Abstract

We present a theoretical and experimental investigation of the THz pulse phase measured by a broadband heterodyne detection method via field-induced second-harmonic generation in ambient air. The dependence of the detected THz phase spectra on the positions of the wire shaped electrodes scanning along the detection plasma is discussed. An additional phase shift around the beam focus is observed. Theoretical deductions reveal that it is caused by the Gouy shift of the optical probe beam and THz beam during the heterodyne detection process.

© 2011 OSA

1. Introduction

In recent years, an efficient broad band THz spectrometry covering the entire THz gap from 0.2 THz to 10 THz has been realized by employing laser-ionized gases as both the generation and detection third-harmonic nonlinear media [18]. In particular, a THz Air-Biased-Coherent-Detection (ABCD) through the introduction of a modulated local bias-induced second harmonic oscillator has been achieved, enabling the coherent detection of THz radiation in gases, which results in a significant improvement on the dynamic range and sensitivity of the system [9]. A systematic study has identified the importance of a number of parameters involved in this detection scheme, such as the power of the fundamental beam, bias field strength, gas pressure, and the third-order nonlinear susceptibility of gases [10,11]. The previous investigations have substantially supported the possibility of applying the coherent pulsed THz spectroscopy with gases as generation and detection media for future broadband spectroscopic imaging and identification measurements.

However, former research works regarding the performance of the coherent THz detection scheme were mainly carried out by focusing on the amplitude of the THz pulse, whereas the phase information was largely ignored. On the other hand, the phase of the THz pulse is indeed a crucial factor which also reflects the intrinsic characteristics of the pulse. As a matter of fact, it has been pointed out that under a certain circumstances the THz absorption fingerprints of a certain material might be extracted by measuring the phase of the THz pulse alone [12,13].

In the THz ABCD setup with the heterodyne detection scheme, the bias field is produced by placing a pair of ac modulated electrodes at the THz focal spot [911]. The detection performance variation influenced by the electrodes position deviation has not been investigated. In this work, the phase of the THz pulse measured by the heterodyne detection based on field induced second-harmonic generation is studied with the electrodes moving along detection air plasma.

2. Theoretical background

The basic principle used to sense the THz pulse is the detection of the THz field induced optical second harmonic waves E2ωTHz through a third order nonlinear process [1,9]. In the inherently coherent broadband ABCD, we use a spatially confined electric field to make the local oscillator. A second harmonic signal E2ωLO can also be produced by the ac bias with the same nonlinear susceptibility. The intensity of the measured second harmonic signal is [9]

I2ω|E2ωTHz+E2ωLO|2(χ(3)EωEω)2(ETHz2+2ETHzEbias+Ebias2)
where ETHz and Ebias are the amplitudes of the THz wave and bias electric field, respectively. Eω is the amplitude of the optical probe beam (ω) and χ(3) is the third-order susceptibility of the ambient air. The first and third incoherent terms can be removed by referring the lock-in amplifier to the modulation frequency of the bias field. In this case, the intensity of the measured second harmonic signal is [9]

I2ωE2ωTHzE2ωLO(χ(3)EωEω)2EbiasETHz

Equation (2) shows that the detected signal is proportional to the THz field, implicating a coherent detection of the THz radiation.

The focuses of the optical probe beam (Eω) and THz beam (ETHz) are overlapped at the same position with each other if we consider a perfectly aligned ABCD system. Every Gaussian beam experiences a π phase shift during a focusing process. The Gouy phase of any Gaussian pulses can be written as [1416]

ϕG=arctan(z/zR)
where z is the propagation direction of the beam and zR is the Rayleigh length. The Rayleigh length can be expressed as
ZR=πw02λ
where w0 is the beam waist and λ is the wavelength. The beam waist can be extracted by the diffraction limit.

In the heterodyne detection, the detected second harmonic is the cross term of the THz-induced second harmonic and bias field-induced second harmonic. The dispersion between the fundamental beam and second harmonic beam will introduce the phase mismatch. The bias electric field doesn’t have a phase shift that can cancel one of the Gouy phases. If we use a short interaction length (thin electrodes), the phase of the output second harmonic has an extra dependence on the position of the electrodes.

Considering the four-wave mixing process in ABCD, the resulted phase of the second harmonic will be determined by a summation of phase shift from all incident waves. Thus, the measured second harmonic beam I2ωEω4EbiasETHz experiences the phase shift of 4ϕR(z)±ϕT(z). Here, ϕR and ϕT are the Gouy phases of the fundamental and THz beams, respectively. The sign “±” corresponds to the processes ω+ω±ΩTHz. In the two possible processes, the process 2ω=ω+ωΩTHz will couple more efficiently to the generated second harmonic than 2ω=ω+ω+ΩTHz. Here, we neglect the latter process altogether [11].

