In this section we discuss the electro-optic sampling technique [44, 45, 46]. It represents a very powerful method to characterize ultrashort pulses as it measures directly the electric field. Hence, it yields the complete spectral information of the field transient. Electro-optic sampling has become a standard technique in terahertz spectroscopy [33, 47, 48] and was applied for ultrafast pulses with wavelength down to the mid-infrared [32, 49]. For shorter wavelengths, however, it can not be applied as it would demand reference pulses to be shorter than available nowadays.
After a discussion of the principle of electro-optic sampling we address questions concern-ing the appropriate polarization directions for the mid-infrared beam and the near-infrared reference pulses. Then we study the wavelength range for which this technique can be ap-plied. Finally, we demonstrate the electro-optic sampling technique by sampling some typical mid-infrared field transients.
Electro-optic sampling The basic idea of electro-optic sampling is the following. The electric field of the mid-infrared pulse induces a change of birefringence in a crystal via the electro-optic effect, i.e., it generates a change in the index ellipsoid of the crystal (appendix B). A short near-infrared pulse measures this electro-optic modulation as a function of time delay via the phase retardance Γ induced by the birefringence.
A suitable sensor crystal is ZnTe since it has sufficient transparency properties in the near-and mid-infrared spectral range near-and a large eletro-optic coefficient [32]. We will see later that the phase retardance Γ is proportional to the amplitude of the mid-infrared electric field: Γ ∼ EM IR. Γ can be measured in the setup shown in Fig. 2.12 (a). The linearly polarized near-infrared beam is focused collinearly with the mid-infrared beam on the ZnTe crystal. This is achieved by leading the near-infrared beam through a hole in the off-axis mirror (OA). Behind the ZnTe crystal the near-infrared beam is slightly elliptically polarized because of the birefringence induced by the electric field of the mid-infrared pulse. It passes successively aλ/4 wave plate and a Wollaston polarizer (WP) to generate two symmetrically arms whose intensity difference is measured with a pair of balanced photo diodes (BPD).
The intensity difference is the phase retardance which will be shown now. To calculate the expected effect we use the matrix representation of optical elements affecting the polarization of the near-infrared pulse (C). The incident, linearly polarized near-infrared beam Ein = (E,0)ei(ω t−k z) is transformed into the elliptically polarized beamEout by passing the ZnTe crystal and theλ/4 waveplate in the following way [see Fig. 2.12 (b)]
Eout = 1
L1 OA ZnTe L2 /4 WP
Figure 2.12: Principle of measurement of the electric field of a mid-infrared pulse using electro-optic sampling. (a) The mid-infrared beam and a near-infrared reference beam are focused onto the ZnTe crystal (fL1 = 200 mm,fOA = 100 mm,fL2 = 100 mm). The reference beam passes a λ/4 wave plate and a Wollaston polarizer (WP) before it is measured with a pair of balanced photo diodes (BPD). (b) Evolution of the polarization of the near-infrared reference beam. The incoming beam is polarized along thexaxis. Under the influence of the mid-infrared induced index-ellipsoid [forn1 and n2 see formula (D.6)] of the ZnTe crystal it becomes slightly elliptically polarized. The index-ellipsoid of the λ/4 wave plate transforms the slightly elliptically polarized beam into a slightly ”non-circularly”polarized beam which is necessary to get two symmetric arms after passing the Wollaston polarizer.
= 1
The diodes measure the time integrated difference signal ∆E
∆E = lim
For Γ1 this results in the normalized difference signal
∆≡∆E/E = Γ∼EM IR (2.12)
By varying the time delay tD between the mid- and the near-infrared pulses the electric field of the mid-infrared pulse is measured directly as illustrated in Fig. 2.13. Thus, the complete information about the spectral phase and amplitude is obtained.
Figure 2.13: The Principle of electro-optic sampling: A short near-infrared reference pulse (NIR) measures the birefringence induced by the electric field of the mid-infrared pulse (MIR) as a function of the time delay tD.
Polarization geometry The non-vanishing components of the electro-optic tensor rZnT e for ZnTe at a mid-infrared wavelength of 10.6µm [50] are
r41 =r52 =r63 = 3.9·10−12m
V (2.13)
We choose a thin ZnTe plate being<110>oriented [32] and both the mid- and near-infrared propagation directions being perpendicular to the ZnTe surface. Under these conditions we show in appendix D that the maximum phase retardance Γ = ωc n3r41EM IRL, where L is the crystal thickness, is obtained for a polarization configuration as depicted in Fig. 2.14.
