The Generation of Ultrashort Laser Pulses

					The Generation of Ultrashort Laser Pulses
 The importance of bandwidth

 More than just a light bulb

 Laser modes and mode-locking

 Making shorter and shorter pulses

         Q-switching and distributed-feedback lasers
         Passive mode-locking and the saturable absorber
         Kerr-lensing and Ti:Sapphire
         Active mode-locking
         Other mode-locking techniques

 Limiting factors

 Commercial lasers
But first: the progress has been amazing!


                                                 S-P Dye        Diode


                                          CW Dye

                                                                    Color                                    The shortest
                                                                                                             pulse vs. year
                              100fs                                                  Cr:LiS(C)AF
                                                  CP M Dye                           Er:fiber                (for different
                                                                                                Cr:YAG       media)

                               10fs               w/Compression


                                  1965    1970      1975     1980   1985     1990   1995       2000   2005
Continuous vs. ultrashort pulses of light
 A constant and a delta-function are a Fourier-Transform pair.

                     Irradiance vs. time              Spectrum

 Continuous beam:

                           time                       frequency

 Ultrashort pulse:

                            time                      frequency
 Long vs. short pulses of light
 The uncertainty principle says that the product of the temporal and
 spectral pulse widths is greater than ~1.

                   Irradiance vs. time               Spectrum

Long pulse

                          time                      frequency

Short pulse

                          time                      frequency
For many years, dyes have been the broadband
media that have generated ultrashort laser
Ultrafast solid-state laser media have
recently replaced dyes in most labs.
Solid-state laser media have broad bandwidths and are convenient.
  Laser power
Light bulbs,
lasers, and
ultrashort pulses

But a light bulb is also broadband.

What exactly is required to make an ultrashort pulse?

Answer:    A Mode-locked Laser

Okay, what’s a laser, what are modes, and what does it mean
to lock them?
              Before               After

                       Unexcited           Excited
                       molecule            molecule


Stimulated emission leads to a chain
reaction and laser emission.
If a medium has many excited molecules, one photon can become
                          Excited medium

This is the essence of the laser. The factor by which an input beam is
amplified by a medium is called the gain and is represented by G.
The laser
A laser is a medium that stores energy, surrounded by two mirrors.
A partially reflecting output mirror lets some light out.

           I0                                          I1

            I3             Laser medium                I2
R = 100%                    with gain, G                    R < 100%

A laser will lase if the beam increases in intensity during a round trip:
that is, if I 3  I 0

Usually, additional losses in intensity occur, such as absorption, scat-
tering, and reflections. In general, the laser will lase if, in a round trip:

         Gain > Loss                This called achieving Threshold.
Calculating the gain:
Einstein A and B coefficients                              1

In 1916, Einstein considered the various transition rates between
molecular states (say, 1 and 2) involving light of irradiance, I:

             Absorption rate = B N1 I

                  Spontaneous emission rate = A N2

          Stimulated emission rate = B N2 I
Laser gain
Neglecting spontaneous emission:

          dI     dI
              c     BN 2 I - BN1I              [Stimulated emission minus absorption]
          dt     dz
                     B  N 2 - N1  I
The solution is:                                  Proportionality constant is the
                                                  absorption/gain cross-section, 

        I ( z )  I (0) exp   N2  N1  z

There can be exponential gain or loss in irradiance. Normally, N2 < N1,
and there is loss (absorption). But if N2 > N1, there’s gain, and we
define the gain, G:
                                                If N2 > N1:    g   N2  N1 
     G  exp   N2  N1  z
                                                If N2 < N1 :      N1  N2 
How to achieve laser threshold
In order to achieve threshold, G > 1, that is, stimulated emission
must exceed absorption:

             B N2 I > B N1 I                       Inversion

Or, equivalently,

                    N2 > N1                                    temperature”

This condition is called Inversion.
It does not occur naturally.                            Molecules

In order to achieve inversion, we must hit the laser medium very
hard in some way and choose our medium correctly.
Why inversion is impossible                                   2           N2
in a two-level system
                                                              1           N1
Write rate equations for the densities of the two states.

