Continuum electron giving birth to terahertz emission

Kaixuan Zhang; Yizhu Zhang; Xincheng Wang; Tian-Min Yan; Y. H. Jiang

doi:10.1364/PRJ.377408

Abstract

The origin of terahertz (THz) generation in a gas-phase medium is still in controversy, although the THz sources have been applied across many disciplines. Herein, the THz generation in a dual-color field is investigated experimentally by precisely controlling the relative phase and polarization of dual-color lasers, where the accompanying third-harmonic generation is employed for in situ determination of the relative phase up to sub-wavelength accuracy. Joint studies with the strong approximation (SFA) theory reveal that the continuum-continuum (CC) transition within an escaped electron wave packet in the single atom gives birth to THz emission, without the necessity of considering the plasma effect. Meanwhile, we develop the analytic form from SFA-based CC description, which is able to reproduce and decompose the classical photocurrent model from the viewpoint of microscopic quantum theory, establishing the quantum-classical correspondence and bringing a novel insight into the mechanism of THz generation. Present studies leave open the possibility for probing the ultrafast dynamics of continuum electrons and a new dimension for the study of THz-related science and methodology.

Terahertz (THz) wave generation (TWG) using dual-color femtosecond pulse, typically focusing 800 nm and 400 nm beams into a gas-phase medium, allows for the convenient and efficient access to moderately strong ultra-broadband THz pulses [1]. Although the approach is widely applied in several disciplines, the underlying generation mechanism is still an ongoing debate [2, 3].

The controversy is partially due to the ambiguity of the single-atom and plasma ensemble contributions. Due to the ambiguity, interpretations of TWG are rather distinct in their appearances, and the intrinsic physical pictures are completely different. From a plasma perspective, the photocurrent (PC) model formulates the emission process classically [4, 5]. The propagation effects in plasma can modulate the spectra and polarization of the TWG [6 - 9]. From the single-atom perspective, the four-wave mixing (FWM), as in crystal nonlinear optics [1], explains the nonlinear THz emission based on the quantum perturbation theory. Based on the nonperturbative theory, the TWG has been numerically studied by solving the time-dependent Schrödinger equation [10 - 13], accounting for the strong-field dynamics of a single atom. However, the time-dependent Schrödinger equation hardly provides a transparent physical picture. Comparatively, the strong-field approximation (SFA), which has been extensively developed to treat various strong-field phenomena, including above threshold ionization, high-order above-threshold ionization, multiple ionization, and high-harmonic generation (HHG), was also developed to provide a clear physical insight [14, 15].

Although the laser plasma has been considered the source of the TWG since the first observation from the laser-gas interaction [16], it is still unclear whether the plasma effect is a necessary ingredient. Similar debate arose in the early days when HHG was studied. Nowadays it is widely accepted that the HHG is a nonperturbative strong-field process dominated by the continuum-bound transition within a single atom or molecule, i.e., recombination of a released electron with its parent ion after ionization. Questions thus arise automatically. Is the THz radiation generated from the individual atom, analogous to HHG, or from the plasma ensemble effect? Can the strong-field TWG, analogous to other strong-field phenomena, be generalized to the strong-field theory framework, and can a unified theory for the TWG be established for integration of apparently distinct interpretations of FWM and the classical PC model?

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In this paper, the joint measurement between the TWG and third-harmonic generation (THG) is performed for in situ determination of phase delay zero of the dual-color fields, which allows us to select the most relevant mechanism in a direct comparison with different theoretical models. The dependence of the THz signal on the delay phase and relative polarization angles is measured, which shows remarkable consistencies with the continuum-continuum (CC) transition model. Our experimental and theoretical investigation confirms that the TWG is dominated by the CC transition within the escaped electronic wave packets, rather than the continuum-bound (CB) recollision, and the single-atom behavior is deconvoluted from the plasma ensemble. The schematic is illustrated in Fig. 1 . Our work has manifold implications. From the theoretical aspect, the application scenario of the SFA is further expanded, bringing the TWG explicably under the framework of strong-field physics similar to the HHG. From application aspect, it implies that the TWG is still obtainable through the CC transition, even when the neutral atoms are fully depleted by the strong pump laser, showing the possibility to achieve intense THz fields by pumping the gas-phase medium with an extremely strong laser. Moreover, since the TWG is encoded by the time-dependent information of the continuum electron, it can be used as a spatial-temporal probe in the microscopic scale, complementary to HHG spectral line shape and photoelectron momentum distributions, to trace ultrafast dynamics of continuum electrons in atoms and molecules [17].

