16. Theory of Ultrashort Laser Pulse Generation

Transcription

1 16. Theory of Ultrashort Laser Pulse Generation Active mode-lockin Passive mode-lockin Build-up of modelockin: The Landau- Ginzber Equation loss ain modulator transmission cos(ω t) time The Nonlinear Schrodiner Equation ain > loss Solitons time Reference: Hermann Haus, Short pulse eneration, in Compact Sources of Ultrashort Pulses, Irl N. Dulin, ed. (Cambride University Press, 1995). 1

2 An actual quotation "The Laser is numbered amon the most miraculous ifts of nature and lends itself to a variety of applications." - Pliny (the elder) Natural History XXII, 49 (1st Century AD) The number of uses of compounds made with laser is immeasurable," said Pliny. Amon them were the followin: diuretic, healin ointment for sores, antidote for wounds caused by poison-tipped weapons, snakebites, and scorpion stins, for shrinkin corns and carbuncles, healin do bites, soothin chilblains, alleviatin couhin and wheezin, and as a cure for out, cramps, pleurisy, and tetanus. silphium laciniatum, a modern relative of laser.

3 The burnin question: aussian or sech? After all this talk about Gaussian pulses what does Ti:sapphire really produce? sech 10-4 aussian aussian sech e ( t ) ( t ) τ or sech?? τ 3

4 ode-lockin yields ultrashort pulses losses multiple oscillatin cavity modes laser ain profile Recall that many frequencies ( modes ) oscillate simultaneously in a laser, and when their phases are locked, an ultrashort pulse results. ω q ω q 1 ω q ω q+1 ω q+ ω q+3 possible cavity modes Sum of ten modes with the same relative phase Sum of ten modes w/ random phase 4

5 ode Lockin modulator transmission cos(ω t) Active ode Lockin time Insert somethin into the laser cavity that sinusoidally modulates the amplitude of the pulse. mode competition couples each mode to modulation sidebands eventually, all the modes are coupled and phase-locked Passive ode Lockin Insert somethin into the laser cavity that favors hih intensities. stron maxima will row stroner at the expense of weaker ones eventually, all of the enery is concentrated in one packet Saturable absorber loss ain ain > loss time 5

6 Active mode-lockin: the electro-optic modulator Applyin a voltae to a crystal chanes its refractive indices and introduces birefrinence. A few kv can turn a crystal into a half- or quarter-wave plate. Pockels cell (voltae may be transverse or lonitudinal) V Polarizer If V = 0, the pulse polarization doesn t chane. If V = V p, the pulse polarization switches to its orthoonal state. Applyin a sinusoidal voltae yields sinusoidal modulation to the beam. An electro-optic modulator can also be used without a polarizer to simply introduce a phase modulation, which works by sinusoidally shiftin the modes into and out of the actual cavity modes. 6

7 Active mode-lockin: the acousto-optic modulator An acoustic wave induces sinusoidal density, and hence sinusoidal refractive-index, variations in a medium. This will diffract away some of a liht wave s enery. Pressure, density, and refractive-index variations due to acoustic wave Input beam ω Quartz Acoustic transducer ω Output beam Diffracted Beam (Loss) Such diffraction can be quite stron: ~70%. Sinusoidally modulatin the acoustic wave amplitude yields sinusoidal modulation of the transmitted beam. 7

8 The odulation Theorem: The Fourier Transform of E(t)cos(ω t) Y {E(t)cos(ω t)} = E( t)cos( ω t) exp( iω t) dt 1 = E( t) [ exp( iω t) + exp( iω t) ] exp( iω t) dt 1 1 E( t)exp( i[ ω ω ] t) dt E( t)exp( i[ ω ω ] t) dt = + + Y {E(t)cos(ω t)} = 1 1 Eɶ ( ω ω ) + Eɶ ( ω + ω ) ultiplication by cos(ω t) introduces side-bands. If E(t) = sinc (t)exp(iω 0 t): ω 0 Y {E(t))} ω Y {E(t)cos(ω t)} ω 0 -ω ω 0 ω 0 +ω ω 8

9 Active modelockin modulator transmission cos(ω t) Time In the frequency domain, a modulator introduces side-bands of every mode. ω n ω ω 0 ω n +ω For mode-lockin, adjust ω so that ω = mode spacin. cavity modes Frequency πc/l This means that: ω = π/cavity round-trip time = π/(l/c) = πc/l Each mode competes for ain with adjacent modes. ost efficient operation is for phases to lock. Result is lobal phase lockin. N coupled equations: E n E n+1, E n-1 9

10 odelin laser modes and ain Lasers have a mode spacin: π π c ω = = T L R Gain profile and resultin laser modes ω 0 ω Μ ω Let the zero th mode be at the center of the ain, ω 0. The n th mode frequency is then: ω = ω + nω where n =, -1, 0, 1, n 0 Let a n be the amplitude of the n th mode and assume a Lorentzian ain profile, G(n): G n a = + + nω Ω [ ] ( ) n 1 1 ( ) / + ( nω ) 1 1 Ω a a n n Ω = ain bandwidth 10

