Fast retrieval of temporal characteristics of FEL pulses using streaking by THz field

A. K. Kazansky; I. P. Sazhina; N. M. Kabachnik

doi:10.1364/OE.27.012939

1. Introduction

One of the most perspective applications of x-ray Free-Electron Lasers (FELs) is a direct study of time evolution of various processes in atomic systems on femtosecond and subfemtosecond scale by a pump-probe method [1–3]. However, to achieve this goal one should overcome the principle obstacle, namely stochastic nature of FEL radiation in non-seeded FEL which leads to the strong scatter in the pulse arrival time, pulse shape and duration. These parameters are critical for successful analysis of the pump-probe experiments. In recent years great efforts have been made to develop methods of spectral and temporal characterization of individual XUV and x-ray pulses generated by FELs [4–9] (for early references see review [10]). The ideal method of temporal characterization of FEL pulses should be fast enough to produce non-invasive measurement of each single pulse on-line. In principle, such possibility can be provided by streaking method [11,12] in which Đř femtosecond or subfemtosecond linearly polarized FEL pulse ionizes atoms and produces a pulse of photoelectrons, a replica of the FEL pulse. The photoelectrons are then streaked by far-infrared or THz pulses linearly polarized in the same direction [6,7,13–16]. Due to interaction of electrons with the slowly changing THz field the time dependence of the original pulse becomes imprinted into the spectral distribution of electrons which is measured by a detector. A necessary condition of application of streaking is that during the measured pulse the streaking field is a monotonous function of time. The streaking method is rather universal and can be used for any photon energy from XUV [13] to x-rays [16] including the energy interval where other methods are invalid. With a suitable adjustment of the streaking field frequency it can be used in a broad range of FEL pulse durations. Therefore, development of ultra-fast methods for on-line analysis of streaking spectra is a necessary and timely task.

Usually for retrieving the FEL pulse shape and pulse duration from the streaked spectrum a classical description of the electron interaction with the THz pulse is used [6,7,15,16]. In this case, the final energy of the photoelectron E is related to its initial energy before the interaction with the THz field, E₀, by the relation (atomic units are used throughout unless otherwise indicated)

E = E_{0} + q A_{THz} (t) cos θ + {(A_{THz} (t))}^{2}

where

q = \sqrt{2 E}

is the final electron momentum, θ is the emission angle counted from the direction of THz linear polarization and A_THz(t) is the vector potential of the THz field at the moment t of electron emission. Besides, usually it is supposed that vector potential linearly depends on time in the time interval when the FEL pulse operates (linear ramp). With this assumptions, retrieving is fast, but its accuracy is limited due to the jitter of the arrival times, especially for long pulses, comparable with the half-period of THz field (or ramp duration for a single-cycle THz pulse).

2. Theoretical background

In this communication we suggest a simple analytical formulas for fast retrieving of the FEL pulse temporal shape from the THz streaking spectra which is free from the above mentioned limitations and may be applied for any shape of the THz field, the only requirement is monotonous dependence of the THz pulse on time during the FEL pulse. Our approach is based on quantum mechanical strong field approximation (SFA) [17] which is valid for relatively fast electrons (kinetic energies of about 30 eV and more). Within SFA, the double differential cross section (DDCS) for photoionization of an atom in the presence of THz field can be presented as follows [18]

\begin{matrix} \frac{d σ (E, θ)}{d E d θ} = \sum_{m_{0}} | \int_{- \infty}^{\infty} d t {\tilde{𝒠}}_{XUV} (t) 𝒝_{l_{0}, m_{0}} (\vec{q} - {\vec{A}}_{THz} (t)) \\ {\times exp (- \frac{i}{2} [\int_{- \infty}^{t} [{(q cos θ - A_{THz} (t^{'}))}^{2} + q^{2} {sin}^{2} θ - q_{0}^{2}] d t^{'}]) |}^{2} \end{matrix}

where

q_{0} = \sqrt{2 E_{0}}

, and ℰ̃_XUV (t) is the envelope of the ionizing XUV pulse. The quantity ℬ_l₀,m₀(q⃗ − A⃗_THz(t)) is the conventional amplitude of primary ionization of the initial atomic state with the orbital quantum numbers l₀, m₀, with ejection of the electron with the momentum q⃗ − A⃗_THz(t). This quantity can be computed with knowing the dipole matrix elements and the scattering phases in the continuum states with angular momenta l = l₀ ± 1, which are conventionally well known. Further, we apply the semiclassical approach. The phase in the integral in Eq. (2) is a fast varying function of time, and the integral can be computed with the stationary phase method as suggested in [11]. The stationary point t_s (the time of ionization providing the final energy E = q²/2) is given by the relation

{(q cos θ - A_{THz} (t_{s}))}^{2} + q^{2} {sin}^{2} θ - q_{0}^{2} = 0 .

