Characterizing isolated attosecond pulses with angular streaking

Siqi Li; Zhaoheng Guo; Ryan N. Coffee; Kareem Hegazy; Zhirong Huang; Adi Natan; Timur Osipov; Dipanwita Ray; Agostino Marinelli; James P. Cryan

doi:10.1364/OE.26.004531

1. Introduction

We present a technique to fully characterize the temporal and spectral profile of sub-femtosecond x-ray and XUV pulses on a single-shot basis. Our technique is a variant of the well-known “attosecond streak camera” technique that has been used to measure the temporal profile of isolated attosecond pulses (IAPs) [1]. Briefly, the attosecond streak camera relies on ionization in the presence of an infrared (IR) laser field. An IAP with sufficiently high photon energy will ionize an atomic system, releasing photoelectrons with a kinetic energy equal to the difference between the photon energy and the ionization potential. Due to the presence of the IR field, the energy of the ionized photoelectron will be modulated, or streaked, depending on the phase of the IR laser field at the time of ionization [2]. Through the streaking interaction, information about the temporal profile of the IAP will be encoded in the energy (and momentum) distribution of the emitted photoelectrons. More specifically, the streaked electron momentum distribution contains real-time information of ultrafast processes which affect the outgoing electron wavepacket (EWP) on ultrafast timescales [3]. Examples of ultrafast processes that delay and reshape the outgoing EWP include the interaction between the EWP and the atomic potential [4–6], and electron-electron correlation effects [7, 8]. Streaking measurements have also been applied to solid-state systems observing similar effects [9–11]. In the event that the outgoing EWP is highly energetic, > 50 eV, and away from any resonance structures, the delay and reshaping of the EWP is minimal, so the primary contribution to the temporal profile of the photoelectron wavepacket is due to the temporal profile of the IAP [12, 13].

In a number of previous works, a linearly polarized streaking laser was used to produce modulations in the measured photoelectron energy distribution. When the delay between the streaking laser and an isolated attosecond pulse or a train of attosecond pulses is varied, a spectrogram of the EWP is recorded. A number of different algorithms have been developed to reconstruct the EWP from the recorded spectrogram [14–22]. However, recording the requisite spectrogram requires shot-to-shot reproducibility of the IAP and accurate timing synchronization between the two pulses over the collection time. If the jitter in either the IAP profile or relative arrival time of the streaking laser is large, a single-shot technique is needed. Assuming the IAP arrives at the zero-crossing of the streaking laser cycle, where the energy shift imparted to the outgoing electron is linear as a function of time, it is possible to use single-shot information to reconstruct the IAP [1]. Such a method has recently been used at a free electron laser (FEL) facility to measure the temporal profile of self-amplified spontaneous emission (SASE) pulses (stochastic spikes across the x-ray pulse duration) [23].

In contrast to a linearly polarized field, which has a time varying amplitude with a constant direction, a circularly polarized field has nearly constant amplitude with a varying direction. This interesting property has lead to the development of angular streaking, which replaces the linearly polarized streaking field with a circularly polarized one, and encodes the EWP temporal profile into the angular distribution of the photoelectrons momentum distribution, instead of the 1-D photoelectron energy distribution [4, 24–27]. If the IAP duration is shorter than the period of the streaking laser field, it is possible to measure IAP profile with single shot experiments, since the IAP profile is encoded in the angle-resolved photoelectron momentum distribution. In this work, we demonstrate that complex photoelectrons momentum distribution recorded in an angular streaking experiment can be used to reconstruct the initial EWP in a highly robust way.

The organization of this paper is as follows. In the next section we present the theoretical model of the streaking process and show simulated photoelectrons momentum distributions. In Sec. 3 we explain the reconstruction algorithm in detail, and present reconstruction results with simulated detector images. In Sec. 4 we study the performance of the reconstruction algorithm under experimental considerations. In Sec. 5 we discuss advantages of our reconstruction technique and potential for improvements.

2. Streaking simulation

In order to simulate the effect of streaking, we analytically solve the time-dependent Schrodinger equation for the laser-atom interaction in the Strong-Field Approximation (SFA), which neglects the effect of the Coulomb potential on the liberated electron [28]. In this approximation, the probability amplitude for finding an electron with momentum $\vec{p}$ is given by,

b (\vec{p}) = - i \int_{- \infty}^{+ \infty} d t [\vec{E} (t) \cdot \vec{d} (\vec{p} + \vec{A} (t))] \exp [- i \int_{t}^{+ \infty} d t^{'} {{(\vec{p} + \vec{A} (t^{'}))}^{2} / 2 + I_{p}}]

where

\vec{E}

is the total electric field,

\vec{d}

is the dipole moment for the transition to the continuum states,

\vec{A}

is the vector potential of the streaking laser, and I_p is the ionization potential of the atom. The integral in the exponential starts from the time of ionization, t, and continues until the measurement is made at t → ∞. The outer integral sums over all possible ionization times. Due to the high frequency oscillations in the carrier frequency of the IAP, only the IAP field contributes to the outer integral to compensate for the fast oscillating phase in the exponential. Therefore, we can replace the total electric field

\vec{E}

with the electric field of the IAP,

{\vec{E}}_{X}

. In the results presented below, the dipole moment is approximated as a momentum independent function with a cos²θ distribution, where θ is the initial emission angle relative to the IAP polarization. For the streaking laser vector potential we use the form,

\vec{A} (t) = A_{0 x} (t - t_{d}) \cos [ω_{L} (t - t_{d}) + ϕ_{x}] \hat{x} + A_{0 y} (t - t_{d}) \sin [ω_{L} (t - t_{d}) + ϕ_{y}] \hat{y},

where t_d is the time delay of the streaking laser envelope relative to the IAP, and ω_L is the streaking laser frequency. For a circularly polarized laser pulse, ϕ_x = ϕ_y = ϕ is the initial phase offset, and A₀_x(t) = A₀_y(t) = A₀(t) is the amplitude of the vector potential. In evaluating Eq. (1) we assume that the streaking laser pulse duration, τ_L, is much longer than the streaking laser period, T_L, (τ_L ≫ T_L), and envoke the slowly varying envelope approximation, A₀(t) ~ A₀. We also assume that the IAP arrives at the peak of the streaking laser envelope, t_d = 0. Thus, Eq. (2) simplifies to

\vec{A} (t) = A_{0} \cos [ω_{L} (t) + ϕ] \hat{x} + A_{0} \sin [ω_{L} (t) + ϕ] \hat{y}

.