11 odelin an amplitude modulator An amplitude modulator uses the electro-optic or acousto-optic effect to deliberately cause losses at the laser round-trip frequency, ω. A modulator multiplies the laser liht (i.e., each mode) by [1 cos(ω t)] 1 1 [ 1 cos( ω t) ] a e = exp( i t) 1 exp( i t) a e n ω + ω n ( ω + ω ) ( ω + ω ) i n t i n t o o iω 1 1 o t in t = a e e e e n + ( 1) ω ω ( + 1) i n t i n ω t Notice that this spreads the enery from the n th to the (n+1) st and (n-1) st modes. Includin the passive loss,, we can write this as: l nω an = an + 1 a n lan + a an + a Ω ( k+ 1) ( k ) ( ) ( k ) ( k ) ( k ) ( k ) ( k ) where the superscript indicates the k th round trip. ( ) n+ 1 n 1 11

12 Solve for the steady-state solution nω an = an + 1 a n lan + a an + a Ω ( ) n+ 1 n 1 ( k+ 1) ( k ) ( ) ( k ) ( k ) ( k ) ( k ) ( k ) In steady state, ( k+ 1) ( k ) a = a n n Also, the finite difference becomes a second derivative when the modes are many and closely spaced: ( k ) ( k ) ( k ) d an+ 1 an + an 1 ω a d ω ( ω) where, in this continuous limit, a ( k) n where: a( ω) ω = nω Thus we have: ω 0 = 1 + Ω ω d dω l a ( ω) 1

13 Solve for the steady-state solution ω 0 = 1 + Ω ω d dω l a ( ω) This differential equation has the solution: ( ω ) = ( ωτ ) a H e ω τ ν / (Hermite Gaussians) with the constraints: 1 4 τ = ω Ω l = ω τ v + 1 In practice, the lowestorder mode occurs: a ω ( ) = Ae ω τ / A Gaussian spectrum! 13

14 A mode lockin: pulse duration ω 0 = 1 + Ω This differential equation has the solution: ω l ( ω ) = ( ωτ ) a H e ω τ ν d a dω ( ω) / with the constraints: 1 4 τ = ω Ω l = ω τ v + 1 The parameter τ determines the spectral bandwidth, which in turn determines the shortest possible pulse duration: 1 τ = 4 Ω ω In other words, the spectral bandwidth of the mode-locked pulse varies as the square root of the ain bandwidth. A mode-lockin does not exploit the full bandwidth of an inhomoeneous medium!

15 Fourier transformin to the time domain Recallin that multiplication by -ω in the frequency domain is just a second derivative in the time domain (and vice versa): becomes: ( k + 1) ( k ) ω ω d an an = 1 a l + Ω dω ( k + 1 ) ( k ( ) ) 1 d k a t a ( t) = 1+ ω l t a t Ω dt which (in the continuous limit) has the solution: v ( i) t a ( t) = H e π ν τ This makes sense because Hermite-Gaussians are their own Fourier transforms. t / τ ( ω) ( ) ( ) The time-domain will prove to be a better domain for modelin passive mode-lockin. 15

16 Other active mode-lockin techniques F mode-lockin produce a phase shift per round trip implementation: electro-optic modulator similar results in terms of steady-state pulse duration Synchronous pumpin ain medium is pumped with a pulsed laser, at a rate of 1 pulse per round trip requires an actively mode-locked laser to pump your laser ($$) requires the two cavity lenths to be accurately matched useful for convertin lon A pulses into short A pulses (e.., 150 psec aron-ion pulses sub-psec dye laser pulses) Additive-pulse or coupled-cavity mode-lockin external cavity that feeds pulses back into main cavity synchronously requires two cavity lenths to be matched can be used to form sub-100-fsec pulses 16

17 Passive mode-lockin Saturable absorption: absorption saturates durin the passae of the pulse leadin ede is selectively eroded Saturable ain: ain saturates durin the passae of the pulse leadin ede is selectively amplified loss ain Fast absorber loss ain Slow absorber ain > loss ain > loss time time 17

18 Kerr lensin is a type of saturable absorber If a pulse experiences additional focusin due to the Kerr lens nonlinearity, and we alin the laser for this extra focusin, then a hih-intensity beam will have better overlap with the ain medium. Hih-intensity pulse Ti:Sapph Low-intensity pulse irror Additional focusin optics can arrane for perfect overlap of the hih-intensity beam back in the Ti:Sapphire crystal. But not the lowintensity beam! This is a type of saturable loss, with the same saturation behavior we ve seen before: I I 1 ~ 1 + I Isat 18

19 Saturable-absorber mode-lockin Nelect ain saturation, and model a fast saturable absorber: α ( ( ) ) a t = α 0 1+ ( ) a t I sat Intensity The transmission throuh a fast saturable absorber: L a a t α e 1 α La 1 α0l a 1 = 1 α0la + γ a Isat α0 La ( ) where: γ = Includin saturable absorption in the mode-amplitude equation: Saturation intensity I sat 1 d + a a = 1+ + γ a a l Ω dt ( k 1) ( k ) ( k ) ( k ) Lumpin the constant loss into l 19

