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Ultrafast Laser Physics Ursula Keller / Lukas Gallmann ETH Zurich, Physics Department, Switzerland www.ulp.ethz.ch Chapter 7: Active modelocking Ultrafast Laser Physics ETH Zurich

Mode locking by forcing all modes in a laser to operate phase-locked, noise is turned into ideal ultrashort pulses τ 1 Δν ~ I (ω) I (t) ~ I (ω) I (t) +π +π ~ φ (ω) 0 -π φ (t) axial modes in laser not phase- locked " noise" ~ φ (ω) 0 -π φ (t) axial modes in laser phase- locked " ultrashort pulse" inverse proportional to phase- locked spectrum"

Active Modelocking

Passive Modelocking

Frequency comb T R = 1 f rep f rep = Δν ax I T ( ω ) = I ( ω )e i φ ( ω ) n=1 δ ( ω nδω ax ) f rep = Δν ax = Δω ax π = 1 T R I T ( t) = F 1 IT ω { } = I ( t) δ ( t nt R ) = I t nt R n=1 n=1 Pulse train: Spectrum is a frequency comb Single pulse: Spectrum is continuous

Frequency comb T R = 1 f rep f rep = Δν ax I T ( ω ) = I ( ω )e i φ ( ω ) n=1 δ ( ω nδω ax ) f rep = Δν ax = Δω ax π = 1 T R I T ( t) = F 1 IT ω { } = I ( t) δ ( t nt R ) = I t nt R n=1 n=1 Frequency comb: f n = f CEO + nf rep f rep : pulse repetition rate frequency f CEO : carrier envelope offset frequency H.R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, U. Keller Appl. Phys. B 69, 37 (1999)#

Carrier-Envelope Offset (CEO) Phase H.R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, U. Keller Appl. Phys. B 69, 37 (1999) F. W. Helbing, G. Steinmeyer, U. Keller IEEE J. of Sel. Top. In Quantum Electron. 9, 1030, 003 Mode-locked pulse train Pulse envelope A(t) T R CEO phase Δϕ 0 f CEO = Δϕ 0 πt R t Electric field: λ/c =.7 fs @800 nm E t = A t exp iω c t + iϕ 0 (t) CEO phase controlled in laser oscillator

Steady-state condition: axial wavelength: Axial mode spacing e g l e iφ = 1 g = l and φ λ m φ ( λ m ) = k n ( λ m ) L = 4π n( λ m ) L λ m λ m = n ( λ m ) L, m = 1,,3,... m = π m, m = 0,1,,... axial mode spacing: Δφ = φ ( λ m+1 ) φ ( λ m ) = π Δφ dφ dω ω =ω m Δφ dφ dω ω =ω m Δω ax = π Δω ax = dk dω LΔω = L ax Δω ax = π υ g Δω ax = π υ g L, Δν = υ g ax L, Δλ ax = λ c υ g L f rep = Δν ax = υ g λ 0 L

τ p << T R, and f rep = 1 T R Different modes of operation cw: continuous wave Q-switching: single axial mode τ p >> T R, and f rep << 1 T R (fundamental) modelocking: one pulse per cavity roundtrip, multi axial modes, phase-locked harmonic modelocking: f rep = n 1 T R, n =, 3,...

Balance between loss modulation and gain

Acousto-optic amplitude loss modulator U. Keller et al, Opt. Lett. 15, 45, 1990 and Ph.D. thesis U. Keller, Stanford University, Appl. Phys. 1989

Acousto-optic amplitude loss modulator t amplitude loss : E 0 e i ω 0 +ω a + E 0 e i ( ω 0 ω a )t intensity loss : E 0 e i ( ω 0 +ω a )t + E 0 e i ( ω 0 ω a )t L ( t) = E 0 [ 1+ cosω a t] U. Keller et al, Opt. Lett. 15, 45, 1990

AOM modelocked Nd:YLF laser at GHz Diode-pumped Nd:YLF laser: actively modelocked with an acousto-optic modelocker (AOM) AOM: Sapphire substrate, 0.5% loss modulation per watt U. Keller et al, Opt. Lett. 15, 45, 1990 K. J. Weingarten et al, Opt. Lett. 15, 96, 1990 Average output power: 135 mw Pulse duration: 7.1 ps Pulse repetition rate: GHz Wavelength: 1.047 µm

