Chapter 6. Laser: Theory and Applications

Chapter 6.

Laser: Theory and Applications

Reading: Sigman, Chapter 6, 7, and 26 Bransden & Joachain, Chapter 15 Laser Basics Light Amplification by Stimulated Emission of Radiation hν = E − E E1 1 0 Stimulated emission hν

E0 Population inversion (N1 > N0) ⇒ laser, maser 2 2 4π ⎛ e ⎞ I(ω ) 2 Transition rate W = ⎜ ⎟ 01 M (ω ) ∝ I(ω ) for stimulated emission 01 2 ⎜ ⎟ 2 01 01 01 m c ⎝ 4πε0 ⎠ ω01 Incident light intensity

Pumping (optical, electrical, etc.) for population inversion

Gain medium High reflector Out coupler Optical cavity Longitudinal Modes in an Optical Cavity EM wave in a cavity Boundary condition: λ cτ πc L = m = m = m m = 1, 2, 3, … 2 2 ω 2L c π c ⇒ λ = ,ν = m, ω = m m 2L L 2L Round-trip time of flight: T = = mτ c Typical laser cavity: L = 1.5 m, λ = 0.75 µm

2L 3 m : T = = =10−8 sec =10 nsec : c 3×108 m / sec 1 ⇒ ν = =108 Hz =100 MHz R T 2L 3 m L m = = = 4×106 = 4 milion !! λ 0.75×10−6 m Single mode

2 2 I(t) = cosω0t = cost

Spectrum Intensity 1 Frequency Intensity

-100 -50 0 50 100 Time Cavity Quality Factors, Qc End mirror L Out coupler R ≈ 1 T = 1 ~ 5 % Energy loss by reflection, transmission, etc.

∞ −δct /T −iω0t E(ω) = E0e e dt ∫0 E e−δct /T e−iω0t 0 E = 0 i(ω −ω0 ) +δc /T t

E(ω) 2 Lorentzian Emission spectrum 2 2 E δ c ω E(ω) = 0 ~ = 2 2 2 T Q (ω −ω ) +δ /T c 0 c ω ω0 Energy of circulating EM wave, Icirc(t) ⎡ t ⎤ 2L T = Icirc (t) = Icirc (0)×exp⎢−δ c ⎥, : round-trip time of flight ⎣ T ⎦ c Number of round trips in t ⎡ ω ⎤ ⇒ Icirc (t) = Icirc (0)×exp⎢− t⎥ ⎣ Qc ⎦ Q-factor of a RLC circuit ωT 4π L 1 ω ωL Q = = Q = = where c ∆ω R δ c λ δ c

Typical laser cavity: L = 1 m, λ = 0.8 µm, δc = 0.01 (~1% loss/RT) 4π 1 Q = ≈1.6×109 ∆ω ≈106 Hz c 0.8×10−6 0.01 Two Level Rate Equations and Saturation

N2 E2 Two Level Rate Equation W N γ N W N dN (t) dN (t) 21 2 21 2 12 1 1 = − 2 dt dt

E1 = −W12 N1(t) + (W21 + γ 21)N2 (t) N1 1 = −W12 []N1(t) − N2 (t) + γ 21N2 (t) γ 21 = : non-radiative decay rate T1

Total number of atoms: N = N1(t) + N2 (t) = constant

Population difference: ∆N(t) = N1(t) − N2 (t) d ∆N(t) = −2W []N (t) − N (t) + 2γ N (t) dt 12 1 2 21 2

= −2W12 []N1(t) − N2 (t) −γ 21[N1(t) − N2 (t) − N1(t) − N2 (t)]

d ∆N(t) − N ∆N(t) = −2W12∆N(t) − dt T1 Steady-State Atomic Response: Saturation d ∆N(t) − N ∆N(t) = 0 = −2W12∆N(t) − dt T1

