Electro-optic sampling

To measure the THz-pulse amplitude as a function of time, i.e. an electromagnetic waveform, experiments in this thesis relied on electro-optic sampling (EOS). The used EOS-scheme is depicted in Figure 3.1 and consists of an EO-crystal - (110)- ZnTe in this case - a quarter-wave-plate (QWP), a Wollaston prism (WL) and two photodiodes (PDs). Other detection methods can be found in Refs. [83, 81].

Calibration. In calibrating the EOS detection, the linear polarisation of the gate beam, after transmitting through the ZnTe and the QWP, is made circular by a proper rotation of the QWP. Wave-plates are birefringent materials characterised by anextraordinary axis (or fast axis) - with refractive indexne - and an ordinary axis - with refractive index no. The difference between ne and no results in dif- ferent propagation speeds for the light components along the respective axes and,

Figure 3.1: Experimental setup used for the characterisation of sGe-QWs in Chapter 4. Femtoseconds pulses are generated from a SpectraPhysics MaiTai Ti:Sapphire laser. A beam splitter is used in order to generate and detect THz-pulses (see text). The blue box (top-left) depicts an example of THz-waveform (blue line) together with the first reflection. The gate, or probe, beam is depicted as red line. The waveform is probed at different time by means of a delay stage. THz-pulses are generated by an interdigitated PC emitter and detected via the electro-optic sampling, which used a<110>-ZnTe (see text).

therefore, in a phase mismatch [83]: ∆φQWP=

ω

c(ne−no)d, (3.1) where d is the wave-plate thickness. Half-wave plates (HWPs) are designed such that their thickness dHWP results in ∆φHWP = π, therefore in a 90

◦ _{rotation of}

the light polarisation when the polarisation of the incident light makes an angle θfast = π/4 with the fast axis. Conversely, for θfast = π/4, a dQWP-thick QWP

leads to ∆φQWP = π/2, making the polarisation of the incident light circular after transmitting through the QWP. Hence, this leads to the same intensity for the two orthogonal components along thex- andy-axis, i.e. Ix =Iy =I0/2, whereI0 is the

total intensity of the gate beam, and results in a voltage difference ∆VPDs = 0 at

THz-detection. In the presented setup, the detection of THz-radiation exploits the Pockels effect (or electro-optic effect), i.e. a second-order, electrically induced, susceptibility which results in the birefringent behaviour of EO-crystals [42]. The electric-field provided by the THz-pulse affects the susceptibility of the ZnTe crystal (with thickness dZnTe) and leads - similar to the case of wave-plates - to a phase

mismatch [83]:

∆φTHz

(xds)∝ dZnTeETHz(xds), (3.2)

which depends on the path difference between THz- and gate-pulses (xds), controlled

by the delay stage. If the gate-pulse travels within the ZnTe when ETHz 6= 0, the gate-beam polarisation is slightly changed due to ∆φTHz. The two orthogonal components, split by the WP, result in a different intensity,Ixy =I0[1±∆φTHz(xds)],

which is measured as the voltage difference between the two PDs, i.e.:

∆VPDs(xds)∝∆φTHz(xds)∝ETHz(xds). (3.3)

Since ∆φTHz is small, EOS requires a lock-in amplifier to detect small changes in ∆VPD. For this reason, the voltage applied on the PC-emitter is modulated with

a frequency ∼ 10 kHz, which is used as a reference for the lock-in detection and ensures a high signal-to-noise ratio. The delay stage is moved in order to obtain the entire THz-waveform, as depicted in the blue box of Figure 3.1. By considering that moving the delay stage byxds leads to double the path change for the gate-beam,

the conversion from delay-distance (in mm) to delay-time (in ps) is obtained with t(ps) = 2xds/c∼xds/0.1499.

ZnTe response. More precisely, the measured THz-signal for a gate beam an- gular frequencyω,ETHzS , differs from the realETHz and is given by the convolution with the ZnTe response functionFZnTe(ω, ωTHz) as [83, 84]:

ETHzS (t) = Z +∞

−∞

ETHz(ωTHz)FZnTe(ω, ωTHz)e−iωTHzt_dω

THz, (3.4)

whose Fourier transform leads to:

ETHzS (ωTHz) =ETHz(ωTHz)FZnTe(ω, ωTHz). (3.5) As the experimental setup just described used a 2mm-thick (110)-ZnTe,ETHz(ωTHz) is obtained as follows. The ZnTe response is given by three different frequency factors: the autocorrelation of the optical electric field, the second order nonlinear

susceptibility and the frequency-dependent velocity mismatch [83, 84]. Within this thesis, the response is considered as merely due to the latter factor which gives rise to a frequency filter function [84]:

FZnTe(ω, ωTHz) =e i∆k(ω,ωTHz)dZnTe−1 i∆k(ω, ωTHz) , (3.6) where ∆k(ω, ωTHz)' ˜n THz ZnTeωTHz c − ωTHz vg(ω) (3.7) and vg(ω) = c/[nopt(ω)−∂nopt/∂ω] is the optical group-velocity. Here dZnTe is the

ZnTe thicnkess, nopt(ω) is the optical refractive index given in Ref. [85] and ˜nTHz ZnTe is the complex refractive index at THz-frequencies, which is given in Ref. [84] and takes into account the TO-phonon resonance in ZnTe (ωTO

ZnTe/2π = 5.32 THz). By considering ω = 2πc/λg - where λg = 800 nm is the wavelength of the gate beam - the real THz spectrumETHz(ω) is obtained from Equation (3.5). The inverse Fourier transform then yieldsETHz(t).