Figure 1.Illustrative description of the TWG in dual-color fields and the mechanism. In panel (a), the dual-color laser ionizes the gas-phase medium and accelerates the electrons. According to the photocurrent model from the plasma perspective, the TWG originates from the rapidly varying macroscopic residual current as indicated by the giant arrow. However, the strong-field theory provides an alternative explanation from the microscopic aspect that each atom behaves as an individual THz emitter. As illustrated by the swarm of particles, each with an arrow indicates the THz radiation from a single atom. Panel (b) presents the radiation mechanisms under the strong-field approximation. After the photoelectron is released by the external light fields from the distorted atomic potential, it may recollide with its parent core, yielding the HHG; or within the continuum wave packet, the transition between continuum states of similar energies leads to the TWG. According to the derived SFA-CC, the involved continuum states are via different quantum paths as indicated by the white solid and dashed lines on the potential surface in (b). Panel (c) shows the origin of the two paths. The photoelectrons of initial state

| Ψ (0) ⟩

are released at different ionization times

t^{'}

and

t^{''}

but arrive at the same intermediate momentum state

k^{'} (t^{'}, t^{''})

. The coherence between the continuum states yields the TWG.

d (t) = ⟨ Ψ (t) | \hat{r} | Ψ (t) ⟩ = ⟨ Ψ_{0} | \hat{U} (t_{0}, t) \hat{r} \hat{U} (t, t_{0}) | Ψ_{0} ⟩

\ddot{d} (t)

\sim 1.75 mJ

2 ω

Figure 2.Comparison of PP distributions,

S_{THz, s} (τ, θ)

(upper row) and

S_{THz, p} (τ, θ)

(lower row), respectively, for the

s

and

p

components of

E_{THz} (t)

obtained from (a) experiment and (b)–(d) theoretical models of the (b) SFA-CB, (c) SFA-CC, and (d) SPC.

τ

τ

E_{THz} (t) \propto \partial j (t) / \partial t = e^{2} N (t) E (t) / m

w (t^{'})

Figure 3.

w (t^{'}; t)

of the SFA-CC and the comparison with

w (t^{'})

of the SPC when

θ = 0 °

and

τ = 0.33 fs

. In collinear dual-color laser fields, the SFA-CC derived

w_{1} (t^{'}; t)

and

w_{2} (t^{'}; t)

are shown in (a) and (b), respectively, with their detailed zoom-in around

t^{'} = 0

in insets (c) and (d). The total contribution,

w (t^{'}; t) = w_{1} (t^{'}; t) + w_{2} (t^{'}; t)

, is presented in inset (e), showing that

w_{2} (t^{'}; t)

almost contributes at

t^{'} = t

only. In (f),

w (t^{'}; t \to \infty)

of the SFA-CC (solid line) is compared with

w (t^{'})

of the SPC (dashed line), showing the correspondence between the SFA-based quantum model and the semi-classical PC model.

w_{1} (t^{'}; t)

w_{2} (t^{'}; t)

a_{1} (t)

In conclusion, the joint measurement between the TWGs and THGs is implemented for in situ determination of the phase delay of the dual-color fields, allowing us to distinguish the different theoretical models. Our research exhibits that the underlying origin of the TWG resides within the scope of nonperturbative single-atomic strong-field processes, although the ensemble behavior of the laser-induced plasma may influence the radiation yields. In contrast to the HHG emitted by the CB transition of a recolliding electron, the TWG originates from the CC transition of a released electron after ionization. The TWG mechanism of the CC—the “soft” transition—beyond the HHG mechanism of “hard” recollision, is a complement to the radiation theory of strong-field physics. Meanwhile, it is shown that the classical PC model can be derived from the quantum SFA-CC method, bridging between the classical and quantum-mechanical interpretations. Also, the FWM can be reached from the SFA-CC by presenting the explicit third-order dependence on the electric field. Hence, the SFA-CC serves to unify the FWM and PC models, while it offers more microscopic details compared to the latter coarse-grained models. Our research of the TWG mechanism opens up the possibility to extract the ultrafast dynamics of continuum electrons from the ionization-induced THz emission.