N ∆Nss 1 1 ∆N = ∆Nss = = , Wsat ≡ 1+ 2W12T N 1+W12 /Wsat 2T1

1.2 Gain coefficient 1.0 in laser materials 0.8 α m ∝ ∆N ∆N ss N 0.6 α α (I) = m0 0.4 m Saturation 1+ I / I sat Intensity, I=I 0.2 sat

0.0 0.01 0.1 1 10

Signal strength, 2W12T Transient Two-Level Solutions d ∆N(t) − N ⎛ 1 ⎞ N ⎜ ⎟ ∆N(t) = −2W12∆N(t) − = −⎜2W12 + ⎟∆N(t) + dt T1 ⎝ T1 ⎠ T1 ⎡ ⎛ 1 ⎞ ⎤ N ∆N(t) = ∆N ss + Aexp⎢− ⎜2W12 + ⎟t⎥, ∆N = ⎜ T ⎟ ss ⎣ ⎝ 1 ⎠ ⎦ 1+ 2W12T

Initial population difference at t = 0: ∆N(0) = ∆N ss + A N ⎡ N ⎤ ⎡ t ⎤ ∆N(t) = + ⎢∆N(0) − ⎥ exp⎢− ()1+ 2W12T1 ⎥ 1+ 2W12T ⎣ 1+ 2W12T ⎦ ⎣ T1 ⎦ 1.0 2W12T1 = 0 0.5 Transient saturation behavior ∆N(t) 1 0.5 following sudden turn-on of an N 2 applied signal 4

0.0 0123 t /T1 Two-Level Systems with Degeneracy

Stimulated transition rates: g1W12 = g2W21 N2 g Rate equation: E2 2 dN (t) dN (t) 1 = − 2 = −W N (t) + (W + γ )N (t) dt dt 12 1 21 21 2 E1 g1 Population ⎛ g ⎞ N ∆N(t) ≡ ⎜ 2 ⎟N (t) − N (t) 1 difference: ⎜ ⎟ 1 2 ⎝ g1 ⎠ 1 Effective signal-stimulated transition probability: W ≡ ()W +W eff 2 12 21 d ∆N(t) − N Rate equation: ∆N(t) = −2Weff ∆N(t) − dt T1

Atomic time constants: T1 and T2

T1 : longitudinal (on-diagonal) relaxation time population recovery or energy decay time

T2 : dephasing time, transverse (off-diagonal) relaxation time time constant for dephasing of coherent macroscopic polarization Steady-State Laser Pumping and Population Inversion

Four-level pumping analysis Rate equation for level 4 E4 τ43 (fast) E 3 pump transition probability, W14 = W41 = Wp pumping τ 32 dN W (slow) 4 = W (N − N ) − (γ + γ + γ )N p dt p 1 4 43 42 41 4

= Wp (N1 − N4 ) − N4 /τ 4 , E2 1 τ (fast) where ≡ γ 4 = γ 43 + γ 42 + γ 41 E1 21 τ 4

Steady state population: W τ N = p 4 N ≈ W τ N , if W τ <<1 4 1+W τ 1 p 4 1 p 4 p 4 Normalized pumping rate Rate equations for level 2 and 3 at steady state

dN3 N4 N3 τ 3 = γ 43 N4 − (γ 32 + γ 31)N3 = − N3 = N4 dt τ 43 τ 3 τ 43

E4 τ43 (fast) τ >>τ so that N >> N In a good laser system, 3 43 3 4 E3 pumping τ32 Wp (slow) dN N N N 2 = γ N + γ N −γ N = 4 + 3 − 2 E dt 42 4 32 3 21 2 τ τ τ 2 42 32 21 τ (fast) E1 21 ⎛τ τ τ ⎞ τ τ τ ⎜ 21 43 21 ⎟ 21 43 21 N2 = ⎜ + ⎟N3 = β N3 where β ≡ + ⎝τ 32 τ 42τ 3 ⎠ τ 32 τ 42τ 3 : population inversion on the 3 → 2 transition If β < 1, N2 < N3