If the THz vector potential is a monotonous function of time when the FEL pulse is operative, Eq. (3) can have only one solution. In this case the retrieval of the FEL pulse is possible. If the FEL pulse arrives at a non-monotonous part of the time-dependence of the THz vector potential (overlapping maximum or minimum), several stationary points can provide complex contributions to the integral in Eq. (2) with varying phases. This leads to the oscillatory interference in the energy dependence of the DDCS and makes unique retrieval of the FEL pulse principally impossible.

Equation (3) in turn gives the final momentum of the photoelectron q as a function of t_s(assuming q > 0):

q = A_{THz} (t_{s}) cos θ + \sqrt{q_{0}^{2} - A_{THz} {(t_{s})}^{2} {sin}^{2} θ} .

For a single stationary point t_s, the evaluation of the integral in Eq. (2) is done using the standard method of stationary phase and taking into account that

A_{THz} (t) = - \int_{t}^{\infty} E_{THz} (t) d t

, which gives

\begin{array}{l} \frac{d σ (E, θ)}{d E d θ} = \sum_{m_{0}} \frac{2 π 𝒠_{XUV}^{2} (t_{s})}{E_{THz} (t_{s}) \sqrt{q_{0}^{2} - 2 E {sin}^{2} θ}} {| 𝒝_{l_{0}, m_{0}} (\vec{q} - {\vec{A}}_{THz} (t_{s})) |}^{2} \\ = \frac{2 π 𝒠_{XUV}^{2} (t_{s})}{E_{THz} (t_{s}) \sqrt{q_{0}^{2} - 2 E {sin}^{2} θ}} \frac{d σ^{(0)} (\tilde{E}, \tilde{θ})}{d E d θ}, \end{array}

where the energy Ẽ and angle θ̃ are defined as

\tilde{E} = \frac{1}{2} {| \vec{q} - {\vec{A}}_{THz} (t_{s}) |}^{2},

\tilde{θ} = arccos (cos θ - A_{THz} (t_{s}) / q) .

These quantities have the meaning of the electron energy and emission angle before entering the THz field. The last factor in Eq. (5) is the usual DDCS of photoionization of the l₀ shell of the atom by the XUV pulse alone which can be presented in a standard form

\frac{d σ^{(0)} ({\tilde{E}}_{s}, {\tilde{θ}}_{s})}{d Ω} = \frac{σ_{l_{0}}^{(0)} ({\tilde{E}}_{s})}{4 π} (1 + β ({\tilde{E}}_{s}) P_{2} (cos {\tilde{θ}}_{s})) .

Here

σ_{l_{0}}^{(0)} ({\tilde{E}}_{s})

and β(Ẽ_s) are the total cross section and anisotropy parameter, P₂(x) is the second Legendre polynomial. We suppose that information on

σ_{l_{0}}^{(0)} ({\tilde{E}}_{s})

and β(Ẽ_s) is available [19]. The cross section

\frac{d σ^{(0)}}{d Ω} ({\tilde{E}}_{s}, {\tilde{θ}}_{s})

is a slowly varying function of energy (out of resonance region) and in a narrow interval of energies and angles can be considered as constant.

Thus the DDCS for ionization by the XUV pulse in the presence of a THz field is factorized into the product of the DDCS of ionization of the atom by the XUV pulse alone, taken at shifted energy and angle, and the factor which depends on characteristics of the pulses. Note, that the XUV pulse is not assumed to be short unlike the case in [11]. The FEL pulse duration is only limited by the general requirement that during this pulse the THz streaking field is a monotonous function of time.

Equation (5) can be directly used to retrieve the XUV pulse from a given electron energy spectrum:

ℰ_{XUV}^{2} (t_{s}) = \frac{E_{THz} (t_{s}) \sqrt{q_{0}^{2} - 2 E {sin}^{2} θ}}{2 π} \frac{d σ (E, θ)}{d E d θ} {[\frac{d σ^{0} (\tilde{E}, \tilde{θ})}{d E d θ}]}^{- 1} .

The scheme of pulse retrieval is as follows. For each value of energy E = q² and a given time-dependence of the THz vector potential A_THz(t), the emission moment t_s is determined with Eq. (4). Then the energy Ẽ and angle θ̃ are calculated according to Eqs. (6) and (7), respectively. Finally using Eq. (9) the XUV pulse is evaluated. This procedure is extremely fast and can be easily applied for shot-to-shot on-line pulse retrieval. It is worthy to note that for a reliable FEL pulse retrieval, accurate information on the time dependence of the THz electric field is quite important.

