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OPTICS LETTERS / Vol. 39, No. 14 / July 15, 2014
Possibility of direct estimation of terahertz pulse electric field Alexander V. Borodin,1,2,* Mikhail N. Esaulkov,1,2 Alexander A. Frolov,3 Alexander P. Shkurinov,1,2 and Vladislav Ya. Panchenko1,2 1
Department of Physics, Lomonosov Moscow State University, Leninskie gory St., 1, Bd. 52, Moscow 119992, Russia 2
3
Institute of Laser and Information Technologies of the Russian Academy of Sciences, Svyatoozerskaya St., 1, Shatura 140700, Russia
Joint Institute for High Temperatures of the Russian Academy of Sciences, Izhorskaya St., 13, Bd. 2, Moscow 125412, Russia *Corresponding author: [email protected] Received April 2, 2014; revised May 22, 2014; accepted May 23, 2014; posted June 9, 2014 (Doc. ID 209274); published July 3, 2014 In this Letter, we introduce a new method of estimation of the terahertz (THz) field amplitude. This method uses second-harmonic generation (SHG) in the presence of THz and DC fields in gaseous media. We take into account contributions from both nonionized molecules and free plasma electrons to the nonlinear process of SHG. We analyze the applicability of this method of detection to obtaining correct information on the waveform and amplitude of broadband THz pulses. © 2014 Optical Society of America OCIS codes: (190.4380) Nonlinear optics, four-wave mixing; (040.2235) Far infrared or terahertz; (120.1880) Detection. http://dx.doi.org/10.1364/OL.39.004092
Generation of broadband THz radiation by means of a four-wave-mixing process in a femtosecond air breakdown plasma allows one to obtain a broadband THz spectrum reaching the limit set by the optical pulse duration. Until recently, it was impossible to detect effectively such a broadband emitted spectrum. Solid-state detectors for time-domain measurements of THz field (semiconductor antennas and electro-optical crystals) have poor sensitivity near phonon lines and thus are ineffective in a part of the THz domain [1]. It was demonstrated recently that gases are an appropriate media for efficient detection of broadband THz pulses [2]. In this Letter, we focus on detection of pulsed THz radiation in gases by means of second-harmonic generation (SHG) in the presence of a DC field (electric field induced second harmonic generation process, EFISH) [3,4]. The SHG in the THz detector is usually treated as a nonlinear process resulting from the response of the neutral molecules of the medium and free electrons which appear in the process of ionization. For the case of SHG due to neutral molecules and atoms, the appearance of second harmonic (SH) and other even harmonics is predetermined by the lack of the inversion symmetry, which is broken by the presence of the THz and DC field. In this case, the SH power contains a coherent summand responsible for THz field detection. As shown in [5], a coherent term appears also for the nonlinear response of free electrons. The spectral sensitivity of the detection process based on the nonlinearity of the free electrons was derived in [5]. In our Letter, we suggest a new approach for estimation of the THz field value, which is suitable for both possible mechanisms of the EFISH process. The THz detection process involves nonlinear interaction of THz field E THz and optical laser field Eω . An additional DC field ensures coherent detection of the THz radiation [2]. First, we consider registration of THz radiation in the case of SHG due to the response of free electrons appear0146-9592/14/144092-04$15.00/0
ing in the beam waist in the process of ionization. For this case, the full energy of radiation at SH consists of three terms, W 2ω Δ W DC W THz;DC Δ W THz Δ;
(1)
where Δ is the time delay between laser and THz pulses. The first summand W DC contains the square of the DC field, W THz;DC Δ is the cross term which depends on both THz and DC field, and W THz Δ contains the THz field amplitude squared. The three terms can be written in the following form (see [5]): W DC GDC · ω2p τ2 k2p d2
p V 4E E 2DC π R2L cτ p J DC ; 16c4 16 2
(2)
V4 W THz;DC Δ GTHz;DC · ω2p τ2 k2p d2 E4 16c Z∞ 2 2 E R c −2ξ−Δ × DC L dξE T ξe τ2 J THz;DC ; (3) 4 Δ V 4 R2 c W THz Δ GTHz · ω2p τ2 k2p d2 E4 L2 16c 2τ 2 Z ∞ Z ξ −2ξ−Δ2 × dξ dξ0 E T ξ0 e τ2 J THz : Δ
Δ
(4)