Note that Eq. (1) calculates the transition amplitude as a function of the photoelectrons final momentum, b(p_x, p_y, p_z), in a coordinate system where z denotes the longitudinal coordinate parallel to beam propagation, and x and y denote the transverse coordinates. The experimentally relevant observable is the probability distribution for the photoelectrons momenta,

B_{3 D} (p_{x}, p_{y}, p_{z}) = {| b (p_{x}, p_{y}, p_{z}) |}^{2} .

We consider two possible measurements techniques that could be achieved by different experimental geometries. A common geometry for photoelectron experiments uses a set of electrostatic lenses to project the 3-D electron momentum distribution to a 2-D plane [30]. In the so-called velocity map imaging spectrometer (VMI), the recorded photoelectron momentum distribution is given by

B_{p} (P_{x}, P_{y}) = \int d p_{z} {| b (p_{x}, p_{y}, p_{z}) |}^{2},

where

\vec{P}

represents a 2-D vector in the plane of projection. Another possible experimental configuration uses a circular array of electron time-of-flight detectors to record a 2-D slice of the 3-D momentum distribution. Then the measured momentum distribution is given by

B_{s} (p_{x}, p_{y}) = {| b (p_{x}, p_{y}, p_{z} = 0) |}^{2} .

The important feature of angular streaking is that the circularly polarized streaking laser imposes a time-dependent momentum kick to the photoelectrons. The angle of the kick is determined by the relative phase between the IAP and the streaking laser pulse. In Fig. 1 we demonstrate the simulated photoelectron momentum distributions recorded in both the projected and sliced detection geometries, for various phases between the the IAP and the streaking laser pulse and for various FWHM durations of the IAP. The momentum distribution begins to widen and shear as the IAP duration increases. As the IAP duration approaches a significant fraction of the streaking laser period, circular fringes begin to appear. These patterns originate from the quantum interferences between photoelectrons released at different times experiencing different momentum kicks but having the same final momentum [31], similar to what has been observed in strong-field driven electron rescattering [32]. This effect is particularlly visible when the EWP is strongly chirped as shown in Fig. 2. These interferences are intra-cycle interferences and are characteristically different from side-band formation (seen in panels (b.3) and (b.4), which is due to interferences between photoelectrons delayed by multiples of streaking laser cycles [26, 33]. This intra-cycle interference effect is particularly apparent when two time-delayed pulses produce overlapping momentum features, as shown in Fig. 2.

Fig. 1 Rows (a) and (b) show simulated photoelectron momentum distributions for various IAPs projected along the laser propagation direction as described by Eq. (4). Panel (a.1) shows the projected electron momentum distribution for a Gaussian IAP with a FHWM duration of 300 as in the absence of the streaking field. Panels (a.2-4) demonstrate the effect of the streaking laser field on the projected photoelectron momentum distribution from (a.1) for different values of ϕ (from Eq. (2)). The red arrow indicates the direction of the instantaneous vector potential at the peak of IAP. Row (b) demonstrates the effect of the IAP duration on the projected momentum distribution when the streaking laser field is directed along the P_y axis (ϕ = π/2): (b.1) 600 as, (b.2) 1.2 fs, (b.3) 2.4 fs, (b.4) 4.8 fs. Row (c) shows simulated slices of the photoelectron momentum distribution according to Eq. (5), as a function of IAP duration. Panel (c.1) shows the p_z = 0 slice of the photoelectron momentum distribution for a Gaussian IAP with a 300 as FWHM duration, in the absence of the streaking laser field. Panels (c.2–4) show the sliced momentum distribution for Gaussian IAPs with FWHM duration of 300 as, 600 as, and 1.2 fs, when the streaking laser field is directed along the p_y direction, i.e. ϕ = π/2. The simulation considers 25 eV photoelectrons interacting with a 1.3 µm laser field with U_p = 4 eV (where U_p = |A₀|²/4 is the ponderomotive potential of the streaking laser field). The photoelectron momentum distribution intensity is normalized to 1.

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Fig. 2 Photoelectron momentum distributions for more complicated IAP structures. Panels (a) and (b) simulate the interaction of a chirped IAP (300 as FWHM) of 1.2 fs FWHM duration with the same streaking laser field as Fig. 1. Panels (c) and (d) simulate the streaked photoelectron momentum distribution from two 600 as FWHM IAPs separated by a quarter of the streaking laser period. Panels (a) and (c) show the projected momentum distribution (Eq. (4)). Panels (b) and (d) show sliced momentum distributions (Eq. (5)).

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3. Reconstruction algorithm

In the following sections we walk through the reconstruction algorithm in detail and discuss its robustness and limitations.

