Introduction to nonlinear optics
Assignment 5 in the course Ultrafast
Science
Due end of May
1. Second harmonic generation of short pulses in a KD*P crystal
A laser engineer wants to frequency double a laser beam from a 30 fs (FWHM)
Ti:Sapphire laser with wavelength 800 nm using a KD*P crystal (refractive index 1.5).
He reads for example in http://www.clevelandcrystals.com/KDP.htm the following info
on KD*P.
0.266µm
no=1.5539
ne=1.5071
0.3547µm
no=1.5250
ne=1.4834
0.532µm
no=1.5071
ne=1.4683
0.5893µm
no=1.5045
ne=1.4661
0.6328µm
no=1.50285
ne=1.4648
0.6943µm
no=1.50095
ne=1.4633
1.064µm
no=1.4934
ne=1.4583
He wants to calculate the phase matching angle and the optimum length of the crystal
to keep good efficiency.
1) Fit a curve to the indices indicated above and plot the extraordinary and ordinary
indexes as a function of wavelength.
2) Calculate the angle between the laser beam (wavelength 800 nm) and the crystal’s
optical axis for phase matching second harmonic generation of type I.
3) The nonlinear coefficient for frequency doubling is 0.5 10-12 m/V. Calculate the
efficiency knowing that the beam diameter is 2 cm, the medium’s length 100 m
and the energy per pulse 5 mJ. Use Eq. 2.43 converted in SI units:
2 2 d 2 I ( ) 2  2
 sin c(k / 2)2
I (2 ) 
3 3
n c 0
4) Using this equation to compare the doubling efficiency for lengths equal to 1mm,
100 m and 10 m as a function of wavelength. (Plot the SHG intensity, in
arbitrary units, as a function of wavelength in the 3 cases).
5) Show formula 2.45 with the parenthesis up to the power -1, knowing that the sinc
function decreases by a factor of 2 when its argument is between -1.39 and 1.39.
Calculate the phase matching bandwidth and compare it with the results of 3)
6) Calculate the group velocity mismatch and the pulse lengthening due to pulse
velocity mismatch for a 100 m long crystal.
2. High harmonic generation in gases- Attosecond pulses
When an intense laser field is focused into an atomic or molecular gas medium, odd
harmonics are generated. Since harmonic generation is a highly non linear process, i.e a
process that strongly depends on the intensity, short laser pulses (few tens of
femtoseconds) are usually used. The specs of the kHz laser used in Lund to generate
harmonics are 0.5 mJ, 35 fs at 800 nm central wavelength at a repetition rate of 1 kHz.
The beam diameter of is D= 8 mm. (Use formulas for Gaussian beams.)
1) Because of the highly energetic photons created during the harmonic generation
process, electron volts (eV) are used as unit of energy. Calculate the energy of a
fundamental laser photon in eV.
2) An atom is exposed to a laser field E(t)=E0 cos(t) where E0 is the field amplitude and
the fundamental laser frequency. The electron tunnels through the Coulomb barrier and
is released into the continuum with zero velocity at time t=ti. Assume that the only force
experienced by the electron is the force from the electric field F(t)=-eE(t) where e is the
electron charge and calculate analytically the velocity and position of the electron as a
function of time t> ti.
3) Calculate the tunneling times ti for which the electron is able to return to the ion core.
The easiest way to do this is to numerically look for zero-crossings of the function for a
range of tunneling times. Note that it is only necessary to examine tunneling times in the
range 0<t < since the process is periodic with the frequency 2.
4) Calculate the return times and return energies for the electrons. The return time is
simply the times t=tr for which the electron returns to the core and the return energy is the
kinetic energy of the electron at time tr. Make a plot of the return energy and the return
time as a function of the tunneling time (Hint: express the return energy in units of Up,
the ponderomotive energy). Make a plot of return energy as a function of return times.
Verify the cutoff law that is Ip+3.2Up.
5) When focusing a 0.5 mJ, 35 fs Gaussian laser beam (D=8 mm) with a focal length of 1
m onto a argon gas jet of atom (Ip Argon 15.6 eV), what is the highest harmonic reached?
6) We use an aluminium filter which cuts harmonics lower than (including) the 11th. We
assume that all of the harmonics selected have the same intensity and that they are
monochromatic. What is the attosecond pulse structure between the 13th harmonic and the
highest harmonic seen in the following conditions:
- When all of these harmonics have the same phase?
- When we account for the delay of the return of the electron for the short trajectory? (we
neglect the influence of the filter and of phase matching effects on the attosecond pulse
structure).