Theory: active modelocking without SPM and GDD Ansatz: Gaussian pulse E ( t) = exp Γ t + iω 0 t Steady-state condition: pulse envelope is not changing after one cavity roundtrip: Γ Γ = 0 Γ Γ = 0 = 16g Δω Γ s + Mω m g Γ s = M g ω m Δω g 4 τ p,s = ln = ln π Re Γ s 4 g M 1 f m Δ f g = 0.4457 4 g M 1 f m Δ f g D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum. Electron. 6, 694, 1970

Ansatz: Gaussian pulse Gaussian pulse after amplification ( E ( t) = exp Γ t + iω 0 t E ( ω ) = exp ω ω 0 ) 4Γ E ( ω ) = exp g ω E ( ω ) = exp g 1 + 4 ω ω 0 Δω g E ( ω ) frequency dependent gain parabolic approximation: Gaussian pulse stays Gaussian pulse after amplification g Δω g 1 + 4 ω ω 0 g 1 and 4( ω ω 0 ) Δω g 1 g 4g Δω g ω ω 0 E 4g ( ω ) exp ω ω 0 Δω g ( exp ω ω 0 ) E ω 4 Γ D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum. Electron. 6, 694, 1970 1 Γ = 1 Γ + 16g Δω g Γ Γ = 16g Δω g Γ Γ 16g Δω g Γ

Gaussian pulse after modulator Amplitude loss modulator: m AM ( t) = exp l t = exp M 1 cosω mt Phase modulator: m FM ( t) = exp[ imcosω m t]

Gaussian pulse after amplitude loss modulator m AM ( t) = exp l t = exp M 1 cosω mt parabolic approximation cosx 1 x + O( x 4 ) m AM ( t) exp Mω m t, t << T m Ansatz: Gaussian pulse E ( t) = exp Γ t + iω 0 t E ( t) = m AM t E ( t) exp Mω m t E ( t) exp Γ t Γ Γ Mω m

Balance between gain and modulator At steady state the pulse shortening of the modulator is compensated by the pulse broadening of the gain. Parabolic approximation: curvature of loss modulation is compared to curvature of gain Γ Γ Mω m Γ Γ 16g Δω Γ g τ p = ln Re Γ pulse shortening pulse broadening

p. 1 and 13 Example Nd:YAG and N:YLF laser

Haus master equation A( T,t) T R T = ΔA i = 0 i Gain: ΔA 1 = g 1+ 1 Ω g d dt A T,t Loss modulator: ΔA M ( 1 cosω m t) A( T,t) Constant loss: A( T,t) T R T ΔA 3 la( T,t) = g 1+ 1 Ω g d dt l M 1 cosω mt A( T,t) parabolic approximation: M ω mt M 1 cosω m t Schrödinger equation for harmonic oscillator H. A. Haus, "Short pulse generation," in Compact Sources of Ultrashort Pulses, I. I. N. Duling, Eds. (Cambridge University Press 1995, New York, 1995) pp. 1-56.

Linearized operators (Appendix 5): Gain Gain dispersion: Ω g Δω g g( ω ) = g( z) L g Δω g 1+ ω ω 0 Ωg = g 1+ ω ω 0 g 1 ω ω 0 Ω g exp g ω A ( ω ) exp g ω A ω 1+ g 1 Δω Ω g = 1+ g g A ω Ω Δω g A ( ω ) Δω = ω ω 0 ΔA 1 = g 1+ 1 Ω g d dt A T,t

Linearized operators (Appendix 5): Modulator A out ( t) = exp M 1 cosω m t A in t e x 1 + x A out ( t) A in ( t) M ( 1 cosω m t) A in ( t) ( t) 1 M 1 cosω m t A in t ΔA = A out ΔA M ( 1 cosω m t) A( T,t)

Linearized operators (Appendix 5): Loss A out ( t) = e l A in ( t) e x 1 + x A out ( t) = e l A in t ΔA 3 = A out ( 1 l) A in ( t) ( t) A in ( t) la in ( T,t) ΔA 3 la( T,t)

Solution to the Master equation g 1+ 1 Ω g d dt l M 1 cosω t m A( T,t) = 0 g Ω g d dt M 1 cosω m t A = l g + λ A Schrödinger equation with a periodic potential (e.g. crystal): m x + V x Ψ = EΨ Does not possess a bound state (Bloch wave functions). However, if periodic potential is deep enough (e.g. for core electrons in crystal), electron wave packet is localized. For a deep cosine-shaped potential we can make a parabolic approximation: M ω mt M 1 cosω m t Schrödinger equation for a harmonic oscillator: solution A n ( t) = W n n π n!τ H n t τ e t τ

