now sum assuming equal amplitudes ( similar to cavity studied before) [ ] e(t) = E 0 e jω0t sin(nωct/2) sin(ω ct/2) Intensity is given by the square [ ] 2 I(t) = e(t) 2 = E2 0 sin(nωct/2) 2η 0 η 0 sin(ω ct/2) Lasers - Fall 2017 Lecture 15 13

14 ing Peak is N 2 times the intensity in one mode Therefore if we assume each mode contributes equally then P peak = N P ave Now if the modes have unequal contributions, estimate the relationship between the peak and the average power using equal area argument P peak t pulse = P aveτ RT or t pulse = τrt N where N is the number of modes that are phase-locked Now N can be estimated by linewidth of gain profile so that or N = ν (c/2d) = ντrt t pulse τrt N 1 ν Lasers - Fall 2017 Lecture 15 14

15 ing Active mode locking - force the phase relationship by use modulator allows the mode-locking homogeneously broadened medium Passive mode locking - use a saturable absorber that only passes a high amplitude field. Lasers - Fall 2017 Lecture 15 15