3. Experimental results and discussion

The schematic diagram of the experimental setup is shown in Fig. 1 . The input laser pulse is generated from a Ti:sapphire regenerative amplifier (Spectra-physics Spitfire) with a repetition rate of 1KHz, central wavelength of 800nm, and pulse duration of 40fs. The pump beam power is 1.2W and focused by a convex lens with a focal length of 100mm and passes through a 100μm type-I BBO crystal, where it generates the second harmonic beam. The superposed fundamental and second harmonic optical fields tunnel-ionize the air and drive a time-dependent current, leading to THz emission in the forward direction [1719]. The THz emission is collected and then focused by two pairs of off-axis parabolic mirrors. The probe beam is 70mW to avoid breakdown. It is focused by a 125mm convex lens, and co-propagates with the THz beam by passing through a hole drilled on the back of the 4th parabolic mirror (PM4) in the THz beam path. The THz and optical beams are then focused collinearly to the same spot where the THz field induces another second-harmonic field E2ωTHz. A pair of wire shaped electrodes with a 1mm gap is placed across the focusing spot with a ~2kV, 500Hz ac bias (synchronized with the laser repetition rate), which induces the second-harmonic field E2ωLO used as a local oscillator. The second-harmonic radiation (E2ωTHz+E2ωLO) is then filtered by a pair of 400nm band pass filters and collected into a photomultiplier tube (PMT). The signal from the PMT is measured by a lock-in amplifier referenced to the 500Hz bias modulation frequency.

 

Fig. 1 Schematic diagram of the experimental setup. L1-2: convex lenses; P1-4: parabolic mirrors; PMT: photomultiplier tube.

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The thin wire electrodes were mounted on a translation stage and able to move along the beam propagation direction. This enables us to detect the small region of THz field at different electrodes positions. The time-domain waveforms of the measured THz pulses versus electrodes position near the beam focus are illustrated in Fig. 2 . The THz waveforms show a significant phase variation along the detection region. The waveform is the most distinct with the electrodes placed at 8mm. It has to be pointed out that the polarities of the measured THz pulses at 6mm and 10mm are reversed which means the second harmonic photons have opposite phases.

 

Fig. 2 The THz time-domain waveforms at different scanning positions of the electrodes near the beam focus.

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The Rayleigh length of the broadband THz beam is frequency-dependent. Since high frequency components exhibit relatively smaller Rayleigh range, the Gouy phase effect is more significant for these components [20]. Figure 3 gives the extracted phase at 2 THz from the measured THz waveforms together with a theoretical simulation versus electrodes position. The scanning step is 0.2mm. The calculated Rayleigh length of the optical probe and THz beams are 1.9mm and 0.7mm, respectively. The fitted line of the total phase shift 4ϕR(z)ϕT(z) agrees with the experimental results very well. The zero phase shift with the electrodes placed at 7mm corresponds to the exactly overlapped focal spots of the optical probe and THz beams.

 

Fig. 3 The phase of the measured second harmonic at different electrodes positions. The black dots are the experimental results and the red solid curve is the fitted line.

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Figure 4 plots the dynamic range of the THz signals versus electrodes position. The dynamic range is defined as DR=20log10ETHzmaxETHzminENoise. It shows a maximum value when the pair of electrodes is placed at the focal spot of the THz beam. The dynamic range decreases at least 3dB when the electrodes moved 2mm from the focal spot. The drop-off of the dynamic range is contributed by the phase mismatch induced by the additional phase shift, together with the amplitudes decrease of the fundamental and THz electric fields.

 

Fig. 4 The dynamic range of the THz signals at different electrodes positions.

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Understanding the phase of THz pulses in ABCD provides an important guidance of electrodes design and detected efficiency optimization. The long electrodes have an average effect since the generated second harmonic along the beam propagation direction will add up coherently. For those second harmonic photons with opposite phases, a cancellation is introduced due to the destructive interference. In order to maintain a better phase match condition, thin electrodes is recommended in ABCD. When the thin electrodes deviate from the focal spot of the THz beam, an additional phase shift is introduced. It causes a temporal reshaping of the detected second harmonic. Although the additional phase shift does not introduce experimental errors to the spectroscopic and imaging measurements, it does influence the performance of the system by bringing down the dynamic range of the THz signal. Therefore, it is suggested that the electrodes be placed as close as possible to the focal spot of the THz beam during the alignment to achieve optimized detection efficiency.

