A direct method to calculate second-order two-dimensional terahertz spectroscopy in frequency-domain based on classical theory

Abstract

Previous theoretical researches on the two-dimensional terahertz spectroscopy (2DTS), which are conducted via inefficiently time-consuming numerical simulation, deal with only single-mode system. To overcome the limitations, we derive a classical-theory-based analytical solution which is applicable to multi-modes system. Three typical weak sources of nonlinearities are introduced. The findings suggest that the analytical results correspond well with those obtained by the traditional numerical simulation. Thus the study provides a more efficient and practical method to directly calculate 2DTS, and, in a broader sense, sheds new light on the theory of 2DTS.

Key words： two-dimensional spectroscopy; terahertz; classical method

Introduction

Two-dimensional (2D) spectroscopy is an important tool to study the dynamics of vibrational excitation of matter [¹]. It provides richer information than one-dimensional (1D) spectroscopy by which it is difficult to obtain the nonlinear information of materials. In 1D time-resolved spectroscopy, the signal emitted from the interaction of a single light pulse with materials is measured in time-domain, and the sample’s intrinsic absorption spectrum can be acquired via Fourier transform (FT). If another pulse with an interval T to the former one is also incident on the sample, a nonlinear signal is emitted, which is determined as a function of detection time t and interval T. If we scan t and T, the 2D-FT of the signal yields two frequency dimensions, detection frequency {\omega }_t and excitation frequency {\omega }_T [²], thus the spectroscopy is a 2D one.

2D spectroscopy has been widely applied in infrared region to study inhomogeneous distribution of resonance frequencies, anharmonicity of intra- and intermolecular vibrational modes, and the coupling between different modes, among other things [¹,³–⁹]. Because of the advent of intense terahertz (THz) pulse sources [¹⁰–¹²], two-dimensional terahertz spectroscopy (2DTS) is now available, revealing more nonlinear characteristics of materials in THz region [¹,³,⁴,¹³–¹⁵].

The physical model of 2DTS can be briefly explained as follows. As is shown in Fig. 1, two successive light pulses E_{\mathrm{1}} and E_{\mathrm{2}} are collinearly incident on samples at time t=-T and t=\mathrm{0} respectively, thus inducing second-order nonlinear polarization P_{\mathrm{NL}}(t,T) which in turn leads to a nonlinear electric field E_{\mathrm{NL}}(t,T). Using 2D-FT in t and T dimensions, we can acquire the corresponding spectrum in frequency-domain as a function of detection frequency {\omega }_t and excitation frequency {\omega }_T.

In experiments, the interval T between E_{\mathrm{1}} and E_{\mathrm{2}} can be easily controlled by an optical delay line. For each certain interval T, the nonlinear signal E_{\mathrm{NL}} is measured as a function of real time t, thus the measurements for different T yield a 2D function E_{\mathrm{NL}}(t,T). After 2D-FT, the experimental 2DTS is obtained [⁴,¹⁴,¹⁵].

Fig.1 Timing schematic diagram of 2DTS

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To distinguish the contributions from different sources of nonlinearities, one needs to perform theoretical calculation to fit the measured results. In THz region, the THz photon energy \hslash \omega is so low that (k_B is the Boltzmann constant and T_s is the temperature of the sample). In another word, 2DTS system has a classical nature. Therefore, current theoretical analysis on 2DTS has been majorly based on Lorentz model in the framework of classical theory [¹,¹⁶–¹⁸]. Despite their significance, these researches are narrow-ranged as they deal with only single-mode systems, and the derivation results are given as nonlinear polarization P_{\mathrm{NL}}(t,T) in terms of multiple integral. Specifically, there are three limitations: first, the multiple integral is time consuming; secondly, numerical 2D-FT needs to be performed to transform the results into frequency-domain; and thirdly, excessive effort is required to obtain the nonlinear electric field.

To overcome the limitations, we propose a classical theory based analytical method which not only can be used to calculate 2DTS directly and efficiently, but also is applicable to multi-mode system.

Theoretical derivation

The derivation of our analytical solution is explained as follows.