               Stimulated emission   Spontaneous
                                     emission    If the total number of
  dN 2
        BI ( N1  N 2 )  AN 2                   molecules is N:
   dt                                                 N  N1  N 2
       BI ( N 2  N1 )  AN 2                      N  N1  N 2
                                            2 N 2  ( N1  N 2 )  ( N1  N 2 )
     d N
          2 BI N  2 AN 2                      N  N
     d N
          2 BI N  AN  AN
Why inversion is impossible                                 2           N2
in a two-level system (cont’d)
                                                            1           N1
                 d N
                        2 BI N  AN  AN
In steady-state:    0  2BI N  AN  AN
                    ( A  2BI )N  AN
                    N  AN /( A  2BI )
                    N  N /(1  2BI / A)

                   N                where:    I sat  A / 2 B
         N 
               1  I / I sat        Isat is the saturation intensity.

N is always positive, no matter how high I is!
It’s impossible to achieve an inversion in a two-level system!
Why inversion is possible                         3
                                                              Fast decay
in a three-level system                           2

                                                Pump          Laser
Assume we pump to a state 3 that             Transition       Transition
rapidly decays to level 2.
                           Spontaneous            1
     dN 2
           BIN1  AN 2
      dt                                 The total number       Level 3
                     Absorption          of molecules is N:     decays
                                                                fast and
     dN1                                  N  N1  N 2
           BIN1  AN 2                                        so is zero.
      dt                                 N  N1  N 2
    d N
          2 BIN1  2 AN 2              2N 2  N  N
                                         2N1  N  N
   d N
         BIN  BI N  AN  AN
Why inversion is possible                       3
                                                            Fast decay
in a three-level system                         2

(cont’d)                                      Pump          Laser
                                           Transition       Transition
  d N
          BIN  BI N  AN  AN        1
In steady-state: 0  BIN  BI N  AN  AN

            ( A  BI )N  ( A  BI ) N

            N  N ( A  BI ) /( A  BI )

             1  I / I sat      where:     I sat  A / B
     N  N
             1  I / I sat      Isat is the saturation intensity.

             Now if I > Isat, N is negative!
Why inversion is easy                        3
                                                       Fast decay
in a four-level system                       2

                                           Pump         Laser
Now assume the lower laser level 1      Transition      Transition
rapidly decays to the ground level 0.
              dN 2                                      Fast decay
As before:          BIN 0  AN 2            0
       dN 2                                 The total number
             BI ( N  N 2 )  AN 2         of molecules is N :
                                              N  N0  N2
Because   N1  0,    N   N 2
                                              N0  N  N2
      d N
           BIN  BI N  AN
At steady state:    0  BIN  BI N  AN
Why inversion is easy                                      Fast decay
in a four-level system
(cont’d)                                     Pump

    0  BIN  BI N  AN                      1
                                                           Fast decay
     ( A  BI )N   BIN

     N   BIN /( A  BI )

     N  ( BIN / A) /(1  BI / A)

                  I / I sat     where:     I sat  A / B
      N   N
                1  I / I sat   Isat is the saturation intensity.

            Now, N is negative—always!
What about the                                                      Fast decay
saturation intensity?
                                                   Pump              Laser
                                                Transition           Transition
             I sat  A / B
                                                                    Fast decay
A is the excited-state relaxation rate: 1/t           0
B is the absorption cross-section, , divided by
the energy per photon, ħw:  / ħw
                                          ħw ~10-19 J for visible/near IR light
Both  and t
depend on the                      w      t ~10-12 to 10-8 s for molecules
molecule, the           I sat   
frequency, and                    t       ~10-20 to 10-16 cm2 for molecules (on
the various                               resonance)
states involved.    105 to 1013 W/cm2

The saturation intensity plays a key role in laser theory.
   Two-, three-, and four-level systems
    It took laser physicists a while to realize that four-level systems are

        Two-level               Three-level                         Four-level
         system       Molecules   system                             system
                      in this level.                                       Fast decay
                                               Fast decay
   Pump         Laser                                        Transition     Laser
Transition      Transition                                                  Transition
                                                                           Fast decay
    At best, you get                                      fast!
                                   If you hit it hard,
   equal populations.                                             Lasing is easy!
                                       you get lasing.
       No lasing.
A dye’s energy levels
Dyes are big molecules, and they have complex energy level structure.

