Distortion analysis of pulsed terahertz signal measured with spectral-encodingtechniqueXiao-Yu Peng, Jing-Hua Teng, Xin-Hai Zhang, and Yong-Lim Foo Citation: Journal of Applied Physics 108, 093112 (2010); doi: 10.1063/1.3499639 View online: http://dx.doi.org/10.1063/1.3499639 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/108/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in To realize the optimal probe pulse length for detection of pulsed terahertz signal with spectral-encodingtechnique Appl. Phys. Lett. 98, 231111 (2011); 10.1063/1.3598405 Single-shot measurement of terahertz temporal waveform using pulse-front tilting by a direct vision dispersionprism Rev. Sci. Instrum. 80, 113703 (2009); 10.1063/1.3257617 Distortion of the intense terahertz signal measured by spectral-encoding technique Appl. Phys. Lett. 94, 221107 (2009); 10.1063/1.3148674 Differential electronic gating: A method to measure the shape of short THz pulses with a poorly defined triggersignal Appl. Phys. Lett. 77, 1762 (2000); 10.1063/1.1311601 Analysis of terahertz pulse measurement with a chirped probe beam Appl. Phys. Lett. 73, 2233 (1998); 10.1063/1.121685

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

131.111.164.128 On: Mon, 22 Dec 2014 08:33:17

Distortion analysis of pulsed terahertz signal measuredwith spectral-encoding technique

Xiao-Yu Peng,a� Jing-Hua Teng, Xin-Hai Zhang, and Yong-Lim FooInstitute of Materials Research and Engineering, A*STAR (Agency for Science, Technology and Research),3 Research Link, Singapore 117602

�Received 23 February 2010; accepted 10 September 2010; published online 9 November 2010�

The principle of the spectral-encoding technique, one of the single-shot techniques for measurementof arbitrary pulsed terahertz �THz� signals, or T-rays, is demonstrated theoretically and throughsimulation. For single-cycle input THz pulse �bipolar wave form�, THz signal could be retrievedfaithfully under certain conditions, while for the multicycle input case, the THz signal cannot beretrieved without any distortion. However, there is an optimal length of the chirped-probe-pulse,corresponding to the characteristic time of the multicycle THz field, to minimize the distortion in theretrieved THz signal. Three main possible sources of distortions from both bipolar and multicycleTHz signals are analyzed systematically. The first distortion originates from the mismatch of theoptimal duration Tco of the chirped-probe-pulse and the T-ray length T. The second one is relevantto the spectrum bandwidth of the probe pulse. The third one occurs when the modulation depth k�1is not satisfied. It comes from the neglect of the quadratic term of the modulation depth k, whichreflects the strength of the THz signal in the retrieving process. Possible strategies are proposed toreduce these distortions. © 2010 American Institute of Physics. �doi:10.1063/1.3499639�

I. INTRODUCTION

Terahertz �THz� radiation have generated tremendous in-terest over past two decades due to its immense potentials inapplications such as THz spectroscopy,1 bio and medicalimaging,2 homeland security,3–5 nondestructive testing,6

tomography,7 wireless communication,8 etc. One technicalproblem in THz application is the effective distortion-freeTHz detection. A typical detection technique for T-ray is theelectro-optic sampling �EOS�.9 For the detection of the T-raywith high repetition rate, generally the scanning EOS tech-nique is used. Scanning EOS is a standard and key techniquefor the THz spectroscopy because both the phase and theamplitude of a THz pulse can be measured. The main advan-tages of this technique include its high sensitivity, high signalto noise ratio, and high temporal resolution depending on theshort probe pulse duration T0. However, this technique needsat least several minutes or more for scanning an entire THzpulse and the measured THz profiles represent averagedwave forms. Thus, it cannot be applied in cases such as theT-rays with low repetition rate or with strong shot-to-shotfluctuations as well as some experiments that may have lowduty cycles. To overcome this problem, some single-shotEOS techniques were introduced.

The first single-shot EOS technique called spectral-encoding technique was first proposed in 1998.10 In thissimple technique, the temporal wave form of the THz pulseis linearly encoded onto the frequency spectrum of an opticalchirped-probe-pulse and then decoded by dispersing theprobe beam in a grating spectrometer. But it was thought tohave low temporal resolution ��T0Tc�1/2, where T0 is theFourier transform limited probe-pulse length and Tc is thechirped-probe-pulse length, and the distortion of the mea-

sured signals in previous work was ascribed to this reason.Afterward, many attempts have been made to improve thetemporal resolution. One method is using an optical streakcamera11 instead of the spectrometer. This configuration is aparallel sampling technique in the time domain. However,this technique requires the streak camera to be synchronizedprecisely to the probe pulse and the temporal resolution islimited by that of the optical streak camera. Another methodis using an interferometric retrieval algorithm12,13 in whichthe temporal resolution could reach the transform-limitedprobe pulse width T0. Nevertheless, a complicated matrixinversion equation needs to be resolved numerically and thiscomplex algorithm often produces numerical artifacts on theretrieved THz wave form.14

Other two single-shot EOS methods are noncollinear ge-ometry, namely, the spatial encoding technique15 and thecross-correlation technique.16 The latter is a combination ofthe spatial and the spectral-encoding method. In both tech-niques, the temporal THz properties are transposed to thetransverse-spatial laser probe profile recorded by a multi-channel detector array or a charge coupled device �CCD�camera. The THz wave form can be obtained through thespace-to-time coordinate conversion. One of the main disad-vantages for both techniques is that both the THz beam andthe probe beam are required to be uniform in intensity dis-tribution across the beams and to have flat wave fronts, oth-erwise distortion will be introduced. To resolve these prob-lems, a single-shot technique via two-dimensional EOimaging with dual echelons was introduced.14 In this tech-nique, different parts of the THz field modulate differentbeamlet that is delayed incrementally since the probe laserbeam is split by the dual echelons. The main disadvantage ofthis method is that the manufacturing defects such as thebubbles trapped in the epoxy layer and the inhomogeneity ina�Electronic mail: pengxy@imre.a-star.edu.sg.

JOURNAL OF APPLIED PHYSICS 108, 093112 �2010�

0021-8979/2010/108�9�/093112/10/$30.00 © 2010 American Institute of Physics108, 093112-1

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

131.111.164.128 On: Mon, 22 Dec 2014 08:33:17

epoxy layer thickness across the prism substrate of the dualechelons can introduce artifacts in the time axis calibration.

Among all the single-shot EOS T-ray detection tech-niques, the spectral-encoding technique is the simplest andmost sensitive one. It requires only several microjoule ofenergy for the probe pulse and can observe the THz waveform in real time. Though the principle of spectral-encodingtechnique had been demonstrated and some distortions of theretrieved THz signal were discussed before,17–20 only thesingle-cycle T-ray signal was discussed in previous worksand there is no report in the multicycle T-ray signals that aremore general in real cases. Furthermore, no systematic analy-sis about the distortions has been reported by now, especiallyfor the multicycle T-ray signals. These distortions signifi-cantly affect the real application of the spectral-encodingtechnique. In this paper, we will demonstrate the principle ofthis technique theoretically. Based on our model, main pos-sible distortions for both the bipolar and the multicycle T-raysignals are analyzed in detail. Accordingly, some feasiblestrategies to reduce the distortions are proposed.

II. PRINCIPLE OF THE SPECTRAL-ENCODINGTECHNIQUE

Spectral-encoding technique is based on the Pockels ef-fect of the EO crystal. A typical setup consists of an electro-optic �EO� crystal placed between two crossed polarizers.For the sake of simplicity, the following theoretical deduc-tion of our model is based on typical bipolar THz wave form,the assumption of a Gaussian shape of the probe pulse, andthe assumption of the EO crystal having broad enough THzbandwidth to cover the entire spectrum of the THz signal tobe measured. This deduction is helpful to understand theretrieving process of arbitrary multicycle T-ray signals. Someresults limited to the case of input bipolar T-ray signals hadbeen published in previous work.17 Here a modified demon-stration including the discussion of input multicycle T-raywave forms is presented.