Acknowledgment. We acknowledge the support from the Shanghai-XFEL Beamline Project (SBP) and Shanghai HIgh repetitioN rate XFEL and Extreme light facility(SHINE).

The expected value of dipole moment is

d (t) = Ψ (t) | \hat{r} | Ψ (t) = Ψ_{0} (t_{i}) | \hat{U} (t_{i}, t) \hat{r} \hat{U} (t, t_{i}) | Ψ_{0} (t_{i})

. With the time-evolution operator

\hat{U} (t, t_{i})

expanded by Dyson series

\hat{U} (t, t_{i}) = {\hat{U}}_{0} (t, t_{i}) i \int_{t_{i}}^{t} d t^{'} \hat{U} (t, t^{'}) \hat{W} (t^{'}) {\hat{U}}_{0} (t^{'}, t_{i})

, where

{\hat{U}}_{0} (t^{'}, t_{i})

is the interaction-free time-evolution operator and

\hat{W}

is the interaction operator, we find that

d (t) = d^{(0)} (t) + d^{(1)} (t) + d^{(2)} (t)

, where

d^{(0)} (t) = Ψ_{0} (t) | \hat{r} | Ψ_{0} (t),

(A1)

d^{(1)} (t) = (i) \int_{t_{i}}^{t} d t^{'} Ψ_{0} (t) | \hat{r} \hat{U} (t, t^{'}) \hat{W} (t^{'}) | Ψ_{0} (t^{'}) + c.c.,

(A2)

d^{(2)} (t) = \int_{t_{i}}^{t} d t^{''} Ψ_{0} (t^{''}) | \hat{W} (t^{''}) \times \int_{t_{i}}^{t} d t^{'} \hat{U} (t^{''}, t) \hat{r} \hat{U} (t, t^{'}) \hat{W} (t^{'}) | Ψ_{0} (t^{'}) .

(A3)

The

d^{(0)} (t)

vanishes in a spherically symmetric system. The

d^{(1)} (t)

, referred to as the CB transition dipole moment, depicts the coherent emission at time

t

induced by the transition of electrons from the continuum, which accumulates over all possible ionization events at time

t^{'}

, to the bound state. The

d^{(1)} (t)

is widely recognized to dominate the HHG process. Similarly, the

d^{(2)} (t)

, referred to as the CC transition dipole moment, describes the coherent emission induced by the transition between the states of the continuum, which is often negligible for the HHG calculation. Here, however, we emphasize the role of

d^{(2)} (t)

in TWG. In the following, the details of

d^{(1)} (t)

and

d^{(2)} (t)

as used to evaluate the emission are presented.

The CB transition

d^{(1)} (t)

is given by Eq. (A2). Under the SFA, which neglects the interaction between the photoelectron and the parent ion, the full time-evolution operator is substituted by one with the external light field only:

\hat{U} (t, t^{'}) \to {\hat{U}}_{I} (t, t^{'}) = \int d k | Ψ_{k}^{(V)} (t) Ψ_{k}^{(V)} (t^{'}) |

, yielding

d^{(1)} (t) = (i) \int_{t_{i}}^{t} d t^{'} Ψ_{0} (t) | \hat{r} {\hat{U}}_{I} (t, t^{'}) \hat{W} (t^{'}) | Ψ_{0} (t^{'}) + c.c. .

The Volkov state

| Ψ_{k}^{(V)} (t) = | k + A (t) e^{i S_{k} (t)}

describes the free electron in the presence of a time-dependent electric field of vector potential

A (t)

with

S_{k} (t) = \frac{1}{2} \int d t^{'} {[k + A (t^{'})]}^{2}

. Assuming

W (t^{'}) = μ [k^{'} + A (t^{'})] \cdot E (t^{'})

, the interaction between the incident electric field

E (t^{'})

and the electron of initial state

| Ψ_{0} (t^{'}) = | ψ_{0} e^{i E_{0} t^{'}}

, it is shown that

d^{(1)} (t) = i \int_{t_{i}}^{t} d t^{'} \cdot \int d k e^{i S_{k, I_{p}} (t, t^{'})} μ^{*} [k + A (t)] (μ [k + A (t^{'})] \cdot E^{*} (t^{'})) + c.c.