 S2: 2nd excited
 electronic state
                                         Lowest vibrational and
                                         rotational level of this
                     Energy              electronic “manifold”
  S1: 1st excited
                                         Excited vibrational and
 electronic state
                                         rotational level

             Pump Transition     Laser Transition

    S0: Ground                           Dyes can lase into any (or
 electronic state                        all!) of the vibrational/
                                         rotational levels of the S0
                                         state, and so can lase very
Lasers modes: The Shah function
 The Shah function, III(t), is an infinitely long train of equally
 spaced delta-functions.


                          III(t )      (t  m)
                                      m 

The symbol III is pronounced shah after the Cyrillic character III, which is
said to have been modeled on the Hebrew letter        (shin) which, in turn,
may derive from the Egyptian      a hieroglyph depicting papyrus plants
along the Nile.
The Fourier transform of the Shah function
                                                 III(t)
       t  m) exp(iwt )dt
     m 
                                                                        t
      t  m) exp(iwt )dt
    m  
                       If w = 2np, where n is an integer, every term is
    exp(iwm)
    m 
                       exp(2mnp i) = 1, and the sum diverges;
                       otherwise, cancellation occurs. So:

                F {III(t)}                     F {III(t )}  III(wp 

The Shah function
and a pulse train
An infinite train of identical pulses
(from a laser!) can be written:         E (t )  III(t / T )  f (t )

where f(t) is the shape of each pulse and T is the time between

                    
                 (t / T  m) f (t  t) dt
               m 

   Set t’ /T = m or t’ = mT                 
                                             m 
                                                     f (t  mT )
The Fourier transform of an infinite train of pulses
An infinite train of identical pulses can be written:

E(t) = III(t/T) * f(t)

where f(t) represents a single pulse and T is the time between pulses.
The Convolution Theorem states that the Fourier Transform of a
convolution is the product of the Fourier Transforms. So:

E (w ) 
  III(wT / 2p ) F (w 

A train of pulses results from a single pulse bouncing back and forth
inside a laser cavity of round-trip time T. The spacing between
frequencies—called laser modes—is then w = p/T or n = 1/T.
Mode-locked vs. non-mode-locked light

 Mode-locked pulse train:
     E (w )     F (w )  (w  2p m / T )
                m 
                                                                    A train of
                                                                    short pulses
            F (w )           (w  2p m / T )  F (w  III(wT / 2p )
                         m 

 Non-mode-locked pulse train:
                                              Random phase for each mode
         E (w )         F (w ) exp(i
                     m 
                                          m   )  (w  2p m / T )

                   F (w )  exp(i m )  (w  2p m / T )             A mess…
                                m 
Generating short pulses = mode-locking
  Locking the phases of the laser modes yields an ultrashort pulse.
Locked modes

Numerical simulation of mode-locking

      Ultrafast lasers often have thousands of modes.
A generic ultrashort-pulse laser

A generic ultrafast laser has a broadband gain medium, a pulse-
shortening device, and two or more mirrors:


Many pulse-shortening devices have been proposed and used.
Pulsed Pumping
 Pumping a laser medium with a short-pulse flash lamp yields a
 short pulse. Flash lamp pulses as short as ~1 µs exist.

 Unfortunately, this yields a pulse as long as the excited-state
 lifetime of the laser medium, which can be considerably longer
 than the pump pulse.

 Since solid-state laser media have lifetimes in the microsecond
 range, it yields pulses microseconds to milliseconds long.

                                        Long and potentially
                                          complex pulse

Q-switching involves:                                  Output intensity

  Preventing the laser
  from lasing until the                         100%

                                                                          Cavity Gain
                                  Cavity Loss
  flash lamp is finished
  flashing, and

  Abruptly allowing the                         0%
  laser to lase.                                              Time

The pulse length is limited by how fast we can switch and the
round-trip time of the laser and yields pulses 10 - 100 ns long.
 How do we Q-switch a laser?
 Q-switching involves preventing lasing until we’re ready.
 A Pockels’ cell switches (in a few nanoseconds) from a quarter-
 wave plate to nothing.