Assuming E_{\mathrm{1}} and E_{\mathrm{2}} are parallel, the total incident field is the sum of two pulses:

(1)E(t,T)=E_{\mathrm{2}}(t)+E_{\mathrm{1}}(t+T).

The system is regarded as summation of a series of oscillators described by Lorentz model. For a N-mode system, the linear equation of motion of the ith vibrational mode is

(2){\displaystyle \frac{d^{\mathrm{2}}{r}_i}{d{t}^{\mathrm{2}}}}+{\gamma }_i{\displaystyle \frac{d{r}_i}{dt}}+{\omega }_i^{\mathrm{2}}{r}_i={\displaystyle \frac{\mathrm{1}}{m_i}}{q}_{i}E(t,T),

where r_i, {\gamma }_i, m_i, {\omega }_i and q_i are respectively the vibrational coordinate, damping constant, mass, resonance frequency and effective charge of the ith mode. The solution of this equation is

(3)r_i(t,T)=q_i{S}_i(t)\ast E(t,T),

where S_i(t) is

(4)S_i(t)={\displaystyle \frac{\mathrm{1}}{m_i{\epsilon }_i}}{e}^{-\frac{\mathrm{1}}{\mathrm{2}}{\gamma }_{i}t}\sin ({\epsilon }_{i}t),\hspace{1em}{\epsilon }_i=\sqrt{{\omega }_i^{\mathrm{2}}-{\gamma }_i^{\mathrm{2}}/\mathrm{4}},\hspace{1em}(t>\mathrm{0}),

and \ast is convolution in t dimension. The FT of S_i(t) is

(5){\tilde{S}}_i({\omega }_t)={\displaystyle \frac{\mathrm{1}}{m_i}}{\displaystyle \frac{\mathrm{1}}{{\omega }_i^{\mathrm{2}}-j{\gamma }_i{\omega }_t-{\omega }_t^{\mathrm{2}}}},

where j is the imaginary unit.

The induced polarization can be expressed as

(6)P(t)=\rho {\displaystyle \sum _{i=\mathrm{1}}^N{p}_i}=\rho {\displaystyle \sum _{i=\mathrm{1}}^N{q}_i{r}_i},

where \rho is the density of the molecule and p_i is the effective dipole moment of the ith mode.

According to the perturbation method, r_i and P are expanded in the order of E as

(7)r_i=r_i^{(\mathrm{1})}+r_i^{(\mathrm{2})},\hspace{0.17em}\hspace{0.17em}P=P^{(\mathrm{1})}+P^{(\mathrm{2})},\hspace{0.17em}\hspace{0.17em}E=E^{(\mathrm{1})}+E^{(\mathrm{2})},

where r_i^{(\mathrm{1})} and P^{(\mathrm{1})} are the linear terms, while r_i^{(\mathrm{2})} and P^{(\mathrm{2})} are the second-order nonlinear terms, and r_i^{(n)},\hspace{0.17em}\hspace{0.17em}{P}^{(n)}\propto {(E^{(\mathrm{1})})}^n. The electric fields are determined by

(8)\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial z^{\mathrm{2}}}}+{\mu }_{\mathrm{0}}{\omega }_t^{\mathrm{2}}\left({\epsilon }_{\mathrm{0}}+\rho {\displaystyle \sum _{i=\mathrm{1}}^N{q}_i{\tilde{S}}_i({\omega }_t)}\right)\right]{\tilde{E}}^{(\mathrm{1})}(z,{\omega }_t)=\mathrm{0},

(9)\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial z^{\mathrm{2}}}}+{\mu }_{\mathrm{0}}{\omega }_t^{\mathrm{2}}\left({\epsilon }_{\mathrm{0}}+\rho {\displaystyle \sum _{i=\mathrm{1}}^N{q}_i{\tilde{S}}_i({\omega }_t)}\right)\right]{\tilde{E}}^{(\mathrm{2})}(z,{\omega }_t,{\omega }_T)=-{\mu }_{\mathrm{0}}{\omega }_t^{\mathrm{2}}{\tilde{P}}^{(\mathrm{2})}(z,{\omega }_t,{\omega }_T),

where {\epsilon }_{\mathrm{0}}, {\mu }_{\mathrm{0}} and z are respectively the permittivity of vacuum, permeability of vacuum and propagation distance in the medium.