4. Conclusion

In conclusion, the phases of the THz pulses are investigated experimentally and theoretically with the electrodes moving along the detection air plasma in the broadband THz heterodyne detection. Temporal reshaping and polarity reversal of the measured second harmonic are observed. The additional phase shift of the measured THz signal is caused by the position deviation of the electrodes. This additional phase shift reduces the dynamic range of the ABCD system.

Acknowledgments

This work was funded by the National Natural Science Foundation of China under Grant No. 11004140, Science and Technology Projects of Beijing Municipal Commission of Education under Grant No. 11224010011, the National Keystone Basic Research Program (973 Program) under Grants No.2007CB310408 and No.2006CB302901. It is also supported by the Support Program for Outstanding Ph.D. Advisors with Grant No. YB20101000701.

References and links

1. J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006). [CrossRef]   [PubMed]  

2. X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006). [CrossRef]   [PubMed]  

3. D. J. Cook and R. M. Hochstrasser, “Intense terahertz pulses by four-wave rectification in air,” Opt. Lett. 25(16), 1210–1212 (2000). [CrossRef]   [PubMed]  

4. J. Dai, X. Xie, and X.-C. Zhang, “Terahertz wave amplification in gases with the excitation of femtosecond laser pulses,” Appl. Phys. Lett. 91(21), 211102 (2007). [CrossRef]  

5. J. Dai and X.-C. Zhang, “Terahertz wave generation from gas plasma using a phase compensator with attosecond phase-control accuracy,” Appl. Phys. Lett. 94(2), 021117 (2009). [CrossRef]  

6. H. Wen and A. M. Lindenberg, “Coherent terahertz polarization control through manipulation of electron trajectories,” Phys. Rev. Lett. 103(2), 023902 (2009). [CrossRef]   [PubMed]  

7. J. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent polarization control of terahertz waves generated from two-color laser-induced gas plasma,” Phys. Rev. Lett. 103(2), 023001 (2009). [CrossRef]   [PubMed]  

8. Y. Liu, A. Houard, B. Prade, S. Akturk, A. Mysyrowicz, and V. T. Tikhonchuk, “Terahertz radiation source in air based on bifilamentation of femtosecond laser pulses,” Phys. Rev. Lett. 99(13), 135002 (2007). [CrossRef]   [PubMed]  

9. N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008). [CrossRef]  

10. X. Lu, N. Karpowicz, Y. Chen, and X.-C. Zhang, “Systematic study of broadband terahertz gas sensor,” Appl. Phys. Lett. 93(26), 261106 (2008). [CrossRef]  

11. X. Lu, N. Karpowicz, and X.-C. Zhang, “Broadband terahertz detection with selected gases,” J. Opt. Soc. Am. B 26(9), A66–A73 (2009). [CrossRef]  

12. L. Zhang, H. Zhong, C. Deng, C. Zhang, and Y. Zhao, “Terahertz wave reference-free phase imaging for identification of explosives,” Appl. Phys. Lett. 92(9), 091117 (2008). [CrossRef]  

13. H. Zhong, C. Zhang, L. Zhang, Y. Zhao, and X.-C. Zhang, “A phase feature extraction technique for terahertz reflection spectroscopy,” Appl. Phys. Lett. 92(22), 221106 (2008). [CrossRef]  

14. A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83(17), 3410–3413 (1999). [CrossRef]  

15. R. McGowan, R. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulses ranging,” Appl. Phys. Lett. 76(6), 670–672 (2000). [CrossRef]  

16. W. Zhu, A. Agrawal, and A. Nahata, “Direct measurement of the Gouy phase shift for surface plasmon-polaritons,” Opt. Express 15(16), 9995–10001 (2007). [CrossRef]   [PubMed]  

17. H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71(17), 2725–2728 (1993). [CrossRef]   [PubMed]  

18. M. Kress, T. Löffler, S. Eden, M. Thomson, and H. G. Roskos, “Terahertz-pulse generation by photoionization of air with laser pulses composed of both fundamental and second-harmonic waves,” Opt. Lett. 29(10), 1120–1122 (2004). [CrossRef]   [PubMed]  

19. A. Houard, Y. Liu, B. Prade, V. T. Tikhonchuk, and A. Mysyrowicz, “Strong enhancement of terahertz radiation from laser filaments in air by a static electric field,” Phys. Rev. Lett. 100(25), 255006 (2008). [CrossRef]   [PubMed]  