         Before switching                      After switching
        0° Polarizer      Mirror             0° Polarizer       Mirror

               Pockels’ cell as                       Pockels’ cell as
                wave plate w/                          an isotropic
                axes at ±45°                             medium

Light becomes circular on the first        Light is unaffected by the
pass and then horizontal on the next       Pockels’ cell and hence is
and is then rejected by the polarizer.     passed by the polarizer.
Passive mode-locking:                      N 0  N1 
the saturable absorber                             N
                                                            
                         For a two-            1  I / I sat
                         level system

              Like a sponge, an absorbing medium can
              only absorb so much. High-intensity spikes
              burn through; low-intensity light is absorbed.

                                   0                  0  N
                       (I) 
                                1  I Isat
The effect of a saturable absorber
First, imagine raster-scanning the pulse vs. time like this:

                  Short time (fs)


                                    Notice that the weak pulses are suppressed,
                                    and the strong pulse shortens and is amplified.

After many round trips, even a slightly saturable absorber can yield
a very short pulse.

       High-intensity spikes
    (i.e., short pulses) see
        less loss and hence
         can lase while low-
     intensity backgrounds
  (i.e., long pulses) won’t.
Passive mode-locking with a slow
saturable absorber
What if the absorber responds slowly (more slowly than the pulse)?

Then only the leading edge will experience pulse shortening.

This is the most common situation, unless the pulse is many ps long.
Gain saturation shortens the pulse
trailing edge.

 The intense spike uses up the laser gain-medium energy,
 reducing the gain available for the trailing edge of the pulse
 (and for later pulses).
 gain and

  Lasers lase when
  the gain exceeds
           the loss.

 The combination of
saturable absorption
  and saturable gain
  yields short pulses
      even when the
  absorber is slower
      than the pulse.
The Passively Mode-locked Dye Laser

                               Pump                    Saturable
                               beam                    absorber

               Gain medium

 Passively mode-locked dye lasers yield pulses as short as a few
 hundred fs.
 They’re limited by our ability to saturate the absorber.
Some common dyes and their
corresponding saturable absorbers
Colliding pulses have a higher peak intensity.

                                            Two pulses colliding

                                                   Single pulse

                               Longitudinal position, z

 And higher intensity in the saturable absorber is what CPM lasers require.
The colliding-pulse mode-
locked (CPM) laser
 A Sagnac interferometer is ideal for
 creating colliding pulses.


               Gain medium


CPM dye lasers produce even shorter pulses: ~30 fs.
A lens and a lens

  A lens is a lens             x
  because the phase
  delay seen by a       L(x)
  beam varies with x:

    f(x) = n k L(x)                In both cases, a
                                   quadratic variation
                                   of the phase with x
                                   yields a lens.

  Now what if L is             x
  constant, but n
  varies with x:

    f(x) = n(x) k L
Kerr-lens mode-locking
A medium’s refractive index
depends on the intensity.

    n(I) = n0 + n2I

If the pulse is more intense in
the center, it induces a lens.
Placing an aperture at the
focus favors a short pulse.

                                     Losses are too high for a low-
                                     intensity cw mode to lase, but not
                                     for high-intensity fs pulse.

Kerr-lensing is the mode-locking mechanism of the Ti:Sapphire laser.
Kerr-lensing is a type of saturable absorber.

 If a pulse experiences additional focusing due to high intensity and
 the nonlinear refractive index, 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!

 This is a type of saturable absorption.
Modeling Kerr-lens mode-locking
Titanium Sapphire (Ti:Sapphire)

Ti:Sapphire is
currently the
workhorse laser
of the ultrafast
emitting pulses as
short as a few fs
and average
power in excess
of a Watt.

                     Al2O3 lattice   oxygen
Titanium Sapphire
                    Absorption and emission
                     spectra of Ti:Sapphire
It can be
pumped with a
Argon laser
(~450-515 nm)
or a doubled-
Nd laser (~532


Upper level
lifetime:             Ti:Sapphire lases from
3.2 msec              ~700 nm to ~1000 nm.
Mechanisms that limit pulse shortening

 The universe conspires to lengthen pulses.