A. Single-cycle T-ray signal

The input T-ray signal with a bipolar THz wave form canbe expressed as ETHz�t�=−�t /T�exp�−t2 /T2�, where T is theTHz pulse length related to the characteristic time �2T,which is the interval between the maximum and the mini-mum as defined by Sun.18 For the sake of simplicity, theentire THz pulse length is roughly defined as Te�4�2T. Theelectric field of the chirped probe pulse is written as Ec�t�=exp�−t2 /Tc

2− i�t2− i�0t�, where Tc is the pulse length and2� is the chirp rate which is approximated by the laser band-width divided by the chirped laser pulse length, namely 2�� ���0 /Tc and �0 is the central frequency. Here “+” and“�” represent the positive and the negative chirp, respec-tively. Generally this probe pulse can be obtained by stretch-ing an original short pulse E0�t�=exp�−t2 /T0

2− i�0t� thatpassing through a grating pair ���0� or a dispersive opticalglass ��0�. T0 is the Fourier transform-limited pulselength, which is related to the laser spectral bandwidth ��0

through T0=2 /��0.

When the chirped probe pulse and a THz pulse copropa-gate in an EO crystal, the chirped pulse is modulated by thepulsed THz field through the Pockels effect. The THz in-duced leakage of the probe is detected in a spectrometer.Some background probe light before the modulation by theTHz field is necessary for coherent detection. It can be ob-tained by the inherent residual birefringence of the EO crys-tal, or analyzer detuning, or through a wave plate after theEO crystal. Considering the small bandwidth limit, in theabove cases the electric field of the chirped probe pulsemodulated by the THz field can be written as21 Em�t�=Ec�t��1+k exp�i��ETHz�t−���. Here, k is the modulation depthor modulation constant which reflects the strength of the THzfield, � is the relative time delay between the probe pulse andthe THz pulse, � is the phase induced by the wave plate oranalyzer detuning. In reality, by increasing the intensity ofthe probe beam, the residual birefringence can be sufficientlystrong to provide enough background probe light. One canconsider a simple case that the polarizer and the analyzer arecrossed exactly to each other, without the wave plate or ana-lyzer detuning, i.e., �=0. In principle, the probe pulse is ableto be completely synchronized with the THz pulse by adjust-ing the time delay, i.e., �=0. Thus, a simple expression of theelectric field of the modulated chirped probe pulse then canbe expressed as

Em�t� = Ec�t��1 + kETHz�t�� . �1�

The spectral modulation is spatially separated on theCCD camera of the spectrometer. The measured signal on aCCD pixel corresponds to the optical frequency �1. Thespectrum of the background chirped pulse Ib �without THzfield modulation� and the modulated spectrum Im by THzfield can be expressed as the convolution of the spectral res-olution function g��1−�� and the square of the Fouriertransform of Ec�t� and Em�t�, respectively. Assuming thespectral resolution of the spectrometer is high enough,g��1−�� can be expressed as a function. Thus Ib and Im

can be expressed, respectively, as follows:

Ib � Ec��1�2 = �−�

�

Ec�t�exp�i�1t�dt2

, �2�

Im � Em��1�2 = �−�

�

Em�t�exp�i�1t�dt2

. �3�

Define a function S��1� relevant to the difference betweenmodulated probe spectrum and the background probe spec-trum as

S��1� = �I = Im − Ib. �4�

As mentioned before, T0=2 /��0 and 2�� ���0 /Tc,one has �� � �T0Tc�−1. For a linear chirp, the probe instan-taneous frequency �1 measured on the CCD pixel is propor-tional to the time t�. The THz field profile thus can be di-rectly obtained through measurement of the spectralmodulation S��1� using following coordinate transformation:

�1 − �0 = 2�t�. �5�

093112-2 Peng et al. J. Appl. Phys. 108, 093112 �2010�

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

131.111.164.128 On: Mon, 22 Dec 2014 08:33:17

Introducing dimensionless pulse lengths m=T /T0 and n=Tc /T0, after deduction one obtains the THz signal relevantto the difference between the THz field modified probe spec-trum and the background probe spectrum as follows:

S�t�� � k�1f�t�/T0� �−t�

T exp��−

t�2

T2 �1�+ k2�2��−

t�

T exp�−

t�2

T2 �2 �2

, �6�

with

�1 = 2�T02n−1�n−4 + n−2�−1/2��m−2 + n−2�2

+ n−2�−3/2��m−2n−2 + n−4 + n−2�2 + m−4n−2�1/4, �7�

f�t�/T0� = n−1 cos�1

2arctg� n3

m2n2 + m2 + n2 −

n3�2m2 + n2��1 + n2���m2 + n2�2 + m4n2�

� t�

T0 2�

− �m−2 + n−2�sin�1

2arctg� n3

m2n2 + m2 + n2 −

n3�2m2 + n2��1 + n2���m2 + n2�2 + m4n2�

� t�

T0 2� , �8�

�1 =m2

1 + n2 +m4n2 + m6

�m2 + n2�2 + m4n2 , �9�

�2 = �T0−4/3n−2��m−2 + n−2�2 + n−2�−3/2, �10�

�2 =n−2�m2n−2 + 1�

�m−2 + n−2�2 + n−2 . �11�

Considering a small amplitude of THz field, i.e., k�1, thesecond item with k2 of Eq. �6� can be neglected, one has

S�t�� � k�1f�t�/T0� �−t�

T exp��−

t�2

T2 �1� . �12�

Giving T and Tc, �1 is a constant. k is the modulationconstant. The function f�t� /T0� has a plateau around the cen-ter of the time window. It should be noted that nearly theentire detected THz pulse falls into this central time windowif the probe pulse is synchronized perfectly with the THzpulse. Thus the value of f�t� /T0� around the central timewindow could be seen roughly as a constant. Therefore �1 isthe main factor which affects the profile of the retrieved THzwave form.

If �1=1 one has S�t���ETHz�t��.This means that the realTHz profile can be faithfully retrieved if �1=1. When m1.4, the equation �1=1 can be resolved and one obtains therelationship between the dimensionless optimal chirped-probe-pulse length no and the dimensionless THz pulselength m as follows:17

no = �1

3�2�6m6 + 4m4 − 8m2 + 1�1/2cos��

3

− �m2 − 1���1/2, �13�

with �=arccos�−�3 /54��18m8+61m6+6m4+24m2−2� / �6m6

+4m4−8m2+1�3/2�.The optimal chirped pulse length then can be expressed

as

Tco = �1

3�2�6m6 + 4m4 − 8m2 + 1�1/2cos��

3

− �m2 − 1���1/2T0. �14�

The optimal chirp rate is

2�o � � 2T0−2�1

3�2�6m6 + 4m4 − 8m2 + 1�1/2cos��

3

− �m2 − 1���−1/2. �15�

Theoretically, it is possible to retrieve the THz signalwithout distortion if one use a chirped-probe-pulse with theoptimal duration Tco �=noT0� matched to the input THz pulselength T �=mT0� according to expression �14� Our simulationresults, including the THz wave forms and the THz spectrashown in Fig. 1 for the T-ray retrieving process, prove theabove theory is right. The parameters used in this simulationare k=0.03, T0=10 fs, and T=0.25 ps. The optimal chirped-probe-pulse duration Tco=1.4795 ps is calculated accordingto expression �14�. From Figs. 1�a� and 1�b�, one can seeclearly that the retrieved THz wave form and its spectrum arecoincident with the original ones very well. The modulated

-2 -1 0 1 2

-1.0

-0.5

0.0

0.5

1.0

TH

zfie

ld(a

.u.)

t (ps)

Original

Tc=Tco

0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.u

.)

