Microjoule mode-locked oscillators: issues of stability and noise Vladimir L. Kalashnikov Institut für Photonik, TU Wien, Gusshausstr. 7/387, A-14 Vienna, Austria Alexander Apolonski Department für Physik der Ludwig-Maximilians-Universität München, Am Coulombwall 1, 85748, Germany SPIE Photonics Europe (Brussels, April 1-16, 1)
Motivation and problem definition Requirements over-1 µj pulse energy* pulse width of 1 picosecond or lower MHz repetition rate Aims high-harmonics generation promising the table-top VUV/XUV sources table-top high-field physics approaching PW/cm intensity level micro-machining Oscillator vs. CPA compactness, simplicity and cost higher repetition rate (MHz vs. khz) reduced noise Issues stability? noise properties? *S.V.Marchese, et al., Optics Express 16, 6397 (8); J.Neuhaus, et al., Optics Express 16, 53 (8) SPIE Photonics Europe (Brussels, April 1-16, 1)
Mechanisms of pulse formation I: Soliton Negative GDD SPM pulse profile pulse profile frequency deviation frequency deviation faster components slower components slower components faster components expansion See, for instance, H.A.Haus, et al. JOSA B 8, 68 (1991) SPIE Photonics Europe (Brussels, April 1-16, 1) contraction
Mechanisms of pulse formation II: CSP Pulse shortening Wigner functions soliton chirped pulse spectral cutoff 1.5 1.5-4.5 -.5 5 1.5 1.5-4 -4 -.5 - -.5 - -5-5 4 4 t t t.5 5 ω ω ω.5 time profiles 1.5 1.5.5 4-1.5 1.5 1.5 1.5 1.5 1.5-4 - 4-4 - 4-4 - 4 t t t See, for instance, H.A.Haus, et al., JOSA B 8, 68 (1991); V.L.Kalashnikov, A.Apolonski, PRA 79, 4389 (9) SPIE Photonics Europe (Brussels, April 1-16, 1)
Characteristics of thin-disk oscillators Comparatively narrow gainband 6 nm for Yb:YAG, that is α=36 fs T 1 ps Large group-delay dispersion β T SPM has to be reduced (e.g., airless resonator) E βp/ γ SAM prevails over SPM κ T EA / r s eff γ ΑΜ.5 GW -1, γ air 3 GW -1 κ/γ 5 or μκ/γ 1 5 for the modulation depth μ=.5 Dispersion of gain contributes to dynamics! * Ω g =5.3 THz ( ) A ω = g ( 1 + iω / Ω ) 1 + ω / Ω g ( ) SPIE Photonics Europe (Brussels, April 1-16, 1) g A ω T.Südmeyer, et al., Nat. Photonics, 599 (8); * R.Paschotta, Appl. Phys. B 79, 163 (4)
Basic regimes: anomalous vs. normal dispersion regimes α is the squared inverse gain bandwidth β is the GDD coefficient 1 - Control parameter c = αγ β κ -1. γ is the SPM coefficient κ is the SAM coefficient 8 c = αγ/βκ 1-3 1-4 1-5 ε -.8 1 6 -.6 Variational approach: for overview see, for instance, D.Anderson, et al. Pramana 57, 917 (1) SPIE Photonics Europe (Brussels, April 1-16, 1) 1 E out =5 μj (1% OC) τ -.4 4.1 ps -. 1. ps. 1 1 1 1 1 5 1 15 5 3 a = α/β κ γ, ε = Eκ α, τ = T α ε τ
Model and principal factors * hν S t S t t t δt () ( ) = θ( + κ) δ ( ) μ L = exp 1+ κ P t () ˆ H = iβ iγ air/ platep t t () LH ˆ ˆ HL ˆ ˆ Ak+ 1 t = G Ak t () ˆ () ( ) ˆ At G[ A] = iβg iγ gp() t A() t + t ( g +Δg) Ωg exp g N Ω + 1+ P( t ) dt t g E s Energy scaling: g ( κ )( ) ( t t) A( t ) dt S( t) g g = + 1 + NP T E, E T A av cav s s cav eff SPIE Photonics Europe (Brussels, April 1-16, 1)
