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Optics Express

  • Vol. 16, Iss. 16 — Aug. 4, 2008
  • pp: 12342–12349
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Optimal chirped probe pulse length for terahertz pulse measurement

Xiao-Yu Peng, Oswald Willi, Min Chen, and Alexander Pukhov  »View Author Affiliations


Optics Express, Vol. 16, Issue 16, pp. 12342-12349 (2008)
http://dx.doi.org/10.1364/OE.16.012342


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Abstract

A detailed analysis of the relationship between the duration of the chirped probe pulse and the bipolar terahertz (THz) pulse length in the spectral encoding technique is carried out. We prove that there is an optimal chirped probe pulse length (or an optimal chirp rate of the chirped probe pulse) matched to the input THz pulse length and derive a rigorous relationship between them. We find that only under this restricted condition the THz signal can be correctly retrieved.

© 2008 Optical Society of America

The spectral encoding technique not only has been applied to many experiments but also has been analyzed theoretically by some authors 9, 10 after it was proposed. In their detailed analysis the temporal resolution Tmin of the detection system was obtained as T min≈ (T 0 T c)½ where T 0 is the original probe pulse length and T c is the chirped probe pulse length. It was proposed that the THz signal could be retrieved without distortion if the pulse length T of the detected THz field satisfies the condition TT min≈ (T 0 T c)½. Though their theory can explain some experimental results, we found that this condition is a relatively inaccurate description. According to Sun’s analysis 9, to reduce the distortion, T c must be reduced so that the temporal resolution could be improved while the smallest T c should be larger than the THz duration. Hence a compromise between the temporal resolution and the chirp rate is desirable. In other words, there would be an optimal chirp rate or an optimal chirped probe pulse length that can be applied to measure the THz signal without distortion if the THz pulse length is given. In this paper we present a detailed analysis and a rigorous deduction of the relationship between the chirped probe pulse length T c and the THz pulse length T. We prove that there is an optimal chirped probe pulse length T co (or an optimal chirp rate) that matches with the input THz pulse length T and only under this restricted condition the THz signal can be retrieved properly.

When the chirped probe pulse and a THz pulse co-propagate in an electro-optic crystal, the chirped pulse is modulated by the pulsed THz field through the Pockels effect. Considering the case that the polarizer and the analyzer are crossed to each other, the electric field component of the chirped probe beam modulated by the THz field can be written as 3, 11: Em (t) = Ec (t)[1 + kETHz (t - τ)] = Ec (t) + kETHz (t - τ)Ec (t) with ETHz(t) the electric field of the THz waveform, τ is the relative time delay between the probe pulse and the THz pulse, and k is the modulation constant. Here we assume τ = 0 for simplicity due to the fact that the probe pulse can be synchronized with the THz pulse by adjusting the time delay in principle. Thus we have Em(t) = Ec(t) + kETHz (t)Ec(t) = Ec (t) + ES (t).

The spectral modulation is spatially separated on the CCD of the spectrometer. Assuming that the measured signal on a CCD pixel corresponds to the optical frequency ω 1 and the spectral resolution function of the spectrometer is g(ω 1-ω), the spectrum of the background chirped pulse Ic in case of no THz field modulation and the modulated spectrum Im by THz pulse can be expressed as the convolution of the spectral function and the square of the Fourier transform of Ec(t) and Em(t), respectively. They are: Icg(ω 1 -ω)*∣Ec (ω)∣2 and Img(ω -ω)*∣Em(ω)∣2. Here Ec(ω) and Em(ω) are the Fourier transform of Ec(t) and Em(t), respectively. If the spectral resolution is large enough, the spectral function of the spectrometer can be expressed as a δ function. Thus we have Ic ∝ δ(ω 1-ω)*∣Ec(ω)∣2 = ∫ -∞ δ(ω 1-ω)∣Ec(ω)∣2 = ∣Ec(ω 1)∣2 and Imδ(ω 1 - ω)*∣Em(ω)∣2 = ∫ -∞ δ(ω 1 - ω) ∣Em(ω)∣2 = ∣Em(ω 1)2. The difference between Im and Ic then can be expressed as Δ∣ ∝ Im - Ic =∣Em(ω 1)∣2 - ∣Ec(ω 1)∣2.

Considering Em(ω 1)= Ec(ω 1)+ Es(ω 1), we have ΔI ∝ ∣Ec(ω 1) + Es(ω 1)∣2-∣Ec(ω 1)∣2 = Ec(ω 1)∙E*s(ω 1)+E*c(ω 1)∙Es(ω 1)+∣Es(ω 1)∣2 , where E*' (ω 1) and E*s(ω 1) are the conjugate of Ec(ω 1)=∫+∞ -∞ Ec(t)exp( 1 t)dt and Es(ω 1) = ∫+∞ -∞ Es(t)exp( 1 t)dt, respectively.