In general, there are three typical second-order nonlinearities: anharmonicity (AH), nonlinear coupling (NC), and nonlinear damping (ND) in 2DTS [¹⁷]. AH corresponds to the nonlinearity rising from vibrational coordinate dependence of the resonance frequency.

Mathematically, AH is introduced by the addition of a third-order term to the harmonic potential function of the vibrational mode as

(10)V(r)={\displaystyle \sum _{i=\mathrm{1}}^N{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{m}_i{\omega }_i^{\mathrm{2}}{r}_i^{\mathrm{2}}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}!}}{\displaystyle \sum _{i,j,k=\mathrm{1}}^N{B}_{ijk}{r}_i{r}_j{r}_k},

where B_{ijk}={\partial }^{\mathrm{3}}V(r)/\partial r_i\partial r_j\partial r_k is the coupling parameter. The r_i{r}_j term does not exist since it can be removed by simply changing the set of modes [¹⁹]. Thus, the {\omega }_i^{\mathrm{2}}{r}_i term in Eq. (2) becomes

(11){\omega }_i^{\mathrm{2}}{r}_i\to {\displaystyle \frac{\mathrm{1}}{m_i}}{\displaystyle \frac{\partial }{\partial r_i}}V={\omega }_i^{\mathrm{2}}{r}_i+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}{m}_i}}{\displaystyle \sum _{j,k=\mathrm{1}}^N{B}_{ijk}{r}_j{r}_k}.

By selecting terms proportional to the same order of E, we obtain the equations for them as

(12){\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}{r}_i^{(\mathrm{1})}}{\mathrm{d}{t}^{\mathrm{2}}}}+{\gamma }_i{\displaystyle \frac{\mathrm{d}{r}_i^{(\mathrm{1})}}{\mathrm{d}t}}+{\omega }_i^{\mathrm{2}}{r}_i^{(\mathrm{1})}={\displaystyle \frac{\mathrm{1}}{m_i}}{q}_i{E}^{(\mathrm{1})}(t,T),

(13){\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}{r}_i^{(2)}}{\mathrm{d}{t}^{\mathrm{2}}}}+{\gamma }_i{\displaystyle \frac{\mathrm{d}{r}_i^{(2)}}{\mathrm{d}t}}+{\omega }_i^{\mathrm{2}}{r}_i^{(2)}={\displaystyle \frac{\mathrm{1}}{2m_i}}\underset{j,k=1}{\overset{N}{\mathrm{\Sigma }}}{B}_{ijk}{r}_i^{(1)}{r}_k^{(1)}\mathrm{.}

By solving Eqs. (12) and (13) with only cross terms of E_{\mathrm{1}} and E_{\mathrm{2}} kept, we achieve the solution of the second-order polarization in time-domain as

(14)P_{\mathrm{AH}}^{(2)}(t,T)=-\rho {\displaystyle \sum _{i,j,k=\mathrm{1}}^N{B}_{ijk}{q}_i{q}_j{q}_k{S}_i(t)\{[S_j(t)\ast E_{\mathrm{2}}(t)][S_k(t)\ast E_{\mathrm{1}}(t+T)]\}}.

This expression agrees with those in Refs. [¹,¹⁶–¹⁸] when N=\mathrm{1}. According to Eqs. (8) and (9), the induced second-order electric field is

(15){\tilde{E}}_{\mathrm{AH}}^{(2)}(z,{\omega }_t,{\omega }_T)=-{\tilde{R}}^{(\mathrm{2})}{\displaystyle \sum _{i,j,k=\mathrm{1}}^N{B}_{ijk}{q}_i{q}_j{q}_k{\tilde{S}}_i({\omega }_t){\tilde{S}}_j({\omega }_t-{\omega }_T){\tilde{S}}_k({\omega }_T)},

where

(16){\tilde{R}}^{(\mathrm{2})}={\displaystyle \frac{\rho {\mu }_{\mathrm{0}}{\omega }_t^{\mathrm{2}}}{\Delta k^{(\mathrm{2})}[\Delta k^{(\mathrm{2})}+\mathrm{2}k({\omega }_t)]}}{\tilde{E}}_{\mathrm{2}}({\omega }_t-{\omega }_T){\tilde{E}}_{\mathrm{1}}({\omega }_T)(e^{j\Delta k^{(\mathrm{2})}z}-\mathrm{1})e^{jk({\omega }_t)z},