Here, ωp is a plasma frequency; c is the speed of light; τ, RL are pulse duration and the radius of the focal spot of laser pulse in the waist; 2d is a typical plasma longitudinal size; V E is the velocity of the oscillatory motion of the electron due to the laser field; E T is the temporal profile of the THz pulse electric field; E DC is the DC field strength; and kp ω∕c. Equations (2)–(4) contain the following factors describing polarization and diffraction effects: © 2014 Optical Society of America
July 15, 2014 / Vol. 39, No. 14 / OPTICS LETTERS
GDC e0 eL 2 1∕8;
sensitivity of the plasma electron response can be neglected, and in this case we obtain from Eq. (3)
GTHz;DC eT eL e0 eL 1∕8e0 eT ; GTHz eT eL 2 1∕8;
J DC J THz
J THz;DC
(5)
Z 2 1 d dze−3iωp z∕4cω 2 2 ; 4d −d 1 iz∕zL Z 2 2 1 d dze−3iωp z∕4cω ; 2 4d −d 1 iz∕zL 1 iz∕zT Z 2 d dze−3iωp z∕4cω 1 2 8d −d 1 iz∕zL Zd 2 dze3iωp z∕4cω c:c: : × −d 1 − iz∕zL 1 − iz∕zT
(6)
(7)
Here, e0 is the unit vector along the direction of the DC field; eL , eT are unit vectors characterizing laser and THz field polarizations; and zL , zT are Rayleigh lengths for laser and THz pulses. Excluding plasma frequency from the first ratio in Eq. (2), the second ratio in Eq. (3) can be presented in the form G J W THz;DC Δ THz;DC THz;DC GDC J DC p Z∞ 2 4 2W −2ξ−Δ × p DC dξE T ξe τ2 : τ π E DC Δ
(8)
W W DC W THz;DC Δ 0 W THz Δ 0; (9)
Thus, W THz;DC Δ 0 W 0
W − W− : 2
(10)
The energy of SH for an arbitrary time delay between the pulses will be as follows: W THz;DC Δ
W − W− FΔ: 2
E T Δ
GDC J DC E DC W − W − FΔ: GTHz;DC J THz;DC 2 2W DC
(11)
If the duration of laser pulses τ is considerably smaller than that of the THz pulse τT , the distortion of the detected THz waveform due to nonuniform spectral
(12)
Here, the first two factors are a constant related to the beam geometry and polarization, and E DC is considered as known. The last factor is the THz waveform normalized to unity, and W , W − , W DC are values to be measured. Equation (12) allows us to obtain the THz waveform using values E DC and W DC , which can be directly measured. For the purpose of describing the EFISH process caused by the nonlinear response of the neutral atoms of the medium, we will start from the following equation for SHG process [6]: E 2ω
4πiω 3 2 χ E ω αE DC βE T ; nc
(13)
where Z α Z β
We are interested in the cross-component W THz;DC Δ because it is proportional to the THz field strength, while W THz Δ depends on the THz amplitude squared. W THz;DC Δ can be represented as W THz;DC Δ W 0 FΔ, where FΔ is dimensionless function describing the temporal profile of the detected signal and W 0 is the value characterizing THz signal amplitude at its peak, where Δ 0 and F0 1. The absolute value W 0 can be calculated by comparing the full power for two cases when vectors e0 and eT have the same directions and the opposite directions:
W − W DC − W THz;DC Δ 0 W THz Δ 0:
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∞ −∞ ∞
−∞
dz
eiΔkz ; 1 iz∕zL
dz
eiΔkz : 1 iz∕zL 1 iz∕zT
(14)
Here, Δk 2kω − k2ω is the wave vector mismatch due to dispersion of the gas. We should pay attention to the limits of the integral. In this case, it is the whole area of interaction, while in plasma it is restricted to the longitudinal size of the plasma cloud 2d [see Eqs. (6) and 7)]. Moreover, in plasma, kω ω∕cεω q ω∕c 1 − ω2p ∕ω2 and therefore Δk −3∕4ω2p ∕ωc. Using Eq. (13), we can find SH radiation intensity I 2ω
c 4πω 3 2 4 χ jE ω j 8π nc × jαj2 E 2DC jβj2 E 2T αβ α βE DC E T :
(15)
Radiation energy at SH can be obtained by integration of Eq. (15) over time and the cross section. Taking into account the Gaussian profile of the laser pulse for energy, we get an expression which is similar to Eq. (1): 2 2 4 Z 2 ∞ 4πω −2ξ−Δ 3 cRL E 0L χ dξe τ2 W 2ω Δ 16 nc −∞ × jαj2 E 2DC jβj2 E 2T ξ αβ α βE DC E T ξ:
(16)
In this case, the component depending only on the DC electric field in process of SHG is 2 r 4πω π cR2 E 4 E 2 χ 3 jαj2 τ L 0L DC : W DC 16 nc 2
(17)
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OPTICS LETTERS / Vol. 39, No. 14 / July 15, 2014
Using Eqs. (16) and (17) and considering τ ≪ τT , we obtain W THz;DC Δ
αβ α β W DC E Δ: E DC T jαj2
(18)
Finally, we get an expression similar to Eq. (12), Fig. 1. Schematic representation of the experimental setup.