, where

S_{k, I_{p}} (t, t^{'}) = \int_{t^{'}}^{t} d t^{''} (\frac{1}{2} {[k + A (t^{''})]}^{2} + I_{p})

with

k

the intermediate momentum and

I_{p} = E_{0}

the ionization energy.

The integration over momentum can be approximated with the stationary phase, and the stationary point

k_{s}

is the solution to the equation

_{k} S_{k, I_{p}} (t, t^{'}) \overset{!}{=} 0

. Therefore, we obtain the CB transition dipole

d^{(1)} (t) = i \int_{t_{i}}^{t} d t^{'} {[\frac{2 π}{i (t t^{'})}]}^{\frac{3}{2}} e^{i S_{k_{s}, I_{p}} (t, t^{'})} \times μ^{*} [k_{s} + A (t)] (μ [k_{s} + A (t^{'})] \cdot E^{*} (t^{'})) + c.c.,

(B1)

recovering the widely used Lewenstein model for the HHG.

Under the strong-field approximation, the substitution

\hat{U} (t, t^{'}) \to {\hat{U}}_{I} (t, t^{'})

yields

d^{(2)} (t) (+ i) (i) \int_{t_{i}}^{t} d t^{''} Ψ_{0} (t^{''}) | \hat{W} (t^{''}) \int_{t_{i}}^{t} d t^{'} {\hat{U}}_{I} (t^{''}, t) \hat{r} {\hat{U}}_{I} (t, t^{'}) \hat{W} (t^{'}) | Ψ_{0} (t^{'}) .

(C1)

Applying the similar procedure as considered for the CB transition, the expansion arrives at

d^{(2)} (t) = (+ i) (i) \int_{t_{i}}^{t} d t^{''} \int_{t_{i}}^{t} d t^{'} \int d k^{'} (μ [k^{'} + A (t^{'})] \times E (t^{'})) e^{i S_{k^{'}, I_{p}} (t^{'}, t)} \times \int d k^{''} k^{''} + A (t) | \hat{r} | k^{'} + A (t) \times (μ^{*} [k^{''} + A (t^{''})] \cdot E (t^{''})) e^{i S_{k^{''}, I_{p}} (t^{''}, t)} .

(C2)

With

k^{''} + A (t) | \hat{r} | k^{'} + A (t) = i_{k^{''}} δ (k^{''} k^{'})

and applying integration by parts, the last integration over

k^{''}

reads

{_{k^{'}} i_{k^{'}} S_{k^{'}, I_{p}} (t^{''}, t)} (μ^{*} [k^{'} + A (t^{''})] \cdot E (t^{''})) e^{i S_{k^{'}, I_{p}} (t^{''}, t)},

yielding

d^{(2)} (t) = (+ i) (i) (i) (1) \int_{t_{i}}^{t} d t^{''} \int_{t_{i}}^{t} d t^{'} \int d k^{'} \times (μ [k^{'} + A (t^{'})] \cdot E (t^{'})) {_{k^{'}} i_{k^{'}} S_{k^{'}, I_{p}} (t^{''}, t)} \times (μ^{*} [k^{'} + A (t^{''})] \cdot E (t^{''})) e^{i S_{k^{'}, I_{p}} (t^{'}, t^{''})} .

(C3)

The integration over

k^{'}

can also be treated by the stationary phase approximation. Solving the saddle point equation

_{k^{'}} S_{k^{'}, I_{p}} (t^{'}, t^{''}) = 0

, we obtain

k_{s}^{'} \equiv k_{s}^{'} (t^{'}, t^{''}) = [α (t^{'}) α (t^{''})] / (t^{'} t^{''})

with the excursion of the oscillating electron

α (t) = \int d t^{'} A (t^{'})

. Within the curly bracket of Eq. (C3), the first term is negligible, and dipole is determined by the second term

d^{(2)} (t) \int_{t_{i}}^{t} d t^{''} \int_{t_{i}}^{t} d t^{'} {[\frac{2 π}{i (t^{'} t^{''})}]}^{3 / 2} (μ [k_{s}^{'} + A (t^{'})] \times E (t^{'})) (μ^{*} [k_{s}^{'} + A (t^{''})] \cdot E (t^{''})) \times [k_{s}^{'} (t^{''} t) + α (t^{''}) α (t)] e^{i S_{k_{s}^{'}, I_{p}} (t^{'}, t^{''})}