3.1. Reconstructing attosecond x-ray pulses from angular streaking

Our reconstruction method is based on the forward transformation of a set of basis waveforms. We suppose that the original EWP can be decomposed into a set of basis functions α_n(t) according to

\vec{E} (t) \cdot \vec{d} \equiv EWP (t) = \sum_{n} c_{n} α_{n} (t),

where c_n is a complex number. Recall that in the limit of highly energetic outgoing electrons away from any resonance structures, the major contribution to the temporal profile of the EWP is due to the electric field temporal profile

\vec{E} (t)

. We simulate the streaking pattern

f_{n} (\vec{p})

associated with each basis function, α_n(t), using Eq. (1). Then the probability amplitude for finding an electron with momentum

\vec{p}

is given by

b (\vec{p}) = \sum_{n} c_{n} f_{n} (\vec{p}),

and thus the measured momentum distribution is given by

B (\vec{p}) = \sum_{n} \sum_{m} c_{n}^{*} c_{m} F_{n m} (\vec{p}),

where

F_{n m} = f_{n}^{*} f_{m}

. Using this methodology, the reconstruction problem is reduced to solving for the complex coefficients c_n. To solve the over-constrained, non-linear system of equations, we minimize the figure of merit defined as

χ^{2} = \sum_{p_{x}, p_{y}} {| M (p_{x}, p_{y}) - B (p_{x}, p_{y}) |}^{2},

where M(p_x, p_y) is the measured photoelectron 2-D momentum distribution (either projected or sliced), and B(p_x, p_y) is constructed from Eq. (8) using either Eq. (4) or (5). We use Matlab’s fminunc function to solve for the complex coefficients c_n.

3.2. von Neumann representation of the EWP

The choice of basis functions, α_n(t), is flexible. Initially one may be tempted to consider basis functions well-defined in time, such as Dirac delta functions centered at various time points, or possibly a Fourier basis. However, a non-linear fitting algorithm will perform most reliably for a basis set where the EWP exhibits the most sparsity, i.e. the EWP can be represented with very few basis functions. We have found that the ideal basis for representing the wavepackets created by short-pulse ionization is the von-Neumann basis [34]. It is worth explicitly recapturing the important features of the von-Neumann representation in the following:

The von-Neumann representation is a joint time-frequency representation for wavepackets, meaning that the von Neumann basis functions are defined by lattice points in both time and frequency.
It has been shown in Ref. [34] that the information content of either a frequency or time basis with N equally spaced points is identical to the von Neumann representation with a lattice of $\sqrt{N}$ points in time and $\sqrt{N}$ points in frequency.
The von-Neumann basis function evaluated in the time domain is given by $α_{i j} (t) = {(\frac{1}{2 α π})}^{\frac{1}{4}} \exp [- \frac{1}{4 α} {(t - t_{j})}^{2} - i t ω_{i}],$ where t_j(ω_i) are the temporal (frequency) lattice points in the von Neumann representation, and α is a constant specified below.
It is straightforward to obtain the von-Neumann basis function in the frequency domain by taking the Fourier transform of Eq. (10), $α_{i j} (ω) = {(\frac{2 α}{π})}^{\frac{1}{4}} \exp [- α {(ω + ω_{i})}^{2} - i t_{j} (ω + ω_{i})] .$
It is clear from Eqs. 10 and 11 that the von-Neumann basis functions are localized in both time and frequency.
To obey the Fourier relation between time and frequency, the normalization constant, α must be equal to $\frac{T}{2 Ω}$ , where T is the time window and Ω is the frequency window for the lattice.
Given the time domain representation of an EWP, ψ(t), the von Neumann representation of the wavepacket, Q_ij, is given by $\begin{matrix} Q_{i j} = \sum_{k, l} {(S_{i j, k l})}^{- 1} \int d t α_{k l}^{*} (t) ψ (t), \\ S_{i j, k l} = \int d t α_{i j}^{*} (t) α_{k l} (t) . \end{matrix}$
Likewise, the time domain representation of the EWP can be retrieved from the von Neumann representation according to $ψ (t) = \sum_{i j} Q_{i j} α_{i j} (t)$

To demonstrate the utility of von Neumann representation for IAPs, we show the decomposition of a characteristic IAP, with N = 6 × 6, N = 8 × 8, and N = 10 × 10 lattice points in Fig. 3. The sparsity of the von Neumann representation can be seen in the 2-D plot of |Q_ij| (top row of Fig. 3) – only a few pixels contribute significantly to the overall amplitude of the coefficients. For example, in the case of N = 8 × 8, only 6 out of the 64 coefficients contain more than 20% of the maximum amplitude. Through empirical search, we have found that for the purpose of characterizing most IAPs it is sufficient to use an N = 8 × 8 von Neumann basis.

Fig. 3 The von Neumann representation (|Q_ij|) of a characteristic IAP (top row) using N = 6 × 6 (left), N = 8 × 8 (middle), and N =10 × 10 (right) lattice points. The vertical axis Δω denotes the frequency difference from the central frequency. The bottom row shows the electric field envelope amplitude and phase as a function of time, using Eq. (13).

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3.3. Results

To demonstrate the algorithm, we simulate angular streaking of a complex EWP, with central energy near 85 eV, streaked by a circularly polarized 1.3 µm laser pulse with $U_{p} = | {\vec{A}}_{0} |^{2} / 4 = 2 eV$ according to Eq. (1). We then employ our reconstruction algorithm to obtain the von Neumann representation of the unstreaked EWP. Figure 4 shows the reconstruction result for both experimental geometries considered here, a projected momentum distribution (Eq. (4), top row) and a sliced momentum distribution (Eq. (5), bottom row). This result demonstrates that this algorithm works well regardless of the method used to collect the data. We have intentionally used an EWP with an intricate substructure in the time and frequency domain to showcase the power of our technique. The secondary bump in the time domain profile gives rise to quantum interferences between photoelectrons generated at different phases of the streaking laser field that arrive at the detector with the same final momentum. Since the particle momentum is linearly proportional to the measured radius on the detector, this interference between the photoelectrons wavepackets appear in the momentum distribution as circular fringes. The fringes implicitly constrain the possible values for the coefficients Q_i,j, and our reconstruction algorithm utilizes such constraints to successfully and completely reconstruct these complicated quantum mechanical interference structures.