Frequency (THz)

Original

Tc=Tco

(a) (b)

(c) (d)

700 800 9000

1

2

3

4

� >0

Inte

nsity

(a.u

.)

Wavelength (nm)

Ib

Im

700 800 900

Wavelength (nm)

I�

700 800 9000

1

2

3

4

� <0

Inte

nsity

(a.u

.)

Wavelength (nm)

Ib

Im

700 800 900

Wavelength (nm)

I�

FIG. 1. �Color online� Retrieved bipolar THz wave form and its spectrumby simulation compared with the original THz field in �a� and with theoriginal THz spectrum in �b�, respectively. The modulated probe spectrumIm vs its background spectrum Ib with positive chirp rate in �c� and negativechirp rate in �d�, respectively. The insets in �c� and �d� show the difference�I �=Im− Ib�. The simulation conditions include k=0.03, T0=10 fs, and T=0.25 ps. The optimal chirped-probe pulse duration Tco=1.4795 ps is cal-culated according to expression �14�.

093112-3 Peng et al. J. Appl. Phys. 108, 093112 �2010�

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

131.111.164.128 On: Mon, 22 Dec 2014 08:33:17

probe spectrum Im versus its background spectrum Ib withpositive chirp rate and negative chirp rate are also shown inFigs. 1�c� and 1�d�, respectively. The insets in Figs. 1�c� and1�d� show the spectrum difference �I �=Im− Ib�. The differ-ence of �I between using positive and negative chirp rate isthat they are antiphase to each other: the former has an an-tiphase shape to the THz wave form while the later has asimilar shape of the THz wave form.

In previous pioneer work10,18 on spectral-encoding tech-nique, a normalized differential intensity relevant to THz sig-nal was defined as S��1�= �Im− Ib� / Ib. However, it was foundthat in reality this normalization would bring noise on thetwo sides of the time window due to the spectrum normal-ization with a small background.10 We have also tried thisnormalization by both simulation and experiment and foundit brought many spikes even larger than the amplitude of theTHz field. After Fourier transformation, this noise wouldcause many spikes in the THz spectrum, which differs muchfrom the original THz spectrum. Besides, one cannot obtainthe expressions �12� and �13� if this normalization is per-formed in the theoretical deduction. From our theory, onefinds that the normalization to the background of probe spec-trum is not necessary.

B. Multicycle T-ray signal

It should be noted that above discussions are based onthe single-cycle THz wave form as an input THz pulse. Forthe multicycle THz wave forms, the situation is more com-plicated. It is difficult to represent an arbitrary multicycleTHz wave form with a simple equation. However, for thesake of simplicity and convenience of the discussion, wemay use the following expression to mimic an arbitrary mul-ticycle THz field

ETHz�t� = −t − T/x

Texp�−

�t − T/x�2

T2 �exp�−2

xT�t − T/x��

cos�2�

xT�t − T/x�� , �16�

where x is a positive real number. We define the entire pulselength Te�4�2T as the same as the bipolar field case. Onemay change the value of x to obtain the THz wave formswith different cycles but Te does not change with the changein x if T does not change. Figures 2�a� and 2�d� shows fourexamples of multicycle THz fields with T=0.5 ps when x=1, 1.5, 5, and 9, respectively. If x is large enough, say x=9, the multicycle THz field tends to resemble a bipolarwave form as shown in Fig. 2�d� but the strength of twopolars are not equal to each other. Specially, if x→�, the Eq.�16� tends toward a standard bipolar field, as described in Eq.�1�. Therefore, in Eq. �16� the characteristic time is still de-fined as the interval between the maximum and the minimum�2T, when x�1.

For multicycle THz wave forms, unfortunately, one can-not deduce a simple result as Eq. �6� including the separateoscillation function f�t� /T0� described in the Eq. �8� and thepart like the original THz field with the factor �1 described inthe Eq. �9�. Consequently one cannot obtain an exact rela-

tionship between the chirped-probe-pulse duration Tc and thecharacteristic time T to retrieve the T-ray signal without dis-tortion, as the Eqs. �13� and �14�. This means that, in anycase of the multicycle THz wave forms, the distortions al-ways exist to some extent. One cannot eliminate them totally.However, our simulations results indicate that there are stillsome optimal chirped-probe-pulse lengths match to the char-acteristic time of the multicycle THz fields. Under this con-dition, the retrieved THz signals have minimum distortion asshown in Fig. 3. We found that it is hard to define when thedistortion is the minimum in the case of multicycle THzwave forms. For the sake of simplicity, we may define thatthe distortion tends to be the minimum �namely, the retrievedTHz wave form is closest to the original THz signal� whenboth full widths of the half-maximum intensity of the re-trieved THz spectrum and the original one are equal to eachother. In addition, because the larger the x, the smaller thecycle number, one may find that the distortion would be re-duced more if the optimal durations of probe pulse are usedas shown in Fig. 3. When x=9, the distortion looks like verysmall and close to the case of standard bipolar THz waveform, as shown in Fig. 3�d�. One can also observe that theoptimal chirped-probe-pulse length Tco increases with the in-crease in x �i.e., the decrease in the cycle number of the THzwave form� as shown in Fig. 3. When x→�, Tco tends to-ward the optimal chirped-probe-pulse length in the case ofstandard bipolar THz field.

III. THREE MAIN POSSIBLE DISTORTIONS

Due to the intrinsic nature of the spectral-encoding tech-nique, principally there are three kinds of possible distortionsfor both the single-cycle and the multicycle T-ray signalsaccording to above theory.

A. The first distortion: from the mismatch of T andTco

In the single-cycle T-ray case, from expressions �13� and�14�, one can find that this is a rigorous relation between the

-2 -1 0 1 2

-0.3

0.0

0.3

TH

zfie

ld(a

.u.)

t (ps)

x=9

(d)

-2 -1 0 1 2

-1

0

1

TH

zfie

ld(a

.u.)

t (ps)

(b)

x=1.5

-2 -1 0 1 2

-0.2

0.0

0.2

0.4

TH

zfie

ld(a

.u.)

t (ps)

(c)

x=5

(a)

-2 -1 0 1 2-4

-2

0

2

TH

zfie

ld(a

.u.)

t(ps)

x=1

FIG. 2. Simulated arbitrary T-ray signal with T=0.5 ps and with multicyclewave forms dependent on the parameter x. The T-ray signal tends to astandard bipolar wave form when x→�.

093112-4 Peng et al. J. Appl. Phys. 108, 093112 �2010�

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

131.111.164.128 On: Mon, 22 Dec 2014 08:33:17

chirped-probe-pulse length Tco and the input bipolar T-raylength T. Only under this rigorous condition can the T-raysignal be possibly retrieved without distortion. Otherwise thefirst distortion would be introduced if TcTco or �Tco. Dueto that �1 is the main factor affecting the profile of the re-trieved THz wave form in Eq. �12�, the first distortion be-comes more important than other ones. For the simplicityand clarity, we only consider this distortion by assuming thefunction f�t� /T0� as a constant. According to Eq. �12�, wecalculate some examples with different chirped-probe-duration Tc, i.e., with different discrepancy from the optimalchirped-probe-pulse length �Tc �=Tc−Tco�, such as �Tc

=0 ps, �Tc=−0.5 ps, �Tc=−0.3 ps, �Tc=0.5 ps, and �Tc

=1 ps. The results are shown in Figs. 4�a� and 4�b�, wherethe parameters used in calculation are k=0.02, T0=25 fs, T=0.25 ps, and Tco=0.93 ps as calculated from the expres-sion �13�. One can clearly see that the larger the �Tc is, theseverer the distortion of the retrieved THz field will be. Indetail, if Tc�Tco, the retrieved THz wave form is com-pressed while its spectrum is broaden comparing to the inputoriginal THz field. Whereas, if TcTco, the retrieved THzwave form is broaden while its spectrum is compressed.