jαj2 W − W− E T Δ FΔ: E DC αβ α β 2W DC
(19)
This means that the estimation method does not depend on the existence of breakdown plasma in the detection region. The appearance of plasma can make it necessary to take into account changes in the THz radiation propagation. It may be considered in a first approximation by diffraction integrals in Eqs. (6), (7), and (14). In the case when electron density is much less than the critical density for THz radiation, the ratio of diffraction integrals is approximately 0.5, the same as for the case of no plasma in the detection region. In the opposite case of high electron density, the detected waveform can be considerably distorted due to propagation of THz radiation through the dense plasma cloud. For the THz radiation from the two-color optical air breakdown plasma, the spatial pattern of the radiation is conical and can be described in the simplified form E T ξ r ⊥ − jzjtgα2 E T r ⊥ ; ξ; z eT ; exp − 2 1 iz∕zT 2RT 1 iz∕zT (20) where r ⊥ is a transversal coordinate, z is a longitudinal coordinate, and α is the cone angle. For this case, the diffraction integral in Eq. (7) should be multiplied by the factor exp−d · tgα2 ∕2R2T which, for experimentally observed cone angles [7,8] is approximately 0.3–0.5. Therefore, this beam profile does not limit the application of the estimation method [Eq. (19)]. For experimental testing of the THz field calibration method, we used the setup shown in Fig. 1. We used 120 fs pulses from the Ti:Sa regenerative amplifier (Spectra Physics SpitFire Pro), with central wavelength 797 nm and energy up to 2.5 mJ. In the generation arm, the pulses with energy 0.9 mJ were focused with a 200 mm lens through a 0.1 mm thick I type Beta Barium Borate crystal oriented for the maximum THz emission efficiency. A 0.35 mm thick Si plate was used to block the optical beam. The THz beam was collimated and refocused with a pair of off-axis parabolic mirrors with focal lengths 150 and 100 mm correspondingly. The probe beam was focused through the hole in the focusing parabolic mirror collinearly with the THz beam, with a 200 mm lens into the 1 mm space between two flat copper electrodes. A 1.5 kV voltage bias was applied to the electrodes so that the peak electric field reached 15 kV∕cm. Typical energy of the detection beam was 0.07 mJ per pulse. The SH radiation was separated from the fundamental beam with a bandpass filter and detected with a photomultiplier tube (PMT) (Hamamatsu R106).
We investigated the detected signal at different energies of the THz pulse to make sure that the SH pulse intensity is not saturated and the detector performs in the linear regime. We controlled the THz beam energy by varying the energy of the pump beam in the generation arm. The energy of the THz beam was measured with a calibrated Golay cell (Tydex Inc.) placed in the focal plane of PM2 on Fig. 1. The data acquired by the Golay cell was compared with the time-integrated profile of the THz pulse electrical field, obtained using the SH intensity measurement. A good correlation between the data can be seen from Fig. 2. To calibrate the THz field amplitude, we measured the SH beam energy in the maximum of the detected waveform for different values and polarities of the DC bias voltage (Fig. 3). For this, we do not need to switch the DC field direction for each THz pulse as in [2], and use the lock-in amplifier at the laser repetition rate. The averaged THz field amplitude for our experimental conditions was estimated using Eq. (19), and turned out to be 2.5 0.3 kV∕cm. Knowing the THz field amplitude at the maximum, we can estimate the average power of the detected THz signal to be P av
c 8π
Z
E 2THz dtS
1 ; τrep:rate
(21)
where E THz is the THz field strength, τrep:rate 1 ms is the interval between the subsequent pulses, S is the THz beam area at the focal plane, and the integral is taken over the THz pulse duration. The average THz power calculated with Eq. (21) was P EFISH 0.15 0.04 μW. We used 1 ps for the THz pulse duration and 500 μm for the THz waist diameter [7]. The average power of the THz radiation measured by the Golay cell was
J Fig. 2. Spectral integrated signal acquired by EFISH method compared with THz signal intensity measured by the Golay cell.
