Terahertz aqueous photonics

Abstract

Developing efficient and robust terahertz (THz) sources is of incessant interest in the THz community for their wide applications. With successive effort in past decades, numerous groups have achieved THz wave generation from solids, gases, and plasmas. However, liquid, especially liquid water has never been demonstrated as a THz source. One main reason leading the impediment is that water has strong absorption characteristics in the THz frequency regime.

A thin water film under intense laser excitation was introduced as the THz source to mitigate the considerable loss of THz waves from the absorption. Laser-induced plasma formation associated with a ponderomotive force-induced dipole model was proposed to explain the generation process. For the one-color excitation scheme, the water film generates a higher THz electric field than the air does under the identical experimental condition. Unlike the case of air, THz wave generation from liquid water prefers a sub-picosecond (200−800 fs) laser pulse rather than a femtosecond pulse (~50 fs). This observation results from the plasma generation process in water.

For the two-color excitation scheme, the THz electric field is enhanced by one-order of magnitude in comparison with the one-color case. Meanwhile, coherent control of the THz field is achieved by adjusting the relative phase between the fundamental pulse and the second-harmonic pulse.

To eliminate the total internal reflection of THz waves at the water-air interface of a water film, a water line produced by a syringe needle was used to emit THz waves. As expected, more THz radiation can be coupled out and detected. THz wave generation from other liquids were also tested.

Graphical abstract

Keywords

terahertz (THz) wave generation / liquid water / laser-induced plasma

1 Introduction

Quantum cryptography communication is a promising technology using the quantum natures of light to achieve the security in telecommunication [1]. Like single photon as quantum key distribution (QKD), continuous variable coherent optical beam can be used to quantum cryptography communication. Compared with the single photon QKD scheme, continuous variable coherent schemes are more attractive because of their high efficiency and compatibility [2]. The continuous variable quantum cryptography setups can easily encode and transmit. Gaussian cryptographic protocols for coherent states have shown its potential on the applications of quantum communication. These protocols are robust due to it can overcome the quantum channel noise.

In the continuous variables coherent optical communication systems, the signal light was injected into a M-Z interferometer. In the Mach-Zehnder (M-Z) interferometer, one beam linearly polarized light was divided into two quadrature coherent components and propagate along the two branches [3]. One of them can be used as the local oscillator (LO), and the amplitude and phase of the other one is Gaussian modulated to carry information. In the output of the M-Z interferometer, two components are coupled for detection. According to the Heisenberg uncertainty principle and quantum no-cloning theorem [4], the two components of coherent light can guarantee the security of signal transmission. In the M-Z interferometer based free-space transmission schemes, the optical coupling and phase synchronization is unstable in the receiver of the systems. Elser et al. [5] proposed a single-mode spatial optical signal transmission scheme to simplify the setup of the continuous variables coherent optical communication scheme. But this scheme cannot achieve the base selection for the Stokes components ${\widehat{S}}_{\mathrm{2}}$ and ${\widehat{S}}_{\mathrm{3}}$ at the receiver end, which is required by the applications of quantum cryptography communication.

In this paper, we demonstrated a scheme using Stokes parameter coding for the continuous variable coherent optical communication scheme to stabilize optical coupling and phase synchronization. The corresponding experimental setup is also demonstrated, which uses SU(2) converter in the detector setup to achieve the base selection.

2 Theoretical analyses

The state of polarization (SOP) of the input light can be presented by Jones matrix as follows [6-8]:

(1)$$E={\left(\begin{array}{ll}{E}_x& E_y\end{array}\right)}^{\mathrm{T}}.$$

When passing through the polarizer and two polarized beam splitters (PBS), the light is transformed to a pure horizontal polarization. The horizontal polarized light is presented as

(2)$${\mathit{E}}_{\mathrm{1}}={\left(\begin{array}{ll}{E}_x& \mathrm{0}\end{array}\right)}^{\mathrm{T}}.$$

In this scheme, an electro-optical amplitude modulator (EOM) is used to adjust the phase delay of the horizontal polarized light. Here, electro-optical crystal acts as a phase delay adjustable wave plate, which can be written as follows [9]:

(3)$${\mathit{E}}_{\mathrm{e}\mathrm{o}\mathrm{m}}=\left(\begin{array}{cc}{\mathrm{e}}^{-\mathrm{i}\tau /\mathrm{2}}{\cos }^{\mathrm{2}}\phi +{\mathrm{e}}^{\mathrm{i}\tau /\mathrm{2}}{\sin }^{\mathrm{2}}\phi & -\mathrm{i}\sin (\tau /\mathrm{2})\sin (\mathrm{2}\phi )\\ -\mathrm{i}\sin (\tau /\mathrm{2})\sin (\mathrm{2}\phi )& {\mathrm{e}}^{-\mathrm{i}(\tau /\mathrm{2})}{\sin }^{\mathrm{2}}\phi +{\mathrm{e}}^{\mathrm{i}(\tau /\mathrm{2})}{\cos }^{\mathrm{2}}\phi \end{array}\right),$$

where $\phi$ is the angle between the crystal axis and the horizontal direction; $\tau$ is the phase delay of the crystal. After passing through the EOM, the Jones matrix of the signal light becomes

(4)$${\mathit{E}}_{\mathrm{o}\mathrm{u}\mathrm{t}}=E_x\left(\begin{array}{c}{\mathrm{e}}^{-\mathrm{i}(\tau /\mathrm{2})}{\cos }^{\mathrm{2}}\phi +e^{\mathrm{i}(\tau /\mathrm{2})}{\sin }^{\mathrm{2}}\phi \\ -\mathrm{i}\sin (\tau /\mathrm{2})\sin (\mathrm{2}\phi )\end{array}\right).$$

As we know,, the SOP of light can be represented by Stokes parameters, which can be mapped to a Poincaré sphere face as shown in Fig. 1.