For the multicycle T-ray signals, the same rule as in thecase of single-cycle T-ray signal can be observed. Figure 5shows the simulation results of an example of retrieved mul-ticycle T-ray signals at different discrepancy from the opti-mal chirped-probe-pulse length �Tc=0 ps, �Tc=−0.8 ps,�Tc=−0.6 ps, �Tc=0.8 ps, and �Tc=1.5 ps. The param-eters used in simulation include x=1, T=0.5 ps, and T0

=10 fs. The corresponding optimal chirped-probe-pulselength is Tco�1.25 ps. One can see that the larger the �Tcis, the severer the distortion of the retrieved THz field willbe. If Tc�Tco, the retrieved THz wave form shown in Fig.5�a� is compressed while its spectrum shown in Fig. 5�b� isbroaden comparing to the input original THz field. Whereas,if TcTco, the retrieved THz wave form is broaden while itsspectrum is compressed. Only when Tc�Tco, the first distor-tion tends to be the minimum.

The physical origin of the first distortion is not difficultto be understood. Optimal chirped-probe-pulse length Tco

means an optimal time window for the T-ray detection. If thedetection time window is too small �Tc�Tco�, only a smallpart of the THz pulse modulates the probe pulse and is de-tected. If the detection time window is too large �TcTco�,only the small part of the probe pulse is modulated, corre-sponding to a small part of the entire spectrum bandwidth ofthe chirped-probe-pulse being used, which means that thetemporal resolution is degraded. Therefore the first distortionis generated when Tc�Tco.

B. The second distortion: relevant to the spectrumbandwidth of the probe pulse

For the multicycle T-ray signals, a distortion still existseven an optimal chirped-probe-pulse length is used as shown

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

x=1

T=0.5 ps

Tco~1.25 ps

TH

zfie

ld(a

.u.)

t (ps)

Original

Tc~Tco

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.u

.)

Frequency (THz)

Original

Tc~Tco

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

-0.5

0.0

0.5

1.0

x=1.5

T=0.5 ps

Tc~1.42 ps

TH

zfie

ld(a

.u)

t (ps)

Original

Tc~Tco

0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.u

.)

Frequency (THz)

Original

Tc~Tco

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

-0.5

0.0

0.5

1.0

x=5

T=0.5 ps

Tco~3.4 ps

TH

zfie

ld(a

.u.)

t (ps)

Original

Tc~Tco

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0 Original

Tc~Tco

Inte

nsity

(a.u

.)

Frequency (THz)

0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0 Original

Tc~Tco

Inte

nsity

(a.u

.)

Frequency (THz)

(a) (b)

(c) (d)

(e) (f)

(g) (h)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

-0.5

0.0

0.5

1.0

x=9

T=0.5 ps

Tco~3.5 ps

TH

zfie

ld(a

.u.)

t (ps)

Original

Tc~Tco

FIG. 3. �Color online� Retrieved multicycle THz wave forms in �a�, �c�, �e�,and �g� and their corresponding spectra in �b�, �d�, �f�, �h� when x=1, 1.5, 5,and 9, respectively. The simulation conditions include k=0.05, �0, T0

=10 fs, and T=0.5 ps. The optimal chirped-probe pulse durations of Tco

�1.25 ps, 1.42 ps, 3.4 ps, and 3.5 ps when x=1, 1.5, 5, and 9, respectively,are found by simulations.

(a) (b)T0=25fs T0=25fs

-0.8 -0.4 0.0 0.4 0.8

-1.0

-0.5

0.0

0.5

1.0Original

�Tc=0ps

�Tc=-0.5ps

�Tc=-0.3ps

�Tc=0.5ps

�Tc=1ps

TH

zfie

ld(a

.u.)

t (ps)

0 1 2 30.0

0.2

0.4

0.6

0.8

1.0 Original

�Tc=0ps

�Tc=-0.5ps

�Tc=-0.3ps

�Tc=0.5ps

�Tc=1ps

Inte

nsity

(a.u

.)

Frequency (THz)

FIG. 4. �Color online� The first distortion of bipolar THz signal due to themismatch �Tc �=Tc−Tco� compared with the case of optimal chirped-probeduration Tco ��Tc=0� in �a� and their spectra in �b�, respectively. The cal-culation parameters include k=0.05, �0, T=0.25 ps, Tco=0.93 ps, T0

=25 fs. The value of Tco is calculated according to the formula �14�.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

-0.5

0.0

0.5

1.0

TH

zfield

(a.u

.)

t (ps)

Original

�Tc=0ps

�Tc=-0.8ps

�Tc=-0.6ps

�Tc=0.8ps

�Tc=1.5ps

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0Original

�Tc=0ps

�Tc=-0.8ps

�Tc=-0.6ps

�Tc=0.8ps

�Tc=1.5ps

Inte

nsity

(a.u

.)

Frequency (THz)

(a) (b)

FIG. 5. �Color online� The first distortion of multicycle THz signals whenTc�Tco /4, Tco /2 in �a� and their spectra in �b�, respectively, compared withthe case of optimal chirped-probe duration Tco�1.25 ps. The simulationparameters include k=0.05, �0, T=0.5 ps, x=1, and T0=25 fs.

093112-5 Peng et al. J. Appl. Phys. 108, 093112 �2010�

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

131.111.164.128 On: Mon, 22 Dec 2014 08:33:17

in Figs. 3 and 5, This is the second distortion which is rel-evant to the spectrum bandwidth of the probe pulse. How-ever, we cannot discuss it separately from the first distortionbecause they cannot be detached from each other theoreti-cally in the case of multicycle T-ray wave forms. Fortunately,the first and the second distortion can be separated theoreti-cally and discussed separately in the case of single-cycleT-ray signal. This quantitative analysis is helpful for under-standing the same rule existing in the multicycle T-ray cases,which can be only discussed through simulation method.

For the single-cycle T-ray signal, the second distortionmathematically comes from the function f�t� /T0� as shownin Eq. �12�. Actually only a small part of it �central part� canbe seen roughly as a constant �the plateau� due to its fierceoscillation on the two sides of the time window. An exampleof this function versus an original THz signal when m=10 isshown in Fig. 6�a�. It is the width of this plateau that deter-mines if it brings in the second distortion. To describe thewidth of this plateau, let f�t��=0, we define roughly a dimen-sionless plateau width ted of f�t� /T0� as

ted = 2�1

2arctg� n3

m2n2 + m2 + n2 + arctg� m2n

m2 + n2 n3�2m2 + n2�

�1 + n2���m2 + n2�2 + m4n2��

1/2

.

�17�

The plateau width then can be expressed as te= tedT0. Thedependence of ted on m is shown in Fig. 6�b�. One can findthat ted broadens with increase in m and it is only related to mand n but not related to T0 directly. This means that if givingthe same m and no, which is related to optimal chirped-pulse-length Tco, the retrieved THz profile would be the same. Dueto the entire length of THz signal is approximately as Te

�4�2T, the dimensionless entire THz signal length can bewritten as Ted=Te /T0�4�2m. Both ted and Ted as well as theoriginal THz field at m=10 are also shown in Fig. 6�a�. Tomeasure the THz pulse without the second distortion, ted

should be larger than the whole THz pulse length. From Fig.6�b�, one finds that ted increases more quickly with m thanTed when m20. This means that the larger the m �i.e., thelonger the THz pulse�, the smaller the second distortion ofthe retrieved THz signal will be. When m �41, ted is largerthan the entire THz pulse length and the second distortiondisappears.

