## Optimal chirped probe pulse length for terahertz pulse measurement

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

^{1}of THz science including material characterization, biomedical imaging, and tomography. One of the typical T-ray detection techniques is electro-optic sampling (EOS)

^{2}, a scanning technique. Through EOS, the electric field profile including phase and amplitude of a THz pulse can be measured. The main benefit of this technique is of its high temporal resolution depending on the short probe pulse length. However, the measured THz profiles represent averaged waveforms and a long time is required to obtain each profile. In addition, EOS is not suitable to be used in some cases such as the T-rays with low repetition rate or with strong shot-to-shot fluctuations and experiments that may have low duty cycles. To solve these problems three different single-shot detection techniques

^{3, 4, 5}were developed. Among these techniques are the popular and practical spectral encoding

^{3}and the cross-correlation

^{5}techniques which normally are used for different cases. Compared to each other, the spectral encoding technique is more convenient to use due to its simple optical arrangement, capability of measuring the THz signal in realtime and providing THz spatiotemporal imaging with its disadvantage of limited temporal resolution. The cross-correlation technique has higher resolution which only depends on the duration of the short probe pulse while it has some disadvantages such as its more complicated optical arrangement and its absence of the ability of spatiotemporal imaging. To enhance the temporal resolution of the spectral encoding technique, an interferometric retrieval algorithm was proposed

^{6}and was applied in experiments

^{7}recently. This technique can provide transform-limited temporal resolution which is mainly limited by the spectral bandwidth of the optical probe pulse, regardless of its chirp. However, with this technique a complicated matrix inversion equation needs to be resolved numerically to retrieve the THz waveform

^{6}. Furthermore, this algorithm needs to be improved due to the poor signal-to-noise ratio and the dominant presence of algorithm artifacts such as the fast oscillations when it was applied in some experiments

^{8}.

^{3, 11}:

*E*(

_{m}*t*) =

*E*(

_{c}*t*)[1 +

*kE*(

_{THz}*t*-

*τ*)] =

*E*(

_{c}*t*) +

*k*(

_{ETHz}*t*-

*τ*)

*E*(

_{c}*t*) with

*E*(

_{THz}*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

*E*(

_{m}*t*) =

*E*(

_{c}*t*) +

*k*(

_{ETHz}*t*)

*E*(

_{c}*t*) =

*E*(

_{c}*t*) +

*E*(

_{S}*t*).

*ω*

_{1}and the spectral resolution function of the spectrometer is

*g*(

*ω*

_{1}-

*ω*), the spectrum of the background chirped pulse

*I*in case of no THz field modulation and the modulated spectrum

_{c}*I*by THz pulse can be expressed as the convolution of the spectral function and the square of the Fourier transform of

_{m}*E*(

_{c}*t*) and

*E*(

_{m}*t*), respectively. They are:

*I*∝

_{c}*g*(

*ω*

_{1}-

*ω*)*∣

*E*(

_{c}*ω*)∣

^{2}and

*I*∝

_{m}*g*(

*ω*-

*ω*)*∣

*E*(

_{m}*ω*)∣

^{2}. Here

*E*(

_{c}*ω*) and

*E*(

_{m}*ω*) are the Fourier transform of

*E*(

_{c}*t*) and

*E*(

_{m}*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

*I*∝ δ(

_{c}*ω*

_{1}-

*ω*)*∣

*E*(

_{c}*ω*)∣

^{2}= ∫

^{∞}

_{-∞}

*δ*(

*ω*

_{1}-

*ω*)∣

*E*(

_{c}*ω*)∣

^{2}

*dω*= ∣

*E*(

_{c}*ω*

_{1})∣

^{2}and

*I*∝

_{m}*δ*(

*ω*

_{1}-

*ω*)*∣

*E*(

_{m}*ω*)∣

^{2}= ∫

^{∞}

_{-∞}

*δ*(

*ω*

_{1}-

*ω*) ∣

*E*(

_{m}*ω*)∣

^{2}

*dω*= ∣

*E*(

_{m}*ω*

_{1})

^{2}. The difference between

*I*and

_{m}*I*then can be expressed as Δ∣ ∝

_{c}*I*-

_{m}*I*=∣

_{c}*E*(

_{m}*ω*

_{1})∣

^{2}- ∣

*E*(

_{c}*ω*

_{1})∣

^{2}.

*E*(

_{m}*ω*

_{1})=

*E*(

_{c}*ω*

_{1})+

*E*(

_{s}*ω*

_{1}), we have Δ

*I*∝ ∣

*E*(

_{c}*ω*

_{1}) +

*E*(

_{s}*ω*

_{1})∣

^{2}-∣

*E*(

_{c}*ω*

_{1})∣

^{2}=

*E*(

_{c}*ω*

_{1})∙

*E**

*(*

_{s}*ω*

_{1})+

*E**

*(*

_{c}*ω*

_{1})∙

*E*(

_{s}*ω*

_{1})+∣

*E*(

_{s}*ω*

_{1})∣

^{2}, where

*E**

*(*

_{'}*ω*

_{1}) and

*E**

*(*

_{s}*ω*

_{1}) are the conjugate of

*E*(

_{c}*ω*

_{1})=∫

^{+∞}

_{-∞}

*E*(

_{c}*t*)exp(

*iω*

_{1}

*t*)

*dt*and

*E*(

_{s}*ω*

_{1}) = ∫

^{+∞}

_{-∞}

*E*(

_{s}*t*)exp(

*iω*

_{1}

*t*)

*dt*, respectively.

