† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11547183 and 11547241) and the Fundamental Research Funds for Central Universities, China (Grant Nos. JD1517 and 2652014012).
Using temporal and spectral methods, the effects of dispersion and filtering induced by Mo/Si multilayer mirrors reflection on incident attosecond pulses were studied. First, two temporal parameters, the pulse broadening factor, and the energy loss factor, were defined to evaluate the effects of dispersion and filtering. Then, by analyzing these temporal parameters, we investigated and compared the dispersion and filtering effects on attosecond pulses. In addition, we explored the origins of pulse broadening and energy loss by analyzing the spectral and temporal characteristics of periodic Mo/Si multilayer mirrors. The results indicate that the filtering effect induced by Mo/Si multilayer mirrors reflection is the dominant reason for pulse broadening and energy loss.
With the rapid progress of attosecond science in recent years,[1–3] multilayer mirrors in the extreme ultraviolet (EUV) and x-ray region for high-efficiency spectral and temporal control of attosecond pulses have attracted a great deal of interest.[4] As an efficient filter and reflector for attosecond sources, e.g., high-order harmonics (HOH),[5–8] the periodic Mo/Si multilayer mirror has been widely used in attosecond science for years.
In 2001, Dresher et al.[9] used a Mo/Si spherical periodic multilayer mirror with a 5-eV band centered at 90 eV to filter and focus EUV pulses. In 2002, Kienberger et al.[10] employed a high-reflectance Mo/Si multilayer mirror with a 15-eV band centered at 93 eV to select radiation for the photoionization experiments. In 2009, Abel et al.[11] made use of a Mo/Si multilayer mirror with a 4-eV band centered at 93 eV in the CEP-scanning experiments. In 2013, Zhan et al.[12] used a Mo/Si mirror for an isolated attosecond pulse reflection.
When using a multilayer mirror to reflect an attosecond pulse, the preservations of pulse duration and photon flux are usually the paramount requirements.[13] However, these requirements are hard to meet for a periodic multilayer mirror, due to its dispersion and filtering effects on the incident attosecond pulse, which can lead to pulse broadening and pulse energy loss. Although temporal pulse responses of periodic Mo/Si multilayer mirrors have been studied for years,[14–18] the effects of dispersion and filtering induced by Mo/Si multilayer mirrors reflection on attosecond pulses have not been investigated so far.
In this paper, we numerically studied the dispersion and filtering effects induced by Mo/Si multilayer mirrors reflection on attosecond pulses. First, two temporal parameters were defined to evaluate pulse broadening and energy loss of incident attosecond pulses quantitatively. Then, the effects of dispersion and filtering on attosecond pulses were investigated respectively. Finally, we compared the dispersion and filtering effects by analyzing the ratios of the pulse broadening factor and the energy loss factor of two effects. Research into the effects of dispersion and filtering induced by periodic Mo/Si multilayer mirrors reflection on attosecond pulses can improve the understanding of the origins of pulse broadening and pulse energy loss induced by periodic multilayer mirrors reflection, and provide a guide for attosecond multilayer mirrors design.
In Fig.
To analyze the effects of dispersion and filtering induced by Mo/Si multilayer mirrors reflection on incident attosecond pulses in the frequency domain, the spectral complex amplitude reflection coefficient r(ω) = |r(ω)|exp[iϕ(ω)], which is the main parameter describing reflective performances of a multilayer mirror, needs to be determined firstly by using a standard matrix method based on the Fresnel equations as follows:[19]
Using r(ω), we can calculate the spectral reflectivity R and the reflective phase ϕ(ω) of a multilayer mirror by
In addition, the group delay dispersion GDD of a multilayer mirror, which plays an important role in the pulse broadening, can be obtained by
The optical constants of Mo and Si used in the simulation were derived from the handbook edited by Henke et al.[20] To provide a reasonable prediction of the performances of a multilayer mirror, the inter-diffusion effect between Mo and Si layers in the multilayer was considered in all cases, following a proven model used in the realistic design of Mo/Si multilayers.[21]
To determine the effects of dispersion and filtering on an incident pulse in the time domain, a typical temporal method based on Fourier transform, which has been used in temporal analysis of the performances of multilayer mirrors by Ksenzov et al.,[22,23] Wonisch et al.,[24] and Suman et al.[25] previously, was employed in our simulation.