July 15, 2014 / Vol. 39, No. 14 / OPTICS LETTERS
Fig. 3. Total SH intensity versus DC-bias voltage for blocked THz beam (black squares) and open THz beam with two polarities of DC field (open and filled gray circles). Solid line is an approximation of experimental data using Eq. (17).
P Golay 2.0 0.3 μW, which is one order higher than the value obtained with the time-domain technique. This discrepancy between the two independent THz power measurements can be caused by overestimation of the THz beam waist diameter or the initial conical structure of the THz beam, which leads to division of the estimated value by the corrected diffraction integral in Eq. (7) (see above). Another reason is due to the nonuniform spectral sensitivity of the gas sensor, which decreases above THz frequency ν < 1∕τ. In our previous work [9], we have shown that the lower-frequency part of the THz spectrum is dominated by the photocurrent emission mechanism, and the higher-frequency part is defined mostly by the nonlinear response of the bound electrons. The Golay cell, in contrast, has almost uniform sensitivity in the THz spectral region, so it can detect both photocurrent and bound electron contribution to the THz radiation with equal efficiency. Thus, the higher detected values of the THz power detected with a Golay cell can be interpreted as evidence of the bound electron contribution which cannot be effectively detected by gas sensor or other time-domain measurement techniques. We intentionally left outside the scope of this work the analysis of the origin of the background signal at the SH
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frequency, which in any case accompanies the THz field detection process. We are very well aware that in the plasma formed in a strong field, the SHG due to contribution of the THz field always accompanies that of SHG due to other mechanisms [3], which will probably have an influence on the effect considered here. For the case when their intensity is less than the sensitivity of the detectors of the experimental setup, we can ignore them when determining field strength of the THz pulse. The analysis of contributions of these signals to SH intensity, which is detected value, is still to be studied. In conclusion, we have introduced a method to estimate THz field amplitude, using known DC field for calibration, that is suitable both for Gaussian and cone-like spatial profiles. The advantage of this method is that it does not require any material constants but uses only values which can be measured directly. On the other hand, it is also does not involve the precise dimensions of the THz beam waist and pulse duration to calculate the value of THz field strength. Moreover, we have shown that this method can be applied both in cases of breakdown in the detection region as well as in cases when no plasma is ignited. We thank N.A. Panov and O.G. Kosareva for fruitful discussion. This work was supported by the RSCF (Grant 14-29-00161). References 1. Y.-S. Lee, Principles of Terahertz Science and Technology (Springer Science, 2008). 2. X. Lu, N. Karpowicz, and X.-C. Zhang, J. Opt. Soc. Am. B 26, A66 (2009). 3. R. S. Finn and J. F. Ward, Phys. Rev. Lett. 26, 285 (1971). 4. P. D. Maker and R. W. Terhune, Phys. Rev. 137, A801 (1965). 5. A. A. Frolov, A. V. Borodin, M. N. Esaulkov, I. I. Kuritsyn, and A. P. Shkurinov, J. Exp. Theor. Phys. 141, 893 (2012). 6. R. W. Boyd, Nonlinear Optics (Academic, 2003). 7. A. V. Borodin, M. N. Esaulkov, I. I. Kuritsyn, I. A. Kotelnikov, and A. P. Shkurinov, J. Opt. Soc. Am. B 29, 1911 (2012). 8. P. Klarskov, A. Strikwerda, K. Iwaszczuk, and P. U. Jepsen, New J. Phys. 15, 075012 (2013). 9. A. V. Borodin, N. A. Panov, O. G. Kosareva, V. A. Andreeva, M. N. Esaulkov, V. A. Makarov, A. P. Shkurinov, S. L. Chin, and X.-C. Zhang, Opt. Lett. 38, 1906 (2013).