Low energy regime: stability thresholds Parameters: SESAM E sat =1 μj/cm, T rel =5 ps; mode size =.5 mm 4 stable relative SPM=1-3, SAM depth =.5% relative SPM=x1-3, SAM depth =.5% relative SPM=x1-3, SAM depth =1% m of air normal dispersion regime Scaling laws E β μ γα 3 β, fs -5-1 -15 - -5-3 stable anomalous dispersion regime E β P γ 6 8 1 1 14 16 18 intracavity energy, μj SPIE Photonics Europe (Brussels, April 1-16, 1)
anomalous dispersion regime Low energy regime: temporal and T β γ P spectral widths Scaling laws T Δ β μ μ α normal dispersion regime T, ps 5. 4.5 4. 3.5 3..5 relative SPM=1-3, SAM depth =.5% relative SPM=x1-3, SAM depth =.5% relative SPM=x1-3, SAM depth =1% PDR Δ, nm 4.4 4. 3.6 3..8 relative SPM=1-3, SAM depth =.5% relative SPM=x1-3, SAM depth =.5% relative SPM=x1-3, SAM depth =1% PDR..4 1.5 1..5 NDR 6 8 1 1 14 16 18 intracavity energy, μj. 1.6 1. 6 8 1 1 14 16 18 intracavity energy, μj NDR SPIE Photonics Europe (Brussels, April 1-16, 1)
Normal dispersion regime: SPM impact Parameters: SESAM E sat =1 μj/cm, T rel =.6 ps, mode size = 1. mm; 4 W AM mode size = 3 mm; average cavity mode size =.4 mm threshold β, fs 1 intracavity: 4 μj intracavity: 8 μj 11 MHz, air spectral power, arb. un. SPM=6e-4 SPM=1e-3 SPM=e-3 SPM=4e-3 SPM=8e-3 1.1..3.4.5.6.7.8 relative SPM -.4 -....4 ω, fs -1 SPIE Photonics Europe (Brussels, April 1-16, 1)
High-energy regime Parameters: SESAM E sat =9 μj/cm, T rel =.5 ps, mode size = 1 mm, 1%; air; 13% AM mode size =.4 mm; average cavity mode size =. mm Normal dispersion regime spectral power, arb. un. Output energy: 6 μj 7 μj 15 μj 48 μj 98 μj power, arb. un. Output energy: 6 μj 7 μj 15 μj 48 μj 98 μj -5. -.5..5 5. ω, ps -1-175 -15-15 -1 t, ps SPIE Photonics Europe (Brussels, April 1-16, 1)
High-energy regime Anomalous dispersion regime spectral power, arb. un. Output energy: 6 μj 13 μj 5 μj 5 μj power, arb. un. Output energy: 6 μj 13 μj 5 μj 5 μj - -1 1 ω, ps -1-75 -5-5 - -175-15 -15 t, ps SPIE Photonics Europe (Brussels, April 1-16, 1)
High-energy regime: pulse width and delay 7. T, ps 1 1 normal dispersion regime anomalous dispersion regime group delay, fs per round trip 6.8 6.6 6. 5.6 5. 4.8 normal dispersion regime anomalous dispersion regime 4 6 8 1 E out, μj 4 6 8 1 E out, μj SPIE Photonics Europe (Brussels, April 1-16, 1)
Timing jitter: dependence on energy σ ( P 4) ( P 4) i= 64 k= 1 k= 1 i= 1 max max 64 Normal dispersion regime Anomalous dispersion regime β, ps.1.1.8.6.4 stable 1 1 8 6 4 σ, fs β, ps -.35 -.3 -.5 -. -.15 -.1 stable 44 4 4 38 36 34 3 σ, fs. -.5 3 4 6 8. 1 15 5 3 35 4 45 5 8 E out, μj E out, μj SPIE Photonics Europe (Brussels, April 1-16, 1)