(17)\Delta k^{(\mathrm{2})}=k({\omega }_t-{\omega }_T)+k({\omega }_T)-k({\omega }_t),

(18)k({\omega }_t)=\sqrt{{\mu }_{\mathrm{0}}{\omega }_t^{\mathrm{2}}\left({\epsilon }_{\mathrm{0}}+\rho {\displaystyle \sum _{i=\mathrm{1}}^N{q}_i{\tilde{S}}_i({\omega }_t)}\right)}.

For NC, the dipole moment is expressed as

(19)p={\displaystyle \sum _{i=\mathrm{1}}^N{q}_i{r}_i}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \sum _{i,j=\mathrm{1}}^N{q}_{ij}{r}_i{r}_j},

where q_{ij}=-{\partial }^{\mathrm{2}}pE/\partial r_i\partial r_j is the coupling parameter. Thus, the q_i{E}_{\mathrm{ex}}/m_i term in Eq. (2) becomes

(20){\displaystyle \frac{\mathrm{1}}{m_i}}{q}_i{E}_{\mathrm{ex}}\to {\displaystyle \frac{\mathrm{1}}{m_i}}{\displaystyle \frac{\partial }{\partial r_i}}pE={\displaystyle \frac{\mathrm{1}}{m_i}}{q}_{i}E+{\displaystyle \frac{\mathrm{1}}{m_i}}{\displaystyle \sum _{j=\mathrm{1}}^N{q}_{ij}{r}_j^{(\mathrm{1})}E}.

By using perturbation method, we obtain the second-order polarization in time-domain as

(21)\begin{array}{l}{P}_{\mathrm{NC}}^{(\mathrm{2})}(t,T)=\rho {\displaystyle \sum _{i,j=\mathrm{1}}^N{q}_{ij}{q}_i{q}_j{S}_i(t)\ast \left\{\begin{array}{l}[S_j(t)\ast E_{\mathrm{2}}(t)]E_{\mathrm{1}}(t+T)\\ +[S_j(t)\ast E_{\mathrm{1}}(t+T)]E_{\mathrm{2}}(t)\end{array}\right\}}\\ +\rho {\displaystyle \sum _{i,j=\mathrm{1}}^N{q}_{ij}{q}_i{q}_j[S_i(t)\ast E_{\mathrm{2}}(t)][S_j(t)\ast E_{\mathrm{1}}(t+T)]}.\end{array}

This expression matches that in Refs. [¹,¹⁶–¹⁸] when N=\mathrm{1}. The induced second-order electric field is

(22){\tilde{E}}_{\mathrm{NC}}^{(2)}(z,{\omega }_t,{\omega }_T)={\tilde{R}}^{(\mathrm{2})}{\displaystyle \sum _{i,j=\mathrm{1}}^N{q}_{ij}{q}_i{q}_j\left\{\begin{array}{l}{\tilde{S}}_i({\omega }_t)\left[{\tilde{S}}_j({\omega }_t-{\omega }_T)+{\tilde{S}}_j({\omega }_T)\right]\\ +{\tilde{S}}_i({\omega }_t-{\omega }_T){\tilde{S}}_j({\omega }_T)\end{array}\right\}}.

