Lecture 8 Con,nuous- Wave Laser*

Lecture 8 Con,nuous- Wave Laser* Min Yan Op,cs and Photonics, KTH 24/04/15 1 * Some figures and texts belong to: O. Svelto, Principles of Lasers, 5th Ed., Springer.

Reading Principles of Lasers (5th Ed.): Chapter 7. Skip: 7.3.2, 7.4.2, 7.8.2.2. Squeeze: 7.9, 7.10. 24/04/15 2

Laser Pump E in E out Mirror 1 Gain medium Mirror 2 Rate equa,on (interplay between N and φ) Threshold condi,ons Steady- state N, φ, P out, η s R 2 for op,mum P out Single- mode selec,on, and tuning 24/04/15 3

Contents Content 1. Rate equa,ons 1. Four- level; 2. Quasi- three- level 2. Threshold and steady states 1. Four- level; 2. Quasi- three- level Time 25 15 3. Op,mum output coupling 5 4. Single- mode selec,on and tuning 30 5. Others Frequency- pulling, fluctua,ons 5 Total: 80 24/04/15 4

Rate equa,ons (4L) S,mulated emission N 2 3 2 N 1 =0 1 0 S,mulated emission Assump*ons: Single- mode Pump & laser mode are uniform (space- independent) N 2 : Popula,on inversion (per unit volume) φ: Total photon number B: S,mulated transi,on rate per photon per mode τ: Effec,ve upper- level life,me [radia,ve (Spon.E.)+nonradia,ve] V a : Volume of the mode in the ac,ve region τ c : Cavity photon life,me 24/04/15 6

B Laser intensity change aler one round trip L Define single- trip logarithmic loss as l Aler round trip, ΔI becomes A b =V a /l If Divide by Δt (round- trip,me) Round- trip,me: Δt=2L e /c, where L e =L+(n- 1)l Since I φ, by comparison with 2 nd rate eq. V: cavity mode volume 24/04/15 7

Rate equa,ons so what? 1. CW characteris,cs Threshold- state condi,on: φ 0, N c Steady- state condi,on: dφ/dt=0, dn/dt=0 2. Transient characteris,cs φ(t) and N(t) can be derived if φ(t=0) and R p (t) are given 3. Output power P out if φ(t) is known 4. Slope efficiency η s, i.e. dp out /dp p To get P out : 24/04/15 8

Rate equa,ons (q3l) N 2 3 2 N 1 0 1 0 If we define f=σ a /σ e and popula,on inversion N=N 2 - fn 1 4L- case Difference: - Popula,on inversion - S,mulated emission term Similarity: - Same 2 nd equa,on 24/04/15 9

Threshold state: N c and R cp (4L) Note: Small amount of photons φ i exist due to spontaneous emission In 2 nd equa,on, let dφ/dt=0: Physically: gain=loss In 1 st equa,on, let dn/dt=0, φ 0, and N=N c : 24/04/15 11

Steady state: N 0, φ 0, P out, η s (4L) Steady lasing state: constant R p >R cp N In 2 nd equa,on, let dφ/dt=0: In 1 st equa,on, let dn/dt=0, and N=N 0 : N c φ 0 R p =N/τ R p R cp R p The output power N 0 and τ c are known I s : satura,on intensity A b : laser beam cross- sec,on area 24/04/15 12

Steady state: N 0, φ 0, P out, η s (4L) Slope efficiency P out dp out =η s dp p Special case: lamp and diode (transverse) pumping (ac,ve medium is uniformly pumped) Since p th P p Transverse efficiency Quantum efficiency Output coupling efficiency Pump efficiency 24/04/15 13 A: cross- sec,onal area of the pumped region of the ac,ve medium

Nd:YAG example 1% atomic doping, lamp- pumped Mul,- mode Space- independent model approximately valid P th =2.2kW η s =2.4% N c 5.7 10 16 ions/cm 3 ; N tot =4.1 10 20 ions/cm 3 ; PI frac,on: 0.04% Q: how to calculate N c, from the figure and other parameters? 24/04/15 14

Threshold and steady states (q3l) 24/04/15 15

Spa,al- dependent case Pump and laser mode densi,es are not uniform Consequence: Rp, u 2, N are no longer uniform. Threshold condi*ons: P th : depends on w 0, and w p (if longitudinal- diode pumping) or a (if transverse pumping) P out : depends on w 0, and w p or a η s : (especially η t ) depends on w 0, and w p or a 24/04/15 16

R 2 for op,mum coupling (4L) R 2 γ 2 =- lnr 2 Condi*on: or where Op,mum state: where P mth is the minimum threshold pump power, i.e. when γ 2 =0 24/04/15 18

Mul,modeness (l,m,n) Fact: Δν << Δν 0 Reason: L=1m Δν=150MHz while Δν 0 =1~300GHz Homogeneously- broadened gain line g=σ(n 2 -N 1 ) Δν Δν 0 ν φ loss R p R cp N Spa,al hole burning R p <R cp Inhomogeneously- broadened gain line Comments: o Spa,al hole burning does not apply for inhomogeneous- line case o Homogeneous- line case: a few modes around the gain center survive Spectral hole burning 24/04/15 20

Single- mode selec,on Single transverse- mode selec*on (l,m) o Aperture (Fresnel no. a 2 /(cλ)< 2) o Unstable resonators (if ac,ve rod has large Φ) Single longitudinal- mode selec*on (n) o Shorter cavity? ( ) o Fabry- Pérot etalon o Ring resonators Condi*on: Mode discrimina*on Align by tuning θ Necessary condi,on: L can be relaxed by 2F Mul,ple FPs can be used 24/04/15 21

Laser tuning Mode discrimina*on Mo*va*on: To harness wide gain linewidth Δν 0 (dye or vibronic solid- state lasers) To lase at one of the many transi,on lines MIR lasers Tuning: gra,ng rota,on VIS- NIR lasers Tuning: prism rota,on 24/04/15 22

Laser tuning Φ=0 24/04/15 23

Laser tuning Φ=0 24/04/15 24

Laser tuning Φ=0 24/04/15 25

Laser tuning Φ=15 24/04/15 26

Laser tuning Φ=30 24/04/15 27

Laser tuning Φ=45 24/04/15 28

Laser tuning Φ=60 24/04/15 29

Laser tuning Φ=75 24/04/15 30

Laser tuning Φ=90 24/04/15 31

Laser tuning Φ=45 Aler 1 st polarizer Incidence is first split into e- and o- rays Aler prop. through plate n e =1.46 n o =1.45 L e =1mm Aler 2 nd polarizer 24/04/15 32

Others Frequency pulling o Formula is exact for homogeneous line o Very small: ν L ν 0 /1000+ν c Frequency fluctua*on [cavity length: L e =n(l- 1)+n a l] o Long- term (>1s): T, ambient pressure o Short- term (<1s): mirror vibra,on, n or n a change, acous,c wave o Stabiliza,on: passive (isola,on) or ac,ve (feedback system) Intensity noise o Gas: Pp, discharge, cavity o Dye: jet density, bubbles o Solid- state: Pp, cavity o Semiconductor: I bias, E- H recombina,on noise o Reduc,on: feedback system 24/04/15 34

Intensity noise 24/04/15 35

Contents Content 1. Rate equa,ons 1. Four- level; 2. Quasi- three- level 2. Threshold and steady states 1. Four- level; 2. Quasi- three- level Time 25 15 3. Op,mum output coupling 5 4. Tuning and single- mode selec,on 30 5. Others Frequency- pulling, fluctua,ons 5 Total: 80 24/04/15 36