Fig.1 Poincaré sphere

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So that one point on the sphere face present one polarization states. As shown in Fig. 1 [10], the following formulas can be written.

(5)$$\begin{array}{c}{S}_{\mathrm{0}}=\sqrt{S_{\mathrm{1}}^{\mathrm{2}}+S_{\mathrm{2}}^{\mathrm{2}}+S_{\mathrm{3}}^{\mathrm{2}}},\\ S_{\mathrm{1}}=S_{\mathrm{0}}\cos (\mathrm{2}\alpha )\cos (2\beta ),\\ S_{\mathrm{2}}=S_{\mathrm{0}}\cos (\mathrm{2}\alpha )\sin (\mathrm{2}\beta ),\\ S_{\mathrm{3}}=S_{\mathrm{0}}\sin (\mathrm{2}\alpha ).\end{array}$$

In our proposed Stokes parameter coding schemes, $S_{\mathrm{2}}$ or $S_{\mathrm{3}}$ can be encoded by modulating angels of $\alpha$ and $\beta$, through EOM and magneto-optical modulation (MOM).

Now, the Stokes components of signal light after the EOM can be seen as [11]

(6)$$\begin{array}{l}{S}_{\mathrm{2}}={\widehat{\mathit{E}}}_{\mathrm{E}\mathrm{O}\mathrm{M}}{\mathit{P}}_{\mathrm{3}}{\mathit{E}}_{\mathrm{E}\mathrm{O}\mathrm{M}},\\ S_{\mathrm{3}}={\widehat{\mathit{E}}}_{\mathrm{E}\mathrm{O}\mathrm{M}}{\mathit{P}}_{\mathrm{4}}{\mathit{E}}_{\mathrm{E}\mathrm{O}\mathrm{M}},\end{array}$$

where ${\mathit{P}}_{\mathrm{3}}$, ${\mathit{P}}_{\mathrm{4}}$ are the sandwich matrixes, which are

(7)$${\mathit{P}}_{\mathrm{3}}=\left[\begin{array}{ll}\mathrm{0}& \mathrm{1}\\ \mathrm{1}& \mathrm{0}\end{array}\right],{\mathit{P}}_{\mathrm{4}}=\left[\begin{array}{ll}\mathrm{0}& \mathrm{i}\\ -\mathrm{i}& \mathrm{0}\end{array}\right].$$

So $S_{\mathrm{2}}$ and $S_{\mathrm{3}}$ can be described as follows:

(8)$$\begin{array}{l}{S}_{\mathrm{2}}=-E_x^{\mathrm{2}}{\sin }^{\mathrm{2}}(\tau /\mathrm{2})\sin \mathrm{4}\phi ,\\ S_{\mathrm{3}}=E_x^{\mathrm{2}}\sin (\mathrm{2}\phi )\sin \tau .\end{array}$$

Here, it is noted that when $\phi ={\mathrm{45}}^{\circ }$, we can set $S_{\mathrm{2}}$ component to zero and modulate the $S_{\mathrm{3}}$ component independently. Then, we will get $S_{\mathrm{3}}=E_x^{\mathrm{2}}\sin \tau$. The experimental setup is shown in Fig. 2.

Fig.2 Position of electro-optic crystal

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Taking $\phi ={\mathrm{45}}^{\circ }$ into Eq. (8), the SOP of the signal after EOM can be derived as

(9)$${\mathit{E}}_{\mathrm{o}\mathrm{u}\mathrm{t}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{E}_{\mathrm{0}}\left(\begin{array}{c}\cos (\tau /\mathrm{2})\\ -\mathrm{i}\sin (\tau /\mathrm{2})\end{array}\right).$$

According to Eq. (9), when $\tau$ is small, the intensity of the horizontal component will be much larger than that of the vertical one. In our scheme, we use the stronger horizontal polarized component as the LO light, and use the vertical component as the signal light. The signal and the LO light were then transmitted along on a single spatial transmission mode.

After the EOM, $S_{\mathrm{2}}$ components can be modulated by a MOM to carry the signal.

The simulation shows that when half-wave plate (HWP) rotates on the azimuth, the $S_{\mathrm{2}}$ component of polar light will be changed such as the MOM. When modulating the signal, we consider the delay between the EOM and HWP. Thus the signal and LO are synchronized at the output of HWP. Since the Jones matrix of HWP is [12]

(10)${\mathit{E}}_{\mathrm{H}}=-\mathrm{i}\left(\begin{array}{ll}\cos (\mathrm{2}\varphi )& \sin (\mathrm{2}\varphi )\\ \sin (\mathrm{2}\varphi )& -\cos (\mathrm{2}\varphi )\end{array}\right),$

where $\varphi$ presents the included angle between the crystal axis and horizontal component of signal light, the optical signal after the MOM can be written as

(11)$${\mathit{E}}_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{E}_x\left(\begin{array}{l}-\mathrm{i}\cos (\mathrm{2}\varphi )\cos (\tau /\mathrm{2})-\sin (\mathrm{2}\varphi )\sin (\tau /\mathrm{2})\\ -\mathrm{i}\sin (\mathrm{2}\varphi )\cos (\tau /\mathrm{2})+\cos (\mathrm{2}\varphi )\sin (\tau /\mathrm{2})\end{array}\right).$$

The Stokes components of the output light are

(12)$$S_{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{E}_x^{\mathrm{2}}\cos \tau \sin (\mathrm{4}\varphi ),$$

(13)$$S_{\mathrm{3}}={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{E}_x^{\mathrm{2}}\sin \tau .$$

According to Eq. (12), when the value of $\tau$ is fixed, we can modulate the $S_{\mathrm{2}}$ by changing the angle $\varphi$ of MOM. Equation (13) indicates that the MOM will not affect the value of $S_{\mathrm{3}}$.