20. R. W. Boyd, Nonlinear Optics (Academic, 1992).

References

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  1. J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006).
    [CrossRef] [PubMed]
  2. X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006).
    [CrossRef] [PubMed]
  3. D. J. Cook and R. M. Hochstrasser, “Intense terahertz pulses by four-wave rectification in air,” Opt. Lett. 25(16), 1210–1212 (2000).
    [CrossRef] [PubMed]
  4. J. Dai, X. Xie, and X.-C. Zhang, “Terahertz wave amplification in gases with the excitation of femtosecond laser pulses,” Appl. Phys. Lett. 91(21), 211102 (2007).
    [CrossRef]
  5. J. Dai and X.-C. Zhang, “Terahertz wave generation from gas plasma using a phase compensator with attosecond phase-control accuracy,” Appl. Phys. Lett. 94(2), 021117 (2009).
    [CrossRef]
  6. H. Wen and A. M. Lindenberg, “Coherent terahertz polarization control through manipulation of electron trajectories,” Phys. Rev. Lett. 103(2), 023902 (2009).
    [CrossRef] [PubMed]
  7. J. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent polarization control of terahertz waves generated from two-color laser-induced gas plasma,” Phys. Rev. Lett. 103(2), 023001 (2009).
    [CrossRef] [PubMed]
  8. Y. Liu, A. Houard, B. Prade, S. Akturk, A. Mysyrowicz, and V. T. Tikhonchuk, “Terahertz radiation source in air based on bifilamentation of femtosecond laser pulses,” Phys. Rev. Lett. 99(13), 135002 (2007).
    [CrossRef] [PubMed]
  9. N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
    [CrossRef]
  10. X. Lu, N. Karpowicz, Y. Chen, and X.-C. Zhang, “Systematic study of broadband terahertz gas sensor,” Appl. Phys. Lett. 93(26), 261106 (2008).
    [CrossRef]
  11. X. Lu, N. Karpowicz, and X.-C. Zhang, “Broadband terahertz detection with selected gases,” J. Opt. Soc. Am. B 26(9), A66–A73 (2009).
    [CrossRef]
  12. L. Zhang, H. Zhong, C. Deng, C. Zhang, and Y. Zhao, “Terahertz wave reference-free phase imaging for identification of explosives,” Appl. Phys. Lett. 92(9), 091117 (2008).
    [CrossRef]
  13. H. Zhong, C. Zhang, L. Zhang, Y. Zhao, and X.-C. Zhang, “A phase feature extraction technique for terahertz reflection spectroscopy,” Appl. Phys. Lett. 92(22), 221106 (2008).
    [CrossRef]
  14. A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83(17), 3410–3413 (1999).
    [CrossRef]
  15. R. McGowan, R. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulses ranging,” Appl. Phys. Lett. 76(6), 670–672 (2000).
    [CrossRef]
  16. W. Zhu, A. Agrawal, and A. Nahata, “Direct measurement of the Gouy phase shift for surface plasmon-polaritons,” Opt. Express 15(16), 9995–10001 (2007).
    [CrossRef] [PubMed]
  17. H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71(17), 2725–2728 (1993).
    [CrossRef] [PubMed]
  18. M. Kress, T. Löffler, S. Eden, M. Thomson, and H. G. Roskos, “Terahertz-pulse generation by photoionization of air with laser pulses composed of both fundamental and second-harmonic waves,” Opt. Lett. 29(10), 1120–1122 (2004).
    [CrossRef] [PubMed]
  19. A. Houard, Y. Liu, B. Prade, V. T. Tikhonchuk, and A. Mysyrowicz, “Strong enhancement of terahertz radiation from laser filaments in air by a static electric field,” Phys. Rev. Lett. 100(25), 255006 (2008).
    [CrossRef] [PubMed]
  20. R. W. Boyd, Nonlinear Optics (Academic, 1992).

2009 (4)

J. Dai and X.-C. Zhang, “Terahertz wave generation from gas plasma using a phase compensator with attosecond phase-control accuracy,” Appl. Phys. Lett. 94(2), 021117 (2009).
[CrossRef]

H. Wen and A. M. Lindenberg, “Coherent terahertz polarization control through manipulation of electron trajectories,” Phys. Rev. Lett. 103(2), 023902 (2009).
[CrossRef] [PubMed]

J. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent polarization control of terahertz waves generated from two-color laser-induced gas plasma,” Phys. Rev. Lett. 103(2), 023001 (2009).
[CrossRef] [PubMed]

X. Lu, N. Karpowicz, and X.-C. Zhang, “Broadband terahertz detection with selected gases,” J. Opt. Soc. Am. B 26(9), A66–A73 (2009).
[CrossRef]

2008 (5)

L. Zhang, H. Zhong, C. Deng, C. Zhang, and Y. Zhao, “Terahertz wave reference-free phase imaging for identification of explosives,” Appl. Phys. Lett. 92(9), 091117 (2008).
[CrossRef]