Fig. 4 Reconstruction of a characteristic IAP. From left to right: input 2-D projected photoelectron momentum distribution (top) from Eq. (4) and a 2-D slice of 3-D momentum distribution (bottom) from Eq. (5); reconstructed von Neumann representation of the IAP; time domain reconstruction of the IAP intensity profile; frequency domain reconstruction of the IAP intensity.

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3.4. Convergence

In implementing the non-linear fitting reconstruction algorithm, we start with a random initial guess for the coefficients c_n. The minimization process takes the initial guess as a seed parameter and searches for a local minimum by iteratively moving in the direction of the largest negative gradient direction until the gradient is sufficiently small. As explained above, the reconstruction algorithm does not make any assumptions about the attosecond pulse other than the time and frequency window that sets the boundaries of the search. Therefore it is important to study the convergence of the algorithm.

For the convergence study we consider the reconstruction from the projected momentum distribution, Eq. (4). We run a total of 100 iterations on the same characteristic IAP with 100 random Monte Carlo initial guesses. We characterize the convergence by comparing the reconstructed pulse duration and spectral width. They are defined as follows

σ_{x} = \sqrt{\frac{\sum_{j}^{N} {(x_{j} - \frac{\sum_{i}^{N} I (x_{i}) x_{i}}{\sum_{i}^{N} I (x_{i})})}^{2} I (x_{j})}{\sum_{j}^{N} I (x_{j})}},

where I(x) is the intensity of the electric field, and x represents either t or ω, the time or frequency lattice points. The input IAP has σ_t = 176 as and σ_ω = 3.18 eV. Figure 5 shows the convergence of the 100 reconstructed intensities in time and frequency domains. The retrieved temporal widths and spectral widths center around the expected values of the input IAP. This shows that our reconstruction algorithm is insensitive to the initial guess and converges to the correct solution.

Fig. 5 Reconstruction results from 100 iterations of the minimization algorithm. The input field intensity (red dashed) and phase (green dashed) are shown along with the reconstructed field intensities (blue shaded) and phase (orange shaded), in the temporal (left) and spectral (right) domains. Each iteration begins with a different random initial guess. We fix the reconstructed phase at one data point to account for constant phase difference. The inset shows a histogram of σ_t,ω calculated according to Eq. (14). The vertical red dashed line indicates the input σ_t,ω.

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3.5. Limitations

To explore the limitations of this algorithm, we explore the effect of the second and third order spectral phase on the reconstruction. We start from a 250 as FHWM Fourier-transform-limited Gaussian pulse E₀(ω). In the spectral domain, we vary the second and third order spectral phase

E (ω) = E_{0} (ω) \exp {i α_{2} {(ω - ω_{0})}^{2} + i α_{3} {(ω - ω_{0})}^{3}} .

We scan different combinations of α₂ and α₃, and run the reconstruction algorithm for each (α₂, α₃) while keeping the same N = 8×8 von Neumann basis. We calculate the root-mean-square error (RMSE) of the temporal field,

RMSE = {[\frac{1}{N_{k}} \sum_{k} {(| E (t_{k}) | - | \sum_{i, j} Q_{i j} α_{ω_{i} t_{j}} (t_{k}) |)}^{2}]}^{\frac{1}{2}},

where t_k indicates the time lattice points with field intensities greater than 20% of the maximum. Figure 6 shows the results for the 2-D parameter space. We also exemplify four points in the (α₂, α₃) space to show their effects on the temporal pulse structures. When α₂ and α₃ are both close to 0, the pulse is a simple Gaussian structure with a width determined by the input spectrum width. As α₃ starts to move away from 0 (see the upper left plot in Fig. 6), secondary peaks develop next to the main peak, introducing intra-cycle interference in the momentum distribution. These intra-cycle interference fringes not only constrain the coefficients in the fitting process to reach a convergent solution of the IAP, but one can also as a first estimate retrieve an approximate delay between the two pulses from the separation between the fringes. The reconstruction easily resolves these structures, which shows the algorithm is capable of reconstructing satellite IAPs as well. As α₂ gets farther from 0, the pulse duration starts to significantly increase and takes up the entire temporal window. The entire range of the parameter space (α₂, α₃) covers pulse durations from 250 as to 3.9 fs FWHM, reaching a significant portion of the streaking laser period of 4.3 as. The RMSE of the temporal field overall falls under 0.05. This is sufficient for most IAP sources with durations of attosecond or sub-femtosecond scale.

Fig. 6 Time domain reconstruction RMSE for different combinations of α₂ and α₃ (see text). The red arrows identify four specific (α₂, α₃) for comparison in the insets. In the insets, the blue line is the input field intensity, and the red starred dash line is the reconstructed field intensity.

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For the demonstration of our reconstruction algorithm we have considered a 1.3 µm streaking laser wavelength. We have found that the performance of the reconstruction technique only depends on the ratio of the IAP duration to the streaking laser period. In general, in streaking experiments, it is important to have an IAP duration (τ) less than the laser period (T_L). When τ > T_L, inter-cycle interference features begin to appear in the photoelectron momentum distribution, as in (b.4) of Fig. 1. When these inter-cycle interference structures appear in the momentum distribution, the reconstruction algorithm is no longer able to distinguish phase difference of multiples of 2π.

However, as an experimental guide, one can resort to the features of the momentum distribution to identify if the relation τ < T_L is obeyed. Panels (b.1–3) and (b.4) in Fig. 1 provide a comparison. When τ < T_L, panels (b.1–3), the momentum distribution may be point symmetric about a centroid away from the image centroid, or axial symmetric along the ${\vec{A}}_{0}$ direction. When τ > T_L, panel (b.4), the angular kick spans out the entire 2π circle so the momentum distribution is point symmetric about the image centroid. This feature distinguishes the IAP duration longer than the streaking laser period, indicating the reconstruction performed under this condition is no longer reliable.