3 Experiments

3.1 Optical path design

Based on Section 2, we design a free space continuous variables coherent optical communication setup, which is shown in Fig. 3. At Alice end, a light beam at 808 nm was focused by a convex with a focus length of 8.5 mm and passes through an isolator. Then, the horizontal polarizer and two PBS ware used to convert this light into the horizontal polarization status ($$E_{\mathrm{1}}$$). After that, we used an EOM (New Focus 4102M) encodes the $$S_{\mathrm{3}}$$ component. Here, we rotated a HWP to control the $$S_{\mathrm{2}}$$ component, which simulates the function of a MOM.

At the Bob end, the measurement base is first selected by a Q-Q-H SU(2) convert box which composes one HWP and two quarter-wave plates (QWP). Then, a PBS splits the light into a horizontal polarized component ($$E_{\mathrm{2}}$$) and a vertical polarized component ($$E_{\mathrm{3}}$$) for detection. A balance homodyne detection circuit was used to detect the optical signal for further digital signal processing using LabVIEW [13].

Fig.3 Integrated optical path structure

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3.2 Stokes parameters detection system

At the detection end, the measurement base is selected by rotating the Q-Q-H wave plate. ${\widehat{a}}_x$, ${\widehat{a}}_y$ are set as the coherent polarization status of input light, and ${\widehat{c}}_x$, ${\widehat{c}}_y$ as the coherent polarization status of output light. The detecting optical path is shown in Fig. 4.

Fig.4 SU(2) convert box

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Since Stokes component follows the following:

(14)$\begin{array}{l}{S}_{\mathrm{2}}=\langle a_x^{†}{a}_y\rangle +\langle a_y^{†}{a}_x\rangle ,\\ S_{\mathrm{3}}=\mathrm{i}(\langle a_y^{†}{a}_x\rangle -\langle a_x^{†}{a}_y\rangle )\mathrm{.}\end{array}$

The homodyne detection light density is

(15)$$I=\langle c_y^{†}{c}_x\rangle -\langle c_x^{†}{c}_y\rangle .$$

The SU(2) convert box can be regarded as a Jones matrix. The output ${\mathit{E}}_{\mathrm{s}\mathrm{u}}$ is described as

(16)$${\mathit{E}}_{\mathrm{s}\mathrm{u}}={\mathit{E}}_{\mathrm{Q}}{\mathit{E}}_{\mathrm{Q}}{\mathit{E}}_{\mathrm{H}},$$

Where ${\mathit{E}}_{\mathrm{Q}}$ and ${\mathit{E}}_{\mathrm{H}}$ present the Jones matrixes of QWP and HWP, respectively. If the crystal axes of three wave plates set of particular angles different angles with the horizontal direction, respectively, Eq. (16) will satisfy the following equations:

(17)$$I=S_{\mathrm{2}},{\mathit{E}}_{\mathrm{s}\mathrm{u}}={\displaystyle \frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}\left(\begin{array}{ll}\mathrm{1}& \mathrm{1}\\ -\mathrm{1}& \mathrm{1}\end{array}\right),$$

(18)$$I=S_{\mathrm{3}},{\mathit{E}}_{\mathrm{s}\mathrm{u}}={\displaystyle \frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}\left(\begin{array}{ll}\mathrm{1}& \mathrm{i}\\ \mathrm{i}& \mathrm{1}\end{array}\right).$$

The angles between the crystal axis and horizontal direction of three wave plates Q-Q-H set as α, β, ϕ, respectively.

$$\alpha =\mathrm{0},\beta =\mathrm{0},\phi =-{\displaystyle \frac{\mathrm{3}}{\mathrm{8}}}\mathrm{\pi },\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}{S}_2\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t};$$

$$\alpha =\mathrm{0},\beta =-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}\mathrm{\pi },\varphi =-{\displaystyle \frac{\mathrm{1}}{\mathrm{8}}}\mathrm{\pi },\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}{S}_{\mathrm{3}}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}.$$

From selecting the measurement base, the balance homodyne detection is implemented [14,15].

In Eq. (13), $S_{\mathrm{3}}$ parameter is a sine function of $\tau$, here $\tau ={\displaystyle \frac{\mathrm{\pi }}{V_{\mathrm{\pi }}}}V$ and $V$ is the modulation voltage of EOM. The result of sine relationship between $S_{\mathrm{3}}$ and $V$ is shown in Fig. 5, where x-axis and y-axis present the drive voltage $V$ and $S_{\mathrm{3}}$ parameter, respectively. The modulation voltage $V$ of the electro-optical crystal is a 1.6 MHz sinusoidal oscillatory wave with a $V_{\mathrm{0}}$ direct current (DC) bias voltage. It can be written as

(19)$$V=V_{\mathrm{0}}+V_x\sin (\omega t),$$

where $\omega$ is the frequency of sine wave. $V_x$ is the input drive voltage of EOM.