   Gain narrowing:
       G(w) = exp(-aw2), then after N passes, the spectrum will narrow
       by GN(w) = exp(-Naw2), which is narrower by N1/2

   Group-velocity dispersion:
       GVD spreads the pulse in time. And everything has GVD…
       All fs lasers incorporate dispersion-compensating components.
       We’ll spend several lectures discussing GVD!!

   Etalon effects:
        This yields multiple pulses, spreading the energy over time,
        weakening the pulses.
The Ti:Sapphire laser including
dispersion compensation
Adding two prisms compensates for dispersion in the Ti:Sapphire
crystal and mirrors.

                   Ti:Sapphire                         Slit for
                   gain medium      cw pump beam       tuning

                                              Prism dispersion

This is currently the workhorse laser of the ultrafast optics community.
Commercial fs lasers

           Mira (<35 fs pulse length, 1 W ave power),
           Chameleon (Hands-free, ~100 fs pulse length),

           Tsunami (<35 fs pulse length, 1 W ave power)
            Mai Tai (Hands-free, ~100 fs pulse length)
Very-short-pulse commercial fs lasers

    KM Labs
        < 20 fs and < $20K


                             As short as 8 fs!
Commercial fs lasers (cont’d)
Ytterbium Tungstate
(Yb:KGW)                               Ytterbium doped laser
                                       materials can be directly
                                       diode-pumped, eliminating
                                       the need for an intermediate
                                       (green) pump laser used in
                                       Ti:Sapphire lasers.
                                       They also offer other
                                       attractive properties, such as
                                       a very high thermal efficiency
                                       and high average power.
         Amplitude Systemes

 Model                    t-Pulse 20   t-Pulse 100   t-Pulse 200
 Pulse energy (nJ)            20           100            200
 Average power (W)            1             1              2
 Repetition rate (MHz)        50            10            10
Active mode-locking

 Any amplitude modulator can preferentially induce losses for
 times other than that of the intended pulse peak. This produces
 short pulses.

 It can be used to start a Ti:Sapphire laser mode-locking.
Gain switching

Modulating the gain rapidly is essentially the same as active

This method is a common one for mode-locking semiconductor
Synchronous pumping
Pumping the gain medium with a train of            The laser round-
already short pulses yields a train of                trip time must
even shorter pulses.                                precisely match
                            Short pulses (ps)     that of the train of
                                                       pump pulses!

                    Pump                          Saturable
                    beam                          absorber

              Gain medium

Trains of 60 ps pulses from a Nd:YAG laser can yield <1 ps
pulses from a sync-pumped dye laser.
Hybrid mode-locking

 Hybrid mode-locking is any type of mode-locking incorporating
 two or more techniques simultaneously.

        Sync-pumping and passive mode-locking

        Active and passive mode-locking

 However, using two lousy methods together doesn’t really work
 all that much better than one good method.
    Diode lasers use hybrid mode-locking
                                       Autocorrelation   Spectrum

          Autocorrelation   Spectrum

Haneda, et al, UP 2004
Additive-pulse mode-locking
 Nonlinear effects in an external cavity can yield a phase-
 distorted pulse, which can be combined in phase with the
 pulse in the main cavity, yielding cancellation in the wings,
 and hence pulse-shortening.

 Early fiber lasers used this mechanism.
The soliton laser
Nonlinear-optical effects can compensate for dispersion, yielding a
soliton, which can be very short and remain very short, despite
dispersion and nonlinear-optical effects.
Commercial fs fiber lasers

    Menlo Systems
    150 fs; 150 mW

    IMRA America

Ultrafast Q-switching using distributed
When two beams cross at an angle, their intensity is sinusoidal.

                                                    Intensity fringes

When energy is deposited sinusoidally in space, the actual gain (g) goes
quadratically with the energy deposited, yielding a type of very fast Q-
switching. Using several stages, fs pulses have been created this way.
Traveling-wave excitation
Pump lasers for ultrafast lasers
Previously, only the Argon Ion laser was available, but much more stable
intracavity-frequency-doubled solid-state lasers are now available.