S(ω1)=Ec(ω1)Es*(ω1)+Es*(ω1)Es(ω1).
(1)

For a linear chirp, the probe instantaneous frequency ω 1 measured on the CCD is proportional to the time t′. In other words, the THz field profile can be directly obtained through measurements of the spectral modulation s(ω 1) using following coordinate transformation ω 1 - ω 1 0 = 2αt′. Due to T 0=2/Δω 1 0 and 2 α ≈ Δω 1 0/Tc as mentioned before, we have αT -1 0 T -1 c. Substituting E 0(t), Ec(t), and ETHz(t) which we have defined above in Eq. (1), and considering ω 1 -ω 0 = 2αt′ and α ≈ T -1 0 ' -1 c, and introducing two dimensionless pulse lengths m=T/T 0 and n=T c/T 0, one obtains

S(t)=kγf(t/T0)(tT1)exp[(t′2T2)χ]
(2)

with

γ=2πT02n1(n4+n2)1/2[(m2+n2)2+n2]3/2[(m2n2+n4+n2)2+m4n2]1/4
(3)
f(t/T0)=n1cos{12arctgn3m2n2+m2+n2n3(2m2+n2)(1+n2)[(m2+n2)+m4n2]tT02}
(m2+n2)sin{12arctgn3m2n2+m2+n2n3(2m2+n2)(1+n2)[(m2+n2)+m4n2]tT02}
(4)
χ=m21+n2+m4n2+m6(m2+n2)2+m4n2
(5)

Fig. 1. An example of the oscillation function f(t′/T 0) (solid line) as well as an original THz field (dashed line) when m = 6.

window hence introducing distortion to some extent at the two ends of the retrieved THz signal. An example of this function as well as the original THz field when m = 6 are shown in Fig. 1. To eliminate the possible distortion generated from function f(t′/T 0), we define a normalized THz signal Sn(t′) to f(t′/T 0), obtaining

Sn(t)=S(t)f(t/T0)=(tT1)exp[(t′2T2)χ]
(6)

One can see clearly from Eq. (6) that the function Sn(t′) ∝ ETHz(t′)if χ =l. In other words, the real THz profile can be retrieved if χ =1 while severe distortion would be introduced if χ>1 or χ<l. When m>1.4 (this condition can be easily be fulfilled due to the fact that T>>T 0) then the only reasonable solution of equation χ = 1 follows as

n={13[2(6m6+4m48m2+1)1/2cosβ3(m21)]}1/2
(7)

with β=arccos[35418m8+61m6+6m4+24m22(6m6+4m48m2+1)3/2].. Here n is the dimensionless optimal probe pulse length. Thus we obtain the optimal duration T co=nT 0 and the optimal chirp rate 2αo ≈ 2T -l 0 T -2 co = 2T -2 0 n -1. This means that it is possible to retrieve the THz signal without distortion if one uses a chirped probe pulse with the optimal duration T co matched to a input THz pulse length T = mT 0 according to Eq. (7). The relationship between m and n according to Eq. (7) is shown in Fig. 2. One can see that the dimensionless optimal chirped probe pulse length n increases nonlinearly with the input dimensionless THz pulse length m.

Fig. 2. Relationship (solid line) between the dimensionless optimal chirped pulse length n and the input dimensionless THz pulse length m. The restricted condition T cT 2/T 0 or T≥(T 0 T c)½ described by Sun’s theory is also shown as the area under the curve n=m 2 (or T c=T 2/T 0, dashed line) and the curve itself compared with the optimal T c curve.

By comparing between the condition described by Sun 9 and Fletcher 10 and the relationship of Eq. (7), one can find that their description is different from ours. In their theory, the condition to retrieve the original THz field without distortion was described as T ≥ (T 0 T c)½ or T cT 2/T 0 (i.e. nm 2 if we introduce the dimensionless pulse length). The curve of n=m 2 (marked “T c=T 2/T 0”) and the curve of the optimal T co (marked “optimal T c”) obtained from Eq. (7) are shown in Fig. 2. One can find that the condition “T cT 2/T 0” is inaccurate as this condition includes the whole area under the curve n=m 2and the curve itself, while our analysis indicates that only those chirped probe pulses with duration T co satisfy the curve obtained from equation (7) can be applied to reconstruct the original THz signal.

Fig. 3. (Color on line) Retrieved THz waveforms vs the original THz waveforms (black dotted lines) with T 0=25fs in (a) and T 0=10fs in (b). The red solid lines are the retrieved THz waveforms calculated according equation (6) under the conditions of input THz pulse length T = 0.5ps (m =20 when T 0=25fs and m =50 when T '=10fs) and their corresponding optimal chirped pulse lengths T co =2.6437ps when T 0=25fs, T co = 4.1943ps when T 0=10fs, respectively, which are calculated from equation (7). The retrieved THz fields calculated according to the equation (6) with T c = T co/2, T c = T co/4, T c = 2T co, and T c = 4T co are also shown in each figure.