It should be noted that by giving a dimensionless inputTHz pulse length m �or T�, for example, m=20, ted is not thelargest one when n=no �or Tc=Tco�. To address this point, anexample of ted as a function of n when m=20 is shown inFig. 7�a�. Both the position of no �=Tco /T0� and the poison ofnm �=Tcm /T0� corresponding to the largest ted are shown inFig. 7�a�. One can find that ted decreases with n when nnm �or TcTcm�. As a result, the second distortion willbecome severer. Figure 7�b� shows the retrieved THz waveforms detected with the optimal chirped-probe-pulse lengthTco and with Tcm corresponding to the maximum ted versusthe original THz signal. In this example, the parameters usedare: k=0.02, T0=25 fs, T=0.5 ps �m=20�, Tco=2.6437 ps,Tcm=0.69 ps. If only considering the second distortion, theplateau width te with Tcm is much wider than that of with Tco,thus the second distortion from the function f�t� /T0� shoulddisappear when Tcm is used. However, due to the mismatchof Tcm to Tco, the distortion of the retrieved THz signal de-tected with Tcm is much severer than that of detected withTco, as shown apparently in Fig. 7�b�. This result also sup-ports that �1 in Eq. �12� is the main factor for the influence ofthe shape of the retrieved THz wave form.

In real situation, generally the optimal chirped pulselength is not used exactly in the measurement, both the firstand the second distortion can be observed at the same timeeven in the single-cycle T-ray case. For the sake of clarity inboth distortions, assuming T0=25 fs, we simulate the re-trieving process with different Tc �=Tco ,Tco /4,Tco /2,2Tco ,4Tco� from Tco under different input THz pulselength T �or m�. The results are shown in Fig. 8. One can findthat even an optimal-chirped-pulse length Tco is used, thesecond distortion due to the function f�t� /T0� still can beobserved when m is small �i.e., T is short�, as shown in Figs.8�a� and 8�b�. However, this distortion decreases with theincrease in the input THz pulse length T �or m�. For example,when T=0.5 ps �m=20 and Tco=2.6437 ps�, the second dis-tortion is too small to be observed, as shown in Figs. 8�c�and 8�d�. On the other hand, the first distortion due to thediscrepancy of Tc to Tco decreases with the increase in T �orm� under the same �Tc �=Tc−Tco�. In Figs. 8�e� and 8�f�,the distortion of the retrieved THz field and its spectrum isvery small even if the discrepancy �Tc is relatively large��Tc=3.7493 ps in the case of Tc=Tco /2 and =7.4986 ps in

-100 -50 0 50 100

-0.4

-0.2

0.0

0.2

0.4

m=10

f(t'/T

0)

t'/T0

Original

f(t'/T0)

(a)

ted

Ted

0 20 40 60 80 1000

200

400

600

800

Te

da

nd

t ed

m (=T/T0)

Ted

ted

(b)

FIG. 6. �Color online� �a� An example of the function f�t� /T0� when m=10 compared with the original bipolar THz field. �b� The dimensionlesseffective time window ted changes with m vs. the dimensionless entire lengthof the bipolar THz pulse ted changes with m.

0 200 400 600

60

80

100

120

140 m=20nm

(=Tcm

/T0)

no

(=Tco

/T0)

t ed

n (=Tc/T

0)

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

TH

zfie

ld(a

.u.)

t (ps)

Original

Tc=Tco

Tc=Tcm

(a) (b)

FIG. 7. �Color online� �a� The dimensionless plateau width ted of f�t� /T0� asa function of the dimensionless chirped-probe pulse length n when m=20 inthe case of bipolar THz signal and �b� Retrieved bipolar THz wave formsdetected with optimal chirped-probe pulse length Tco and with Tcm corre-sponding to the maximum ted vs the original THz signal. The simulationparameters include k=0.05, �0, T0=25 fs, T=0.5 ps �m=20�, Tco

=2.6437 ps, and Tcm=0.71 ps.

093112-6 Peng et al. J. Appl. Phys. 108, 093112 �2010�

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

131.111.164.128 On: Mon, 22 Dec 2014 08:33:17

the case of Tc=2Tco�, while in Figs. 8�a� and 8�b�, only asmall discrepancy �Tc ��Tc=0.16315 ps in the case of Tc

=Tco /2 and �Tc=0.3263 ps in the case of Tc=2Tco� bringssevere distortion. Considering above two aspects, thespectral-encoding technique is more suitable for the detec-tion of THz pulse with large m.

Same rules can be found in the case of multicycle T-raysignals. Figure 9 shows the simulation results of an exampleof retrieved multicycle �x=1� T-ray signals with the sameT0=10 fs but different T=0.25 ps �m=25�, 0.5 ps �m=50�and 1 ps �m=100�, respectively. The corresponding Tco are�0.45 ps, 1.25 ps, and 3.7 ps, respectively. One can observethat the longer the T-ray pulse or the larger the normalizedT-ray length m, the smaller of the retrieved THz signal willbe, as the same dependence as in the case of bipolar T-raycase. When the T-ray length is long enough and the corre-sponding optimal chirped-probe-pulse length is applied, thedistortion tends to be very small as shown in Figs. 9�a� and9�b�.

In Fig. 8, when TcTco, the retrieved bipolar THz waveform distorts more severely with the increase in Tc, It lookslike an oscillation function as shown in Fig. 8�a� and itsspectrum looks like an interference pattern as shown in Fig.8�b� when Tc=4Tco. This phenomenon cannot be observedwhen Tc�Tco. This is due to the reason that the plateauwidth ted of the function f�t� /T0� increases with the decreasein n �or Tc� between nm �or Tcm� and no �or Tco�, while ted

decreases with the increase in n during a large rang of n, asshown in Fig. 7�a�. Especially in the short THz pulse case,this phenomenon is easier to be observed. When Tc is much

larger than Tco, says Tc=15Tco, the retrieved THz signal os-cillates fiercely, as shown in Fig. 10�a�, and the envelope �notdrawn� of its interference spectrum is very similar to thespectrum of the original THz signal as shown in Fig. 10�b�.Same phenomenon can be observed in the case of multicycleT-ray signals. Figure 11 shows simulation results of an ex-ample of multicycle T-ray signal �x=1, T=0.5 ps, and T0

=25 fs� when Tc�15Tco compared with the case of Tc

�Tco and the original signal. One can find the envelope �notdrawn� of the interference THz spectrum matches the spec-trum of the original T-ray signal very well. This phenomenonhad also been observed in previous experiment22 in whichthe envelope of the retrieved THz spectrum with interferencepattern was used to estimate the original THz spectrum andthe electron pulse length. We also observed this interferenceTHz spectrum in our experiment. However, it should benoted that, either in the single cycle T-ray case or the multi-

-0.8 -0.4 0.0 0.4 0.8 1.2

-1.0

-0.5

0.0

0.5

1.0

T=0.125ps

m=5

Tco=0.3263psTH

zfie

ld(a

.u.)

t (ps)

Original

Tc=Tco

Tc=Tco/4

Tc=Tco/2

Tc=2Tco

Tc=4Tco

(a)

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.u

.)

Frequency (THz)

Original

Tc=Tco

Tc=Tco/4

Tc=Tco/2

Tc=2Tco

Tc=4Tco

-2 -1 0 1 2

-1.0

-0.5

0.0

0.5

1.0

T=0.5ps

m=20

Tco=2.6437psTH

zfie

ld(a

.u.)

t (ps)

Original

Tc=Tco

Tc=Tco/4

Tc=Tco/2

Tc=2Tco

Tc=4Tco

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.u

.)