H. Zhong, C. Zhang, L. Zhang, Y. Zhao, and X.-C. Zhang, “A phase feature extraction technique for terahertz reflection spectroscopy,” Appl. Phys. Lett. 92(22), 221106 (2008).
[CrossRef]

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

X. Lu, N. Karpowicz, Y. Chen, and X.-C. Zhang, “Systematic study of broadband terahertz gas sensor,” Appl. Phys. Lett. 93(26), 261106 (2008).
[CrossRef]

A. Houard, Y. Liu, B. Prade, V. T. Tikhonchuk, and A. Mysyrowicz, “Strong enhancement of terahertz radiation from laser filaments in air by a static electric field,” Phys. Rev. Lett. 100(25), 255006 (2008).
[CrossRef] [PubMed]

2007 (3)

Y. Liu, A. Houard, B. Prade, S. Akturk, A. Mysyrowicz, and V. T. Tikhonchuk, “Terahertz radiation source in air based on bifilamentation of femtosecond laser pulses,” Phys. Rev. Lett. 99(13), 135002 (2007).
[CrossRef] [PubMed]

J. Dai, X. Xie, and X.-C. Zhang, “Terahertz wave amplification in gases with the excitation of femtosecond laser pulses,” Appl. Phys. Lett. 91(21), 211102 (2007).
[CrossRef]

W. Zhu, A. Agrawal, and A. Nahata, “Direct measurement of the Gouy phase shift for surface plasmon-polaritons,” Opt. Express 15(16), 9995–10001 (2007).
[CrossRef] [PubMed]

2006 (2)

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006).
[CrossRef] [PubMed]

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006).
[CrossRef] [PubMed]

2004 (1)

2000 (2)

D. J. Cook and R. M. Hochstrasser, “Intense terahertz pulses by four-wave rectification in air,” Opt. Lett. 25(16), 1210–1212 (2000).
[CrossRef] [PubMed]

R. McGowan, R. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulses ranging,” Appl. Phys. Lett. 76(6), 670–672 (2000).
[CrossRef]

1999 (1)

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83(17), 3410–3413 (1999).
[CrossRef]

1993 (1)

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71(17), 2725–2728 (1993).
[CrossRef] [PubMed]

Agrawal, A.

Akturk, S.

Y. Liu, A. Houard, B. Prade, S. Akturk, A. Mysyrowicz, and V. T. Tikhonchuk, “Terahertz radiation source in air based on bifilamentation of femtosecond laser pulses,” Phys. Rev. Lett. 99(13), 135002 (2007).
[CrossRef] [PubMed]

Chen, Y.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

X. Lu, N. Karpowicz, Y. Chen, and X.-C. Zhang, “Systematic study of broadband terahertz gas sensor,” Appl. Phys. Lett. 93(26), 261106 (2008).
[CrossRef]

Cheville, R.

R. McGowan, R. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulses ranging,” Appl. Phys. Lett. 76(6), 670–672 (2000).
[CrossRef]

Cook, D. J.

Dai, J.

J. Dai and X.-C. Zhang, “Terahertz wave generation from gas plasma using a phase compensator with attosecond phase-control accuracy,” Appl. Phys. Lett. 94(2), 021117 (2009).
[CrossRef]

J. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent polarization control of terahertz waves generated from two-color laser-induced gas plasma,” Phys. Rev. Lett. 103(2), 023001 (2009).
[CrossRef] [PubMed]

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

J. Dai, X. Xie, and X.-C. Zhang, “Terahertz wave amplification in gases with the excitation of femtosecond laser pulses,” Appl. Phys. Lett. 91(21), 211102 (2007).
[CrossRef]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006).
[CrossRef] [PubMed]

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006).
[CrossRef] [PubMed]

Deng, C.

L. Zhang, H. Zhong, C. Deng, C. Zhang, and Y. Zhao, “Terahertz wave reference-free phase imaging for identification of explosives,” Appl. Phys. Lett. 92(9), 091117 (2008).
[CrossRef]

Eden, S.

Falcone, R. W.

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71(17), 2725–2728 (1993).
[CrossRef] [PubMed]

Feng, S.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83(17), 3410–3413 (1999).
[CrossRef]

Fletcher, C.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

Gordon, S.

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71(17), 2725–2728 (1993).
[CrossRef] [PubMed]

Grischkowsky, D.

R. McGowan, R. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulses ranging,” Appl. Phys. Lett. 76(6), 670–672 (2000).
[CrossRef]

Hamster, H.

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71(17), 2725–2728 (1993).
[CrossRef] [PubMed]

Hochstrasser, R. M.

Houard, A.