According to Eqs. (13) and (19), the relationship between $$S_{\mathrm{3}}$$ and $V$ is described as follows:

(20)$S_{\mathrm{3}}={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{E}_x\sin {\displaystyle \frac{\mathrm{\pi }}{V_{\mathrm{\pi }}}}[V_{\mathrm{0}}+V_x\sin (\omega t)].$

When the change of $V_x$ is fairly small:

(21)$$S_{\mathrm{3}}\approx {\displaystyle \frac{k}{\mathrm{4}}}{E}_x{\displaystyle \frac{\mathrm{\pi }}{V_{\mathrm{\pi }}}}[V_{\mathrm{0}}+V_x\sin (\omega t)].$$

Since ${\displaystyle \frac{k}{\mathrm{4}}}{E}_x{\displaystyle \frac{\mathrm{\pi }}{V_{\mathrm{\pi }}}}$ is a constant, let $M={\displaystyle \frac{k}{\mathrm{4}}}{E}_x{\displaystyle \frac{\mathrm{\pi }}{V_{\mathrm{\pi }}}}$, Eq. (21) is simplified as follows:

(22)$$S_{\mathrm{3}}\approx M[V_{\mathrm{0}}+V_x\sin (\omega t)]=M{V}_{\mathrm{0}}+M{V}_x\sin (\omega t).$$

When the detection signal passes through the bandpass filter at the end of the detection circuit, the DC signal is filtered, and then

(23)$$S_{\mathrm{3}}-M{V}_{\mathrm{0}}=M{V}_x\sin (\omega t).$$

4 Results and discussion

When the random selection measurement base is $S_{\mathrm{2}}$, the relative between the horizontal azimuth $\varphi$ of HWP and the value of $S_{\mathrm{2}}$ component can be described as $S_{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{E}_x^{\mathrm{2}}\cos \tau \sin \mathrm{4}\varphi$. The corresponding measurement results are shown in Fig. 6.

Fig.5 Relationship between EOM voltage and $$S_{\mathrm{3}}$$ parameter

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As discussed in Section 3.2, in Eq. (23), we note that the $V_x$ is not linearly relative to the $S_{\mathrm{3}}$. Thus, we carefully choose a linear operation point of EOM according to the relationship as shown in Fig. 5. To find the linear operation point, we measured the relationship between the detective voltage of $S_{\mathrm{3}}$ and the driving voltage of EOM. According to the measurement, we set the operation point at 2 V.

Fig.6 Relationship between HWP $\varphi$ and $S_{\mathrm{2}}$

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Then, we modulate the EOM by using a random code at a bit rate of 1.6 MHz which has 16 equal parts and follows the Gaussian distribution. The length of the random code is $${\mathrm{10}}^{\mathrm{4}}$$ bits. At the receiver, the signal is linearly demodulated. Figure 7 is the probability distributions of the demodulation signal at different part, which follow the Gaussian distribution.

Fig.7 Gaussian distribution of $S_{\mathrm{3}}$ demodulation signal

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Fig.8 Error code analysis of $S_{\mathrm{3}}$ acquisition signal. (a) Coding value; (b) decoding value

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Figure 8 shows the input and the output results. We also measure the bit error rate using the array indexing tool in LabVIEW. For $${\mathrm{10}}^{\mathrm{4}}$$ samples, the sending-to-receiving bit error rate is measured and lower than ${\mathrm{10}}^{-\mathrm{4}}$, which indicates that if a linearly modulated operation point can be properly choose in the EOM, the $S_{\mathrm{3}}$ component can be demodulated in our proposed receiver setup.

At the detection end, if we randomly rotate the SU(2) convert box to modulate the signal and singal has quantum properties, then the commutators of $S_{\mathrm{2}}$ and $S_{\mathrm{3}}$ components follow the uncertainty relations [16]:

(24)$$\mathrm{\Delta }{\widehat{S}}_{\mathrm{2}}*\mathrm{\Delta }{\widehat{S}}_{\mathrm{3}}\ge \left|\langle {\widehat{S}}_{\mathrm{1}}\rangle \right|.$$

Equation (24) indicates if one Stokes operator is nonzero, the other two Stokes operators cannot be simultaneously measured with certainty because Heisenberg uncertainty principle and quantum no-cloning theorem. According to the Heisenberg uncertainty principle, the quantum broadening effect makes a potential eavesdropper cannot accurately measure the values of $S_{\mathrm{2}}$ and $S_{\mathrm{3}}$ components. Thus the security of the communication is guaranteed. However, we can still select two groups of components (${\widehat{S}}_{\mathrm{2}}$ and $${\widehat{S}}_{\mathrm{3}}$$) to carry the signal by using the EOM and MOM. Thus in the receiver, we can decode the information by using the relationship of $S_{\mathrm{2}}=\langle {\widehat{S}}_{\mathrm{2}}\rangle$ and $S_{\mathrm{3}}=\langle {\widehat{S}}_{\mathrm{3}}\rangle$.

5 Conclusions

In this paper, we proposed a scheme using Stokes parameter coding for a spatial continuous variables coherent optical communication scheme to stabilize optical coupling and phase synchronization. In the receiver, we also demonstrated that an SU(2) convert box can be used to achieve the base selection of the $S_{\mathrm{2}}$ or $S_{\mathrm{3}}$ components. Our experimental results also indicate if the $S_{\mathrm{3}}$ component can be linearly modulated by properly choosing the operation point of EOM, the Stokes parameters can be demodulated in our proposed receiver setup. This scheme resolves the problem that the coherence is hard to be guaranteed in the M-Z interferometer optical path, and it has great advantages in continuous variable coherent optical communication.