4. Experimental considerations

4.1. Counting noise

In Sec. 3.3 we demonstrate the successful reconstruction of a characteristic IAP from photoelectron momentum distributions, using both the projected and sliced measurement geometries. For experiments that have stable IAP sources with negligible shot-to-shot noise, it is straightforward to integrate over multiple shots to overcome counting statistics from signal fluctuations. For unstable IAP sources, such as SASE x-ray FELs, it is preferable to make single-shot measurements. We show that our reconstruction algorithm is robust against counting noise.

To simulate the effect of experimental counting statistics, we take the characteristic IAP photoelectron momentum distribution calculated using Eqs. (4) and (5), and sample this distribution according to:

Y_{\vec{p}} (k) = \frac{μ_{\vec{p}}^{k} e^{μ_{\vec{p}}}}{k!},

where

Y_{\vec{p}} (k)

is the probability of measuring k electrons with momentum

\vec{p}

,

μ_{\vec{p}} = N \times B (\vec{p})

is the expected electron yield at momentum

\vec{p}

, and N is the average number of electrons in a recorded momentum spectrum. We run 100 iterations of the reconstruction and plot the reconstructed temporal and spectral intensities in Fig. 7. Each iteration starts with a different 2D momentum spectrum sampled according to Eq. (17). We study the effect of various Poisson noise levels with total electron numbers N = 20 K, 10 K, and 5 K. The standard deviation of the temporal width σ_t distribution is 5 as, 7 as, and 11 as, for 20K, 10K, and 5K respectively. Likewise, the standard deviation of the spectral width σ_ω distribution is 0.08 eV, 0.14 eV, and 0.18 eV, for 20K, 10K, and 5K respectively. We plot the reconstructed phase with a constant shift set by the maximum input phase in the time domain and the minimum input phase in the frequency domain. In center regions of the major and the secondary peaks, the reconstructed phase converges to the expected value. The broadening of the phase begins to show up only when the intensity falls below 5% of the maximum. The effect of the counting noise can be seen in the broadening of the σ_t,ω histograms as well as the broadening of the phase convergence in the low intensity tails. In the case of the noisiest measurements considered, 5K electrons per image, the reconstruction algorithm is able to produce the temporal and spectral widths within 7% uncertainty compared the input pulse. Overall, this shows that our non-linear fitting reconstruction algorithm is robust against counting noise, and can be used as a single-shot diagnostic for IAPs assuming one can record thousands of electrons in a single shot.

Fig. 7 Convergences of the reconstruction algorithm for total electron numbers (from left to right) 20K, 10K, and 5K, are illustrated in the time (top) and spectral (bottom) domains. The red and green dash lines represent the input field intensity and phase respectively, and the shaded blue and orange lines represent the 100 reconstructed field intensity and phase curves respectively. Each iteration starts with a different Poisson sampled momentum distribution with a fixed total number of electrons, and with a different initial guess. We fix the reconstructed phase at one data point to account for constant phase difference. The smaller figures are histograms of σ_t,ω calculated according to Eq. (14). The vertical red dashed line indicates the input σ_t,ω.

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4.2. Properties of streaking laser

As discussed in Sec. 2, our reconstruction method assumes that the vector potential of the streaking laser, A₀(t), is approximately constant over the duration of the laser cycle. This approximation is valid when the streaking laser pulse duration τ_L is large compared to the streaking laser period, T_L. We have also assumed that t_d = 0, which means that the IAP arrives at the peak of the streaking laser envelope. This can be achieved experimentally by scanning the delay between the streaking laser pulse and the IAP until the shift of the momentum distribution is the strongest. However, since τ ≪ τ_L where τ_L is the streaking pulse duration, the exact delay between the IAP and the streaking pulse is not critical, as long as their time overlap produces a significant momentum kick.

For the reconstruction described in Sec. 3, we treat the streaking pulse vector potential amplitude A₀ as a known input to the reconstruction algorithm. Experimentally, the value of A₀ depends on the intensity of the streaking laser and the exact delay between the streaking pulse envelope and the IAP, both of which may not be known. Nevertheless, the experimental value of A₀ can be estimated from the measured momentum distribution and the exact value refined by varying the value of A₀ to minimize χ² (Eq. (9)). An initial estimate of A₀ can be obtained from the offset of the centroid of the streaked photoelectron momentum distribution from the centroid of the unstreaked distribution. Comparing χ² in the neighborhood of this estimate leads to the experimental value of A₀. We demonstrate this process in Fig. 8, which shows the figure of merit χ² for various values of A₀. Clearly, the figure of merit reaches a minimum at U_p = 1.44 eV, which is the value used to generate the initial streaked photoelectron momentum distribution. Figure 8 also shows the temporal RMSE (Eq. (16)) for the reconstruction, which is minimized for the experimental value of A₀. Using the figure of merit it is possible to determine the value of the streaking vector potential used for the experiment. This is analogous to the reconstruction of the streaking laser field which has been demonstrated in spectrogram based reconstruction algorithms [16–22], but in our implementation we are only able to reconstruct the streaking laser field over the a single cycle of the streaking laser when the streaking interaction occurs.

Fig. 8 Figure of merit, χ², as a function of possible vector potential amplitudes $A_{0}^{^{'}}$ . We simulate the momentum distribution with $A_{0} = 2.4 \sqrt{eV}$ , i.e. U_p = 1.44 eV. We are able to find the correct vector potential amplitude as the $A_{0}^{^{'}}$ corresponding to the lowest cost χ².

