Sun et al. [2
2. F. G. Sun, Zhiping Jiang, and X.-C. Zhang, “Analysis of terahertz pulse measurement with a chirped probe beam,” Appl. Phys. Lett. 73, 2233 (1998). [CrossRef]
] have analysed the system using the method of stationary phase and derived the limiting time resolution. In this paper we use a method of analysis appropriate for THz signals in the form of long or short pulses, and show that only under restricted conditions is the detected signal a faithful representation of the THz signal, and ambiguities arise under more general conditions.
The response of the electro-optic detector is linear in the electric field of the THz signal. We can therefore model the signal as a collection of Gaussian pulses and calculate the response to each pulse separately. For a Gaussian pulse of amplitude A and length T, arriving at time t0, the THz electric field is
The transmitted amplitude of the chirped NIR pulse through the electro-optic crystal and crossed polarisers is proportional to (B + kE) where B represents the background transmission in the absence of a THz signal, and k is a modulation constant. We assume that the chirp is long enough to sample the whole of the THz pulse, so that the electric field of the chirp can be written as
The instantaneous frequency, ω, is defined by
with chirp rate 2α. The NIR transmitted amplitude reaching the spectrometer is then
The frequency components of this modulated probe are
The spectrometer output is proportional to ∣ S(ω)∣2, assuming adequate resolution, and this is interpreted as the time-dependence of the THz signal by means of the relation
Consider first the strong signal case, in which the modulation, kA is larger than the background, and B can be neglected.
From (3) and (4)
Carrying out the Gaussian integral and substituting (2αt) for (ω - ω0) gives a detected signal of the form
Allowing for the quadratic dependence of ∣S (ω)∣ on the THz signal, the spectrometer output would be interpreted as being due to a THz pulse proportional to ∣S (ω)∣ and thus the detected pulse length is given by
Figure 1 shows the relation between the detected pulse length and the true pulse length, T.
Fig. 1. Variation of detected pulse length versus input pulse length for strong signals.
For THz pulses with length T longer than the characteristic time Tmin the detected pulse correctly matches the input signal, but for fast pulses, shorter than Tmin, the detected pulse length is
Thus, short pulses are distorted and appear to be increased in length. This introduces ambiguity whereby both long and short THz pulses can produce the same detected pulse length. The effect arises from the uncertainty principle; a section of duration T from the NIR chirp pulse contains a range of frequencies of width at least T
1, so that short pulses give a wide signal in the spectrometer.
The characteristic time, Tmin, is given by
0 is the original NIR pulse length before chirping, and T
C is the length of the chirp, as shown by Sun et al.
For a weak signal, the modulation term in (3) is smaller than the background, the
modulation of the spectrometer signal, Δ ∣S∣2, is proportional to BkA, and the term in k2A2 is negligible,
The resulting expression can be simplified for the cases of pulses with length longer or shorter than Tmin.
For long THz pulses, using Eq. (5
and the spectrometer output matches the input signal.
For short THz pulses, the modulation of the spectrometer output is
Fig. 2. Detected pulse shape for fast weak signals.
The form of this output is shown in Fig. 2
for small values of ∣t
∣. For longer ∣t
∣ the oscillations will be attenuated by the spectrometer resolution and the exponential
envelope factor. It is clear that this output bears little relation to the form of the THz input. The spurious oscillations arise from the interference between the frequency components of the chirp and the frequencies produced as a consequence of the uncertainty principle with a fast pulse.
The characteristic limiting time, Tmin can be estimated for typical values of the system parameters. If the length of the initial NIR pulse, T0, is 50 fs, and the length of the chirp, TC, is sufficient to sample a THz signal of 20 ps duration, then
In this case signal frequencies beyond about 1 THz could not be detected. In order to reach higher frequencies, a shorter initial NIR pulse would be needed, or a shorter chirp used to sample only a part of the THz signal.
In principle it may be possible to distinguish between distorted and undistorted pulses by repeating the measurement with an altered chirp rate. An undistorted pulse would not be affected by the change, but the form of the distorted pulse is dependent on the chirp rate, and could thus be distinguished. However, this would add considerable experimental complication.
In summary, for THz pulses longer than the limiting time, Tmin both strong and weak signals are faithfully reproduced by the spectrometer. Strong pulses shorter than Tmin produce an output that mimics the behaviour of long pulses and leads to ambiguity. Weak short pulses produce a spurious output that gives no information on the form of the input signal.