ND is introduced by adding a second-order term to the damping as

(23){\gamma }_i{\displaystyle \frac{\mathrm{d}{r}_i}{\mathrm{d}t}}\to {\gamma }_i{\displaystyle \frac{\mathrm{d}{r}_i}{\mathrm{d}t}}+{\displaystyle \sum _{j,k=\mathrm{1}}^N{\gamma }_{ij}^k{\displaystyle \frac{\mathrm{d}{r}_k}{\mathrm{d}t}}{r}_j},

where {\gamma }_{ij}^k is the nonlinear damping parameter. Through derivation, the polarization is

(24)P_{\mathrm{ND}}^{(\mathrm{2})}(t,T)=-\rho {\displaystyle \sum _{i,j,k=\mathrm{1}}^N{\gamma }_{ij}^k{q}_i{q}_j{q}_k{S}_i(t)\left\{\begin{array}{l}[S_j(t)\ast E_{\mathrm{2}}(t)]\left[{\displaystyle \frac{d{S}_k(t)}{dt}}\ast E_{\mathrm{1}}(t+T)\right]\\ +[S_j(t)\ast E_{\mathrm{1}}(t+T)]\left[{\displaystyle \frac{d{S}_k(t)}{dt}}\ast E_{\mathrm{1}}(t)\right]\end{array}\right\}}.

And the induced second-order electric field is

(25){\tilde{E}}_{\mathrm{ND}}^{(2)}(z,{\omega }_t,{\omega }_T)=-{\tilde{R}}^{(\mathrm{2})}{\displaystyle \sum _{i,j,k=\mathrm{1}}^N{\gamma }_{ij}^k{q}_i{q}_j{q}_k{\tilde{S}}_i({\omega }_t)\left[\begin{array}{l}{\tilde{S}}_j({\omega }_t-{\omega }_T){\tilde{S}}_k^d({\omega }_T)\\ +{\tilde{S}}_j^d({\omega }_t-{\omega }_T){\tilde{S}}_k({\omega }_T)\end{array}\right]},

where

(26){\tilde{S}}_i^d({\omega }_t)=\left({\epsilon }_i{e}^{-j\pi {\omega }_t/\mathrm{2}{\epsilon }_i}-{\displaystyle \frac{{\gamma }_i}{\mathrm{2}}}\right){\tilde{S}}_i({\omega }_t).

Results

To demonstrate the solution, a 3-mode system is considered, the parameters of which are set as

(27)\begin{array}{l}\begin{array}{lll}{\omega }_{\mathrm{1}}=\mathrm{0.9}\times \mathrm{2}\mathrm{\pi \; THz,}& {\omega }_{\mathrm{2}}=\mathrm{1.2}\times \mathrm{2}\mathrm{\pi \; THz,}& {\omega }_{\mathrm{3}}=\mathrm{1.7}\times \mathrm{2}\mathrm{\pi \; THz}\end{array},\\ {\gamma }_i=\mathrm{0.2}{\omega }_i,i=\mathrm{1},\mathrm{2},\mathrm{3},\\ m_{\mathrm{1}}=m_{\mathrm{2}}=m_{\mathrm{3}},\\ q_{\mathrm{1}}=q_{\mathrm{2}}=q_{\mathrm{3}}.\end{array}

We choose a typical THz shape pulse as incident pulses, as shown in Fig. 2, and assume E_{\mathrm{1}}(t) and E_{\mathrm{2}}(t) are identical.

Fig.2 THz pulse used in calculation shown in time- (a) and frequency- (b) domain

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Given that the purpose of this section is to validate the results, only three cases are considered: 1) all nonlinear parameters are zero except B_{\mathrm{123}}=B_{\mathrm{132}}=\mathrm{...}=B_{\mathrm{321}} (AH), 2) all nonlinear parameters are zero except q_{\mathrm{12}}=q_{\mathrm{21}} (NC), 3) all nonlinear parameters are zero except {\gamma }_{\mathrm{23}}^{\mathrm{1}}={\gamma }_{\mathrm{32}}^{\mathrm{1}} (ND). The nonlinear electric field {\tilde{E}}^{(\mathrm{2})}({\omega }_t,{\omega }_T) is calculated by two methods, i.e., our analytical solution and the traditional simulation method of which the process is: 1) the polarization P^{(\mathrm{2})}(t,T) is calculated in time-domain through numerical integral according to Eqs. (14), (21) and (24) respectively; 2) numerical 2D-FT is performed so that the polarization in frequency-domain {\tilde{P}}^{(\mathrm{2})}({\omega }_t,{\omega }_T) is obtained; 3) the electric field {\tilde{E}}^{(\mathrm{2})}({\omega }_t,{\omega }_T) is numerically calculated according to Eq. (8). Figure 3 illustrates the calculated absolute value of nonlinear electric fields with normalized magnitude. Six sub-graphs show the electric fields corresponding to three different sources of nonlinearities and the two calculation methods. Frequency {\omega }_i+{\omega }_j and {\omega }_i-{\omega }_j are expressed as {\Omega }^{ij}={\omega }_i+{\omega }_j and {\Omega }_j^i={\omega }_i-{\omega }_j in these sub-graphs.