Solution to the Master equation A n ( t) = W n n π n!τ H n t τ e t τ Hermite polynomial of grade n, H 0 = 1: τ = D g 4 M s gain dispersion parameter: D g = g Ω g τ p = 1.66τ = 1.66 4 g M 1 ω m Ω g = 0.445 4 g M 1 f m Δ f g curvature of the loss modulation: M s = Mω m The corresponding eigenvalues are given by: The solution with if all other solutions Re λ = 0 < 0 Re λ n is a stable pulse λ n = g n l Mω m τ n + 1 A n ( T,t) = A n ( t)e λ nt T R

Solution to the Master equation λ n = g n l Mω m τ n + 1 Stable solution: 0 = g l 1 Mω mτ λ 0 = 0 λ n < 0 for n 1 D g = g Ω g g = l + 1 Mω mτ g(ω) M s = Mω m 1 Mω mτ = M s τ eq. 51 g = l + 1 Mω mτ = D g τ << 1 eq. 49 l ω g = l + D g τ steady-state condition: g = l ( gain equal loss )

Active modelocking in the spectral domain ΔA( t) = M ( 1 cosω m t) A( t) = m( t) A( t) E (t) ΔA ( ω ) = m ( ω ) A ( ω ) M m ( ω ) = F { M + Mcosω m t} = π Mδ ω = π Mδ ω + M F eiω mt { } + F e iω mt { } + π M δ ( ω ω m ) + δ ( ω + ω m ) 1 M M/ M/ t f - f m f f + f m für M << 1

Active modelocking with SPM (no GDD) Master equation: A( T,t) T R T = ΔA i = 0 i

Linearized operators: self-phase modulation (SPM) n > 0 I(t) I t φ ( t) = kn I ( t) L K = kn L K A( t) δ A( t) leading edge SPM: red Pulsfront Gaussian Pulse Zeitabhängige Intensität trailing Pulsflanke edge SPM: blue t ω ( t) t Verbreiterung des Spektrums Spectral broadening ω t ω 0 0 t δ kn L K = dφ ( t ) dt = δ di ( t) dt E ( L K,t) = A 0,t exp iω 0 t + iφ ( t) = A( 0,t)exp iω 0t ik n ω 0 A( L K,t) = e iδ A A( 0,t)e ik n ω 0 L K δ A <<1 ( ) A 0,t 1 iδ A t L K iδ A( t) e ik n ω 0 L K ΔA SPM iδ A( T,t)

Active modelocking with SPM (no GDD) T R T A T,t = g l + D g t M ( 1 cosω t m ) iδ A( T,t) + iψ A T,t = 0 For an analytical solution need an additional degree of freedom: carrier envelope offset phase E ( z,t) A z,t e i ( ω 0t k( ω 0 )z+ψ ) A out ( t) = e iψ A in ( t) ΔA 5 iψ A( T,t)

Active modelocking with SPM (no GDD) T R T A T,t = g l + D g t M ( 1 cosω mt) iδ A( T,t) + iψ A T,t = 0 Solution: chirped Gaussian pulse = A 0 exp 1 A t t τ ( 1 iβ) τ p = 1.66τ = 1.66 4 Mω m D g + φ nl 4D g g l Mω m τ φ nl τ = 0 4D g

Example Nd:YLF laser Autocorrelation Spectrum FWHM = 17 ps -40-0 0 0 40 Time [ps] FWHM = 0.10 nm (a) (b) T=.5% flat Output 15 cm HR AO modelocker Nd:YLF crystal 5 mm thick HR 1 µm HT pump 15 cm Pump HR flat -0.10 0.00 0.10 Relative bandwidth [nm] τ p,fwhm = 1.66 4 Mω m D g + φ nl 4D g = 1.66 4 1.005 10 6 s.467 10 17 s + 5.09 10 17 s p. 6 τ p,fwhm = 17.8 ps B. Braun, K. J. Weingarten, F. X. Kärtner, U. Keller, Appl. Phys. B 61, 49, 1995

Example Nd:glass laser D. Kopf, F. X. Kärtner, K. J. Weingarten, U. Keller, Optics Lett.19, 146, 1994 output pump beam cw Ti:Saph 7cm R=10cm Nd:glass 4mm R=10cm AOM PD OC x φ : knife SF10 prisms OC knife edge spectral filter = long-pass wavelength filter