OPTICS LETTERS / Vol. 39, No. 14 / July 15, 2014
Possibility of direct estimation of terahertz pulse electric field Alexander V. Borodin,1,2,* Mikhail N. Esaulkov,1,2 Alexander A. Frolov,3 Alexander P. Shkurinov,1,2 and Vladislav Ya. Panchenko1,2 1
Department of Physics, Lomonosov Moscow State University, Leninskie gory St., 1, Bd. 52, Moscow 119992, Russia 2
3
Institute of Laser and Information Technologies of the Russian Academy of Sciences, Svyatoozerskaya St., 1, Shatura 140700, Russia
Joint Institute for High Temperatures of the Russian Academy of Sciences, Izhorskaya St., 13, Bd. 2, Moscow 125412, Russia *Corresponding author: [email protected] Received April 2, 2014; revised May 22, 2014; accepted May 23, 2014; posted June 9, 2014 (Doc. ID 209274); published July 3, 2014 In this Letter, we introduce a new method of estimation of the terahertz (THz) field amplitude. This method uses second-harmonic generation (SHG) in the presence of THz and DC fields in gaseous media. We take into account contributions from both nonionized molecules and free plasma electrons to the nonlinear process of SHG. We analyze the applicability of this method of detection to obtaining correct information on the waveform and amplitude of broadband THz pulses. © 2014 Optical Society of America OCIS codes: (190.4380) Nonlinear optics, four-wave mixing; (040.2235) Far infrared or terahertz; (120.1880) Detection. http://dx.doi.org/10.1364/OL.39.004092
Generation of broadband THz radiation by means of a four-wave-mixing process in a femtosecond air breakdown plasma allows one to obtain a broadband THz spectrum reaching the limit set by the optical pulse duration. Until recently, it was impossible to detect effectively such a broadband emitted spectrum. Solid-state detectors for time-domain measurements of THz field (semiconductor antennas and electro-optical crystals) have poor sensitivity near phonon lines and thus are ineffective in a part of the THz domain [1]. It was demonstrated recently that gases are an appropriate media for efficient detection of broadband THz pulses [2]. In this Letter, we focus on detection of pulsed THz radiation in gases by means of second-harmonic generation (SHG) in the presence of a DC field (electric field induced second harmonic generation process, EFISH) [3,4]. The SHG in the THz detector is usually treated as a nonlinear process resulting from the response of the neutral molecules of the medium and free electrons which appear in the process of ionization. For the case of SHG due to neutral molecules and atoms, the appearance of second harmonic (SH) and other even harmonics is predetermined by the lack of the inversion symmetry, which is broken by the presence of the THz and DC field. In this case, the SH power contains a coherent summand responsible for THz field detection. As shown in [5], a coherent term appears also for the nonlinear response of free electrons. The spectral sensitivity of the detection process based on the nonlinearity of the free electrons was derived in [5]. In our Letter, we suggest a new approach for estimation of the THz field value, which is suitable for both possible mechanisms of the EFISH process. The THz detection process involves nonlinear interaction of THz field E THz and optical laser field Eω . An additional DC field ensures coherent detection of the THz radiation [2]. First, we consider registration of THz radiation in the case of SHG due to the response of free electrons appear0146-9592/14/144092-04$15.00/0
ing in the beam waist in the process of ionization. For this case, the full energy of radiation at SH consists of three terms, W 2ω Δ W DC W THz;DC Δ W THz Δ;
(1)
where Δ is the time delay between laser and THz pulses. The first summand W DC contains the square of the DC field, W THz;DC Δ is the cross term which depends on both THz and DC field, and W THz Δ contains the THz field amplitude squared. The three terms can be written in the following form (see [5]): W DC GDC · ω2p τ2 k2p d2
p V 4E E 2DC π R2L cτ p J DC ; 16c4 16 2
(2)
V4 W THz;DC Δ GTHz;DC · ω2p τ2 k2p d2 E4 16c Z∞ 2 2 E R c −2ξ−Δ × DC L dξE T ξe τ2 J THz;DC ; (3) 4 Δ V 4 R2 c W THz Δ GTHz · ω2p τ2 k2p d2 E4 L2 16c 2τ 2 Z ∞ Z ξ −2ξ−Δ2 × dξ dξ0 E T ξ0 e τ2 J THz : Δ
Δ
(4)