In a good laser system, γ 42 ≈ 0 so that the level 4 will relax primarily into the level 3. τ N τ β ≈ 21 condition for population inversion β ≡ 2 ≈ 21 <<1 τ 32 N3 τ 32 Fluorescent quantum efficient The number of fluorescent photons spontaneously emitted on the laser transition divided by the number of pump photons absorbed on the pump transitions when the laser material is below threshold γ γ τ τ η = 43 × rad = 4 × 3 γ 4 γ 3 τ 43 τ rad Fraction of the total atoms excited to Fraction of the total decay out of level level 4 relax directly into the level 3 3 is purely radiative decay to level2

Four level population inversion N = N1 + N2 + N3 + N4

N N − N (1− β )ηW τ 4-level system 3 2 = p rad N 1+ []1+ β + 2τ 43 /τ rad ηWpτ rad ∆Nth N 0 ∆ In a good laser system, τ rad >>τ 43 , β →0. 3-level system N − N (1− β )ηW τ ηW τ -N 3 2 ≈ p rad ≈ p rad N 1+ (1+ β )ηWpτ rad 1+ηWpτ rad 0123 Wpτrad Laser gain saturation analysis

N3 E3 γ32 Pumping transition

E2 N 2 dN3 = Wp (N0 − N3 ) ≈ Wp N0 ≈ Wp N pumping stimulated dt pump γ21 W Wp sig

E Effective pumping rate: Rp =η p Wp N0 1 N1

γ quantum efficiency E3 → E2 E0 1 N0≈N

Rate equations for laser levels 1 and 2: γ 2 = γ 21 + γ 20

Wsig + γ 21 dN2 N1 = Rp = Rp −Wsig (N2 − N1) −γ 2 N2 dt Wsig (γ 1 + γ 20 ) + γ 1γ 2

W + γ dN1 sig 1 = Wsig (N2 − N1) + γ 21N2 −γ 1N1 N2 = Rp dt Wsig (γ 1 + γ 20 ) + γ 1γ 2 Gain saturation behavior ⎛ γ −γ ⎞ 1 ⎜ 1 21 ⎟ ∆N21 = N2 − N1 = ⎜ ⎟Rp × ⎝ γ 1γ 2 ⎠ 1+ [](γ 1 + γ 20 ) /γ 1γ 2 Wsig 1 = ∆N0 1+Wsigτ eff ⎛ γ −γ ⎞ ⎛ τ ⎞ Small signal or unsaturated ⎜ 1 21 ⎟ ⎜ 1 ⎟ ∆N0 = ⎜ ⎟Rp = ⎜1− ⎟× Rpτ 2 population inversion ⎝ γ 1γ 2 ⎠ ⎝ τ 21 ⎠ 1 γ γ ⎛ τ ⎞ 1 2 or ⎜ 1 ⎟ Effective recovery time = τ eff =τ 2 ⎜1+ ⎟ τ eff γ 1 + γ 20 ⎝ τ 20 ⎠ 1 If γ20 ≈ 0, γ2 ≈γ21 ∆N21 = Rp (τ 2 −τ1)× 1+Wsigτ 2

• Population inversion requires τ21 > τ1.

• ∆N0 ∝ Rp ×τ 2 (1−τ1 /τ 2 )

• If τ1 → 0, τeff ≈τ2.

• The saturation intensity of the inverted population is independent of Rp. Wave Propagation in an Atomic Medium

Wave equation in a laser medium 2 2 [∇ +ω µε()1+ χat − iσ /ωε ]E(x, y, z) = 0 atomic Ohmic loss susceptibility Plane wave approximation 2 ⎡ d 2 ⎤ ⎢ 2 + β ()1+ χat − iσ /ωε ⎥E(z) = 0 ⎣dz ⎦ ω 2π n “Free-space” propagation constant: β = ω µε = n = c λ −Γz Propagation factor: E(z) = E0e 2 2 Γ = −β ()1+ χat − iσ /ωε