Example Nd:glass laser D. Kopf, F. X. Kärtner, K. J. Weingarten, U. Keller, Optics Lett.19, 146, 1994 Output 70 mw TBP = 0.3 Nd:phosphate (+ knife) Nd:silicate (no knife) output pump beam cw Ti:Saph OC 7cm R=10cm Nd:glass 4mm knife SF10 prisms x R=10cm AOM φ : PD

Homogeneous vs. inhomogeneous broadening Nd:phosphate homogeneous behaviour Nd:silicate inhomogeneously broadened no filter needed TBP=0.3 TBP=0.3

Active modelocking with SPM and negative GDD GDD < 0 T R T A T,t = id iδ A T,t t A T,t + g l + D g t M 1 cosω mt A T,t = 0 nonlinear Schrödinger equation

First and second order dispersion Taylor expansion around the center frequency ω 0 : Δω = ω ω 0 k n ( ω ) k n ( ω 0 ) + k n Δω + 1 k Δω n +... First order dispersion: k n = dk n dω Second order dispersion: k n = d k n dω

Linearized operators: group delay dispersion (GDD) A ( z,δω ) = A ( 0,Δω )e iδknz = A 0,Δω e i k n ω 0 +Δω k n ω 0 z k n ( ω ) k n ( ω 0 ) + k n Δω + 1 k Δω n +... A ( z, Δω ) A 0, Δω Δ AGDD exp i 1 k nδω k n Δω <<1 A 0, Δω ( Δω ) i 1 k nδω A ( Δω ) idδω A ( Δω ) 1 i 1 D 1 k n k nδω ω i t ω t = i t Fourier transformation: Δω t ΔA GDD id t

Soliton-like pulse formation and stabilization A(T,t ) : slowly varying field envelope -> Haus master equation T R T A(T,t ) = id + iδ A(T,t) t A(T,t ) + g l + D g t q(t ) A(T,t ) GDD SPM sat.gain - loss net gain filtering loss modulation Nonlinear Schrödinger Equation T R = cavity roundtrip time g D g = Ω g = gain dispersion Ω g = gain bandwidth active modelocking» % <<

Soliton perturbation theory Ansatz: A(T,t) = Asech t τ T exp i Φ 0 T R soliton + small perturbations continuum only GDD & AOM q(t) = M (1-cos(ω M t)) spreading Continuum pulse spectrum Soliton pulse spectrum Loss Saturated gain GDD GDD GDD GDD Time domain w 0 Frequency domain w Dispersion spreads continuum out where it sees more loss

Example Nd:glass laser Initially interference of soliton pulse with continuum Over >1000 round trips continuum starts to decay Stable soliton pulses are obtained Soliton modelocking τ= 4D δ ep

Dispersion tuning: Pulses behave like ideal solitons GDD negative GVD, fs 550 500 450 400 350 300 τ = 4D δ W t... pulse width D... GDD d... SPM coefficient W... pulse energy Pulse width, ps 1.4 1. 1.0 0.8 0.8 Pulse width Time-bandwidth-product 1.0 1. 1.4 1.6 ideal 1.8 0.35 0.30 0.5 0.0 0.15 0.10 0.05 0.00 Time-Bandwidth-Product Prism Position, mm typical soliton behaviour linear dependence of τ on total dispersion Nd:glass (phosphate), mm thick

Pulse width reduction due to soliton formation pulsewidth reduction R 1 10 8 6 4 without reshaped gain profile (Nd:phosphate) flattened net gain profile R= AML pulsewidth (without SPM, GVD) pulsewidth of soliton 1 10 100 1000 10000 normalized dispersion D n = D D g τ min = 6 D g Φ 0 M S D... group delay dispersion (total intracavity) D g = g Ω g... gain dispersion Ω g... group delay dispersion (total intracavity) R max = 1.66 1.76 D n,min = 9 1 3 (9Φ 0 /) D g M S (9Φ 0 /) D g M S

Conclusion and outlook active mode-locking + soliton Slow absorber + soliton loss gain Fast saturable absorber loss gain Theory & Experiments: Soliton modelocking (AOM to stabilize soliton) Soliton modelocking (slow saturable absorber to stabilize soliton) Fast saturable absorber (e.g. Kerr lens modelocking - KLM)