3. Examples and discussion

In order to illustrate the application of the suggested method we have made pulse restoration for an arbitrary pulse and a model electric field of a single-cycle THz pulse. The latter is shown in Fig. 1(a) in units kV/cm. (For electric field 1 a.u. = 5.142 · 10⁶ kV/cm.) The corresponding vector potential which is defined as $A_{THz} (t) = - \int_{t}^{\infty} E_{THz} (t) d t$ is shown in Fig. 1(b) (black curve) in atomic units. It was presented by the analytical expression:

A_{THz} (t) = \frac{α (t - t_{1}) (t - t_{2})}{1 + β {(t - t_{0})}^{4}} .

The parameters (in atomic units) α = 2.8 × 10⁻⁹, β = 1.5 × 10⁻¹⁷, t₀ = −0.2 × 10⁴, t₁ = −1.5 × 10⁴, t₂ = 2.0 × 10⁴ were chosen in order to imitate a single-cycle THz pulse similar to that used in experiment [7,20]. In the same Fig. 1(b) the dashed (red) curve shows some model FEL pulse. It was chosen, rather arbitrary, as a pulse with three maxima. In a larger scale it is shown in Fig. 2 (circles). The pulse is described as a combination of three Gaussian:

ℰ_{XUV} (t) = \sum_{i = 1}^{3} a_{i} exp [- {(t + t_{i})}^{2} / b_{i}] .

The total duration of the pulse was chosen to be about 300 fs. At such a duration the non-linearity of the vector potential may be important. For narrower pulses the retrieval is simpler since the ramp of the vector potential is almost linear and a standard linear approach is applicable.

Fig. 1 (a) Model THz electric field in kV/cm as a function of time in fs. (b) The corresponding vector potential of the THz field in atomic units as a function of time (black curve). Red dashed curve shows the XUV pulse in arbitrary units.

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Fig. 2 An imitation of the FEL pulse used as an example. Black circles show the pulse calculated using Eq. (11) with parameters (in atomic units) a₁ = 1.5, a₂ = 1.7, a₃ = 2.3, b₁ = 7.0 × 10⁶, b₂ = b₃ = 4.0 × 10⁶, t₁ = 10.5 × 10³, t₂ = 6.0 × 10³, t₃ = 15.0 × 10³. Red curve is a pulse retrieved from the four spectra shown in Fig. 3. The four retrieved pulses coincide with the accuracy of the line width.

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For simplicity we have assumed that the angular distribution of electrons without THz field is isotropic, β(Ẽ) = 0 i.e the cross section $\frac{d σ^{(0)}}{d E d θ} (\tilde{E}, \tilde{θ})$ is a constant independent of θ. Using SFA (Eq. (2)) with the vector potential from Fig. 1, we have calculated the streaking spectra which are shown in Fig. 3 for four different detection angles.

Fig. 3 The electron spectra calculated for different emission angles. The electrons are produced in photoionization by the XUV pulse shown in Fig. 2 assisted by the THz field shown in Fig 1. The curve numbers correspond to the following emission angles: 1: θ = 0; 2: θ = π/6; 3: θ = π/4; 4: θ = π/3.

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Now we suppose that the above calculated spectra are obtained in a streaking experiment and try to reconstruct the FEL pulse from these spectra using Eq. (9). (Naturally, we use the same vector potential (10) shown in Fig. 1.) All four spectra, obtained for four different emission angles θ, have yielded fairly identical pulses shown in Fig. 2 by the solid curve. The four curves coincide within the width of the line and perfectly agree with the initial pulse (circles). Note that the opportunity to retrieve the same pulse from a few measurements at different emission angles enhances the accuracy of the retrieving procedure.

4. Conclusion

In conclusion, we suggest a very simple analytical expression which permits to retrieve the FEL pulse from the THz streaking spectrum. The suggested expression is based on quantum mechanical SFA description of the streaking process and valid for an arbitrary shape of the THz vector potential if this vector potential is monotonous in the time range when the FEL pulse is operative. The suggested expression can be used for on-line single-shot temporal characterization of the FEL pulses. The accuracy of the retrieved results depends on the precision of the information about the time-dependence of the THz field. The pulse retrieval can be made more precise by combining the results of simultaneous measurements at different scattering angles.

Funding

The Spanish Ministerio de Economia y Competividad (FIS2016-76617-P, FIS2016-7647); Donostia International Physics Center (San Sebastian/Donostia Spain).

Acknowledgments

A.K. Kazansky acknowledges the support of the Spanish Ministerio de Economia y Competitividad (grants FIS2016-76617-P and FIS2016-76471-P). N.M. Kabachnik acknowledges hospitality and financial support from DIPC (Donostia/San Sebastian) and from the theory group in cooperation with the SQS work package of European XFEL (Hamburg).

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Fast retrieval of temporal characteristics of FEL pulses using streaking by THz field

Abstract

1. Introduction

2. Theoretical background

3. Examples and discussion

4. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (3)

Equations (11)

Optics Express