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16. Theory of Ultrashort Laser Pulse Generation modulator transmission cos( t) Active mode-locking M Passive mode-locking time Build-up of mode- locking: The Landau- Ginzberg Equation loss gain The Nonlinear gain > loss Schrodinger Equation Solitons time Reference: Hermann Haus, “Short pulse generation,” in Compact Sources of Ultrashort Pulses, Irl N. Duling, ed. 1 (Cambridge University Press, 1995). The burning question: gaussian or sech2? After all this talk about Gaussian pulses… what does Ti:sapphire really produce? 0 10 sech2 -2 10 -4 10 gaussian -6 10 -8 10 2 sech -10 10 gaussian -12 10 -80 -40 0 40 80 -80 -40 0 40 80 2 t e or sech2 t ?? 2 Mode-locking yields ultrashort pulses multiple oscillating cavity modes Recall that many frequencies (“modes”) losses laser oscillate simultaneously in gain a laser, and when their profile phases are locked, an ultrashort pulse results. q q1 q q+1 q+2 q+3 possible cavity modes Sum of ten modes with the same relative phase Sum of ten modes w/ random phase 3 modulator transmission Mode Locking cos(Mt) Active Mode Locking time Insert something 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 Mode Locking Saturable absorber loss Insert something into the laser cavity that favors high intensities. gain gain > loss strong maxima will grow stronger at the expense of weaker ones eventually, all of the energy is time concentrated in one packet 4 Active mode-locking: the electro-optic modulator Applying a voltage to a crystal changes its refractive indices and introduces birefringence. A few kV can turn a crystal into a half- or quarter-wave plate. Polarizer If V = 0, the pulse polarization doesn’t change. “Pockels cell” V If V = V , the pulse (voltage may be p transverse or polarization switches to its longitudinal) orthogonal state. Applying a sinusoidal voltage 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 shifting the modes into and out of the actual cavity modes. 5 Active mode-locking: 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 light wave’s energy. Pressure, density, and Acoustic refractive-index variations transducer due to acoustic wave Output beam Input beam Quartz Diffracted Beam (Loss) Such diffraction can be quite strong: ~70%. Sinusoidally modulating the acoustic wave amplitude yields sinusoidal modulation of the transmitted beam. 6 The Modulation Theorem: The Fourier Transform of E(t)cos(Mt) {E(t)cos( t)} = Et()cos( t )exp( i t ) dt Y M M 1 Et( ) exp( it ) exp( it ) exp( itdt ) 2 MM 11 E( t )exp( i [ ] t ) dt E ( t )exp( i [ ] t ) dt 22MM 11 Y {E(t)cos( t)} = EE()()M M M 22 Multiplication by cos(Mt) introduces side-bands. Y {E(t))} Y {E(t)cos(Mt)} 2 If E(t) = sinc (t)exp(i0t): - 0 0 M 0 0 M 7 Active mode- modulator transmission cos( t) locking M Time In the frequency domain, a modulator introduces side-bands of every mode. For mode-locking, adjust M so nM nM 0 that M = mode spacing. c/L cavity This means that: = 2/cavity round-trip time modes Frequency M = 2/(2L/c) = c/L Each mode competes for gain with adjacent modes. Most efficient operation is for phases to lock. Result is global phase locking. N coupled equations: En En+1, En-1 8 Modeling laser modes Gain profile and and gain resulting laser modes Lasers have a mode spacing: 2 c M TLR 0 th th Let the zero mode be at the center of the gain, 0. The n mode frequency is then: nM 0 n where n = …, -1, 0, 1, … th Let an be the amplitude of the n mode and assume a Lorentzian gain profile, G(n): g Gna() 1 a nn[]2 2 1(nMg )/ 2 n 1 M 1 ga2 n 9 g Modeling an amplitude modulator An amplitude modulator uses the electro-optic or acousto-optic effect to deliberately cause losses at the laser round-trip frequency, M. A modulator multiplies the laser light (i.e., each mode) by M[1cos(Mt)] intoM 11 int oM MtaeMititae1 cos(Mn ) exp( M ) 1 exp( M ) n 22 it 11in11 t in t Ma eo eMM einM t e n 22 Notice that this spreads the energy from the nth to the (n+1)st and (n-1)st modes. Including the passive loss, , we can write this as: 2 n M kk aagkk1 12M aa kk aaa k nn2 nn2 nn11 n g where the superscript indicates the kth round trip. 10 Solve for the steady-state solution 2 n M kk aagkk1 12M aa kk aaa k nn2 nn2 nn11 n g kk1 In steady state, aann Also, the finite difference becomes a second derivative when the modes are