First, we need to obtain the spectral component E0(ω) of an incident pulse from its temporal component E0(t) by Fourier transform. Then, to obtain the spectral component of the reflected pulse E1(ω), simply multiply E0(ω) with |r(ω)|(or exp [iϕ(ω)]), which describes the filtering (or dispersion) effects induced by a multilayer mirror reflection, finally, inverse Fourier transform with the object of E1(ω) to obtain the temporal component E1(t) and intensity I1(t) of the reflected pulse. The whole calculation procedure can be expressed by the following equations:
In addition, to describe the temporal effects of dispersion and filtering on incident attosecond pulses quantitatively, two temporal parameters were induced. The first one is the pulse broadening factor PBF, which describes the pulse broadening induced by a multilayer mirror reflection, and can be defined as the reflected-to-incident pulse duration ratio by
![](cpb_25_9_097802/cpb160198ieqn1.gif)
![](cpb_25_9_097802/cpb160198ieqn2.gif)
Furthermore, all incident isolated attosecond pulses used in our simulation were assumed to be non-chirped Gaussian-shaped pulses, and their temporal component E0(t) can be expressed by
First, we studied the dispersion effect of periodic Mo/Si multilayer mirrors with different bi-layers (N = 1–50) on incident attosecond pulses with various durations (τ0 = 100 as, 200 as, 300 as, 400 as, 500 as (1 as = 10−18 s)) by calculating the pulse broadening factor PBFd and the energy loss factor ELFd (see Fig.
For each incident attosecond pulse, both PBFd and ELFd increase rapidly until reaching their maximums, then decrease slowly with the increase of N. The maximums of PBFd and ELFd show at the same N (named Nmax), and Nmax for each case are listed in Table
![]() | Table 1. The PBFd and ELFd of periodic Mo/Si multilayer mirrors with Nmax in the condition of incident attosecond pulses with τ0 = 100 as, 200 as, 300 as, 400 as, and 500 as. . |
In Fig.
Meanwhile, in Fig.
Besides the dispersion effect, the spectrum filtering of a periodic Mo/Si multilayer mirror is another main effect on an incident attosecond pulse. Hence, we studied the filtering effect of periodic Mo/Si multilayer mirrors with different bi-layers (N = 1–50) on incident attosecond pulses with various durations (τ0 = 100 as, 200 as, 300 as, 400 as, 500 as) by calculating the pulse broadening factor PBFf and the energy loss factor ELFf (Fig.
For each incident pulse, PBFf increases rapidly at first, then slight decreases before remaining constant, while ELFf decreases rapidly at first, then remains constant. To understand the varied trends of PBFf and ELFf with changing N shown in Fig.
![]() | Table 2. Temporal and spectral parameters of periodic Mo/Si multilayers with N = 5, 10, 15, 20, 25 for incident pulse with τ0 = 100 as. . |
In Table
In Fig.
In this section, we compared the dispersion and filtering effects induced by periodic Mo/Si multilayer mirrors reflection by calculating the ratios of pulse broadening factor PBF and the energy loss factor ELF (i.e., PBFd/PBFf and ELFd/ELFf), as shown in Fig.
The small PBFd/PBFf (blue region in Fig.
In contrast with the pulse broadening case, the small ELFd/ELFf (blue region in Fig.
In this paper, we studied the effects of dispersion and filtering induced by Mo/Si multilayer mirrors reflection on attosecond pulses. The pulse broadening factor and the energy loss factor were calculated for incident attosecond pulses with various durations. When only the dispersion effect of the periodic multilayer mirror was considered, the pulse broadening factor and the energy loss factor both increase rapidly at first, then decrease slowly with the increase of the bi-layer number of a periodic multilayer. In addition, both the pulse broadening factor and the energy loss factor decrease with the increase of the duration of an incident pulse. By investigating the temporal and spectral performances of the periodic multilayer mirrors, we concluded that the more and larger group delay dispersion ripple of the multilayer in the spectral region of the incident pulse can cause more serious energy loss and pulse broadening. When only the filtering effect of the multilayer mirror was considered, the periodic Mo/Si multilayer with more bi-layers exhibits a more serious filtering effect for an incident pulse, which leads to larger pulse broadening according to the Fourier theory. In addition, there is a tradeoff between reflective bandwidth and peak reflectivity for pulse energy loss, which leads to the irregular varying of energy loss with the increase of multilayer bi-layers. Furthermore, the pulse broadening factor and the energy loss factor both decrease with the increase of multilayer bi-layers, which indicates that the incident pulse with a smaller duration suffers from more serious energy loss and pulse broadening. Finally, by comparing the dispersion and filtering effects of periodic Mo/Si multilayer mirrors on incident attosecond pulses, we conclude that the filtering effect of a periodic Mo/Si multilayer mirror is the dominant reason for the pulse broadening and energy loss for an incident attosecond pulse which is reflected by this mirror. The research on the effects of dispersion and filtering induced by periodic Mo/Si multilayer mirror reflection on attosecond pulses can improve the understanding of the origins of pulse broadening and pulse energy loss induced by periodic multilayer mirrors reflection, and give a guide for attosecond multilayer mirrors design.
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