*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}= 2

*αt′*. Due to

*T*

_{0}=2/Δ

*ω*

_{1}

_{0}and 2

*α*≈ Δ

*ω*

_{1}

_{0}/

*T*c as mentioned before, we have

*α*≈

*T*

^{-1}

_{0}

*T*

^{-1}

*. Substituting*

_{c}*E*

_{0}(

*t*),

*E*(

_{c}*t*), and

*E*(

_{THz}*t*) which we have defined above in Eq. (1), and considering

*ω*

_{1}-

*ω*

_{0}= 2

*αt′*and α ≈

*T*

^{-1}

_{0}

*'*

^{-1}

*, and introducing two dimensionless pulse lengths*

_{c}*m*=

*T*/

*T*

_{0}and

*n*=

*T*

_{c}/

*T*

_{0}, one obtains

*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

*S*(

_{n}*t′*) to

*f*(

*t′*/

*T*

_{0}), obtaining

*S*(

_{n}*t′*) ∝

*E*(

_{THz}*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*is the dimensionless optimal probe pulse length. Thus we obtain the optimal duration

*T*

_{co}=

*nT*

_{0}and the optimal chirp rate 2

*α*≈ 2

_{o}*T*

^{-l}

_{0}

*T*

^{-2}

*= 2*

_{co}*T*

^{-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*.

^{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*

_{c}≤

*T*

^{2}/

*T*

_{0}(i.e.

*n*≤

*m*

^{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*

_{c}≤

*T*

^{2}/

*T*

_{0}” is inaccurate as this condition includes the whole area under the curve

*n*=

*m*

^{2}and 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.

*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.

*T*

_{c}=

*T*

_{c}-

*T*

_{co}, we calculated four cases with

*T*

_{c}=

*T*

_{co}/2,

*T*

_{c}=

*T*

_{co}/4,

*T*

_{c}= 2

*T*

_{co}, and

*T*

_{c}= 4

*T*

_{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.

*T*

_{0}is, the greater the difference of the retrieved THz waveform to the same discrepancy Δ

*T*

_{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 Δ

*T*

_{c}is relatively large ( Δ

*T*

_{c}=-2.097155ps in case of

*T*

_{c}=

*T*

_{co}/2 and Δ

*T*

_{c}=4.1943ps in the case of

*T*

_{c}= 2

*T*

_{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 Δ

*T*

_{c}(Δ

*T*

_{c}=-1.32185ps in the case of

*T*

_{c}=

*T*

_{co}/2 and Δ

*T*

_{c}=2.6437ps in the case of

*T*

_{c}= 2

*T*

_{co}) is smaller than that of Fig. 3(b).

*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.

*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 2

*α*≈ 2

_{o}*T*

_{0}

^{-1}

_{0}

*T*

^{-1}

*= 2*

_{co}*T*

^{-2}

_{0}

*n*

^{-1}respectively. We find that only under this restricted condition the measured THz signal can be retrieved without distortion.

## Acknowledgment

## References and links

1. | S.P. Mickan and X.-C. Zhang, “T-ray sensing and imaging,” |

2. | Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. |

3. | Zhiping Jiang and X.-C. Zhang, “Electro-optic measurement of THz field pulses with a chirped optical beam,” Appl. Phys. Lett. |

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. |

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. |

6. | B. Yellampalle, K. Y. Kim, G. Rodriguez, J. H. Glownina, and A. J. Taylor, Appl. Phys. Lett. |

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. |

8. | Jeroen van Tilborg, “Coherent terahertz radiation from laser-wakefield-accelerated electron beams”, Doctoral Dissertation, p. |

9. | F. G. Sun, Zhiping Jiang, and X.-C. Zhang, “Analysis of terahertz pulse measurement with a chirped probe beam,” Appl. Phys. Lett. |

10. | J. R. Fletcher, “Distortion and uncertainty in chirped pulse THz spectrometers,” Opt. Express |

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 |

**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

- S. P. Mickan and X.-C. Zhang, "T-ray sensing and imaging," Interrnational Journal of High Speed Electronics and Systems 12, 601 (2003).
- Q. Wu and X.-C. Zhang, "7 terahertz broadband GaP electro-optic sensor," Appl. Phys. Lett. 70, 1784-1786 (1997). [CrossRef]
- 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]
- 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]
- 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]
- B. Yellampalle, K. Y. Kim, G. Rodriguez, J. H. Glownina, and A. J. Taylor, Appl. Phys. Lett. 87, 211109 (2005). [CrossRef]
- 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).
- Jeroen van Tilborg, "Coherent terahertz radiation from laser-wakefield-accelerated electron beams", Doctoral Dissertation, p.76 (2006).
- 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]
- J. R. Fletcher, "Distortion and uncertainty in chirped pulse THz spectrometers," Opt. Express 10, 1425-1430 (2002). [PubMed]
- 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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