Wavelength range Eq. 2.12 is exact for a static mid-infrared field. Here, however, we want to measure the electric field of an ultrashort mid-infrared pulse with center frequency Ω. As A consequence, two points have to be taken into account:
x
y
z E
NIRE
NIRE
MIRZnTe surface
Figure 2.14: ZnTe surface and polarizations of the near- and mid-infrared beam yielding the maximum effect (for the near-infrared beam there are two possibilities).
• The difference between the group velocity of the near-infrared pulse and the phase velocity of the mid-infrared pulse [51]: First we consider the near-infrared reference pulse to be negligibly short and centered at a frequency ω0. δ(x) is the group/phase velocity mismatch time after having passed a distance x in the crystal
δ(x) =
1
vgr(ω0) − 1 vph(Ω)
x
with the group velocity of the near-infraredvgr(ω0) and the mid-infrared phase velocity vph = n(Ω)c . The mid-infrared field EM IR(x) measured by the near-infrared pulse is
EM IR(x) =EM IR cos(Ω·δ(x) +ϕ)
where ϕ is the phase of the mid-infrared wave (which depends on the time delay tD between the near- and mid-infrared pulse). The electro-optic effect induced by the mid-infrared pulse is averaged along the crystal
∆ =˜ ω
c n(ω0)3r41
L
0 EM IR(x)dx
• The finite pulse length of the near-infrared pulse. This effect is modelled by the convolution with the normalized envelope of the near-infrared pulse
IN IR(x0) =Inorme−4 ln 2
n(ω)x0 c τp
2
where the near-infrared pulse length is τp = 14 fs.
Taking into account these two points the measured normalized differential signal reads
∆ef f(Ω, L, ϕ) = ω
c n(ω0)3r41
L
0
∞
−∞ EM IR(x−x0)·IN IR(x0)dx0dx
Figure 2.15: Difference signal ∆max(L) plotted as a function of the crystal thickness L for λM IR = 3µm (dash-dotted line), 10µm (dashed line), 20µm (dotted line) and for a situation without group velocity mismatch.
Both effects distort the signal because they act differently on different frequency components of the mid-infrared pulse. Note that for different crystal thicknesses L the measured signal
∆ef f = ∆ef f(ϕ) experiences a phase shift. ϕmax is the phase where the signal reaches its maximum (∆max ≡∆ef f(ϕmax)).
In Fig. 2.15the difference signal ∆max(L) is plotted as a function of the crystal thickness L for λM IR = 3, 10, and 20µm and for a situation without group velocity mismatch. As can be seen, for shorter mid-infrared wavelengths the effects described above become more serious. Consequently, a thinner ZnTe crystal is required. ForλM IR= 10µm the maximum difference signal is obtained at a crystal thickness of 8µm. As we will sample ultrafast mid-infrared pulses with center frequencies between 10µm and 12µm we have used a crystal thickness of L= 10µm.
Experimental data Typical mid-infrared electric field transients generated via phase-matched difference frequency mixing in a GaSe crystal and measured with electro-optic sampling are shown in Fig. 2.16. In (a) the electro-optic signal is shown as a function of time delay between the mid-infrared pulse and a near-infrared reference pulse. In (b) the mid-infrared pulse has passed a 6 mm thick potassium bromide (KBr) window. The
respec-Figure 2.16: (a) Mid-infrared electric field transients measured with electro-optic sampling.
The mid-infrared pulse was generated via phasematched difference-frequency mixing in a GaSe crystal. (b) Field transient after passing 6 mm KBr. (c), (d) Power spectra (solid line) and spectral phases (circles) gained from the Fourier transform of the respective transients.
Dashed lines: Quadratic fit with a chirp∂2Φ/∂ ω2 of (c) 37 fs2 and (d) 90 fs2.
tive power spectra [Fig. 2.16 (c),(d): solid lines] and the spectral phases [Fig. 2.16 (c),(d):
circles] were obtained by the Fourier transforms of the transients. The pulse in Fig. 2.16 (a) has a linear chirp ∂2Φ/∂ ω2 of 37 fs2. After having passed the KBr window the linear chirp increases to 90 fs2. Hence, 6 mm KBr account for a linear chirp of 53 fs2. With the Sellmeier equation for KBr
n2KBr(λ) = 1.39408 +0.79221·λ2
λ2−0.1462 +0.01981·λ2
λ2 −0.1732 +0.15587·λ2
λ2−0.1872 +0.17673·λ2
λ2 −60.612 +2.06217·λ2 λ2−87.722 (λinµ) a linear chirp of 52 fs2 after the passage of 6 mm KBr is calculated which is in good agreement with the measured one.
In conclusion, we have seen that electro-optic sampling is an excellent method to deter-mine the spectral phase and amplitude of a mid-infrared pulse. This is used in the following to characterize shaped mid-infrared pulses.