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In addition to the magnitude of the streaking laser field, we also consider variations in the ellipticity of the streaking laser field. So far we have only considered the case of perfect circular polarization, i.e. A_x = A_y and Δϕ = ϕ_y − ϕ_x = 0 in Eq. (2). There are many experimental factors that could lead to variations from these conditions, and we will show how these variations affect the reconstruction. First we investigate the effect that an elliptical polarization has on the streaked photoelectron momentum distribution. Figure 9 shows a streaked photoelectron momentum distribution averaged over all possible values of ϕ_x. Such a momentum distribution could be obtained by scanning the delay between the streaking laser and the IAP over a single cycle of the streaking laser, or, in the case of poor synchronization between, the jitter between the two sources could create the same image by averaging many shots. In comparison to the perfect circularly polarized laser (left panel in Fig. 9), errors in the relative amplitude, R_A = A₀_x/A₀_y, result in an overall stretch of the averaged momentum distribution compared to the case when R_A = 1. Phase errors, Δϕ ≠ 0, result in a shearing of the momentum distribution. Therefore, the phase-averaged momentum distribution provides a sensitive tool for the experimentalist to adjust the polarization.

Fig. 9 Phase-averaged projected momentum distribution (see text) generated for the same IAP as in Fig. 4 and 5. From left to right: $R_{A} = \frac{A_{0 y}}{A_{0 x}} = 1$ , Δϕ = 0 (circular polarization); R_A = 1.5, Δϕ = 0 (amplitude error only); R_A = 1, Δϕ = 0.3 rad (phase error only); and R_A = 1.5, Δϕ = 0.3 rad (combined amplitude and phase error). The dashed white circle is a perfect circle to guide the eye.

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In practice, it will be impossible to eliminate all polarization errors, so we study the effect of polarization errors on the reconstruction algorithm. We simulate the photoelectron momentum distribution using the same IAP as in Fig. 4 and 5, but for elliptically polarized streaking laser fields with R_A ∈ (0, 1.5) and Δϕ ∈ (0, 0.3 rad). We run the reconstruction on the IAP using basis functions calculated assuming a perfectly circularly polarized streaking laser. Figure 10 shows the temporal RMSE for each reconstruction. We can see that the effect of amplitude error is not as severe as the phase error, which increases the RMSE significantly. However, the phase error considered here is a rather large angle which produces an easily recognizable shear in the phase-averaged momentum distribution (Fig. 9). Moreover, as can be seen in the reconstructed profiles in Fig. 10, even with such conservative estimates of the polarization error, the reconstruction is able to trace out the overall IAP profile.

Fig. 10 Time domain reconstruciton RMSE for different combinations of polarization errors. The red arrows identify the corner cases of the scan for comparison in the insets. In the insets, the blue line is the input field intensity, and the red starred dash line is the reconstructed field intensity.

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5. Conclusions

In this paper we discussed the application of angular streaking with circularly polarized laser field as a diagnostic tool for isolated attosecond pulses (IAPs). We presented a theoretical background for the streaking process and showed simulation of angular streaking in the strong field approximation. The focus of the paper is the reconstruction of the temporal and spectral profile of an IAP. Our algorithm is generally applicable to any type of streaking experiment where the amplitude of the measurement variable can be written as a linear function of the electric field. We showed that this algorithm converges to the correct solution under Monte Carlo random initial guesses. We explored the limitations of the algorithm by testing it with various pulse shapes, and we showed that the algorithm is able to reconstruct these complicated structures within 0.05 RMS error. We considered various experimental implementations of counting statistics and the streaking laser properties. The reconstruction algorithm is robust against counting statistics in the case that the momentum spectrum is made with upwards of five thousand electrons. We showed that the streaking pulse vector potential amplitude can be found empirically by running the reconstruction algorithm and minimizing the cost function. The imperfect polarization of the streaking pulse may degrade the accuracy of the reconstruction, but still preserves the overall shape of the IAP.

For the eternity of the paper we considered a streaking laser pulse with a wavelength of 1.3 µm. This choice was driven by a desire to reconstruct IAPs with a duration near 1 fs. As mentioned in Sec. 3.5, the important metric for choosing a streaking laser wavelength is the ratio of the IAP duration to the streaking laser duration. A majority of table-top based IAP production is done using 800 nm pulses. Typically IAPs generated with 800 nm pulses are much shorter than 1 fs. The reconstruction technique presented above should be completely applicable to table-top IAPs generated and streaked by 800 nm laser pulses.

In this paper we have focused on 2-D measurement of the 3-D momentum distribution. However, as shown in [29], one can use a reaction microscope to measure the 3-D photoelectron momentum distribution of a streaked electron wavepacket. Although this technique is currently only capable of recording a few electrons per shot, perhaps a modification of the detector to include a time resolving camera [35] could be used for single-shot measurements. In this case, the reconstruction technique can be modified to work with the 3-D measurement by calculating the basis functions directly in the 3-D momentum space.

Although this algorithm is a self-contained method, in that we do not resort to any external temporal or spectral information, such information could provide more stringent constraints and could potentially speed up the reconstruction process. Experimental uncertainties, such as the imperfect circular polarization of the streaking laser, insufficient count rates, large detector background, and/or inaccurate energy calibration, will degrade the performance of the algorithm and likely slow down the reconstruction process. Nevertheless, low resolution diagnostic tools (such as photon spectrometers) can provide additional constraints for the nonlinear fitting process and potentially improve performance.

The algorithm presented in this work is able to reconstruct much more complicated pulse structures than what has been presented above. This holds significant potential in a variety of applications. For example, consider the most basic operation mode of an x-ray FEL, which relies on the self-amplified spontaneous emission (SASE) process. The SASE process produces a stochastic train of femtosecond- and sub-femtosecond-width spikes that vary from shot to shot. The single-shot technique described in section 4.1 is able to reconstruct these structures exploiting the constraints imposed by the quantum interference patterns of photoelectrons released at various times along the pulse. There is also a great deal of interest in pushing FEL facilities beyond the femtosecond barrier, creating attosecond x-ray pulses [36] to study ultrafast electron motion with element specificity [37]. This reconstruction algorithm could be an important tool for the development and implementation of attosecond FELs.