References

[1] Wang T, Klarskov P, Jepsen P U. Ultrabroadband THz time-domain spectroscopy of a free-flowing water film. IEEE Transactions on Terahertz Science and Technology, 2014, 4(4): 425–431

[2] Lee Y S. Principles of Terahertz Science and Technology. Vol. 170. New York: Springer US, 2009

[3] Mittleman D M. Twenty years of terahertz imaging. Optics Express, 2018, 26(8): 9417–9431

[4] Zhao J, e Y, Williams K, Zhang X C, Boyd R W. Spatial sampling of terahertz fields with sub-wavelength accuracy via probe-beam encoding. Light, Science & Applications, 2019, 8(1): 55

[5] Look D C. Molecular beam epitaxial GaAs grown at low temperatures. Thin Solid Films, 1993, 231(1–2): 61–73

[6] Beard M C, Turner G M, Schmuttenmaer C A. Subpicosecond carrier dynamics in low-temperature grown GaAs as measured by time-resolved terahertz spectroscopy. Journal of Applied Physics, 2001, 90(12): 5915–5923

[7] Boyd R W.Nonlinear Optics. 2nd ed. New York: Academic Press, 2003

[8] Hebling J, Almasi G, Kozma I, Kuhl J. Velocity matching by pulse front tilting for large area THz-pulse generation. Optics Express, 2002, 10(21): 1161–1166

[9] Hebling J, Yeh K L, Hoffmann M C, Bartal B, Nelson K A. Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities. Journal of the Optical Society of America B, Optical Physics, 2008, 25(7): B6–B19

[10] Fülöp J A, Pálfalvi L, Klingebiel S, Almási G, Krausz F, Karsch S, Hebling J. Generation of sub-mJ terahertz pulses by optical rectification. Optics Letters, 2012, 37(4): 557–559

[11] Zhang X C, Ma X, Jin Y, Lu T M, Boden E P, Phelps P D, Stewart K R, Yakymyshyn C P. Terahertz optical rectification from a nonlinear organic crystal. Applied Physics Letters, 1992, 61(26): 3080–3082

[12] Hauri C P, Ruchert C, Vicario C, Ardana F. Strong-field single-cycle THz pulses generated in an organic crystal. Applied Physics Letters, 2011, 99(16): 161116

[13] Shalaby M, Hauri C P. Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness. Nature Communications, 2015, 6(1): 5976

[14] Fülöp J A, Tzortzakis S, Kampfrath T. Laser-driven strong-field terahertz sources. Advanced Optical Materials, 2020, 8(3): 1900681

[15] Hamster H, Sullivan A, Gordon S, White W, Falcone R W. Subpicosecond, electromagnetic pulses from intense laser-plasma interaction. Physical Review Letters, 1993, 71(17): 2725–2728

[16] Hamster H, Sullivan A, Gordon S, Falcone R W. Short-pulse terahertz radiation from high-intensity-laser-produced plasmas. Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 1994, 49(1): 671–677

[17] Cook D J, Hochstrasser R M. Intense terahertz pulses by four-wave rectification in air. Optics Letters, 2000, 25(16): 1210–1212

[18] Johnson K, Price-Gallagher M, Mamer O, Lesimple A, Fletcher C, Chen Y, Lu X, Yamaguchi M, Zhang X C. Water vapor: an extraordinary terahertz wave source under optical excitation. Physics Letters, 2008, 372(38): 6037–6040 (Part A)

[19] Xie X, Dai J, Zhang X C. Coherent control of THz wave generation in ambient air. Physical Review Letters, 2006, 96(7): 075005

[20] Kim K Y, Glownia J H, Taylor A J, Rodriguez G. Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields. Optics Express, 2007, 15(8): 4577–4584

[21] Kim K Y, Taylor A, Glownia J, Rodriguez G. Coherent control of terahertz supercontinuum generation in ultrafast laser–gas interactions. Nature Photonics, 2008, 2(10): 605–609

[22] Kim K Y. Generation of coherent terahertz radiation in ultrafast laser-gas interactions. Physics of Plasmas, 2009, 16(5): 056706

[23] Karpowicz N, Zhang X C. Coherent terahertz echo of tunnel ionization in gases. Physical Review Letters, 2009, 102(9): 093001

[24] Ronne C, Thrane L, Åstrand P O, Wallqvist A, Mikkelsen K V, Keiding S R. Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation. Journal of Chemical Physics, 1997, 107(14): 5319–5331

[25] Thrane L, Jacobsen R H, Uhd Jepsen P, Keiding S R. THz reflection spectroscopy of liquid water. Chemical Physics Letters, 1995, 240(4): 330–333

[26] Kotz J C, Treichel P M, Townsend J. Chemistry and Chemical Reactivity. Raleigh, NC: Cengage Learning, 2012

[27] Engels D, Schmid-Burgk J, Walmsley C. Water maser emission from OH/IR stars. Astronomy & Astrophysics, 1986, 167: 129–144

[28] Neufeld D A, Melnick G J. Excitation of millimeter and submillimeter water masers. Astrophysical Journal, 1991, 368: 215–230

[29] Neufeld D A, Maloney P R, Conger S. Water maser emission from X-ray-heated circumnuclear gas in active galaxies. Astrophysical Journal, 1994, 436: 127–130

[30] Alfano R R, Shapiro S. Observation of self-phase modulation and small-scale filaments in crystals and glasses. Physical Review Letters, 1970, 24(11): 592–594

[31] Jimbo T, Caplan V L, Li Q X, Wang Q Z, Ho P P, Alfano R R. Enhancement of ultrafast supercontinuum generation in water by the addition of Zn²⁺ and K⁺ cations. Optics Letters, 1987, 12(7): 477–479

[32] Kandidov V, Kosareva O, Golubtsov I, Liu W, Becker A, Akozbek N, Bowden C M, Chin S L. Self-transformation of a powerful femtosecond laser pulse into a white-light laser pulse in bulk optical media (or supercontinuum generation). Applied Physics B, Lasers and Optics, 2003, 77(2–3): 149–165

[33] Liu W, Petit S, Becker A, Aközbek N, Bowden C M, Chin S L. Intensity clamping of a femtosecond laser pulse in condensed matter. Optics Communications, 2002, 202(1–3): 189–197

