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 q1 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 22MM   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(i0t):     -    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 nM nM 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(nMg )/

2  n  1 M 1 ga2  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[1cos(Mt)]

 int  11 int MtaeMititae1 cos( )oM exp( ) 1 exp( ) oM Mn M M n  22

it11in11 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 kk1  nM  kk M kk k  aagnn12 aa nn   aaann11  n   2 2 g

where the superscript indicates the kth round trip. 10 Solve for the steady-state solution

2 kk1  nM  kk M kk k  aagnn12 aa nn   aaann11  n   2 2 g

kk1   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 aaann112nM   2 a ()k d aan  () where:   nM 

Thus we have: 222 MM d 01ga22  g 2 d  11 Solve for the steady-state solution 

222 MM d 01ga22  g 2 d

This differential equation  22/2 (Hermite Gaussians) has the solution: aHe  

22 1 MM g 221 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 kk1 MM d aagnn1 22  a g 2 d

2 kk1  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 42    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 42  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 0 1  model a fast saturable absorber:  I   sat 

The transmission through a fast saturable absorber: Saturation intensity 2 at 2  La  eLL111aa 00    1 La a  Isat  0 La where:   Isat Including saturable absorption in the mode-amplitude equation:

2 2 Lumping kk1 1 d  kk the constant aag1 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  aLdd,exp  ik   k k  L  a 0, 2

nd Ignore the constant phase and vg, and approximate the 2 -order phase:

1122 expik Ldd a 0,  1 ik  L a 0, 22 Inverse-Fourier-transforming: 1 dd22  aLtdd,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 k   at

The master equation (assuming small effects) becomes:

22 2 kk1 1 dd  kk aag1 22  iD 2   iaa g dt dt kk1 In steady state: aaiat      () where  is the phase shift per round trip.

2 gd 2 ig  22  iD  iaat 0 g dt

This important equation is called the Landau-Ginzberg Equation. 22 Solution to the Master Equation

It is: 1i  t at A0 sech  

where:  1 i g ig 22  iD 0  g 

1 g 22 22iD23 i   i A0 g

The complex exponent yields chirp. 23 The pulse length and chirp parameter

2 g D  D n g

24 The spectral width

The spectral width vs. dispersion for various SPM values.

A broader spectrum is possible if some positive chirp is acceptable. 25 The Nonlinear Schrodinger Equation

Recall the master equation:

22 2 kk1 1 dd  kk aag1 22  iD 2   iaa g dt dt

If you were interested in light pulses not propagating inside a laser cavity, but in a medium outside the laser, then you would change this to a continuous differential equation (rather than a difference equation with a discrete round-trip index k).

You would also ignore gain and loss (both linear loss and saturable loss).

Then you would have the Nonlinear Schrodinger Equation:

2 a 2 iD i a a D – GVD parameter 2  – Kerr nonlinearity zt 26 The Nonlinear 2 a 2 iD2 i a a Schrodinger zt Equation (NLSE)

The solution to the nonlinear Shrodinger equation is:  2  t  A0 azt(,) A0 sech exp  i z  2 

2 1  A where:   0 2 2D

Note that /D < 0, or no solution exists.

But note that, despite dispersion, the pulse length and shape do not vary with distance: solitons 27 Collision of two solitons

28 Evolution of a soliton from a square wave

29 So then do all mode-locked lasers produce sech(t) pulses?

2 a 2 iD2 i a a z zt

The Master Equation assumed that the dispersion is uniform throughout the laser cavity, so that the pulse is always experiencing a certain (constant) GVD as it propagates through one full round-trip.

pump

But that is far from between the prisms: negative GVD being true.

Ti: sapphire crystal: positive GVD 30 Distributed dispersion within the laser cavity

gain medium

distance within the laser cavity the space between the prisms group velocity dispersion, D one round trip

So the dispersion D should depend on position within the laser cavity, D = D(z). In principle, so should the Kerr nonlinearity, = (z) since

this nonlinearity only exists inside the gain medium. 31 Perturbed nonlinear Schrodinger equation

2 a 2 iD z2 i  z a a z zt The pulse experiences (z) = zero except inside the Ti:sapphire crystal these perturbations D(z) = cavity dispersion map (positive inside the } periodically, once per crystal, negative between the prisms) cavity round-trip.

Solution: requires numerical integration except in the asymptotic limit (infinite |D|)

Result: the pulse shape is not invariant! It varies during one cavity the round trip. But it is still round trip stable at any particular location in the laser. “Dispersion-managed solitons” 32 Q. Quraishi et al., Phys. Rev. Lett. 94, 243904 (2005) In real lasers, the pulse shape is complicated

Dispersion management can produce many different pulse shapes.

• small dispersion swing: sech pulses • moderate dispersion: Gaussian pulses • large dispersion: the result depends more sensitively on gain filtering one roundtrip Also: The perturbed NLSE neglects gain 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: g = g(z) and  = (z)

Bottom line: it’s complicated. 33 Y. Chen et al., J. Opt. Soc Am. B 16, 1999 (1999)