Timing jitter: dependence on dispersion 1 α=36 fs, E out =7 μj α=9 fs, E out =7 μj α=36 fs, E out =14 μj α=36 fs, E out =6 μj, κ/ 1 σ, fs 1 pulse width, ps 1 1 -.3 -. -.1.1..3.4 β, ps -.3 -. -.1.1..3.4 β, ps SPIE Photonics Europe (Brussels, April 1-16, 1)
Normal dispersion regime: temporal and spectral coherence power, arb. un..6.5.4.3..1 1..8.6.4. coherence ratio spectral power, arb. un. 1.5 1..5 1..8.6.4. coherence ratio. -13-1 -11-1.. -.5..5 t, ps ω, fs -1 Γ a aa * i j i i i j a j j SPIE Photonics Europe (Brussels, April 1-16, 1)
Anomalous dispersion regime: temporal and spectral coherence 1. 15 1. power, arb. un..6.4..8.6.4. coherence ratio spectral power, arb. un. 1 5.8.6.4. coherence ratio. 1.5 15. 17.5..5 t, ps -.4 -....4 ω, fs -1. SPIE Photonics Europe (Brussels, April 1-16, 1)
Analytical theory of timing noise * ( κ) ( f ) + ( κ) ( f ) 4 + h 4 + h ω ( ω) ω () cav π τg cav π cav β ν ν S( f ) = A d + t A t dt ft ET ET () Δ normal disp. 1 A( ) d E ω ω ω.315.65 anomalous disp. T E t A t dt.59t + τ g Ωg T Ξ=Δ cav = Ξ 1.763 16( + κ ) Ξ= 3T S g ( f ) =.5( + κ ) 1 1 S( f ) = S g f ( π f ) + τ T g cavωg ( ) ( π f ) Θ Θ +Ω + π π π ( Θ +Ω ) + ( f ) ( Θ Ω ) + ( f ) 4 *further development of that in: R.Paschotta, Appl. Phys. B 79, 163 (4) SPIE Photonics Europe (Brussels, April 1-16, 1)
1 log 1 S 1 log 1 S -3-34 -36-38 -4-4 -44-46 -3-34 -36-38 -4-4 Analytical timing noise spectra quantum limit normal dispersion regime anomalous dispersion regime 1 4 1 5 1 6 1 7 f, Hz gainband broadening anomalous dispersion regime 1 log 1 S 1 log 1 S -3-34 -36-38 -4-4 -44-46 -3-34 -36-38 -4-4 normal dispersion regime fluctuating gain anomalous dispersion regime 1 4 1 5 1 6 1 7 f, Hz gainband broadening normal dispersion regime -44-46 1 4 1 5 1 6 1 7 f, Hz -44-46 1 4 1 5 1 6 1 7 f, Hz SPIE Photonics Europe (Brussels, April 1-16, 1)
Conclusions The level of stabilizing dispersion is substantially reduced in the normal dispersion energy scalable regime. The pulse durations in the normal dispersion regime are approximately tenfold of those in the anomalous dispersion regime, but the spectra are broader for the former. The timing jitter due to gain fluctuations and quantum noise is substantially reduced for the normal dispersion regime. The source of translation of gain fluctuations into timing jitter is the gain dispersion, which increases with gainband narrowing. One may assume, that the source of such reduction is the enhanced spectral filtering for the chirped dissipative soliton. Analytical results agree with the numerical ones and, also, demonstrate the timing noise reduction in the positive dispersion regime. Acknowledgements This work is supported by Austrian Fonds zur Förderung der wissenschaftlichen Forschung, Project P93 The main results are surveyed in V.L.Kalashnikov, A.Apolonski, arxiv:13.546 [physics.optics] SPIE Photonics Europe (Brussels, April 1-16, 1)