Γ = iβ 1+ χat − iσ /ωε = iβ 1+ χ′(ω) + iχ′′(ω) − iσ /ωε Usually, χ , − iσ /ωε <<1 a at χ (ω) = at ω 2 −ω 2 + iΓω ⎡ 1 1 σ ⎤ a Γ ≈ iβ ⎢1+ χ′(ω) + i χ′′(ω) − i ⎥ ()ω 2 −ω 2 + iΓω ⎣ 2 2 2ωε ⎦ = a a 2 2 2 2 2 ()ω −ωa + Γ ω 1 1 σ χ′(ω) = iβ + i βχ′(ω) − βχ′′(ω) + ()2 2 2 2 2ε v a ω −ωa = 2 ()ω 2 −ω 2 + Γ2ω 2 1 a = iβ + i ∆βm (ω) −α m (ω) +α0 aγ ω 2 + i 2 2 2 2 2 ()ω −ωa + γ ω Propagation of a +z traveling wave χ′′(ω)

E(z,t) = Re E0 exp{iω t − i[β + ∆βm (ω)]z + [α m (ω) −α0 ]z} Phase shift Gain by atomic transiton by atomic transition + ohmic loss

The effects of ohmic losses and atomic transition are included. Propagation factors [] φtot (z,ω) = β + ∆βm (ω) z “free-space” Contribution β(ω) z 2π total phase shift a(ω 2 −ω 2 ) ∆β ∝ χ′(ω) = a m 2 2 2 2 2 ()ω −ωa + Γ ω

χ0′[]2(ω −ωa ) / Γ ≈ 2 1+ []2()ω −ωa / Γ ω

atomic phase-shift contribution ∆βm (ω)z ω ωa χ′′ atomic gain 0 α m (ω)z α m ∝ χ′′(ω) ≈ 2 1+ []2()ω −ωa / Γ ω ωa Single-Pass Laser Amplification

z = 0 z = L Gain medium Laser gain formulas E(L) Complex amplitude gain: g(ω) ≡ = exp{}− i[]β + ∆β (ω) L + [α (ω) −α ]L E(0) m m 0 Total phase shift amplitude gain or loss I(L) 2 Power or intensity gain: G(ω) ≡ = g()ω = exp{}2[]α m (ω) −α0 L I(0) 1 = βχ′′(ω) 2 Midband value χ0′′ Lorenzian transition line shape: χ′′(ω) = 2 1+ []2(ω −ωa ) / ∆ωa ⎧ω Lχ0′′ 1 ⎫ Power gain: G(ω) = exp⎨ × 2 ⎬ ⎩ v 1+ []2(ω −ωa ) / ∆ωa ⎭ Power gain in decibels (dB): 4.34ω L G (ω) ≡10log G(ω) = 4.34ln G(ω) = a χ′′(ω) dB 10 v Amplification bandwidth and gain narrowing

3 3-dB amplifier bandwidth: ∆ω3dB = ∆ωa GdB (ωa ) − 3

Amplifier phase shift ωL βL Total phase shift: φ (z,ω) = []β + ∆β (ω) L = + χ′(ω) tot m v 2 Atomic transition phase shift: ⎛ ω −ω ⎞ G (ω ) 2(ω −ω ) / ∆ω ∆β (ω)L = ⎜2 a ⎟×α (ω)L = dB a × a a m ⎜ ⎟ m 2 ⎝ ∆ωa ⎠ 20log10 e 1+ []2(ω −ωa ) / ∆ωa Saturation Intensities in Laser Materials

Saturation of the population difference dI Traveling wave: = 2α I = ∆Nσ I dz m Stimulated transition cross-section 1 1 Population difference: ∆N = ∆N0 × = ∆N0 × 1+Wτ eff 1+ I / I sat

1 dI(z) 2α = 2α (z) = m0 m where 2α m0 ≡ ∆N0σ I(z) dz 1+ I(z) / I sat

I =Iout ⎡1 1 ⎤ L ⎢ + ⎥dI = 2α m0 dz ∫I =Iin I I ∫0 ⎣ sat ⎦ unsaturated power gain ⎛ I ⎞ I − I ⎜ out ⎟ out in where G ≡ exp(2α L) ln⎜ ⎟ + = 2α m0 L = lnG0 0 m0 ⎝ Iin ⎠ I sat Overall power gain:

Iout ⎡ Iout − Iin ⎤ G ≡ = G0 ×exp⎢− ⎥ Iin ⎣ I sat ⎦

⎡ (G −1)Iin ⎤ ⎡ (G −1)Iout ⎤ = G0 ×exp⎢− ⎥ = G0 ×exp⎢− ⎥ ⎣ I sat ⎦ ⎣ GI sat ⎦

I 1 ⎛ G ⎞ I G ⎛ G ⎞ out = ln⎜ 0 ⎟ and out = ln⎜ 0 ⎟ I sat G −1 ⎝ G ⎠ I sat G −1 ⎝ G ⎠

Power extraction and available power

⎛ G0 ⎞ Iextr ≡ Iout − Iin = I sat ×ln⎜ ⎟ ⎝ G ⎠

⎡ ⎛ G0 ⎞⎤ Iavail ≡ lim⎢I sat ×ln⎜ ⎟⎥ = I sat ×ln()G0 = 2α m0 L× I sat G→1⎣ ⎝ G ⎠⎦ Specific Laser Systems Laser media: gas, dye, chemical, excimer, solid-state, fiber, semiconductor, free-electron

http://en.wikipedia.org/wiki/Image:Laser_spectral_lines.png Ruby Laser • This system is a three level laser with

lasing transitions between E2 and E1. • The excitation of the Chromium ions is done by light pulses from flash lamps (usually Xenon). •The Chromium ions absorb light at wavelengths around 545 nm (500-600 nm). As a result the ions are transferred to the

excited energy level E3. • From this level the ions are going down to

the metastable energy level E2 in a non- radiative transition. The energy released in this non-radiative transition is transferred to the crystal vibrations and changed into heat that must be removed away from the system.

• The lifetime of the metastable level (E2) is about 5 msec. Energy level diagram of Ruby laser http://web.phys.ksu.edu/vqm/laserweb/Ch-6/C6s2t1p2.htm He-Ne Laser

Energy level diagram

Schematics http://en.wikipedia.org/wiki/Helium-neon_laser Excimer Lasers

Excimer Wavelenth Fe 157 nm ArF 193 nm KrF 248 nm XeCl 308 nm

Applications • Marking • Gain medium: inert gas (Ar, Kr, Xe etc.) • Micromachining + halide (Cl, F etc.) • Laser Ablation • Excited state is induced by an electrical • Laser Annealing discharge or high-energy electron beams. • Surface Structuring • Laser action in an excimer molecule • Laser Vision Correction occurs because it has a bound • Optical Testing and Inspection (associative) excited state, but a repulsive • Pulsed Laser Deposition (disassociative) ground state. • Fiber Bragg Gratings http://tftlcd.khu.ac.kr/research/poly-Si/chapter4.html Semiconductor Lasers – laser diodes

e h e h

forward bias off forward bias on

Band structure near a semiconductor p-n junction

Diode laser structure Vertical cavity surface emitting lasers (VCSEL) structure

en.wikipedia.org/wiki/Laser_diode, www.mtmi.vu.lt/pfk/funkc_dariniai/diod/led.htm Laser Q-Switching Q-switched laser output: short and intense burst of laser output dumping all the accumulated population inversion in a single short laser pulse (~10 ns) loss High initial cavity loss t loss Pumping process builds up a large population inversion gain t loss Cavity loss is suddenly gain “switched” to low value t loss laser “giant pulse” laser action output gain takes place t Laser Q-switching techniques

Rotating mirror

Electro-optic

Acousto-optic

Saturable absorber Active Q-switching: rate-equation analysis

E2 n(t)