Here, ωp is a plasma frequency; c is the speed of light; τ, RL are pulse duration and the radius of the focal spot of laser pulse in the waist; 2d is a typical plasma longitudinal size; V E is the velocity of the oscillatory motion of the electron due to the laser field; E T is the temporal profile of the THz pulse electric field; E DC is the DC field strength; and kp ω∕c. Equations (2)–(4) contain the following factors describing polarization and diffraction effects: © 2014 Optical Society of America
July 15, 2014 / Vol. 39, No. 14 / OPTICS LETTERS
GDC e0 eL 2 1∕8;
sensitivity of the plasma electron response can be neglected, and in this case we obtain from Eq. (3)
GTHz;DC eT eL e0 eL 1∕8e0 eT ; GTHz eT eL 2 1∕8;
J DC J THz
J THz;DC
(5)
Z 2 1 d dze−3iωp z∕4cω 2 2 ; 4d −d 1 iz∕zL Z 2 2 1 d dze−3iωp z∕4cω ; 2 4d −d 1 iz∕zL 1 iz∕zT Z 2 d dze−3iωp z∕4cω 1 2 8d −d 1 iz∕zL Zd 2 dze3iωp z∕4cω c:c: : × −d 1 − iz∕zL 1 − iz∕zT
(6)
(7)
Here, e0 is the unit vector along the direction of the DC field; eL , eT are unit vectors characterizing laser and THz field polarizations; and zL , zT are Rayleigh lengths for laser and THz pulses. Excluding plasma frequency from the first ratio in Eq. (2), the second ratio in Eq. (3) can be presented in the form G J W THz;DC Δ THz;DC THz;DC GDC J DC p Z∞ 2 4 2W −2ξ−Δ × p DC dξE T ξe τ2 : τ π E DC Δ
(8)
W W DC W THz;DC Δ 0 W THz Δ 0; (9)
Thus, W THz;DC Δ 0 W 0
W − W− : 2
(10)
The energy of SH for an arbitrary time delay between the pulses will be as follows: W THz;DC Δ
W − W− FΔ: 2
E T Δ
GDC J DC E DC W − W − FΔ: GTHz;DC J THz;DC 2 2W DC
(11)
If the duration of laser pulses τ is considerably smaller than that of the THz pulse τT , the distortion of the detected THz waveform due to nonuniform spectral
(12)
Here, the first two factors are a constant related to the beam geometry and polarization, and E DC is considered as known. The last factor is the THz waveform normalized to unity, and W , W − , W DC are values to be measured. Equation (12) allows us to obtain the THz waveform using values E DC and W DC , which can be directly measured. For the purpose of describing the EFISH process caused by the nonlinear response of the neutral atoms of the medium, we will start from the following equation for SHG process [6]: E 2ω
4πiω 3 2 χ E ω αE DC βE T ; nc
(13)
where Z α Z β
We are interested in the cross-component W THz;DC Δ because it is proportional to the THz field strength, while W THz Δ depends on the THz amplitude squared. W THz;DC Δ can be represented as W THz;DC Δ W 0 FΔ, where FΔ is dimensionless function describing the temporal profile of the detected signal and W 0 is the value characterizing THz signal amplitude at its peak, where Δ 0 and F0 1. The absolute value W 0 can be calculated by comparing the full power for two cases when vectors e0 and eT have the same directions and the opposite directions:
W − W DC − W THz;DC Δ 0 W THz Δ 0:
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∞ −∞ ∞
−∞
dz
eiΔkz ; 1 iz∕zL
dz
eiΔkz : 1 iz∕zL 1 iz∕zT
(14)
Here, Δk 2kω − k2ω is the wave vector mismatch due to dispersion of the gas. We should pay attention to the limits of the integral. In this case, it is the whole area of interaction, while in plasma it is restricted to the longitudinal size of the plasma cloud 2d [see Eqs. (6) and 7)]. Moreover, in plasma, kω ω∕cεω q ω∕c 1 − ω2p ∕ω2 and therefore Δk −3∕4ω2p ∕ωc. Using Eq. (13), we can find SH radiation intensity I 2ω
c 4πω 3 2 4 χ jE ω j 8π nc × jαj2 E 2DC jβj2 E 2T αβ α βE DC E T :
(15)
Radiation energy at SH can be obtained by integration of Eq. (15) over time and the cross section. Taking into account the Gaussian profile of the laser pulse for energy, we get an expression which is similar to Eq. (1): 2 2 4 Z 2 ∞ 4πω −2ξ−Δ 3 cRL E 0L χ dξe τ2 W 2ω Δ 16 nc −∞ × jαj2 E 2DC jβj2 E 2T ξ αβ α βE DC E T ξ:
(16)
In this case, the component depending only on the DC electric field in process of SHG is 2 r 4πω π cR2 E 4 E 2 χ 3 jαj2 τ L 0L DC : W DC 16 nc 2
(17)
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Using Eqs. (16) and (17) and considering τ ≪ τT , we obtain W THz;DC Δ
αβ α β W DC E Δ: E DC T jαj2
(18)
Finally, we get an expression similar to Eq. (12), Fig. 1. Schematic representation of the experimental setup.