Frequency (THz)

Original

Tc=Tco

Tc=Tco/4

Tc=Tco/2

Tc=2Tco

Tc=4Tco

-4 -3 -2 -1 0 1 2 3 4

-1.0

-0.5

0.0

0.5

1.0

T=1ps

m=40

Tco=7.4986psTH

zfie

ld(a

.u.)

t (ps)

Original

Tc=Tco

Tc=Tco/4

Tc=Tco/2

Tc=2Tco

Tc=4Tco

0.0 0.2 0.4 0.6 0.80.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.u

.)

Frequency (THz)

Original

Tc=Tco

Tc=Tco/4

Tc=Tco/2

Tc=2Tco

Tc=4Tco

(b)

(c) (d)

(e) (f)

FIG. 8. �Color online� Retrieved THz wave forms of the original bipolarTHz pulses with different T detected with corresponding different chirped-probe pulse length Tc in �a�, �c�, and �e� and their corresponding spectra in�b�, �d�, and �f�, respectively. Here, T0=25 fs, k=0.05.

-0.8 -0.4 0.0 0.4 0.8

-1.0

-0.5

0.0

0.5

1.0

T=0.25ps

m=25

Tco~0.45psTH

zfie

ld(a

.u.)

t (ps)

Original

Tc~Tco

Tc~Tco/2

Tc~2Tco

0 1 2 3 4 5 6 7 8 90.0

0.2

0.4

0.6

0.8

1.0 Original

Tc~Tco

Tc~Tco/2

Tc~2Tco

Inte

nsity

(a.u

.)

Frequency (THz)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

-0.5

0.0

0.5

1.0

T=0.5ps

m=50

Tco~1.25ps

Original

Tc~Tco

Tc~Tco/2

Tc~2Tco

TH

zfie

ld(a

.u.)

t (ps)

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0Original

Tc~Tco

Tc~Tco/2

Tc~2Tco

Inte

nsity

(a.u

.)

Frequency (THz)

-3 -2 -1 0 1 2 3

-1.0

-0.5

0.0

0.5

1.0

T=1ps

m=100

Tco~3.7ps

Original

Tc~Tco

Tc~Tco/2

Tc~2Tco

TH

zfie

ld(a

.u.)

t (ps)

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0Original

Tc~Tco

Tc~Tco/2

Tc~2Tco

Inte

nsity

(a.u

.)

Frequency (THz)

(a) (b)

(c) (d)

(e) (f)

FIG. 9. �Color online� Retrieved THz wave forms of the original multicycleTHz pulses with different T detected with corresponding different chirped-probe pulse length Tc in �a�, �c�, and �e� and their corresponding spectra in�b�, �d�, and �f�, respectively. Here, T0=25 fs and k=0.05.

-3 -2 -1 0 1 2 3

-1.0

-0.5

0.0

0.5

1.0

T=0.125ps

m=5

Tco=0.3263psTH

zfie

ld(a

.u.)

t (ps)

Original

Tc=Tco

Tc=15Tco

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.u

.)

Frequency (THz)

Original

Tc=Tco

Tc=15Tco

(a) (b)

FIG. 10. �Color online� Retrieved THz wave forms of an original bipolarTHz signal detected with large Tc and the optimal Tc vs the original ones in�a� and their corresponding spectra in �b�. The simulation parameters includek=0.05, T0=25 fs, T=0.125 ps �m=5�, Tco=0.3263 ps, and �0.

093112-7 Peng et al. J. Appl. Phys. 108, 093112 �2010�

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

131.111.164.128 On: Mon, 22 Dec 2014 08:33:17

cycle T-ray cases, if Tc is only several times lager than theTco, e.g., Tc=4Tco in Fig. 8�b� of the bipolar T-ray case andTc�2Tco in Fig. 9�b� of the multicycle case �x=1�, the en-velope of the interferential pattern can only roughly reflectthe original THz spectrum.

Given an input THz pulse length T, the shorter the T0,the larger the m �m=T /T0� and the smaller the distortion willbe. In order to see clearly the change in second distortionwith T0, we simulated the retrieved THz wave forms andtheir spectra versus the original ones in both the single-cycleand the multicycle �x=1� T-ray cases under the conditions ofdifferent T0 with their optimal chirped-probe-pulse lengths,respectively. The results are shown in Figs. 12 and 13. In thebipolar T-ray case, the effective time window ted of the func-tion f�t� /T0� increase with the decrease in T0, as shown inthe inset of Fig. 12�b�. One can observe that, whether insingle-cycle or multicycle T-ray case, the smaller the T0, thesmaller the second distortion will be. In reality, one can use ashorter original probe pulse �i.e., with broader spectrum� andstretch it to measure the shorter THz signal so as to reducethe second distortion. Another strategy is, instead of stretch-ing a short fs pulse as the probe pulse, using a supercon-tinuum pulse generated by the short fs pulse and then tostretch it as the probe pulse.23

C. The third distortion: when k™1 is not satisfied

We have assumed k�1 in above discuss and the qua-dratic term k2 has been neglected. This means the THz signalto be detected should be small enough. However, in somecases the T-ray signal could be very strong. For example,

T-ray sources based on free electron laser24 or synchrotronradiation from relativistic electron bunches accelerated bylinac25,26 are intense, some potential tabletop T-raysources27–30 based on laser-plasmas are also powerful. Inthese cases the prerequisite condition of k�1 is not satisfied.As a result, the third distortion is introduced.

To see how the third distortion of the retrieved THz sig-nal evolves when the THz strength increases, we performedthe simulation of the THz signal retrieving processes underthe conditions of different k �k=0.02, 0.5, 1, 2, 3, 5, and 7�.Simulation parameters are T0=10 fs, T=0.25 ps, Tco

=1.4795 ps, and �0. The simulation results are shown inFigs. 14�a�–14�c�, which represent the retrieved THz wave

(a)

0 1 2 3 4 5 6 7 80.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.u

.)

Frequency (THz)-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

-1.0

-0.5

0.0

0.5

1.0Original

T0=50fs,m=2

T0=25fs,m=4

T0=10fs,m=10

T0=5fs,m=20

TH

zfield

(a.u

.)

t (ps)

(b)

0 20 40 60250

300

350

400

450

Te

d(f

s)

T0

(fs)

FIG. 12. �Color online� Retrieved bipolar THz signals vs the original onesin �a� and the spectra in �b� under the conditions of different original probepulse length T0 with their optimal chirped-probe pulse lengths, respectively.The inset in �b� shows the effective time window ted of the function f�t� /T0�changes with T0. The simulation parameters include T=0.1 ps, k=0.05, �0, Tco=0.1625 ps, 0.2329 ps, 0.372 ps, and 0.5287 ps when T0=50 fs, 25fs, 10 fs, and 5 fs, respectively.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

-0.5

0.0

0.5

1.0

TH

zfie

ld(a

.u.)

t (ps)

Original

T0=50fs, m=10

T0=25fs, m=20

T0=10fs, m=50

T0=5fs, m=100

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0Original

T0=50fs, m=10

T0=25fs, m=20

T0=10fs, m=50

T0=5fs, m=100

Inte

nsity

(a.u

.)

Frequency (THz)

(a) (b)

FIG. 13. �Color online� Retrieved THz signals vs. the original multicycleTHz wave forms in �a� and the spectra in �b� under the conditions of differ-ent original probe pulse length T0 with their optimal chirped-probe pulselengths, respectively. The simulation parameters include T=0.5 ps, k=0.05, �0, Tco�0.75 ps, 0.82 ps, 1.25 ps, and 1.8 ps when T0=50 fs, 25fs, 10 fs, and 5 fs, respectively.