N1 ≈ 0

Coupled rate equations dn(t) dN(t) = KN(t)n(t) −γ n(t) = R −γ N(t) − Kn(t)N(t) dt c dt p 2 n(t) : cavity photon number N(t) = N2 − N1: inverted population difference γc = 1/τc : total cavity decay rate Rp : pumping rate γ2 : decay rate for N K : coupling coefficient between photons and atoms Pumping interval, and population build-up loss Pumping process builds up gain a large population inversion t

n(t) = 0 and a pump pulse with constant intensity Rp is turned on at t = 0. dN(t) = R − N(t) /τ ⇒ N(t) = R τ []1− exp(−t /τ ) dt p 2 p 2 2

Rpτ 2 Energy storage during the pumping interval for N(t) a fixed pumping rate Typical solid state

lasers: τ2 ≈ 200 µs τ2 2τ2 Pump Pulse Duration Pulse build-up time, Tb loss

Tb Cavity loss is suddenly gain “switched” to low value t Tb (~100 ns) << τ2 (∼100 µs) N N : population inversion just after switching Initial inversion ratio: r ≡ i i Nth Nth : threshold inversion just after switching dn(t) 1 r −1 = KN(t)n(t) − n(t) ≈ K[]Ni − Nth n(t) ≈ n(t) dt τ c τ c

⇒ n(t) = ni exp[](r −1)t /τ c , ni ≈ 1 : initial spontaneous emission noise excitation

Steady state photon number, nss : • Photon number with continuous pumping at a pumping rate r times above its threshold • Stimulate emission (KN(t)n(t)) becomes significant.

• nss << np (photon number at the peak power) Build-up time, Tb from the initial photon density ni to a photon density nss n ⎡(r −1)T ⎤ τ ⎛ n ⎞ ss b c ⎜ ss ⎟ = exp⎢ ⎥ ⇒ Tb = ln⎜ ⎟ ni ⎣ τ c ⎦ r −1 ⎝ ni ⎠ Ratio of initial to final photon numbers: n ⎛ n ⎞ ss ≈108 to1012 ±20 % in ln⎜ ss ⎟ ⎜ n ⎟ ni ⎝ i ⎠ ()25 ± 5 T T 2L Therefore, T ≈ × where τ = and T = b c δ c r −1 δc c Fractional power loss per round trip Examples: 1. Flash-pumped Nd:YAG laser

L = 60 cm (T = 4 ns), R=0.35 (δc = ln(1/R)=1.05), r = 3 ⇒ Tb ≈ 50 ns

2. cw-pumped Nd:YAG laser or low-gain CO2 laser L = 2 m (T = 12 ns), R=0.8 (δc = ln(1/R)=0.22), r = 1.5 ⇒ Tb ≈ 3 µs Pulse output interval dn(t) gain = KN(t)n(t) −γ n(t) = K[]N(t) − N n(t) laser dt c th loss output dN(t) t = Rp −γ 2 N(t) − Kn(t)N(t) ≈ −Kn(t)N(t) dt

Initial conditions: N = Ni = rNth and n = ni =1 at the switching time t = ti dn N − N n(t ) N (t ) ⎛ N ⎞ = th ⇒ dn = ⎜ th −1⎟dN ∫n ∫N dN N i i ⎝ N ⎠

N (t ) ⎛ Nth ⎞ N(t) ⇒ n(t) ≈ ⎜ −1⎟dN = Nth ln − (N(t) − Ni ) ∫N i ⎝ N ⎠ Ni N Ni Ni where r = i ⇒ n(t) ≈ Ni − N(t) − ln N r N(t) th Time evolution of n(t) and N(t) during the pulse output interval

N(t) Ni

n(t)

Nth

Nf ni ≈ 0 nf ≈ 0 t Mode-Locking Mode-locking is a technique in optics by which a laser can be made to produce ultrashort pulses with the pulse width of the order of subpicoseconds (< 10-12). The basis of the technique is to induce a fixed phase relationship between the modes of the laser's resonant cavity. The laser is then said to be phase-locked or mode- locked. Interference between these modes causes the laser light to be produced as a train of pulses. Depending on the properties of the laser, these pulses may be of extremely brief duration, as short as a few femtoseconds. - Wikipedia General description of electric field in a laser cavity E(t) = E ei(ωmt+φm ) ∑ m φm : initial phase of the mth mode m The laser is said to be “mode-locked” when the initial phases are equal.