(C4)

after the substitution

_{k_{s}^{'}} S_{k_{s}^{'}, I_{p}} (t^{''}, t) = k_{s}^{'} (t^{''} t) + α (t^{''}) α (t)

. The emission is given by the acceleration

a = {\ddot{d}}^{(2)} (t)

. Using the Leibniz integral rule, the second-order derivative of Eq. (C4) with respect to

t

is evaluated:

{\ddot{d}}^{(2)} (t) = E (t) \int_{t_{i}}^{t} d t^{''} \int_{t_{i}}^{t} d t^{'} {[\frac{2 π}{i (t^{'} t^{''})}]}^{3 / 2} \times (μ [k_{s}^{'} (t^{'}, t^{''}) + A (t^{'})] \cdot E (t^{'})) \times (μ^{*} [k_{s}^{'} (t^{'}, t^{''}) + A (t^{''})] \cdot E (t^{''})) \times e^{i S_{k_{s}^{'} (t^{'}, t^{''}), I_{p}} (t^{'}, t^{''})} 2 Re \int_{t_{i}}^{t} d t^{'} {[\frac{2 π}{i (t t^{'})}]}^{3 / 2} \times (μ [k_{s}^{'} (t, t^{'}) + A (t)] \cdot E (t)) \times (μ^{*} [k_{s}^{'} (t, t^{'}) + A (t^{'})] \cdot E (t^{'})) \times [k_{s}^{'} (t, t^{'}) + A (t)] e^{i S_{k_{s}^{'} (t, t^{'}), I_{p}} (t, t^{'})},

(C5)

showing the full form of

{\ddot{d}}^{(2)} (t) = a_{1} (t) + a_{2} (t)

as presented by Eqs. (2) and (3) in the main text.

In this work, considering the H(1s) initial state for simplicity, the dipole matrix element reads

μ (k) = i 2^{\frac{7}{2}} k / [π {(k^{2} + 1)}^{3}]

, and

I_{p} = 0.5 a.u.

The femtosecond amplifier (Libra, Coherent Inc.) delivers a laser pulse with

～ 800 nm

center wavelength and

～ 35 fs

pulse duration. The pulse with

～ 1.75 mJ

is guided into the experiment setup (Fig. 4). The beam with 96% pulse energy is reflected as the pump beam for the TWG, and the transmission beam is used as the probe beam for electro-optic sampling (EOS). The pump beam passes through a 200-μm type-I BBO crystal with the double-frequency efficiency of

～ 23 %

. The polarization of the laser beam is horizontal (

p

polarized), and the optical axis of BBO is kept perpendicular to the laser polarization to obtain the maximum efficiency. The outcoming second harmonic beam is

s

polarized. The relative polarization

θ

between the fundamental

ω

and second-harmonic

2 ω

beams can be controlled by rotating the zero-order dual-wavelength wave plate (DWP), which acts as a half-wave plate for the

ω

beam and a full-wave plate for the

2 ω

beam. The definition of observables is schematically illustrated in Fig. 5. Controlling

θ

with a half-wave plate, instead of rotating BBO crystal, avoids the mixture of the polarization of

\hat{o}

ray and

\hat{e}

ray in the BBO crystal. The ellipticities of both

2 ω

and

ω

beams are better than 0.1 when DWP rotates. Throughout the measurement, the

ω

and

2 ω

beams can be considered as linearly polarized. The time delay

τ

of the dual-color fields can be varied by moving the BBO crystal along the propagation direction, because of different refractive indices at

ω

and

2 ω

in air. Because of the collinear propagation geometry, the fluctuation of

τ

can be passively suppressed within the sub-wavelength accuracy.

Figure 4.Experimental setup. BS, beam splitter; CH, chopper;

β

-BBO, beta barium borate; DWP, dual-wavelength plate; PM, parabolic mirror; QWP, quarter-wave plate; GLP, Glan-laser polarizer; WP, Wollaston polarizer; BD, balanced detector.

Figure 5.Definition of observables. The 800 nm (

ω

) and 400 nm (

2 ω

) beams collinearly propagate. The

2 ω

beam is always

s

polarized. The relative time delay

τ

and polarization angle

θ

are controlled in the measurement. Both the

s

- and

p

-polarized terahertz waves are detected.