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

Original k=0.02

k=0.5 k=1

k=2 k=3

k=5 k=7Inte

nsity

(a.u

.)

Frequency (THz)

-0.8-0.40.0 0.4 0.8

-1.0

-0.5

0.0

0.5

1.0

TH

zfield

(a.u

.)

t (ps)

Tc=Tco

700 750 800 850 9000

3

6

9

12

15

Inte

nsity

(a.u

.)

Wavelength (nm)

Ib

k=0.02

k=0.5

k=1

k=2

k=3

k=5

k=7

Tc=Tco

0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

Original k=0.02

k=0.5 k=1

k=2 k=3

k=5 k=7

Inte

nsity

(a.u

)

Frequency (THz)

-0.8-0.40.0 0.4 0.8

-1.0

-0.5

0.0

0.5

1.0

TH

zfie

ld(a

.u.)

t (ps)

700 750 800 850 9000

2

4

6

8

10

12 Ib

k=0.02

k=0.5

k=1

k=2

k=3

k=5

k=7

Inte

nsity

(a.u

.)

Wavelength (nm)

Tc=Tco/4

Tc=4Tco

700 750 800 850 9000

2

4

6

8

10

12 Ib

k=0.02

k=0.5

k=1

k=2

k=3

k=5

k=7

Inte

nsity

(a.u

.)

Wavelength (nm)

(a1)

(b1) (c1)

(a2)

(b2) (c2)

(c3)

Tc=Tco/4

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

Original k=0.02

k=0.5 k=1

k=2 k=3

k=5 k=7Inte

nsity

(a.u

.)

Frequency (THz)

-0.8-0.40.0 0.4 0.8

-1.0

-0.5

0.0

0.5

1.0

TH

zfie

ld(a

.u.)

t (ps)

(a3)

(b3)

Tc=4Tc

o

FIG. 14. �Color online� Retrieved THz wave forms and their spectra vs theoriginal bipolar THz signals, and the modulated probe spectra Im vs thebackground spectrum Ib with different modulation depth k when Tc=Tco in�a1�, �b1�, and �c1�, when Tc=Tco /4 in �a2�, �b2�, and �c2�, and when Tc

=4Tco in �a3�, �b3�, and �c3�, respectively. The simulation parameters includeT0=10 fs, T=0.25 ps, Tco=1.4795 ps, and �0.

-4 -2 0 2 4 6

-1.0

-0.5

0.0

0.5

1.0

x=1

T=0.5 ps

Tco~0.82 psTH

zfie

ld(a

.u.)

t (ps)

Original

Tc~Tco

Tc~15Tco

0 1 2 30.0

0.2

0.4

0.6

0.8

1.0Original

Tc~Tco

Tc~15Tco

Inte

nsity

(a.u

.)

Frequency (THz)

(a) (b)

FIG. 11. �Color online� Retrieved THz wave forms of an original multicycleTHz signal �x=1� detected with large Tc and the optimal Tc vs the originalones in �a� and their corresponding spectra in �b�. The simulation parametersinclude k=0.05, T0=25 fs, T=0.5 ps �m=5�, Tco�0.82 ps, and �0.

093112-8 Peng et al. J. Appl. Phys. 108, 093112 �2010�

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

131.111.164.128 On: Mon, 22 Dec 2014 08:33:17

forms, THz spectra, and the modulated probe spectra versusthe background spectra, respectively. To investigate if it hasthe same dependence on k when the nonoptimal probe pulseduration is used, the cases with Tc=Tco /4 and Tc=4Tco arealso simulated. The suffixes 1, 2, and 3 represent the THzsignal detected with optimal chirped-probe-pulse durationTco, Tco /4, and 4Tco, respectively. From Figs. 14 ��a1�, �a2�,and �a3��, one finds that the negative polar of the T-ray waveform is always suppressed compared with the original onewith the increase in the THz strength, whatever the optimalor nonoptimal probe pulse duration is used. The larger the k,the more the amplitude of the negative polar of the retrievedTHz wave form is reduced. In the case of extremely intenseTHz field, e.g., k3, the negative polar changes to positive.Only under the condition that k�1, e.g., k=0.02, and withoptimal probe duration, the retrieved THz wave formmatches the original THz wave form very well, as shown inFig. 14, �a1�. From Figs. 14 ��b1�, �b2�, and �b3��, we can seethat both sides of the spectrum especially the lower fre-quency part are “raised” and the central part of the spectrumis “suppressed” with the increase in k. The stronger the origi-nal signal is, the larger the distortion of the retrieved signalwill be. When k is very large, e.g., k3, the peak positionshifts to the lower frequency. Only when k�1, e.g., k=0.02, the retrieved THz spectrum matches the original THzspectrum very well. However, the distortion of the retrievedTHz spectrum is too small to affect the spectroscopy signifi-cantly when k�1 and the optimal probe pulse length is used�as shown in Fig. 14, �b1��. Corresponding to the phenom-enon of the suppressed negative polar of the retrieved THzfield in Fig. 14�a�, the part of the probe spectrum modulatedby the positive polar of the input THz field increases morequickly than the part of that modulated by the negative polarwith the increase in k. Particularly, if k is very large, e.g.,k3, the part of the probe spectrum modulated by the nega-tive polar of the input THz field begins to hunches in thesame direction as the part modulated by the positive polar.Comparing Figs. 14 ��c2� and �c3�� with Fig. 14, �c1�, onefinds that under the condition of same k, the width of themodulated part of the probe spectrum detected with shorterprobe pulse length �Tco /4� is broader than that of measuredwith optimal pulse length Tco, whereas the width of themodulated part of the probe spectrum detected with longerprobe pulse length �4Tco� is narrower than that of detectedwith optimal probe pulse length.

For the multicycle T-ray signals, the same phenomenaare observed. Figure 15 shows the simulation results of anexample �x=1� of the THz signal retrieving processes underthe conditions of different k �k=0.02, 0.5, 1, 2, 3, 5, and 7�and the optimal chirped-probe-pulse length is applied. Otherparameters used in the simulation are T0=10 fs, T=0.5 ps,Tco�1.25 ps, and �0. One can see clearly from Fig. 5�a�that all the strengths of the negative polars of the T-ray waveforms are reduced with the increase in k, especially the mainnegative polar with the largest strength is suppressed moreseverely even upheaving to the positive direction when k1. As to the spectra of the retrieved T-ray signals, both thelower frequency part and the higher-frequency part are raisedwhile the central-frequency part is suppressed evidently withthe increase in k, as shown in Fig. 15�b�.

It should be noted that if Tc is much larger than Tco, sayTc=20Tco, the modulated part of the probe spectrum and thespectrum difference �I are much smaller than that of Tc

=Tco as shown in Fig. 16 in which k=0.1 in the case ofsingle-cycle T-ray signals. Same phenomena can also be ob-served in multicycle T-ray signals �not shown here�. In prac-tical case, if the THz field is not strong �k is small� and Tc ismuch larger than Tco, say k�0.03 and Tc=20Tco, the probespectrum would be modulated too weakly to be distinguishedfrom the background due to the fluctuation of the probe spec-trum from pulse to pulse. As a result, one would havethought that there would be no THz signal. An advisablemethod is to estimate the suitable Tc not far from the Tco

before measurement.The dependence of the third distortion on T-ray strength

or k had been observed and proved in previous experiment.20

If the T-ray signal to be measured is found to be very strong,for example, the suppress effect of the negative polar is ob-served, that means k�1 is not satisfied, one has to considerreducing the third distortion. A possible method is to use aneutral density attenuator �NDA� or a set of NDA with flatabsorption coefficient over a wide THz bandwidth to reducethe intense signal to a low level to satisfy the condition ofk�1. After retrieving this attenuated THz signal, the trueintense THz wave form can be obtained by multiplying themeasured signal accordingly.