In addition to attosecond pulse durations, next generation FEL sources have repetition rates up to 1 MHz. It is very challenging to build high resolution detectors with such high repetition rates. However, Fig. 4 employs a 64 × 64-pixel detector image to reconstruct the pulse, which suggests that we only need thousands of pixels worth of information to achieve successful reconstruction.

The robustness of the algorithm with limited resolution detectors indicates that we can likely use low resolution detectors working at high repetition rate to achieve this single-shot streaking experiment and reconstruct the x-ray pulse at these next generation facilities.

The ability to measure and reconstruct complicated pulse shapes will also prove useful in understanding attosecond scale phenomena which occur in photoionization. In the context of current streaking measurements, the reconstruction algorithm actually recovers the temporal profile of the outgoing EWP, $\vec{E} \cdot \vec{d}$ . Throughout the paper we have assumed the most of the structure in this outgoing EWP is due to the temporal profile of the IAP. For high energy photoelectrons sufficiently far away from any resonance structures this is generally a good approximation. However, attosecond streaking experiments are routinely made in highly structured continua. This structure is imprinted onto the electron wavepacket [3], and it should be possible to recover this structure using our same algorithm. In this way, we can gain insight into the intricate atomic or molecular structures that are challenging to probe otherwise.

We have made the code for the reconstruction algorithm available for download at [38].

Funding

US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Chemical Sciences, Geosciences, and Biosciences Division; DOE, Office of Science, BES (DE-AC02-76SF00515); DOE, Office of Science, BES, Scientific User Facilities, Accelerator and Detector Research Division Field Work Proposal (100317); National Science Foundation Grant (PHY-1535215).

Acknowledgments

We would like to acknowledge useful conversations with Nick Hartmann and Phillip Bucksbaum. SL acknowledges support from the Robert H. Siemann Graduate Fellowships in Physics.

References and links

1. J. Itatani, F. Quéré, G. L. Yudin, M. Yu. Ivanov, F. Krausz, and P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88(17), 173903 (2002). [CrossRef] [PubMed]

2. R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and M. Drescher, “Atomic transient recorder,” Nature. 427(6977), 817–821 (2004). [CrossRef] [PubMed]

3. R. Pazourek, S. Nagele, and J. Burgdörfer, “Attosecond chronoscopy of photoemission,” Rev. Mod. Phys. 87(3), 765 (2015). [CrossRef]

4. P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dörner, H. G. Muller, and U. Keller, “Attosecond ionization and tunneling delay time measurements in helium,” Sci. 322(5907), 1525–1529 (2008). [CrossRef]

5. M. Schultze, M. Fieß, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, and C. A. Nicolaides, “Delay in photoemission,” Sci. 328(5986), 1658–1662 (2001). [CrossRef]

6. M. Sabbar, S. Heuser, R. Boge, M. Lucchini, T. Carette, E. Lindroth, and U. Keller, “Resonance effects in photoemission time delays,” Phys. Rev. Lett. 115(13), 133001 (2015). [CrossRef] [PubMed]

7. M. Ossiander, F. Siegrist, V. Shirvanyan, R. Pazourek, A. Sommer, T. Latka, and R. Kienberger, “Attosecond correlation dynamics,” Nat. Phys. 13(3), 280–285 (2017). [CrossRef]

8. V. Gruson, L. Barreau, Á. Jiménez-Galan, F. Risoud, J. Caillat, A. Maquet, B. Carré, F. Lepetit, J. F. Hergott, T. Ruchon, and L. Argenti, “Attosecond dynamics through a Fano resonance: Monitoring the birth of a photoelectron,” Sci. 354(6313), 734–738 (2016). [CrossRef]

9. A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, and M. Drescher, “Attosecond spectroscopy in condensed matter,” Nature. 449(7165), 1029–1032 (2007). [CrossRef] [PubMed]

10. M. Schultze, M. Fieß, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, and C. A. Nicolaides, “Delay in photoemission,” Sci. 328(5986), 1658–1662 (2010). [CrossRef]

11. S. Neppl, R. Ernstorfer, E. M. Bothschafter, A. L. Cavalieri, D. Menzel, J. V. Barth, F. Krausz, R. Kienberger, and P. Feulner, “Attosecond time-resolved photoemission from core and valence states of magnesium,” Phys. Rev. Lett. 109(8), 087401 (2012). [CrossRef] [PubMed]

12. J. M. Dahlström, D. Guénot, K. Klünder, M. Gisselbrecht, J. Mauritsson, A. L’Huillier, A. Maquet, and R. Taïeb, “Theory of attosecond delays in laser-assisted photoionization,” Chem. Phys. 414, 53–64 (2013). [CrossRef]

13. M. Isinger, R. J. Squibb, D. Busto, S. Zhong, A. Harth, D. Kroon, S. Nandi, C. L. Arnold, M. Miranda, J. M. Dahlström, and E. Lindroth, “Photoionization in the time and frequency domain,” Sci. 358(6365), 893–896 (2017). [CrossRef]

14. P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Sci. 292(5522), 1689–1692 (2001). [CrossRef]

15. H. G. Muller, “Reconstruction of attosecond harmonic beating by interference of two-photon transitions,” Appl. Phys. B. 74, 17–21 (2002). [CrossRef]

16. Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A. 71(1) 011401 (2005). [CrossRef]

17. J. Gagnon, E. Goulielmakis, and V. S. Yakovlev, “The accurate FROG characterization of attosecond pulses from streaking measurements,” Appl. Phys. B. 92(1), 25–32 (2008). [CrossRef]

18. M. Chini, S. Gilbertson, S. D. Khan, and Z. Chang, “Characterizing ultrabroadband attosecond lasers,” Opt. Express 18(12), 13006–13016 (2010). [CrossRef] [PubMed]