φm = φ : constant Single mode 2 2 I(t) = cosω0t = cost 120 Spectrum 100 Intensity 1 80 Frequency 60

Intensity 40 20 0 -100 -50 0 50 100 Time Three modes in phase 2 I(t) = cos0.9t + cost + cos1.1t

120 Spectrum 0.9 1.1 100 Intensity 1 80 Frequency 60 δω = 0.1

Intensity 40 2π T = = 20π 20 δω 0 -100 -50 0 50 100 Time Eleven modes with random phases 2 I(t) = cos(0.5t +φ0.5 ) + cos(0.6t +φ0.6 ) + ...cos(1.0t +φ1.0 ) + ... + cos(1.5t +φ1.5 )

120 Spectrum 100 Intensity 0.5 1 1.5 80 Frequency 60

Intensity 40 20 0 -100 -50 0 50 100 Time Eleven modes with the same phase, φ = 0 2 I(t) = cos0.5t + cos0.6t + ...cos1.0t + ...cos1.4t + cos1.5t

120 Spectrum 100 Intensity 0.5 1 1.5 80 Frequency 60

Intensity 40 20 0 -100 -50 0 50 100 Time In the time domain, EM field is a periodic repetition: E(t) = E(t + nT ) mπ c i t iωmt L E(t) = ∑ Eme = ∑ Eme m m mπ c mπ c⎛ 2nL ⎞ mπ c i (t+nT ) i ⎜ t+ ⎟ i t =1 L L ⎝ c ⎠ L 2π mni E(t + nT ) = ∑∑Eme = Eme = ∑ Eme e = E(t) mm m T 1 iωt Spectrum for one period: E(ω) = E(t)e dt Note Em = E(ωm ) T ∫0 Normalized spectrum for N periods:

1 NT 1 N −1 (n+1)T E N (ω) = E(t)eiωt dt = E(t)eiωt dt ∫0 ∑∫nT NT NT n=0 1 N −1 T 1 N −1 1 T = E(t'+nT )eiω(t'+nT )dt' = eiω nT E(t')eiωt'dt' ∑∫0 ∑ ∫0 NT n=0 N n=0 T ⎛ 1 N −1 ⎞ 1 1− eiω NT = eiω nT E(ω) = E(ω) ⎜ ∑ ⎟ iωT ⎝ N n=0 ⎠ N 1− e Normalized power spectrum for N periods: 2 iω NT 2 1 1− e 2 I N (ω) = E N (ω) = E(ω) N 2 1− eiωT 1 sin2 (NωT / 2) = I(ω) N 2 sin2 (ωT / 2) 2π δω = T 1 δ f ≈ NT Periodic comb

sin2 (NωT / 2) ∞ ⎛ 2π n ⎞ When N goes to infinity, ⇒ δ ⎜ω − ⎟ 2 N →∞ ∑ sin (ωT / 2) n=−∞ ⎝ T ⎠ Dirac comb Periodic repetition of EM wave in a laser cavity in the time domain E(t) = E(t + nT )

ω0

T τ p

Laser power spectrum I(ω)

2π π c δω = = T L 1 ∆ω ≈ τ p

δ f ω0 120 Spectrum 100 80

Single ity 60

mode Intens 40 20 Intensity 0 120 0.5 1 1.5 Three 100 80 ity modes 60

Intens 40 in phase 20 0 Intensity 120 0.5 1 1.5 100 80 Eleven modes ity 60 random phases Intens 40 20 Intensity 0 0.5 1.5 120 1 100 80 Eleven modes ity 60 same phase Intens 40 20 0 Intensity -100 -50 0 50 100 0.5 1 1.5 Time Frequency Net gain and cavity modes Net gain spectrum 2π π c 2 2 ln 2 1 δω = = ∆ω = T L π τ p