IV. CONCLUSION

We demonstrated the principle of a single-shot T-ray de-tection method, namely the spectral-encoding technique

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0 x=1

T=0.5 ps

Tco~1.25 ps

TH

zfie

ld(a

.u.)

t (ps)0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0 Original

k=0.02

k=0.5

k=1

k=2

k=3

k=5

k=7

Inte

nsity

(a.u

.)

Frequency (THz)

(a) (b)

FIG. 15. �Color online� Retrieved THz wave forms in �a� and their spectrain �b� vs the original multicycle THz signal with different modulation depthk. Simulation parameters include T0=10 fs, T=0.5 ps, Tco�1.25 ps, and�0.

760 780 800 820 840

-0.08

-0.04

0.00

0.04

0.08

�I(a

.u.)

Wavelength (nm)

k=0.1,Tc=Tco

k=0.1,Tc=20Tco

700 750 800 850 9000.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.u

.)

Wavelength (nm)

Background

k=0.1,Tc=Tco

k=0.1,Tc=20Tco

(a) (b)

FIG. 16. �Color online� The modulated probe spectra vs the backgroundspectrum and the spectrum difference �I �=Im− Ib� in the case of single-cycle T-ray signal when Tc=Tco and Tc=20Tco in �a� and �b�, respectively.The simulation parameters include T0=10 fs, T=0.25 ps, Tco=1.4795 ps,k=0.1, and �0.

093112-9 Peng et al. J. Appl. Phys. 108, 093112 �2010�

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

131.111.164.128 On: Mon, 22 Dec 2014 08:33:17

theoretically and through simulation. For single-cycle inputTHz signal �bipolar wave form�, it is under certain preciseconditions that the THz signal could be retrieved faithfully,while for the multicycle T-ray case, it cannot be retrievedwithout any distortion. However, there is still an optimalchirped-probe-pulse length corresponding to the characteris-tic time of the multicycle THz field. Under this condition, theretrieved THz signal has a minimum distortion. Three mainpossible sources of distortions from both bipolar and multi-cycle T-ray signals measured by this technique are system-atically analyzed through theory and simulation. The firstdistortion originates from the mismatch of the optimalchirped-probe-pulse duration Tco and the T-ray length T ofTHz signal; the second one is relevant to the spectrum band-width of the probe pulse; the third one occurs when k�1 isnot satisfied. It comes from the neglect of the quadratic termof the modulation depth k which reflects the strength of theTHz signal in the retrieving process.

A chirped probe pulse with Tc as closer to Tco as possibleis proposed to reduce the first distortion. A chirped opticalprobe pulse with a broader spectrum is helpful for reducingthe second distortion. The third distortion could be elimi-nated by reducing the THz signal intensity with a THz NDAbefore the signal retrieving process.

ACKNOWLEDGMENTS

This work was supported by the Terahertz Science andTechnology Program �Grant Nos. 0821410038 and0821410039� of A*STAR in Singapore.

1S. P. Mickan and X.-C. Zhang, Int. J. High Speed Electron. Syst. 13, 601�2003�.

2E. Pickwell and V. P. Wallace, J. Phys. D: Appl. Phys. 39, R301 �2006�.3S. R. Murrill, B. Redman, and R. L. Espinolac, Proc. SPIE 6549, 654902�2007�.

4H. Hoshina, Y. Sasaki, A. Hayashi, C. Otani, and K. Kawase, Appl. Spec-trosc., 63, 81 �2009�.

5Y. Hu, P. Huang, L. Guo, X. Wang, and C. Zhang, Phys. Lett. A 359, 728�2006�.

6W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325�2007�.

7S. H. Wang, B. Ferguson, D. Abbott, and X.-C. Zhang, J. Biol. Phys. 29,247 �2003�.

8L. Möller, J. Federici, A. Sinyukov, C. Xie, H. C. Lim, and R. C. Giles,Opt. Lett. 33, 393 �2008�.

9Q. Wu and X.-C. Zhang, Appl. Phys. Lett. 70, 1784 �1997�.10Z. Jiang and X.-C. Zhang, Appl. Phys. Lett. 72, 1945 �1998�.11Z. Jiang, F. G. Sun, and X.-C. Zhang, Opt. Lett. 24, 1245 �1999�.12B. Yellampalle, K. Y. Kim, G. Rodriguez, J. H. Glownina, and A. J.

Taylor, Appl. Phys. Lett. 87, 211109 �2005�.13K. Y. Kim, B. Yellampalle, G. Rodriguez, R. D. Averitt, A. J. Taylor, and

J. H. Glownia, Appl. Phys. Lett. 88, 041123 �2006�.14K. Y. Kim, B. Yellampalle, A. J. Taylor, G. Rodriguez, and J. H. Glownia,

Opt. Lett. 32, 1968 �2007�.15J. Shan, A. S. Weling, E. Knoesel, L. Bartels, M. Bonn, A. Nahata, G. A.

Reider, and T. F. Heinz, Opt. Lett. 25, 426 �2000�.16S. P. Jamison, J. Shen, A. M. Macleod, W. A. Gillespie, and D. A.

Jaroszynski, Opt. Lett. 28, 1710 �2003�.17X.-Y. Peng, O. Willi, M. Chen, and A. Pukhov, Opt. Express 16, 12342

�2008�.18F. G. Sun, Z. Jiang, and X.-C. Zhang, Appl. Phys. Lett. 73, 2233 �1998�.19J. R. Fletcher, Opt. Express 10, 1425 �2002�.20X.-Y. Peng, R. Jung, T. Toncian, O. Willi, and H.-T. Teng, Appl. Phys.

Lett. 94, 221107 �2009�.21B. Yellampalle, K. Y. Kim, G. Rodriguez, J. H. Glownia, and A. J. Taylor,

Opt. Express 15, 1376 �2007�.22J. van Tilborg, C. Toth, N. H. Matlis, G. R. Plateau, and W. P. Leemans,

Opt. Lett. 33, 1186 �2008�.23U. Schmidhammer, V. De Waele, J.-R. Marques, N. Bourgeois, and M.

Mostafavi, Appl. Phys. B: Lasers Opt. 94, 95 �2009�.24G. Ramian, Nucl. Instrum. Methods Phys. Res. A 318, 225 �1992�.25G. P. Williams, Rep. Prog. Phys. 69, 301 �2006�.26G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G.

P. Williams, Nature �London� 420, 153 �2002�.27Z.-M. Sheng, K. Mima, J. Zhang, and H. Sanuki, Phys. Rev. Lett. 94,

095003 �2005�.28V. B. Gildenburg and N. V. Vvedenskii, Phys. Rev. Lett. 98, 245002

�2007�.29W. P. Leemans, C. G. R. Geddes, J. Faure, C. To’th, J. van Tilborg, C. B.

Schroeder, E. Esarey, G. Fubiani, D. Auerbach, B. Marcelis, M. A. Car-nahan, R. A. Kaindl, J. Byrd, and M. C. Martin, Phys. Rev. Lett. 91,074802 �2003�.

30X.-Y. Peng, C. Li, M. Chen, T. Toncian, R. Jung, O. Willi, Y.-T. Li, W.-M.Wang, S.-J. Wang, F. Liu, A. Pukhov, Z.-M. Sheng, and J. Zhang, Appl.Phys. Lett. 94, 101502 �2009�.

093112-10 Peng et al. J. Appl. Phys. 108, 093112 �2010�

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

131.111.164.128 On: Mon, 22 Dec 2014 08:33:17