jαj2 W − W− E T Δ FΔ: E DC αβ α β 2W DC
(19)
This means that the estimation method does not depend on the existence of breakdown plasma in the detection region. The appearance of plasma can make it necessary to take into account changes in the THz radiation propagation. It may be considered in a first approximation by diffraction integrals in Eqs. (6), (7), and (14). In the case when electron density is much less than the critical density for THz radiation, the ratio of diffraction integrals is approximately 0.5, the same as for the case of no plasma in the detection region. In the opposite case of high electron density, the detected waveform can be considerably distorted due to propagation of THz radiation through the dense plasma cloud. For the THz radiation from the two-color optical air breakdown plasma, the spatial pattern of the radiation is conical and can be described in the simplified form E T ξ r ⊥ − jzjtgα2 E T r ⊥ ; ξ; z eT ; exp − 2 1 iz∕zT 2RT 1 iz∕zT (20) where r ⊥ is a transversal coordinate, z is a longitudinal coordinate, and α is the cone angle. For this case, the diffraction integral in Eq. (7) should be multiplied by the factor exp−d · tgα2 ∕2R2T which, for experimentally observed cone angles [7,8] is approximately 0.3–0.5. Therefore, this beam profile does not limit the application of the estimation method [Eq. (19)]. For experimental testing of the THz field calibration method, we used the setup shown in Fig. 1. We used 120 fs pulses from the Ti:Sa regenerative amplifier (Spectra Physics SpitFire Pro), with central wavelength 797 nm and energy up to 2.5 mJ. In the generation arm, the pulses with energy 0.9 mJ were focused with a 200 mm lens through a 0.1 mm thick I type Beta Barium Borate crystal oriented for the maximum THz emission efficiency. A 0.35 mm thick Si plate was used to block the optical beam. The THz beam was collimated and refocused with a pair of off-axis parabolic mirrors with focal lengths 150 and 100 mm correspondingly. The probe beam was focused through the hole in the focusing parabolic mirror collinearly with the THz beam, with a 200 mm lens into the 1 mm space between two flat copper electrodes. A 1.5 kV voltage bias was applied to the electrodes so that the peak electric field reached 15 kV∕cm. Typical energy of the detection beam was 0.07 mJ per pulse. The SH radiation was separated from the fundamental beam with a bandpass filter and detected with a photomultiplier tube (PMT) (Hamamatsu R106).
We investigated the detected signal at different energies of the THz pulse to make sure that the SH pulse intensity is not saturated and the detector performs in the linear regime. We controlled the THz beam energy by varying the energy of the pump beam in the generation arm. The energy of the THz beam was measured with a calibrated Golay cell (Tydex Inc.) placed in the focal plane of PM2 on Fig. 1. The data acquired by the Golay cell was compared with the time-integrated profile of the THz pulse electrical field, obtained using the SH intensity measurement. A good correlation between the data can be seen from Fig. 2. To calibrate the THz field amplitude, we measured the SH beam energy in the maximum of the detected waveform for different values and polarities of the DC bias voltage (Fig. 3). For this, we do not need to switch the DC field direction for each THz pulse as in [2], and use the lock-in amplifier at the laser repetition rate. The averaged THz field amplitude for our experimental conditions was estimated using Eq. (19), and turned out to be 2.5 0.3 kV∕cm. Knowing the THz field amplitude at the maximum, we can estimate the average power of the detected THz signal to be P av
c 8π
Z
E 2THz dtS
1 ; τrep:rate
(21)
where E THz is the THz field strength, τrep:rate 1 ms is the interval between the subsequent pulses, S is the THz beam area at the focal plane, and the integral is taken over the THz pulse duration. The average THz power calculated with Eq. (21) was P EFISH 0.15 0.04 μW. We used 1 ps for the THz pulse duration and 500 μm for the THz waist diameter [7]. The average power of the THz radiation measured by the Golay cell was
J Fig. 2. Spectral integrated signal acquired by EFISH method compared with THz signal intensity measured by the Golay cell.