… … ω−4 ω−3 ω−2 ω−1 ω0 ω1 ω2 ω3 ω4 ω ωn = ω0 + nδω iω t Total electric field with phase-locked modes: n E(t) = ∑ Ene If the gain spectrum is a Gaussian distribution, n ⎡ 2 ⎤ ⎡ 2 ⎤ ⎛ ωn −ω0 ⎞ ⎛ nδω ⎞ En = E(ωn ) = E0 exp⎢− 4ln 2⎜ ⎟ ⎥ = E0 exp⎢− 4ln 2⎜ ⎟ ⎥ ⎣⎢ ⎝ ∆ω ⎠ ⎦⎥ ⎣⎢ ⎝ ∆ω ⎠ ⎦⎥ 2 ⎡ ⎛ t − mT ⎞ ⎤ E(t) = eiω0t E(ω )einδωt = E eiω0t exp⎢− 2ln 2⎜ ⎟ ⎥ ∑ n 0 ∑ ⎢ ⎜ τ ⎟ ⎥ n m ⎣ ⎝ p ⎠ ⎦ ω0 t T τ p Net gain spectrum 2π π c 2 2 ln 2 1 δω = = ∆ω = T L π τ p ωn = ω0 + nδω … … ω−4 ω−3 ω−2 ω−1 ω0 ω1 ω2 ω3 ω4 ω m For a laser cavity, L = 1.5m: 2 2π π c λ δω = = = 6.28×108 Hz = 0.628 GHz, ∆λ ≈ ∆ω T L 2π c Gain Medium Bandwidth Number of modes Minimum Pulse

(wavelength) ∆λ (nm) m = ∆ω / δω duration, τp Ar+-ion (520 nm) ~ 0.007 ~ 80 ~ 150000 fs Rubi (694.3 nm) ~ 0.2 ~ 1300 ~ 6000 fs Nd:YAG (1064 nm) ~ 10 ~ 25000 ~ 120 fs Dye (620 nm) ~ 100 ~ 8×105 ~ 12 fs Ti:Sapphire (800 nm) ~ 400 ~2 ×106 ~ 3 fs Passive and Hybrid Mode-Locking Passive mode-locking is accomplished by interplay between saturable absorber and gain medium without any external modulation.

Saturable Saturable absorber gain medium

) 1 G0 T )

G G /2 1/2 0 Gain ( 1 Transmission ( T0 abs amp Is Intensity (I) Is Intensity (I) Pulse shape modification by saturable absorber and gain medium

Saturable Leading edge absorber absorption

Before After

Leading edge Saturable amplification gain medium

Before After Pulse shortening process due to saturation of the absorption and the gain

Loss

net gain > 0 Gain

Pulse peak is amplified Mode-locked pulse Pulse tail is attenuated t Colliding-pulse mode-locked dye laser

symmetric Pulse amplification compressor G SA laser, 515 nm + Ar

Gain Saturable absorber 620 nm Rh6G DODCI ~50 fs

absorber gain dispersion

t Kerr-Lens Mode-Locking (KLM) Ti:Sapphire femtosecond oscillator

Band width ~ 200 nm

76 MHz 100 fs 13 nJ

M7 OC Ti:Sapphire Low power Slit

High power

self-focusing:n = n0 + n2 I Low power Hard aperture nonlinear medium + hard aperture = Instantaneous saturable absorber : High power Transmission is lower at low intensity Nonlinear compared to high intensity. media Note: Sometimes a hard aperture is not necessary, because larger overlap between the pump beam and the cavity mode in the gain medium at high intensity leads to larger modal gain.

120

ity 100 80 1. Mechanical shocks (e.g., vibrating starter, 60 Intens 40 20 jolting mirror) initiate mode-locking. 0 120

ity 100 80 2. The pulse regime is favored over the 60 Intens 40 20 continuous regime. 0 120

ity 100 80 3. Spectral broadening by SPM supports the 60 Intens 40 20 pulse shortening. 0 Time