The dual-color fields are focused by a silver parabolic mirror with an effective focal length of 100 mm. Atmospheric air is ionized for the TWG, simultaneously emitting THG. Here, we use a tightly focusing scheme to deliberately prevent the propagation effect in plasma. The THz waves are collected and collimated with a gold parabolic mirror with 100 mm focal length, and focused into 1 mm thick (110)-cut ZnTe crystal with a same parabolic mirror. A 500 μm thick polished silicon wafer reflects the residual laser, allowing the transmission of only the THz component. A pellicle beam splitter combines the THz pulse and the probe beam to implement the free-space EOS detection. The signal-to-noise ratio (SNR) of the terahertz time-domain waveforms is better than 100:1. A polarization-sensitive THz-EOS is employed in our experiment. A metal wire-grid THz polarizer filters out the orthogonally polarized components of TWG, and the ZnTe crystal is fixed at the special orientation, where the responses for

s

- and

p

-polarized THz components are the same [23].

In the measurement, both

s

- and

p

-polarized THz electric fields

E_{THz} (t)

are recorded with the EOS method. The THz peak-peak (PP) amplitude is defined as

S_{THz} = \pm | \max [E_{THz} (t)] \min [E_{THz} (t)] |

. The distributions of

S_{THz, s (p)} (τ, θ)

are presented in Fig. 2(a) in the main text. The

S_{THz, s (p)} (τ, θ)

are normalized to the maximum. The positive direction of

S_{THz, s} (τ, θ)

and

S_{THz, p} (τ, θ)

are defined when the maxima of THz waveforms appear along the positive direction of the

s

and

p

axes.

The THG of

～ 266 nm

is coincidently measured with the TWG for the in situ determination of the absolute

τ

of the dual-color fields. The THG signals are reflected by the silicon wafer with residual

ω

and

2 ω

beams, and spectrally separated by a suprasil prism. The

s

and

p

components of THG signals are decomposed with a Glan-laser polarizer and collected into a fiber spectrometer. The THG signals are measured with 50 ms integration time, 10-time average in our measurement.

APPENDIX E. DETERMINATION OF TIME DELAY ZERO BY JOINT MEASUREMENT OF THIRD-ORDER HARMONICS

The dependence of TWG on time delay

τ

between the dual-color fields is critical to determine the TWG mechanisms. For instance, the TWG maximum is predicted to appear at

τ = 0.33 fs

in the PC theory; however, it is predicated at

τ = 0 fs

in the perturbative FWM theory. Only when the

τ

is precisely known, the electric field waveforms for the TWG can be determined for further comparison of the measured data with different theories. However, the precise

τ

is difficult to obtain, since it is nontrivial to directly monitor the electric fields in practical experiments.

In our experiment, the joint measurement of TWG and THG is conducted. The THG yields along the

s

-polarization

I_{3 rd} (τ, θ)

are shown in Fig. 6(a). In addition, we have examined the

I_{3 rd} (τ, θ)

evaluated by different theories, including the CB, CC transitions, and SPC. All results, as shown in Figs. 6(b)–6(d), predict the similar

τ

dependence that the maximum

I_{3 rd} (τ)

appears at

τ = 0 fs

. The time delay zero of dual-color fields in the experiment can therefore be precisely determined by comparison with the theoretical results.

Figure 6.Distribution of the THG along

s

-polarization

I_{3 rd} (τ, θ)

obtained from (a) the measurement, (b)

{\ddot{d}}^{(1)} (t)

of SFA-CB, (c)

{\ddot{d}}^{(2)} (t)

of SFA-CC, and (d) SPC.

In Eq. (5) in the main text, neglecting the neutral depletion is referred to as the SPC model. The

S_{THz, s (p)}

from the SPC model is shown in Fig. 2 of the main text. Here, the

S_{THz, s (p)}

from the traditional PC model is presented in Fig. 7 by comparison. It is shown that there is no significant deviation of the SPC from the PC, which involves the neutral depletion.

Figure 7.(a)

S_{THz, s} (τ, θ)

and (b)

S_{THz, p} (τ, θ)

evaluated from the PC model.

Figure 8.(a)

S_{THz, s} (τ = 0.33 fs, θ)

and (b)

S_{THz, s} (τ, θ = 90 °)

of the experiment (red star), PC (blue dashed), and SPC (magenta dotted) models.