A. Houard, Y. Liu, B. Prade, V. T. Tikhonchuk, and A. Mysyrowicz, “Strong enhancement of terahertz radiation from laser filaments in air by a static electric field,” Phys. Rev. Lett. 100(25), 255006 (2008).
[CrossRef] [PubMed]

Y. Liu, A. Houard, B. Prade, S. Akturk, A. Mysyrowicz, and V. T. Tikhonchuk, “Terahertz radiation source in air based on bifilamentation of femtosecond laser pulses,” Phys. Rev. Lett. 99(13), 135002 (2007).
[CrossRef] [PubMed]

Johnson, K.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

Karpowicz, N.

X. Lu, N. Karpowicz, and X.-C. Zhang, “Broadband terahertz detection with selected gases,” J. Opt. Soc. Am. B 26(9), A66–A73 (2009).
[CrossRef]

J. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent polarization control of terahertz waves generated from two-color laser-induced gas plasma,” Phys. Rev. Lett. 103(2), 023001 (2009).
[CrossRef] [PubMed]

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

X. Lu, N. Karpowicz, Y. Chen, and X.-C. Zhang, “Systematic study of broadband terahertz gas sensor,” Appl. Phys. Lett. 93(26), 261106 (2008).
[CrossRef]

Kress, M.

Lesimple, A.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

Lindenberg, A. M.

H. Wen and A. M. Lindenberg, “Coherent terahertz polarization control through manipulation of electron trajectories,” Phys. Rev. Lett. 103(2), 023902 (2009).
[CrossRef] [PubMed]

Liu, Y.

A. Houard, Y. Liu, B. Prade, V. T. Tikhonchuk, and A. Mysyrowicz, “Strong enhancement of terahertz radiation from laser filaments in air by a static electric field,” Phys. Rev. Lett. 100(25), 255006 (2008).
[CrossRef] [PubMed]

Y. Liu, A. Houard, B. Prade, S. Akturk, A. Mysyrowicz, and V. T. Tikhonchuk, “Terahertz radiation source in air based on bifilamentation of femtosecond laser pulses,” Phys. Rev. Lett. 99(13), 135002 (2007).
[CrossRef] [PubMed]

Löffler, T.

Lu, X.

X. Lu, N. Karpowicz, and X.-C. Zhang, “Broadband terahertz detection with selected gases,” J. Opt. Soc. Am. B 26(9), A66–A73 (2009).
[CrossRef]

X. Lu, N. Karpowicz, Y. Chen, and X.-C. Zhang, “Systematic study of broadband terahertz gas sensor,” Appl. Phys. Lett. 93(26), 261106 (2008).
[CrossRef]

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

Mamer, O.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

McGowan, R.

R. McGowan, R. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulses ranging,” Appl. Phys. Lett. 76(6), 670–672 (2000).
[CrossRef]

Mysyrowicz, A.

A. Houard, Y. Liu, B. Prade, V. T. Tikhonchuk, and A. Mysyrowicz, “Strong enhancement of terahertz radiation from laser filaments in air by a static electric field,” Phys. Rev. Lett. 100(25), 255006 (2008).
[CrossRef] [PubMed]

Y. Liu, A. Houard, B. Prade, S. Akturk, A. Mysyrowicz, and V. T. Tikhonchuk, “Terahertz radiation source in air based on bifilamentation of femtosecond laser pulses,” Phys. Rev. Lett. 99(13), 135002 (2007).
[CrossRef] [PubMed]

Nahata, A.

Prade, B.

A. Houard, Y. Liu, B. Prade, V. T. Tikhonchuk, and A. Mysyrowicz, “Strong enhancement of terahertz radiation from laser filaments in air by a static electric field,” Phys. Rev. Lett. 100(25), 255006 (2008).
[CrossRef] [PubMed]

Y. Liu, A. Houard, B. Prade, S. Akturk, A. Mysyrowicz, and V. T. Tikhonchuk, “Terahertz radiation source in air based on bifilamentation of femtosecond laser pulses,” Phys. Rev. Lett. 99(13), 135002 (2007).
[CrossRef] [PubMed]

Price-Gallagher, M.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

Roskos, H. G.

Rudd, J. V.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83(17), 3410–3413 (1999).
[CrossRef]

Ruffin, A. B.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83(17), 3410–3413 (1999).
[CrossRef]

Sullivan, A.

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71(17), 2725–2728 (1993).
[CrossRef] [PubMed]

Thomson, M.

Tikhonchuk, V. T.