[34] Dharmadhikari A, Rajgara F, Mathur D. Systematic study of highly efficient white light generation in transparent materials using intense femtosecond laser pulses. Applied Physics B, Lasers and Optics, 2005, 80(1): 61–66

[35] Kaya N, Strohaber J, Kolomenskii A A, Kaya G, Schroeder H, Schuessler H A. White-light generation using spatially-structured beams of femtosecond radiation. Optics Express, 2012, 20(12): 13337–13346

[36] Dharmadhikari J A, Steinmeyer G, Gopakumar G, Mathur D, Dharmadhikari A K. Femtosecond supercontinuum generation in water in the vicinity of absorption bands. Optics Letters, 2016, 41(15): 3475–3478

[37] Attwood D, Sakdinawat A. X-rays and Extreme Ultraviolet Radiation: Principles and Applications. Cambridge: Cambridge University Press, 2017

[38] McNaught S, Fan J, Parra E, Milchberg H M. A pump–probe investigation of laser-droplet plasma dynamics. Applied Physics Letters, 2001, 79(25): 4100–4102

[39] Düsterer S, Schwoerer H, Ziegler W, Ziener C, Sauerbrey R. Optimization of EUV radiation yield from laser-produced plasma. Applied Physics B, Lasers and Optics, 2001, 73(7): 693–698 doi:10.1007/s003400100730

[40] Kurz H G, Steingrube D S, Ristau D, Lein M, Morgner U, Kovačev M. High-order-harmonic generation from dense water microdroplets. Physical Review A, 2013, 87(6): 063811 doi:10.1103/PhysRevA.87.063811

[41] Flettner A, Pfeifer T, Walter D, Winterfeldt C, Spielmann C, Gerber G. High-harmonic generation and plasma radiation from water microdroplets. Applied Physics B, Lasers and Optics, 2003, 77(8): 747–751

[42] Donnelly T D, Rust M, Weiner I, Allen M, Smith R A, Steinke C A, Wilks S, Zweiback J, Cowan T E, Ditmire T. Hard X-ray and hot electron production from intense laser irradiation of wavelength-scale particles. Journal of Physics B, Atomic, Molecular, and Optical Physics, 2001, 34(10): L313–L320

[43] Malmqvist L, Rymell L, Hertz H M. Droplet‐target laser‐plasma source for proximity X‐ray lithography. Applied Physics Letters, 1996, 68(19): 2627–2629

[44] Berglund M, Rymell L, Hertz H M. Ultraviolet prepulse for enhanced X‐ray emission and brightness from droplet‐target laser plasmas. Applied Physics Letters, 1996, 69(12): 1683–1685

[45] Rymell L, Hertz H M. Droplet target for low-debris laser-plasma soft X-ray generation. Optics Communications, 1993, 103(1–2): 105–110

[46] Nikogosyan D N, Oraevsky A A, Rupasov V I. Two-photon ionization and dissociation of liquid water by powerful laser UV radiation. Chemical Physics, 1983, 77(1): 131–143

[47] Crowell R A, Bartels D M. Multiphoton ionization of liquid water with 3.0-5.0 eV photons. Journal of Physical Chemistry, 1996, 100(45): 17940–17949

[48] Kennedy P K. A first-order model for computation of laser-induced breakdown thresholds in ocular and aqueous media. I. Theory. IEEE Journal of Quantum Electronics, 1995, 31(12): 2241–2249

[49] Kennedy P K, Hammer D X, Rockwell B A. Laser-induced breakdown in aqueous media. Progress in Quantum Electronics, 1997, 21(3): 155–248

[50] Hirori H, Doi A, Blanchard F, Tanaka K. Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO₃. Applied Physics Letters, 2011, 98(9): 091106

[51] Blanchard F, Razzari L, Bandulet H C, Sharma G, Morandotti R, Kieffer J C, Ozaki T, Reid M, Tiedje H F, Haugen H K, Hegmann F A. Generation of 1.5 μJ single-cycle terahertz pulses by optical rectification from a large aperture ZnTe crystal. Optics Express, 2007, 15(20): 13212–13220

[52] Wu Q, Zhang X C. Free‐space electro‐optic sampling of terahertz beams. Applied Physics Letters, 1995, 67(24): 3523–3525

[53] Jin Q, E Y, Williams K, Dai J, Zhang X C. Observation of broadband terahertz wave generation from liquid water. Applied Physics Letters, 2017, 111(7): 071103

[54] Zhang L L, Wang W M, Wu T, Feng S J, Kang K, Zhang C L, Zhang Y, Li Y T, Sheng Z M, Zhang X C. Strong terahertz radiation from a liquid-water line. Physical Review Applied, 2019, 12(1): 014005

[55] Wang W M, Gibbon P, Sheng Z M, Li Y T. Integrated simulation approach for laser-driven fast ignition. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2015, 91(1): 013101

[56] Buccheri F, Zhang X C. Terahertz emission from laser-induced microplasma in ambient air. Optica, 2015, 2(4): 366–369

[57] Zhang J Z, Lam J K, Wood C F, Chu B T, Chang R K. Explosive vaporization of a large transparent droplet irradiated by a high intensity laser. Applied Optics, 1987, 26(22): 4731–4737

[58] Schaffer C, Nishimura N, Glezer E, Kim A, Mazur E. Dynamics of femtosecond laser-induced breakdown in water from femtoseconds to microseconds. Optics Express, 2002, 10(3): 196–203

[59] Courvoisier F, Boutou V, Favre C, Hill S C, Wolf J P. Plasma formation dynamics within a water microdroplet on femtosecond time scales. Optics Letters, 2003, 28(3): 206–208

[60] Lindinger A, Hagen J, Socaciu L D, Bernhardt T M, Wöste L, Duft D, Leisner T. Time-resolved explosion dynamics of H₂O droplets induced by femtosecond laser pulses. Applied Optics, 2004, 43(27): 5263–5269