19. J. Gagnon and V. S. Yakovlev, “The direct evaluation of attosecond chirp from a streaking measurement,” Appl. Phys. B. 103(2), 303–309 (2011). [CrossRef]

20. M. Lucchini, M. H. Brügmann, A. Ludwig, L. Gallmann, U. Keller, and T. Feurer, “Ptychographic reconstruction of attosecond pulses,” Opt. Express 23(23), 29502–29513 (2015). [CrossRef] [PubMed]

21. P. D. Keathley, S. Bhardwaj, J. Moses, G. Laurent, and F. X. Kärtner, “Volkov transform generalized projection algorithm for attosecond pulse characterization,” New J. Phys. 18(7), 073009 (2016). [CrossRef]

22. H. Wei, T. Morishita, and C. D. Lin, “Critical evaluation of attosecond time delays retrieved from photoelectron streaking measurements,” Phys. Rev. A. 93(5), 053412 (2016). [CrossRef]

23. W. Helml, A. R. Maier, W. Schweinberger, I. Grguraš, P. Radcliffe, G. Doumy, C. Roedig, J. Gagnon, M. Messerschmidt, S. Schorb, C. Bostedt, F. Grüner, L. F. DiMauro, D. Cubaynes, J. D. Bozek, Th. Tschentscher, J. T. Costello, M. Meyer, R. Coffee, S. Düsterer, A. L. Cavalieri, and R. Kienberger, “Measuring the temporal structure of few-femtosecond free-electron laser X-ray pulses directly in the time domain,” Nat. Photon. 8(12), 950–957 (2014). [CrossRef]

24. P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys 4(7), 565–570 (2008). [CrossRef]

25. N. Hartmann, G. Hartmann, and W. Helml, “Angle-Resolved Streaking for Complete Attosecond FEL pulse characterization,” Nat. Photonics (accepted for publication).

26. A. K. Kazansky, A. V. Bozhevolnov, I. P. Sazhina, and N. M. Kabachnik, “Interference effects in angular streaking with a rotating terahertz field,” Phys. Rev. A. 93(1), 013407 (2016). [CrossRef]

27. A. K. Kazansky, I. P. Sazhina, V. L. Nosik, and N. M. Kabachnik, “Angular streaking and sideband formation in rotating terahertz and far-infrared fields,” J. Phys. B. 50(10) 105601 (2017). [CrossRef]

28. M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, and T. Brabec, “Quantum theory of attosecond XUV pulse measurement by laser dressed photoionization,” Phys. Rev. Lett. 88(17), 173904 (2002). [CrossRef] [PubMed]

29. R. Boge, S. Heuser, M. Sabbar, M. Lucchini, L. Gallmann, C. Cirelli, and U. Keller, “Revealing the time-dependent polarization of ultrashort pulses with sub-cycle resolution,” Opt. Express 22(22), 26967–26975, 2014. [CrossRef] [PubMed]

30. A. TJB. Eppink and D. H. Parker, “Velocity map imaging of ions and electrons using electrostatic lenses: Application in photoelectron and photofragment ion imaging of molecular oxygen,” Rev. Sci. Instrum. 68(9) 3477–3488 (1997). [CrossRef]

31. A. K. Kazansky, A. V. Bozhevolnov, I. P. Sazhina, and N. M. Kabachnik, “Interference effects in angular streaking with a rotating terahertz field,” Phys. Rev. A 93(1) 013407 (2016). [CrossRef]

32. M. Spanner, O. Smirnova, P. B. Corkum, and M. Y. Ivanov, “Reading diffraction images in strong field ionization of diatomic molecules,” J. Phys. B. 37(12) L243 (2004) [CrossRef]

33. M. Meyer, P. Radcliffe, T. Tschentscher, J. T. Costello, A. L. Cavalieri, I. Grguras, A. R. Maier, R. Kienberger, J. Bozek, C. Bostedt, S. Schorb, R. Coffee, M. Messerschmidt, C. Roedig, E. Sistrunk, L. F. Di Mauro, G. Doumy, K. Ueda, S. Wada, S. Düsterer, A. K. Kazansky, and N. M. Kabachnik, “Angle-resolved electron spectroscopy of laser-assisted Auger decay induced by a few-femtosecond X-ray pulse,” Phys. Rev. Lett. 108(6) 063007 (2012). [CrossRef] [PubMed]

34. S. Fechner, F. Dimler, T. Brixner, G. Gerber, and D. J. Tannor, “The von Neumann picture: a new representation for ultrashort laser pulses,” Opt. Express 15(23) 15387–15401 (2007). [CrossRef] [PubMed]

35. M. Fisher-Levine and A. Nomerotski, “TimepixCam: a fast optical imager with time-stamping,” J. Instrum. 11(03) c03016 (2016). [CrossRef]

36. J. MacArthur, J. Duris, Z Huang, and A. Marinelli, “High Power Sub-Femtosecond X-Ray Pulse Study for the LCLS,” in Proceedings of the 8th International Particle Accelerator Conference(IPAC’17), pp.2848–2850.

37. S. Mukamel, D. Healion, Y. Zhang, and J. D. Biggs, “Multidimensional attosecond resonant X-ray spectroscopy of molecules: Lessons from the optical regime,” Annu. Rev. Phys. Chem. 64101–127, 2013. [CrossRef]

38. Matlab code for the reconstruction algorithm: https://www.mathworks.com/matlabcentral/fileexchange/65828-characterizing-isolated-attosecond-pulses-with-angular-streaking.

Characterizing isolated attosecond pulses with angular streaking

Abstract

1. Introduction

2. Streaking simulation

3. Reconstruction algorithm

3.1. Reconstructing attosecond x-ray pulses from angular streaking

3.2. von Neumann representation of the EWP

3.3. Results

3.4. Convergence

3.5. Limitations

4. Experimental considerations

4.1. Counting noise

4.2. Properties of streaking laser

5. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (10)

Equations (17)

Optics Express