20 The sech pulse shape In steady state, this equation has the solution: ( ) A0 sech t a t = τ FWH =.6 τ where the conditions on τ and A 0 are: γ A = Ω τ 0 l + = Ω 0 τ 1 Note: the pulse duration is now τ = Ω γ A 0 proportional to the inverse of the ain bandwidth. Passive mode lockin produces shorter pulses! 0

21 The aster Equation: includin GVD Expand k to second order in ω: 1 k k 0 k k ( ω ) = ( ω ) + ω + ω After propaatin a distance L d, the amplitude becomes: 1 a Ld i k k k Ld a (, ω ) = exp ( ω o) + ω + ω ( 0, ω ) Inore the constant phase and v, and approximate the nd -order phase: 1 1 ω ( ω) ω ( ω) exp ik Ld a 0, 1 ik Ld a 0, Inverse-Fourier-transformin: 1 d d a Ld, t = 1+ ik Ld 0, 1 0, a t + id a t dt dt 1 where: D k L d ( ) ( ) ( ) 1

22 The aster Equation (continued): includin the Kerr effect The Kerr Effect: n = n + n I 0 so: Φ = π n L k a t λ The master equation (assumin small effects) becomes: ( ) δ a ( t) ( 1) ( ) 1 d d a a + 1 id ( γ iδ ) a = + + l + a Ω dt dt k k k k ( ) ( ) ( k + 1) ( k ) In steady state: where ψ is the phase shift per round trip. a a = iψ a( t) d iψ + ( l) + + id + ( γ iδ ) a a( t) = Ω dt 0 This important equation is called the Landau-Ginzber Equation.

23 Solution to the aster Equation It is: a t ( ) t = A0 sech τ ( 1+ iβ ) where: ( 1 iβ ) + iψ + l + + id = τ Ω 0 1 τ Ω + id ( + 3iβ β ) = ( γ iδ ) A 0 The complex exponent yields chirp. 3

24 The pulse lenth and chirp parameter Dn = Ω D 4

25 The spectral width The spectral width vs. dispersion for various SP values. A broader spectrum is possible if some positive chirp is acceptable. 5

26 The Nonlinear Schrodiner Equation Recall the master equation: ( 1) ( ) 1 d d a a + 1 id ( γ iδ ) a = + + l + a Ω dt dt k k k k ( ) ( ) If you were interested in liht pulses not propaatin inside a laser cavity, but in a medium outside the laser, then you would chane this to a continuous differential equation (rather than a difference equation with a discrete round-trip index k). You would also inore ain and loss (both linear loss and saturable loss). Then you would have the Nonlinear Schrodiner Equation: a z = id iδ a a t D GVD parameter δ Kerr nonlinearity 6

27 The Nonlinear Schrodiner Equation (NLSE) a z = id iδ a a t The solution to the nonlinear Shrodiner equation is: δ A0 a( z, t) = A0 sech exp i z ( t ) τ where: 1 τ δ A0 = D Note that δ /D < 0, or no solution exists. But note that, despite dispersion, the pulse lenth and shape do not vary with distance: solitons 7

28 Collision of two solitons 8

29 Evolution of a soliton from a square wave 9

30 So then do all mode-locked lasers produce sech(t) pulses? a z = id iδ a a z t ( ) The aster Equation assumed that the dispersion is uniform throuhout the laser cavity, so that the pulse is always experiencin a certain (constant) GVD as it propaates throuh one full round-trip. pump But that is far from bein true. between the prisms: neative GVD Ti: sapphire crystal: positive GVD 30

31 Distributed dispersion within the laser cavity roup velocity dispersion, D ain medium the space between the prisms one round trip distance within the laser cavity So the dispersion D should depend on position within the laser cavity, D = D(z). In principle, so should the Kerr nonlinearity, δ = δ(z) since this nonlinearity only exists inside the ain medium. 31

32 Perturbed nonlinear Schrodiner equation a z = id z i z a a z t ( ) δ ( ) ( ) δ(z) = zero except inside the Ti:sapphire crystal D(z) = cavity dispersion map (positive inside the crystal, neative between the prisms) } The pulse experiences these perturbations periodically, once per cavity round-trip. Solution: requires numerical interation except in the asymptotic limit (infinite D ) Result: the pulse shape is not invariant! It varies durin the round trip. But it is still stable at any particular location in the laser. Q. Quraishi et al., Phys. Rev. Lett. 94, (005) Dispersion-manaed solitons one cavity round trip 3

33 In real lasers, the pulse shape is complicated Dispersion manaement can produce many different pulse shapes. small dispersion swin: sech pulses moderate dispersion: Gaussian pulses lare dispersion: the result depends more sensitively on ain filterin Also: The perturbed NLSE nelects ain and saturable loss, both of which are required for mode-locked operation. In principle, both would also need to be included in a distributed way: = (z) and γ = γ(z) one round trip Bottom line: it s complicated. Y. Chen et al., J. Opt. Soc Am. B 16, 1999 (1999) 33