[61] Stan C A, Milathianaki D, Laksmono H, Sierra R G, McQueen T A, Messerschmidt M, Williams G J, Koglin J E, Lane T J, Hayes M J, Guillet S A H, Liang M, Aquila A L, Willmott P R, Robinson J S, Gumerlock K L, Botha S, Nass K, Schlichting I, Shoeman R L, Stone H A, Boutet S. Liquid explosions induced by X-ray laser pulses. Nature Physics, 2016, 12(10): 966–971

[62] E Y, Jin Q, Tcypkin A, Zhang X C. Terahertz wave generation from liquid water films via laser-induced breakdown. Applied Physics Letters, 2018, 113(18): 181103

[63] Bebb H B, Gold A. Multiphoton ionization of hydrogen and rare-gas atoms. Physical Review, 1966, 143(1): 1–24

[64] DeMichelis C. Laser induced gas breakdown: a bibliographical review. IEEE Journal of Quantum Electronics, 1969, 5(4): 188–202

[65] Shen Y R. The Principles of Nonlinear Optics. New York: Wiley, 1984

[66] Lambropoulos P. Mechanisms for multiple ionization of atoms by strong pulsed lasers. Physical Review Letters, 1985, 55(20): 2141–2144

[67] Perry M D, Landen O L, Szöke A, Campbell E M. Multiphoton ionization of the noble gases by an intense 1014-W/cm2 dye laser. Physical Review A: General Physics, 1988, 37(3): 747–760

[68] Keldysh L. Ionization in the field of a strong electromagnetic wave. Soviet Physics, JETP, 1965, 20(5): 1307–1314

[69] Ammosov M V. Tunnel ionization of complex atoms and of atomic ions in an altemating electromagnetic field. Soviet Physics, JETP, 1987, 64: 1191

[70] Bass M, Barrett H. Avalanche breakdown and the probabilistic nature of laser-induced damage. IEEE Journal of Quantum Electronics, 1972, 8(3): 338–343

[71] Bloembergen N. Laser-induced electric breakdown in solids. IEEE Journal of Quantum Electronics, 1974, 10(3): 375–386

[72] Morgan C G. Laser-induced breakdown of gases. Reports on Progress in Physics, 1975, 38(5): 621–665

[73] Puliafito C, Steinert R. Short-pulsed Nd:YAG laser microsurgery of the eye: biophysical considerations. IEEE Journal of Quantum Electronics, 1984, 20(12): 1442–1448

[74] Noack J, Vogel A. Laser-induced plasma formation in water at nanosecond to femtosecond time scales: calculation of thresholds, absorption coefficients, and energy density. IEEE Journal of Quantum Electronics, 1999, 35(8): 1156–1167

[75] Williams F, Varma S, Hillenius S. Liquid water as a lone‐pair amorphous semiconductor. Journal of Chemical Physics, 1976, 64(4): 1549–1554

[76] Sacchi C. Laser-induced electric breakdown in water. Journal of the Optical Society of America B, Optical Physics, 1991, 8(2): 337–345

[77] Feng Q, Moloney J V, Newell A C, Wright E M, Cook K, Kennedy P K, Hammer D X, Rockwell B A, Thompson C R. Theory and simulation on the threshold of water breakdown induced by focused ultrashort laser pulses. IEEE Journal of Quantum Electronics, 1997, 33(2): 127–137

[78] Raĭzer Y P. Reviews of topical problems: breakdown and heating of gases under the influence of a laser beam. Soviet Physics Uspekhi, 1966, 8(5): 650–673

[79] Hatanaka K, Ida T, Ono H, Matsushima S, Fukumura H, Juodkazis S, Misawa H. Chirp effect in hard X-ray generation from liquid target when irradiated by femtosecond pulses. Optics Express, 2008, 16(17): 12650–12657

[80] Dai J, Liu J, Zhang X C. Terahertz wave air photonics: terahertz wave generation and detection with laser-induced gas plasma. IEEE Journal of Selected Topics in Quantum Electronics, 2011, 17(1): 183–190

[81] Kreß M, Löffler T, Thomson M D, Dörner R, Gimpel H, Zrost K, Ergler T, Moshammer R, Morgner U, Ullrich J, Roskos H G. Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy. Nature Physics, 2006, 2(5): 327–331

[82] Gaal P, Kuehn W, Reimann K, Woerner M, Elsaesser T, Hey R. Internal motions of a quasiparticle governing its ultrafast nonlinear response. Nature, 2007, 450(7173): 1210–1213

[83] Roskos H, Thomson M, Kreß M, Löffler T. Broadband THz emission from gas plasmas induced by femtosecond optical pulses: from fundamentals to applications. Laser & Photonics Reviews, 2007, 1(4): 349–368

[84] Oh T, Yoo Y, You Y, Kim K Y. Generation of strong terahertz fields exceeding 8 MV/cm at 1 kHz and real-time beam profiling. Applied Physics Letters, 2014, 105(4): 041103

[85] Thomson M D, Blank V, Roskos H G. Terahertz white-light pulses from an air plasma photo-induced by incommensurate two-color optical fields. Optics Express, 2010, 18(22): 23173–23182

[86] Zhang X C, Shkurinov A, Zhang Y. Extreme terahertz science. Nature Photonics, 2017, 11(1): 16–18

[87] Dai J, Karpowicz N, Zhang X C. Coherent polarization control of terahertz waves generated from two-color laser-induced gas plasma. Physical Review Letters, 2009, 103(2): 023001

[88] Wen H, Lindenberg A M. Coherent terahertz polarization control through manipulation of electron trajectories. Physical Review Letters, 2009, 103(2): 023902