To verify the validity of our theory, we calculated a representative example with the original THz pulse length T =0.5ps. We assumed in our calculation that the Fourier-limited pulse length T 0=25fs and T 0=10fs. According to Eq. (7) the optimal chirped probe pulse lengths T co matched to above T can be calculated to be as 2.6437ps when T 0 =25fs and 4.1943ps, when T 0 =10fs, respectively. With these T and their T co, we calculated the THz fields according to Eq. (6). The results are shown in Fig. 3 in which T 0=25fs in (a) and T 0=10fs in (b). In Fig. 3, the dotted black lines represent the original THz fields, while the solid red lines are the retrieved THz fields calculated with the optimal probe pulse length Tco. One can find that the retrieved THz fields with the T co match the original ones very well.

To see the behavior of the retrieved THz field due to the discrepancy ΔT c=T c-T co, we calculated four cases with T c = T co/2, T c = T co/4, T c = 2T co, and T c = 4T co which fulfill the condition T ≥ (T 0 T c)½ of Sun’s theory 9 with the same T(=0.5ps).The results are also shown in Fig. 3. One can clearly see that all these cases introduce distortions and the larger the difference of T c to T co is, the more severe the distortion of the retrieved THz field will be In detail, if T c<T co, the retrieved THz waveform will be compressed while its spectrum would be broaden compared to the input original THz field. Whereas, if T c>T co, the retrieved THz waveform will be broaden while its spectrum would be compressed.

In addition, one can find that the larger the T 0 is, the greater the difference of the retrieved THz waveform to the same discrepancy ΔT c will be if one compares Fig. 3(a) with Fig. 3(b). In Fig. 3(b), the distortion of the retrieved THz fields is very small even if the discrepancy ΔT c is relatively large ( ΔT c =-2.097155ps in case of T c = T co/2 and ΔT c=4.1943ps in the case of T c = 2T co), while in Fig. 3(a) the distortion of the retrieved THz fields is more severe than that shown in Fig. 3(b) even if their ΔT cT c=-1.32185ps in the case of T c = T co/2 and ΔT c =2.6437ps in the case of T c = 2T co) is smaller than that of Fig. 3(b).

In other words, if an input THz pulse is measured using different probe beams, the shorter the original probe pulse length (or the broader the spectrum of the original probe pulse) is, with the same discrepancy to their optimal chirped pulse length, the smaller the distortion of the retrieved THz signal will be.

We also simulated the whole retrieving process of the THz waveforms under the condition of k<<1 (for example k = 0.02). We find that the simulation results are the same to the ones shown in Fig. 3. Our simulations confirm our deduction process and prove the validity of Eq. (7). The detail of these simulation results will be published in the future. It should be noted that our analysis is based on a bipolar THz field. For other THz pulses such as with multiple cycles, we prove that there still is an optimal chirped probe pulse duration T co by means of our simulation, while the characteristic time of the THz field should to be redefined and the expression of Eq. (7) needs to be modified. The detailed theoretical deduction will be performed in the future work.

In summary, we conduct a rigorous deduction and a detailed analysis of the relationship between the chirped pulse length T c and the input bipolar THz pulse length T in the spectral-encoding technique. We prove that there is an optimal chirped pulse duration T co or an optimal chirp rate matched to the input THz pulse length T. We derive a relationship between the dimensionless optimal chirped pulse length n and the input dimensionless THz pulse length m. With this relationship the optimal duration and the optimal chirp rate of the chirped probe pulse can be calculated according to T co=nT 0 and 2αo ≈ 2T 0 -1 0 T -1 co = 2T -2 0 n -1 respectively. We find that only under this restricted condition the measured THz signal can be retrieved without distortion.

Acknowledgment

The first author would like to thanks Prof. Dr. Georg Pretzler, Dr. Jiaan Zheng, Dr. Jun Zheng and Dr. Huichun Wu for their helpful discussions. This work was supported by the DFG programs GRK 1203 and TR18. M. C acknowledges support from the Alexander von Humboldt Foundation.

References and links

1.

S.P. Mickan and X.-C. Zhang, “T-ray sensing and imaging,” Interrnational Journal of High Speed Electronics and Systems, 12, 601 (2003).

2.

Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. 70, 1784–1786 (1997). [CrossRef]

3.

Zhiping Jiang and X.-C. Zhang, “Electro-optic measurement of THz field pulses with a chirped optical beam,” Appl. Phys. Lett. 72, 1945–1947 (1998). [CrossRef]

4.

