2.4 Vibrational relaxation
3.1.4 Handling
In general, the optical path length for a beam passing through a medium of length l and refraction index n is given as n l. The refraction index may be intensity-dependent (electrooptical Kerr effect), subjecting short pulses propagating through materials to intensity-dependent nonlinear processes such as self-focusing and self-phase modulation.
The pulses are also broadened in time due to different propagation times for their spectral components, imposing a delay of the higher energetic parts relative to the lower energetic ones (group velocity dispersion).
Self-focusing is an induced lens effect. Assuming a single-mode beam with a Gaussian transverse profile propagating in a medium with refractive index n given by
n = n 0 + ∆ n ( E2 ), (3.1) where ∆n ( E2 ) or ∆n (I) is an optical-field induced refractive index change. If ∆n is positive, the central part of the beam having a higher intensity experiences a larger refractive index than the edge and therefore travel at a slower velocity than the edge. As the
beam travels, the originally plane wavefront gets progressively more distorted, similar to the effect of a positive lens. Self-focusing can cause material damage and limits the focusability of the beam.
Self-phase modulation stems from the same intensity-dependent refractive index change
∆n( E2 ). Consider a time-dependent envelope E(t) of an optical pulse and an instantaneous response ∆n( E(t)2 ) given by n2 E(t)2 = n2 I(t) of the refractive index. In a length l of the nonlinear medium, the pulse is subject to an intensity-dependent phase shift
∆ϕ (t) = (ω / c) n2I(t) l and a corresponding frequency modulation ∆ω (t)= −∂ ϕ
∂ (∆ )
t , which appears as a broadened spectrum [Sieg].
Self-phase modulation is one of the contributing mechanisms for continuum generation, which may be desired for tunability reasons, but is an annoying effect if it takes place in the probe cell or in a nonlinear crystal intended for second harmonic generation (see 2.5).
Therefore, care was taken to avoid self-focusing or self-phase modulation by expanding the amplified beam to one cm diameter with a telescope (f = -50 and f = 200 mm) just behind the third amplification stage. When any focus was required, usually the largest acceptable focused spot size was used. For a Gaussian beam of wavelength λ, the focused spot size d0 behind a lens with focal length f is approximated as d0 ≈2f λ / D [Sieg]. D is the diameter of the beam when passing the lens. The depth (length) of focus is ≈π/2 (d0 / λ) λ.
Group velocity dispersion is a consequence of the frequency dependence of the refraction index. For most transparent dielectric materials in the visible region of the electromagnetic spectrum the following approximation holds :
n0 (ω) = A + B ω . (3.2) The high frequency components of the pulse are more delayed than the low frequency ones, thus the pulse broadens in time. To compensate this so-called "positive dispersion"
(wavelength-dependence) of the group velocities, a prism or grating set-up introducing a net group velocity dispersion with the opposite sign can be used [Bor 84, Fork 84, Treacy 69].
These frequency-dependent delay lines are called pulse compressors, as each can compensate a quadratic phase shift reducing the pulselength of the broadened pulse.
Figure 3.1-4: Prism pair used for pulse compression.
According to a Fourier theorem, the time-bandwidth product of any pulsed signal is constrained by the uncertainty principle ∆frms∆trmsρ 1/2, where ∆frms and ∆trms are the root-mean-square widths of the signal in frequency and in time. As a consequence, the achievable pulsewidth (fwhm) τ for a Gaussian pulse with a frequency width of ∆f is limited to a minimum of τ ≈ 0.44 / ∆f [Sieg].
In a prism compressor, the beam passes twice through two prisms separated by a length L.
The group-velocity dispersion ϕ′′ is defined as the third term of a Taylor expansion of the pulse phase in frequency:
ϕ ω( ) = ϕ ω( 0)+ ( - ω ω ϕ0) ′ + 12 ( - ω ω0) 2 ϕ′′ (3.3) The corresponding instantaneous temporal phase to the third term in equation 3.3 is proportional to the square of time, t2, and the corresponding instanteneous frequency has a linear dependence on t. That is why a quadratic phase shift means a linear frequency shift (linear chirp), a cubic phase shift means a quadratic chirp etc.
For the prism set-up of Figure 3.1-4, with e defined as AB + CD , ϕ′′is given as where n0 is the refraction index of the prism material and n′0 is its first derivative in wavelength (n′0 = (dn / dλ) ) at the wavelength λ0; n′′0 is the second derivative. The negative contribution to the group velocity dispersion scales with the distance L between the prisms, whereas the path e within the prisms adds positive group velocity dispersion. Values
for n0 , n′0, n′′0 and ϕ′′at λ = 620 nm for different materials are presented in Table 3.1.4.
Table 3.1.4 : Values for the refractive index and its wavelength derivatives for different media at 620 nm [Salin 87].
Medium n0 n0' / µm-1 n0'' / µm-2
SF10 1.7244 -0.1079 0.5725
FeD 05-25 1.8011 -0.13326 0.67787
SiO2 1.4572 -0.02984 0.12144
BK7 1.5159 -0.03756 0.16641
H2O 1.33 -0.02729 0.1303
SF10 prisms were chosen for compression of the amplified pulses due to their high value for
′
n0, enabling a more or less compact design of the compressor. When the prisms were polished with ceroxide once a year to avoid reflection losses, the overall energy transmittance of the compressor was ≈90 %, a large value compared to the maximum of approx. 50% transmittance of an even more compact grating compressor.
Propagation through materials does unfortunately not only involve a quadratic phase shift, but also cubic and higher order terms, which means that equation 3.3 is only valid as a first approximation. The cubic phase shift can be compensated by using a combination of prism and grating compressor, as these devices introduce a cubic phase shift with an opposite respective sign [Cruz 88]. Also, mirrors with chirped multilayer coatings have been shown to compensate quadratic and cubic phase shifts [Szip 94]. A monotonic variation of the multilayer period throughout the deposition process leads to a wavelength-dependent penetration depth of the optical field in the coating and to the desired group velocity dispersion.
As a consequence of the Fourier theorem mentioned above, a restriction in spectral width will impose an increase in pulselength. Therefore, only broadband dielectric high reflection mirrors centered at 616 nm or aluminium mirrors were used to propagate the beam.