many and closely spaced: where, in this 2 continuous limit, kkk 2 d aaann112nM 2 a ()k d aan () where: nM Thus we have: 222 MM d 01ga22 g 2 d 11 Solve for the steady-state solution 222 MM d 01ga22 g 2 d This differential equation 22/2 (Hermite Gaussians) has the solution: aHe 22 1 MM g 221 with the constraints: gM M v 4 2g 2 In practice, the lowest- 22 /2 A Gaussian spectrum! order mode occurs: aAe 12 Fourier transforming to the time domain Recalling that multiplication by -2 in the frequency domain is just a second derivative in the time domain (and vice versa): 222 kk1 MM d aagnn1 22 a g 2 d 2 kk1 1 dM22 k becomes: atatg 1 22 M tat g dt 2 which (in the continuous limit) has the solution: v i t t22/2 at H e 2 This makes sense because Hermite-Gaussians are their own Fourier transforms. The time-domain will prove to be a better domain for modeling 13 passive mode-locking. AM mode-locking – a round-trip analysis A simple Gaussian pulse (one round trip) analysis: modulated 2 2 2 4gL g loss gain exp 0 exp exp 0 4'2 2 42 g approximate (Gaussian) form for modified spectral round trip gain. (g = saturated gain) width ' modulator Assuming g = 0: transmission 16gL 2 ' 2 g approximate (Gaussian) form for modulated loss 2 2 2 2 t 2 exp "t exp m exp 't "' m 2 2 14 AM mode-locking (continued) 16gL 2 2 steady-state condition " 2 m 0 g 2 Steady-state pulse duration (FWHM): 1 2 2 ln 2 gL 4 42 number of ~ RT p 2 oscillating m g N modes BUT: with an inhomogeneously broadened gain medium, using the full bandwidth g: 1 RT p ~ g N AM mode-locking does not exploit the full bandwidth of an inhomogeneous medium! 15 Other active mode-locking techniques FM mode-locking produce a phase shift per round trip implementation: electro-optic modulator similar results in terms of steady-state pulse duration Synchronous pumping gain 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 lengths to be accurately matched useful for converting long AM pulses into short AM pulses (e.g., 150 psec argon-ion pulses sub-psec dye laser pulses) Additive-pulse or coupled-cavity mode-locking external cavity that feeds pulses back into main cavity synchronously requires two cavity lengths to be matched can be used to form sub-100-fsec pulses 16 Passive mode-locking Saturable absorption: absorption saturates during the passage of the pulse leading edge is selectively eroded Saturable gain: gain saturates during the passage of the pulse leading edge is selectively amplified Fast absorber Slow absorber loss loss gain gain gain > loss gain > loss time time 17 Kerr lensing is a type of saturable absorber If a pulse experiences additional focusing due to the Kerr lens nonlinearity, and we align the laser for this extra focusing, then a high-intensity beam will have better overlap with the gain medium. High-intensity pulse Mirror Additional focusing optics can arrange for perfect overlap of the high-intensity Ti:Sapph beam back in the Ti:Sapphire crystal. Low-intensity pulse But not the low- intensity beam! I 1 This is a type of saturable loss, with the same ~ saturation behavior we’ve seen before: III1 sat 18 Saturable-absorber mode-locking Intensity 2 at Neglect gain saturation, and 2 at 1 model a fast saturable absorber: 0 I sat The transmission through a fast saturable absorber: Saturation intensity 2 at 2 La eLL111aa 00 1 La a Isat 0 La where: Isat Including saturable absorption in the mode-amplitude equation: 2 2 Lumping kk1 1 d kk the constant aag1 22 aa g dt loss into ℓ 19 The sech pulse shape In steady state, this equation has the solution: t at A0 sech FWHM = 2.62 where the conditions on and A0 are: 2 2g A0 22 g g g 22 0 g 20 The Master Equation: including GVD 1 Expand k to second order in : kk k k 2 0 2 After propagating a distance Ld, the amplitude becomes: 1 2 o aLdd,exp ik k k L a 0, 2 nd Ignore the constant phase and vg, and approximate the 2 -order phase: 1122 expik Ldd a 0, 1 ik L a 0, 22 Inverse-Fourier-transforming: 1 dd22 aLtdd,1 ikL22 at 0,1 iDat 0, 2 dt dt 1 where: DkL 21 2 d The Master Equation (continued): including the Kerr effect 2 22 The Kerr Effect:nn nI so: 02 nLat2 at k The master equation (assuming small effects) becomes: 22 2 kk1 1 dd kk aag1 22 iD 2 iaa g dt dt kk1 In steady state: aaiat () where is the phase shift per round trip.
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