Suite “señora Hilda”

Since Type I SHG was utilized in our setup and our goal was to carry out HHG with parallelly polarized fundamental and second-harmonic pulses, a half-wave (λ/2) waveplate

had to be employed. Our waveplate was a true-zero one, its thickness set such that it behaved as a half-wave plate for our fundamental pulse in a very broad bandwidth, while it was a full-waveplate for the second-harmonic around 400 nm. A reasonable question could be: why was it necessary to rotate the fundamental in the rst place and not the second-harmonic? Perhaps the bandwidth of "perfect" rotation could be thus kept broad, broader than with our chosen method. However, by simple maths, it turns out that if we do so, for the fundamental wave we will actually obtain the performance of a quarter- wave plate. This would entail that the linear polarization of the fundamental would be converted to circular; which is obviously not our intention.

A waveplate can be created from any birefringent material. By making sure that the optic axis is parallel with both the entrance and the exit surfaces of the crystal, the incoming frequency components of a pulse will be split into two waves propagating with dierent velocities. These orthogonal linearly polarized components are the extraordinary

(e-wave) and ordinary wave (o-wave), which can be characterized by their two indices of refractionne andno (for uniaxial crystals, materials having one optic axis). Because they inuence the propagation speeds, depending on which one is smaller, one wave will go through the crystal slower than the other. The axis along which the faster moving wave is polarized is called the fast axis, while in the orthogonal direction lies the slow axis. There exists a phase dierence between them, and it is called retardance ∆. When they

emerge from the crystal, the resultant vector, the polarization state of the light, will be a superposition of these two components. For a half-waveplate the retardance is π, and it can be expressed using the wavenumber evaluated for the e-wave and o-wave, as well as waveplate thickness L:

2π

λ L|ne−no|=π (5.9)

We used crystal quartz in our experiments, because it is a strong material, and thus it lends itself to the fabrication of low-order, eventually even zero-order waveplates. From equation5.9 the necessary crystal thickness can be obtained, which is 42µm.

Beyond this, through further theoretical calculations we simulated to which extent such a thin waveplate rotates the fundamental wave's frequency components, and how well it leaves the polarization of the second-harmonic's spectral components unchanged (more correctly said, it should rotate the latter by 180◦). If the fast axis is along the

x axis, while the slow axis is parallel to the y axis in a cartesian coordinate system the phase-delays can be written as:

ϕy = 2π λ Lne (5.10) ϕx = 2π λ Lno (5.11)

Using the Jones-matrix formalism [142], we can assess what the outgoing wave's two vector components will be. It is possible to see how much cutting on the tails of the spectrum would occur if we had a polarizer after the waveplate, oriented at−45◦, with the incoming

wave's linear polarization being at an angle of 45◦ (the angles in both cases measured

with respect to the x axis). In other words, the bandwidth reduction can be estimated this way. This step is justied, since we use a pellicle afterwards (at Brewster's angle for the polarization we want to keep)) to clean the polarization, to retain only the component that is along the direction at−45◦. The resultant wave's Ex and Ey components can be expressed with Jones matrices as:

" Ex Ey # = " 1/2 −1/2 −1/2 1/2 # " ejϕx ₀ 0 ejϕy # " Ein/ √ 2 Ein/ √ 2 # (5.12)

5000 600 700 800 900 1000 0.2 0.4 0.6 0.8 1 wavelength [nm] Transmission 3500 400 450 500 0.2 0.4 0.6 0.8 1 wavelength [nm] Transmission

Figure 5.7: Theoretical amplitude transmission of ourλ/2waveplate as measured after a perfect polarizer. The thickness of the true zero-order waveplate was 42µm, therefore it rotated the polarization of the fundamental by90◦, and that of the second-harmonic by 180◦. The rotation was not perfect for all wavelengths, and it caused a bandwidth reduction for our pulses, as determined by these transmission curves.

In the above equation the rst matrix after the equal sign is the polarizer's Jones-matrix, the second corresponds to that of the waveplate, while the third is the incoming wave's vector components, with Ein being the magnitude of the vector. If we take Ein equal to 1, the outgoing wave's magnitude, and therefore the transmission of the combination of a waveplate and a polarizer as a function wavelength will be simply p

E2

x+Ey2. The transmission function that we calculated this way can be seen in Fig. 5.7. We can conrm that the bandwidth in which satisfactory rotation is realized is large enough to preserve the spectrum of a near-one-octave fundamental pulse, and that of the second-harmonic, as well. The power that is not transmitted through this system naturally goes somewhere, and for a waveplate it is the perpendicular polarization component that gains some in- tensity. Our pellicles that serve as our polarizers, introducing no extra dispersion due to their almost negligible thickness of below one µm, simply cannot totally extinguish this

detrimental polarization component, and thus some low amount of elliptical polarization will remain.

Apart from the above characterization of the waveplate, we can learn more about this element's eect on our pulse. By taking the rst derivative of the phase from equation 5.12 with respect to the angular frequency we can calculate the propagation times for both the fundamental and the second-harmonic pulses. From this one can deduce the time-delay, which is positive, and this number was mentioned in section 5.3.2 above. If the second derivative is calculated, we can obtain the respective pulses' GDD values. The GDD for the fundamental at 760 nm is 1.9 fs2_{, and for the second-harmonic at 430 nm it}