Fig.3 Normalized 2D diagrams corresponding to (a) AH, (b) NC, and (c) ND obtained by analytical calculation, and those corresponding to (d) AH, (e) NC, and (f) ND obtained by numerical simulation. Only the first quartile is presented to display more details

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Discussion

Obviously, the analytical results agree well with the simulations. For AH (Figs. 3(a) and 3(d)), peaks appear at ({\omega }_t,{\omega }_T)=\pm ({\omega }_i,\pm {\omega }_j),\pm ({\omega }_i,{\omega }_i\pm {\omega }_j),\pm ({\omega }_i\pm {\omega }_j,{\omega }_i), where i,j=\mathrm{1},\mathrm{2},\mathrm{3}(i\ne j), as predicted in Eq. (15). These peaks correspond to the coupling between mode 1, 2, and 3 through the anharmonicity of potential. For NC (Figs. 3(b) and 3(e)), peaks appear at ({\omega }_t,{\omega }_T)=\pm ({\omega }_i,\pm {\omega }_j),\pm ({\omega }_i,{\omega }_i\pm {\omega }_j),\pm ({\omega }_i\pm {\omega }_j,{\omega }_i), where i,j=\mathrm{1},\mathrm{2}(i\ne j), as predicted in Eq. (22). These peaks correspond to the coupling between mode 1 and 2 through the nonlinearity of dipole moment. For ND (Figs. 3(c) and 3(f)), peaks appear at ({\omega }_t,{\omega }_T)=\pm ({\omega }_i,\pm {\omega }_j),\pm ({\omega }_i,{\omega }_i\pm {\omega }_j),\pm ({\omega }_i\pm {\omega }_{\mathrm{3}},{\omega }_i),\pm ({\omega }_{\mathrm{3}}-{\omega }_i,{\omega }_{\mathrm{3}}), where i=\mathrm{1},\mathrm{2},j=\mathrm{1},\mathrm{2},\mathrm{3}(i\ne j), as predicted in Eq. (25). These peaks correspond to the coupling between the vibrational coordinate of mode 1, 2 and the velocity of mode 3 through the nonlinearity of damping.

However, the calculation amount of the analytical solution is much smaller than numerical simulation, since the former only involves multiplication and addition, while the later involves integral. The ratio of the calculation amounts of these two methods is nearly \mathrm{1}:\mathrm{0.19}{F}_w/\Delta f (F_w is frequency window and \Delta f is the spectral resolution). For experiments the frequency window may be over 30 THz [⁴], and if the spectral resolution is 0.1 THz, the calculation amount of numerical simulation is 57 times larger than that of analytical calculation.

Conclusions

Using multi-oscillator model involving three sources of weak nonlinearities, we have achieved a classical-theory-based analytical solution for second-order 2DTS. By calculating the spectrum of a 3-mode system, we have verified that our analytical calculations are consistent with the traditional numerical simulation results. Compared to former theoretical works which only dealt with single-mode systems, our solution has two major advantages: first, it is more wide-ranging in term of application as it can be applied to multi-modes system; secondly, it has much lower computational cost as it directly gives the second-order transmission field instead of polarization and no numerical integral needs to be performed. To conclude, this new analytical approach offers a more efficient and practical method to directly obtain second-order 2DTS in replacement of complex simulation. In a broader sense, our study sheds new light on the theory of 2DTS, which may enable promising future applications.

Acknowledgements

We acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 11574105, 61475054, 61405063 and 61177095), and the Fundamental Research Funds for the Central Universities (No. 2017KFYXJJ029).