July 15, 2014 / Vol. 39, No. 14 / OPTICS LETTERS
Fig. 3. Total SH intensity versus DC-bias voltage for blocked THz beam (black squares) and open THz beam with two polarities of DC field (open and filled gray circles). Solid line is an approximation of experimental data using Eq. (17).
P Golay 2.0 0.3 μW, which is one order higher than the value obtained with the time-domain technique. This discrepancy between the two independent THz power measurements can be caused by overestimation of the THz beam waist diameter or the initial conical structure of the THz beam, which leads to division of the estimated value by the corrected diffraction integral in Eq. (7) (see above). Another reason is due to the nonuniform spectral sensitivity of the gas sensor, which decreases above THz frequency ν < 1∕τ. In our previous work [9], we have shown that the lower-frequency part of the THz spectrum is dominated by the photocurrent emission mechanism, and the higher-frequency part is defined mostly by the nonlinear response of the bound electrons. The Golay cell, in contrast, has almost uniform sensitivity in the THz spectral region, so it can detect both photocurrent and bound electron contribution to the THz radiation with equal efficiency. Thus, the higher detected values of the THz power detected with a Golay cell can be interpreted as evidence of the bound electron contribution which cannot be effectively detected by gas sensor or other time-domain measurement techniques. We intentionally left outside the scope of this work the analysis of the origin of the background signal at the SH
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frequency, which in any case accompanies the THz field detection process. We are very well aware that in the plasma formed in a strong field, the SHG due to contribution of the THz field always accompanies that of SHG due to other mechanisms [3], which will probably have an influence on the effect considered here. For the case when their intensity is less than the sensitivity of the detectors of the experimental setup, we can ignore them when determining field strength of the THz pulse. The analysis of contributions of these signals to SH intensity, which is detected value, is still to be studied. In conclusion, we have introduced a method to estimate THz field amplitude, using known DC field for calibration, that is suitable both for Gaussian and cone-like spatial profiles. The advantage of this method is that it does not require any material constants but uses only values which can be measured directly. On the other hand, it is also does not involve the precise dimensions of the THz beam waist and pulse duration to calculate the value of THz field strength. Moreover, we have shown that this method can be applied both in cases of breakdown in the detection region as well as in cases when no plasma is ignited. We thank N.A. Panov and O.G. Kosareva for fruitful discussion. This work was supported by the RSCF (Grant 14-29-00161). References 1. Y.-S. Lee, Principles of Terahertz Science and Technology (Springer Science, 2008). 2. X. Lu, N. Karpowicz, and X.-C. Zhang, J. Opt. Soc. Am. B 26, A66 (2009). 3. R. S. Finn and J. F. Ward, Phys. Rev. Lett. 26, 285 (1971). 4. P. D. Maker and R. W. Terhune, Phys. Rev. 137, A801 (1965). 5. A. A. Frolov, A. V. Borodin, M. N. Esaulkov, I. I. Kuritsyn, and A. P. Shkurinov, J. Exp. Theor. Phys. 141, 893 (2012). 6. R. W. Boyd, Nonlinear Optics (Academic, 2003). 7. A. V. Borodin, M. N. Esaulkov, I. I. Kuritsyn, I. A. Kotelnikov, and A. P. Shkurinov, J. Opt. Soc. Am. B 29, 1911 (2012). 8. P. Klarskov, A. Strikwerda, K. Iwaszczuk, and P. U. Jepsen, New J. Phys. 15, 075012 (2013). 9. A. V. Borodin, N. A. Panov, O. G. Kosareva, V. A. Andreeva, M. N. Esaulkov, V. A. Makarov, A. P. Shkurinov, S. L. Chin, and X.-C. Zhang, Opt. Lett. 38, 1906 (2013).
