Ultrafast Laser Physics

Transcription

1 Ultrafast Laser Physics Ursula Keller / Lukas Gallmann ETH Zurich, Physics Department, Switzerland Chapter 7: Active modelocking Ultrafast Laser Physics ETH Zurich

2 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"

3 Active Modelocking

4 Passive Modelocking

5 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

6 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)#

7 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 nm E t = A t exp iω c t + iϕ 0 (t) CEO phase controlled in laser oscillator

8 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

9 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,...

10 Balance between loss modulation and gain

11 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

12 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

13 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: µm

14 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 = g M 1 f m Δ f g D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum. Electron. 6, 694, 1970

15 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 = g M 1 f m Δ f g D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum. Electron. 6, 694, 1970

16 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 ω ω 0 Δω g E ( ω ) frequency dependent gain parabolic approximation: Gaussian pulse stays Gaussian pulse after amplification g Δω g 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, Γ = 1 Γ + 16g Δω g Γ Γ = 16g Δω g Γ Γ 16g Δω g Γ

17 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]

18 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

19 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

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

21 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

22 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

23 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)

24 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)

25 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

26 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

27 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 = g M 1 ω m Ω g = 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

28 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 )

29 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

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

31 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)

32 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)

33 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 = Mω m D g + φ nl 4D g g l Mω m τ φ nl τ = 0 4D g

34 Example Nd:YLF laser Autocorrelation Spectrum FWHM = 17 ps 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 Relative bandwidth [nm] τ p,fwhm = Mω m D g + φ nl 4D g = s s 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

35 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

36 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

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

38 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

39 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

40 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

41 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» % <<

42 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

43 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

44 Dispersion tuning: Pulses behave like ideal solitons GDD negative GVD, fs τ = 4D δ W t... pulse width D... GDD d... SPM coefficient W... pulse energy Pulse width, ps Pulse width Time-bandwidth-product ideal Time-Bandwidth-Product Prism Position, mm typical soliton behaviour linear dependence of τ on total dispersion Nd:glass (phosphate), mm thick

45 Pulse width reduction due to soliton formation pulsewidth reduction R without reshaped gain profile (Nd:phosphate) flattened net gain profile R= AML pulsewidth (without SPM, GVD) pulsewidth of soliton 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 = D n,min = (9Φ 0 /) D g M S (9Φ 0 /) D g M S

46 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)