References

1 Hattori T. Classical theory of two-dimensional time-domain terahertz spectroscopy. Journal of Chemical Physics, 2010, 133(20): 204503

PMID

2 Cervetto V, Helbing J, Bredenbeck J, Hamm P. Double-resonance versus pulsed Fourier transform two-dimensional infrared spectroscopy: an experimental and theoretical comparison. Journal of Chemical Physics, 2004, 121(12): 5935–5942

PMID

3 Okumura K, Tanimura Y. Two-dimensional THz spectroscopy of liquids: non-linear vibrational response to a series of THz laser pulses. Chemical Physics Letters, 1998, 295(4): 298–304

4 Woerner M, Kuehn W, Bowlan P, Reimann K, Elsaesser T. Ultrafast two-dimensional terahertz spectroscopy of elementary excitations in solids. New Journal of Physics, 2013, 15(2): 025039

5 Zanni M T, Gnanakaran S, Stenger J, Hochstrasser R M. Heterodyned two-dimensional infrared spectroscopy of solvent-dependent conformations of acetylproline-NH₂. Journal of Physical Chemistry B, 2001, 105(28): 6520–6535

6 Woutersen S, Hamm P. Nonlinear two-dimensional vibrational spectroscopy of peptides. Journal of Physics Condensed Matter, 2002, 14(39): R1035–R1062

7 Woutersen S, Hamm P. Structure determination of trialanine in water using polarization sensitive two-dimensional vibrational spectroscopy. Journal of Physical Chemistry B, 2000, 104(47): 11316–11320

8 Hamm P, Lim M, DeGrado W F, Hochstrasser R M. The two-dimensional IR nonlinear spectroscopy of a cyclic penta-peptide in relation to its three-dimensional structure. Proceedings of the National Academy of Sciences of the United States of America, 1999, 96(5): 2036–2041

PMID

9 Bredenbeck J, Helbing J, Behrendt R, Renner C, Moroder L, Wachtveitl J, Hamm P. Transient 2D-IR spectroscopy: snapshots of the nonequilibrium ensemble during the picosecond conformational transition of a small peptide. Journal of Physical Chemistry B, 2003, 107(33): 8654–8660

10 Jewariya M, Nagai M, Tanaka K. Enhancement of terahertz wave generation by cascaded c⁽²⁾ processes in LiNbO₃. Journal of the Optical Society of America. B, Optical Physics, 2009, 26(9): A101–A106

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

PMID

12 Yeh K L, Hoffmann M C, Hebling J, Nelson K A. Generation of 10 mJ ultrashort terahertz pulses by optical rectification. Applied Physics Letters, 2007, 90(17): 171121

13 Elsaesser T, Reimann K, Woerner M. Focus: phase-resolved nonlinear terahertz spectroscopy--from charge dynamics in solids to molecular excitations in liquids. Journal of Chemical Physics, 2015, 142(21): 212301

PMID

14 Kuehn W, Reimann K, Woerner M, Elsaesser T. Phase-resolved two-dimensional spectroscopy based on collinear n-wave mixing in the ultrafast time domain. Journal of Chemical Physics, 2009, 130(16): 164503

PMID

15 Pashkin A, Sell A, Kampfrath T, Huber R. Electric and magnetic terahertz nonlinearities resolved on the sub-cycle scale. New Journal of Physics, 2013, 15(6): 065003

16 Hu J, Liu J, Li H, Wang K, Yang Z, Wang S. Influence of the amplitude ratio between two terahertz pulses on two-dimensional spectroscopy. Chinese Science Bulletin, 2014, 59(2): 138–146

17 Li H, Liu J, Wang K, Yang Z. A classical iterative theory based on the Langevin equation for two-dimensional nonlinear terahertz spectroscopy. Journal of Modern Optics, 2013, 60(10): 773–780

18 Li H, Liu J, Wang K, Yang Z, Du Z. Influence of terahertz pulse width on two-dimensional terahertz spectroscopy. Journal of Modern Optics, 2012, 59(10): 923–929

19 Okumura K, Tanimura Y. Sensitivity of two-dimensional fifth-order Raman response to the mechanism of vibrational mode-mode coupling in liquid molecules. Chemical Physics Letters, 1997, 278(1-3): 175–183