[89] Dai J, Zhang X C. Terahertz wave generation from thin metal films excited by asymmetrical optical fields. Optics Letters, 2014, 39(4): 777–780

[90] Dey I, Jana K, Fedorov V Y, Koulouklidis A D, Mondal A, Shaikh M, Sarkar D, Lad A D, Tzortzakis S, Couairon A, Kumar G R. Highly efficient broadband terahertz generation from ultrashort laser filamentation in liquids. Nature Communications, 2017, 8(1): 1184

[91] Shen Y, Watanabe T, Arena D A, Kao C C, Murphy J B, Tsang T Y, Wang X J, Carr G L. Nonlinear cross-phase modulation with intense single-cycle terahertz pulses. Physical Review Letters, 2007, 99(4): 043901

[92] Turchinovich D, Hvam J M, Hoffmann M C. Self-phase modulation of a single-cycle terahertz pulse by nonlinear free-carrier response in a semiconductor. Physical Review B, 2012, 85(20): 201304

[93] Nanni E A, Huang W R, Hong K H, Ravi K, Fallahi A, Moriena G, Dwayne Miller R J, Kärtner F X. Terahertz-driven linear electron acceleration. Nature Communications, 2015, 6(1): 8486

[94] Zhang D, Fallahi A, Hemmer M, Wu X, Fakhari M, Hua Y, Cankaya H, Calendron A L, Zapata L E, Matlis N H, Kärtner F X. Segmented terahertz electron accelerator and manipulator (STEAM). Nature Photonics, 2018, 12(6): 336–342

[95] Jin Q, Dai J, E Y, Zhang X C. Terahertz wave emission from a liquid water film under the excitation of asymmetric optical fields. Applied Physics Letters, 2018, 113(26): 261101

[96] Kiran P P, Bagchi S, Krishnan S R, Arnold C L, Kumar G R, Couairon A. Focal dynamics of multiple filaments: Microscopic imaging and reconstruction. Physical Review A., 2010, 82(1): 013805

[97] Liu X L, Lu X, Liu X, Xi T T, Liu F, Ma J L, Zhang J. Tightly focused femtosecond laser pulse in air: from filamentation to breakdown. Optics Express, 2010, 18(25): 26007–26017

[98] Jin Q, E Y, Gao S, Zhang X C. Preference of subpicosecond laser pulses for terahertz wave generation from liquids. Advanced Photonics, 2020, 2(1): 015001

[99] Chin S L. Femtosecond Laser Filamentation. Vol. 55. New York: Springer US, 2010

[100] Docchio F. Lifetimes of plasmas induced in liquids and ocular media by single Nd:YAG laser pulses of different duration. EPL, 1988, 6(5): 407–412 (Europhysics Letters)

[101] Feng Q, Wright E M, Moloney J V, Newell A C. Laser-induced breakdown versus self-focusing for focused picosecond pulses in water. Optics Letters, 1995, 20(19): 1958–1960

[102] Dai J, Zhang X C. Terahertz wave generation from gas plasma using a phase compensator with attosecond phase-control accuracy. Applied Physics Letters, 2009, 94(2): 021117

[103] Dorney T D, Baraniuk R G, Mittleman D M. Material parameter estimation with terahertz time-domain spectroscopy. Journal of the Optical Society of America A, Optics, Image Science, and Vision, 2001, 18(7): 1562–1571

[104] Babushkin I, Kuehn W, Köhler C, Skupin S, Bergé L, Reimann K, Woerner M, Herrmann J, Elsaesser T. Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases. Physical Review Letters, 2010, 105(5): 053903

[105] Bergé L, Skupin S, Köhler C, Babushkin I, Herrmann J. 3D numerical simulations of THz generation by two-color laser filaments. Physical Review Letters, 2013, 110(7): 073901

[106] Sprangle P, Peñano J R, Hafizi B, Kapetanakos C A. Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 2004, 69(6): 066415

[107] Ponomareva E A, Stumpf S A, Tcypkin A N, Kozlov S A. Impact of laser-ionized liquid nonlinear characteristics on the efficiency of terahertz wave generation. Optics Letters, 2019, 44(22): 5485–5488

[108] Tcypkin A N, Ponomareva E A, Putilin S E, Smirnov S V, Shtumpf S A, Melnik M V, E Y, Kozlov S A, Zhang X C. Flat liquid jet as a highly efficient source of terahertz radiation. Optics Express, 2019, 27(11): 15485–15494

[109] E Y, Jin Q, Zhang X C. Enhancement of terahertz emission by a preformed plasma in liquid water. Applied Physics Letters, 2019, 115(10): 101101 doi:10.1063/1.5119812

[110] Ponomareva E A, Tcypkin A N, Smirnov S V, Putilin S E, Yiwen E, Kozlov S A, Zhang X C. Double-pump technique-one step closer towards efficient liquid-based THz sources. Optics Express, 2019, 27(22): 32855–32862

[111] Huang H H, Nagashima T, Hsu W H, Juodkazis S, Hatanaka K. Dual THz wave and X-ray generation from a water film under femtosecond laser excitation. Nanomaterials (Basel, Switzerland), 2018, 8(7): 523

[112] Huang H H, Nagashima T, Yonezawa T, Matsuo Y , Ng S H, Juodkazis S, Hatanaka K. Giant enhancement of THz wave emission under double-pulse excitation of thin water flow. Applied Sciences (Basel, Switzerland), 2020, 10(6): 2031

Acknowledgements

This research was sponsored by the Army Research Office under Grant No. W911NF-17-1-0428, Air Force Office of Scientific Research under Grant No. FA9550-18-1-0357, and National Science Foundation under Grant No. ECCS-1916068.

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