A. Houard, Y. Liu, B. Prade, V. T. Tikhonchuk, and A. Mysyrowicz, “Strong enhancement of terahertz radiation from laser filaments in air by a static electric field,” Phys. Rev. Lett. 100(25), 255006 (2008).
[CrossRef] [PubMed]

Y. Liu, A. Houard, B. Prade, S. Akturk, A. Mysyrowicz, and V. T. Tikhonchuk, “Terahertz radiation source in air based on bifilamentation of femtosecond laser pulses,” Phys. Rev. Lett. 99(13), 135002 (2007).
[CrossRef] [PubMed]

Wen, H.

H. Wen and A. M. Lindenberg, “Coherent terahertz polarization control through manipulation of electron trajectories,” Phys. Rev. Lett. 103(2), 023902 (2009).
[CrossRef] [PubMed]

Whitaker, J. F.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83(17), 3410–3413 (1999).
[CrossRef]

White, W.

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71(17), 2725–2728 (1993).
[CrossRef] [PubMed]

Winful, H. G.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83(17), 3410–3413 (1999).
[CrossRef]

Xie, X.

J. Dai, X. Xie, and X.-C. Zhang, “Terahertz wave amplification in gases with the excitation of femtosecond laser pulses,” Appl. Phys. Lett. 91(21), 211102 (2007).
[CrossRef]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006).
[CrossRef] [PubMed]

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006).
[CrossRef] [PubMed]

Yamaguchi, M.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

Zhang, C.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

L. Zhang, H. Zhong, C. Deng, C. Zhang, and Y. Zhao, “Terahertz wave reference-free phase imaging for identification of explosives,” Appl. Phys. Lett. 92(9), 091117 (2008).
[CrossRef]

H. Zhong, C. Zhang, L. Zhang, Y. Zhao, and X.-C. Zhang, “A phase feature extraction technique for terahertz reflection spectroscopy,” Appl. Phys. Lett. 92(22), 221106 (2008).
[CrossRef]

Zhang, L.

H. Zhong, C. Zhang, L. Zhang, Y. Zhao, and X.-C. Zhang, “A phase feature extraction technique for terahertz reflection spectroscopy,” Appl. Phys. Lett. 92(22), 221106 (2008).
[CrossRef]

L. Zhang, H. Zhong, C. Deng, C. Zhang, and Y. Zhao, “Terahertz wave reference-free phase imaging for identification of explosives,” Appl. Phys. Lett. 92(9), 091117 (2008).
[CrossRef]

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

Zhang, X.-C.

J. Dai and X.-C. Zhang, “Terahertz wave generation from gas plasma using a phase compensator with attosecond phase-control accuracy,” Appl. Phys. Lett. 94(2), 021117 (2009).
[CrossRef]

J. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent polarization control of terahertz waves generated from two-color laser-induced gas plasma,” Phys. Rev. Lett. 103(2), 023001 (2009).
[CrossRef] [PubMed]

X. Lu, N. Karpowicz, and X.-C. Zhang, “Broadband terahertz detection with selected gases,” J. Opt. Soc. Am. B 26(9), A66–A73 (2009).
[CrossRef]

X. Lu, N. Karpowicz, Y. Chen, and X.-C. Zhang, “Systematic study of broadband terahertz gas sensor,” Appl. Phys. Lett. 93(26), 261106 (2008).
[CrossRef]

H. Zhong, C. Zhang, L. Zhang, Y. Zhao, and X.-C. Zhang, “A phase feature extraction technique for terahertz reflection spectroscopy,” Appl. Phys. Lett. 92(22), 221106 (2008).
[CrossRef]

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

J. Dai, X. Xie, and X.-C. Zhang, “Terahertz wave amplification in gases with the excitation of femtosecond laser pulses,” Appl. Phys. Lett. 91(21), 211102 (2007).
[CrossRef]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006).
[CrossRef] [PubMed]

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006).
[CrossRef] [PubMed]

Zhao, H.

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

Zhao, Y.

H. Zhong, C. Zhang, L. Zhang, Y. Zhao, and X.-C. Zhang, “A phase feature extraction technique for terahertz reflection spectroscopy,” Appl. Phys. Lett. 92(22), 221106 (2008).
[CrossRef]

L. Zhang, H. Zhong, C. Deng, C. Zhang, and Y. Zhao, “Terahertz wave reference-free phase imaging for identification of explosives,” Appl. Phys. Lett. 92(9), 091117 (2008).
[CrossRef]

Zhong, H.

H. Zhong, C. Zhang, L. Zhang, Y. Zhao, and X.-C. Zhang, “A phase feature extraction technique for terahertz reflection spectroscopy,” Appl. Phys. Lett. 92(22), 221106 (2008).
[CrossRef]

L. Zhang, H. Zhong, C. Deng, C. Zhang, and Y. Zhao, “Terahertz wave reference-free phase imaging for identification of explosives,” Appl. Phys. Lett. 92(9), 091117 (2008).
[CrossRef]

Zhu, W.