Jie. Shan, Aniruddha S. Weling, Ernst Knoesel, Ludwig Bartels, Mischa Bonn, Ajay Nahata, Georg A. Reider, and Tony F. Heinz, “single-shot measurement of terahertz electromagnetic pulses by use of electro-optic sampling,” Opt. Lett. 25, 426–428 (2000). [CrossRef]

5.

Steven P. Jamison, Jingling Shen, A. M. Macleod, W. A. Gillespie, and D. A. Jaroszynski, “High-temporal-resolution, single-shot characterization of terahertz pulses,” Opt. Lett. 28, 1710–1712 (2003). [CrossRef]

6.

B. Yellampalle, K. Y. Kim, G. Rodriguez, J. H. Glownina, and A. J. Taylor, Appl. Phys. Lett. 87, 211109 (2005). [CrossRef]

7.

K. Y. Kim, B. Yellampalle, G. Rodriguez, R. D. Averitt, A. J. Taylor, and J. H. Glownia, “Single-shot, interferometric, high-resolution, terahertz field diagnostic,” Appl. Phys. Lett. 88, 041123 (2006).

8.

Jeroen van Tilborg, “Coherent terahertz radiation from laser-wakefield-accelerated electron beams”, Doctoral Dissertation, p.76 (2006).

9.

F. G. Sun, Zhiping Jiang, and X.-C. Zhang, “Analysis of terahertz pulse measurement with a chirped probe beam,” Appl. Phys. Lett. 73, 2233–2235 (1998). [CrossRef]

10.

J. R. Fletcher, “Distortion and uncertainty in chirped pulse THz spectrometers,” Opt. Express 10, 1425–1430 (2002). [PubMed]

11.

Yellampalle, K. Y. Kim, G. Rodriguez, J. H. Glownia, and A. J. Taylor, “Details of electro-optic terahertz detection with a chirped probe pulse,” Opt. Express 15, 1376–1383 (2007). [CrossRef] [PubMed]

OCIS Codes
(110.0110) Imaging systems : Imaging systems
(040.2235) Detectors : Far infrared or terahertz

ToC Category:
Detectors

History
Original Manuscript: May 13, 2008
Revised Manuscript: July 2, 2008
Manuscript Accepted: July 2, 2008
Published: August 1, 2008

Citation
Xiao-Yu Peng, Oswald Willi, Min Chen, and Alexander Pukhov, "Optimal chirped probe pulse length for terahertz pulse measurement," Opt. Express 16, 12342-12349 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-12342


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References

  1. S. P. Mickan and X.-C. Zhang, "T-ray sensing and imaging," Interrnational Journal of High Speed Electronics and Systems 12, 601 (2003).
  2. Q. Wu and X.-C. Zhang, "7 terahertz broadband GaP electro-optic sensor," Appl. Phys. Lett. 70, 1784-1786 (1997). [CrossRef]
  3. Zhiping Jiang and X.-C. Zhang, "Electro-optic measurement of THz field pulses with a chirped optical beam," Appl. Phys. Lett. 72, 1945-1947 (1998). [CrossRef]
  4. J. Shan, A. S. Weling, E. Knoesel, L. Bartels, M. Bonn, A. Nahata, G. A. Reider, and T. F. Heinz, "Single-shot measurement of terahertz electromagnetic pulses by use of electro-optic sampling,"Opt. Lett. 25, 426-428 (2000). [CrossRef]
  5. S. P. Jamison, J. Shen, A. M. Macleod and W. A. Gillespie, and D. A. Jaroszynski, "High-temporal-resolution, single-shot characterization of terahertz pulses," Opt. Lett. 28, 1710-1712 (2003). [CrossRef]
  6. B. Yellampalle, K. Y. Kim, G. Rodriguez, J. H. Glownina, and A. J. Taylor, Appl. Phys. Lett. 87, 211109 (2005). [CrossRef]
  7. K. Y. Kim, B. Yellampalle, G. Rodriguez, R. D. Averitt, A. J. Taylor, and J. H. Glownia, "Single-shot, interferometric, high-resolution, terahertz field diagnostic," Appl. Phys. Lett. 88, 041123 (2006).
  8. Jeroen van Tilborg, "Coherent terahertz radiation from laser-wakefield-accelerated electron beams", Doctoral Dissertation, p.76 (2006).
  9. F. G. Sun, Zhiping Jiang, and X.-C. Zhang, "Analysis of terahertz pulse measurement with a chirped probe beam,"Appl. Phys. Lett. 73, 2233-2235 (1998). [CrossRef]
  10. J. R. Fletcher, "Distortion and uncertainty in chirped pulse THz spectrometers," Opt. Express 10, 1425-1430 (2002). [PubMed]
  11. Yellampalle, K. Y. Kim, G. Rodriguez, J. H. Glownia, and A. J. Taylor, "Details of electro-optic terahertz detection with a chirped probe pulse," Opt. Express 15, 1376-1383 (2007). [CrossRef] [PubMed]

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