Appl. Phys. Lett. (7)

J. Dai, X. Xie, and X.-C. Zhang, “Terahertz wave amplification in gases with the excitation of femtosecond laser pulses,” Appl. Phys. Lett. 91(21), 211102 (2007).
[CrossRef]

J. Dai and X.-C. Zhang, “Terahertz wave generation from gas plasma using a phase compensator with attosecond phase-control accuracy,” Appl. Phys. Lett. 94(2), 021117 (2009).
[CrossRef]

N. Karpowicz, J. Dai, X. Lu, Y. Chen, M. Yamaguchi, H. Zhao, X.-C. Zhang, L. Zhang, C. Zhang, M. Price-Gallagher, C. Fletcher, O. Mamer, A. Lesimple, and K. Johnson, “Coherent heterodyne time-domain spectrometry covering the entire ‘terahertz gap’,” Appl. Phys. Lett. 92(1), 011131 (2008).
[CrossRef]

X. Lu, N. Karpowicz, Y. Chen, and X.-C. Zhang, “Systematic study of broadband terahertz gas sensor,” Appl. Phys. Lett. 93(26), 261106 (2008).
[CrossRef]

L. Zhang, H. Zhong, C. Deng, C. Zhang, and Y. Zhao, “Terahertz wave reference-free phase imaging for identification of explosives,” Appl. Phys. Lett. 92(9), 091117 (2008).
[CrossRef]

H. Zhong, C. Zhang, L. Zhang, Y. Zhao, and X.-C. Zhang, “A phase feature extraction technique for terahertz reflection spectroscopy,” Appl. Phys. Lett. 92(22), 221106 (2008).
[CrossRef]

R. McGowan, R. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulses ranging,” Appl. Phys. Lett. 76(6), 670–672 (2000).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (8)

A. Houard, Y. Liu, B. Prade, V. T. Tikhonchuk, and A. Mysyrowicz, “Strong enhancement of terahertz radiation from laser filaments in air by a static electric field,” Phys. Rev. Lett. 100(25), 255006 (2008).
[CrossRef] [PubMed]

H. Hamster, A. Sullivan, S. Gordon, W. White, and R. W. Falcone, “Subpicosecond, electromagnetic pulses from intense laser-plasma interaction,” Phys. Rev. Lett. 71(17), 2725–2728 (1993).
[CrossRef] [PubMed]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83(17), 3410–3413 (1999).
[CrossRef]

J. Dai, X. Xie, and X.-C. Zhang, “Detection of broadband terahertz waves with a laser-induced plasma in gases,” Phys. Rev. Lett. 97(10), 103903 (2006).
[CrossRef] [PubMed]

X. Xie, J. Dai, and X.-C. Zhang, “Coherent control of THz wave generation in ambient air,” Phys. Rev. Lett. 96(7), 075005 (2006).
[CrossRef] [PubMed]

H. Wen and A. M. Lindenberg, “Coherent terahertz polarization control through manipulation of electron trajectories,” Phys. Rev. Lett. 103(2), 023902 (2009).
[CrossRef] [PubMed]

J. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent polarization control of terahertz waves generated from two-color laser-induced gas plasma,” Phys. Rev. Lett. 103(2), 023001 (2009).
[CrossRef] [PubMed]

Y. Liu, A. Houard, B. Prade, S. Akturk, A. Mysyrowicz, and V. T. Tikhonchuk, “Terahertz radiation source in air based on bifilamentation of femtosecond laser pulses,” Phys. Rev. Lett. 99(13), 135002 (2007).
[CrossRef] [PubMed]

Other (1)

R. W. Boyd, Nonlinear Optics (Academic, 1992).

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup. L1-2: convex lenses; P1-4: parabolic mirrors; PMT: photomultiplier tube.

Fig. 2
Fig. 2

The THz time-domain waveforms at different scanning positions of the electrodes near the beam focus.

Fig. 3
Fig. 3

The phase of the measured second harmonic at different electrodes positions. The black dots are the experimental results and the red solid curve is the fitted line.

Fig. 4
Fig. 4

The dynamic range of the THz signals at different electrodes positions.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

I 2ω | E 2ω THz + E 2ω LO | 2 ( χ (3) E ω E ω ) 2 ( E THz 2 +2 E THz E bias + E bias 2 )
I 2ω E 2ω THz E 2ω LO ( χ (3) E ω E ω ) 2 E bias E THz
ϕ G =arctan(z/ z R )
Z R = π w 0 2 λ

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