The compression of high-energy, radially polarized pulses in a gas-filled hollow-core fiber (HCF) is theoretically studied. The simulation results indicate that a 40-fs input pulse can be compressed to a full-width at half-maximum of less than 9 fs when the pulse energy reaches 7.0 mJ with a transmission efficiency of more than 67% after propagating through a 1-m-long, 500-μm diameter HCF filled with neon. Furthermore, the spatio-temporal intensity distributions of the compressed pulses with different initial input energies are studied, and the numerical results indicate that the spatio-temporal intensity distributions are more uniform for lower input pulse energies.
© 2017 Optical Society of America
In the past two decades, many researchers have been focused on generating energetic few-cycle sources [1–5] because of the significant applications in several fields such as ultrafast spectroscopy [6–8], high-order harmonic generation , attosecond physics [10,11], and particle acceleration. Among the various methods of generating intense ultrashort pulse, the hollow-core fiber (HCF) compressor filled with noble gas is an established method and serves as the workhorse in many labs around the world.
The polarization state of a pulse to be compressed is usually spatially uniform, such as linearly polarized (LP) or circularly polarized (CP) pulses. According to [12–19], LP or CP pulses have been achieved with few-cycle pulse durations and energies less than 3 mJ. Meanwhile, there has been an increasing interest in light sources with spatially varying polarizations, such as radially polarized (RP) pulses. Compared with LP pulses, RP pulses have unique characteristics. For example [20–22], RP pulses can be more tightly focused, there is a significant longitudinally polarized component at the focus, and the transversely and longitudinally polarized components are spatially separated at the focus. Because of these unique properties, RP sources have been used in several significant applications, such as high-resolution imaging, remote sensing, plasmonic focusing, material processing, and optical tweezers [21–26]. There are many methods of generating RP sources. However, the peak power and pulse duration of such sources have not yet achieved similar levels to those of LP sources. Generation of energetic few-cycle RP pulses would open new possibilities in high-field laser physics.
Recently, Carbajo et al.  pushed the pulse duration and peak power of RP pulses to a few optical cycles and a few gigawatts based on the gas-filled HCF compressor. They first generated a conventional LP few-cycle pulse with a HCF compressor and converted the polarization state from LP to RP with a segmented waveplate. Later, Wang et al.  numerically studied the possibility of direct spectral broadening of the RP pulses in a HCF directly and then compressing the RP pulse, and argued that it is advantageous to adopt this method compared to that in .
In this work, we theoretically study the dynamic propagation and compression of higher energy few-cycle pulses with radial polarization in a gas-filled, HCF based on the three-dimensional g-UPPE model [29, 30]. Then, we compare the pulse compression results in the time domain and transmission efficiency after the RP pulses have propagated through a meter-long, 500-μm core-diameter, gas-filled HCF with different input energies. Efficient pulse spectral broadening of 40-fs RP pulses is illustrated in a neon-filled HCF and an initial input pulse energy of about 5.0–7.0 mJ after subsequent group-delay dispersion (GDD) compensation. The output energy is about 4.0–5.0 mJ, implying peak power tens of gigawatts higher than that reported in . We can see that higher energy RP pulses undergo spatio-temporal splitting after propagating through the HCF and the main reason of the phenomenon is that high-order chirp is difficult to eliminate only by GDD compensation. Finally, the spatial distribution of the energetic RP pulses after propagating through the HCF is presented.
2. Theoretical model and method
We assume that the pulse propagation in the atomic gas-filled HCF follows a cylindrical symmetric model; thus, the propagation dynamics of RP pulses in the HCF is modeled by the generalized unidirectional pulse propagation equations with lossy boundary conditions (gUPPE-b) [29–31], which can be described as follows :32] developed and used forward Maxwell equation (FME) which is a paraxial form of the unidirectional pulse propagation equation (UPPE), and the model in our manuscript is also a form of UPPE with paraxial approximation. In a word, although the linear part of the propagation is performed in (r,ω) domain, it is shown in  that this form is equivalent to the UPPE in (k,ω) domain in isotropic medium. Where the pulse envelopein Eq. (1) is the optical field distribution in the frequency domain.describes the dispersion and diffraction effect,is the frequency-dependent wavenumber, vg is the group velocity of the propagating RP pulse.andare the nonlinear polarizations related to the bounded and free electrons respectively., and are the vacuum permittivity, angular frequency, and speed of light, respectively. For the cubic Kerr effect, the nonlinear polarization in the time domain is , the value is the third-order nonlinear susceptibility tensor component and the is the vectorial expression of optical field distribution in the time domain.
The gas ionization effect is modeled by, where is the ionization rate calculated according to the Perelomov–Popov–Terent’ev model ,is the electron density,is the neutral density of gas, is the pulse intensity envelope, andis the ionization potential. represents the plasma effect model, where e, me, and τc describe the electron charge, mass, and collision time, respectively. Assuming the ionized electrons are born at rest, the electron densityevolves as
In the experimental configuration, the initial pulse is first focused and coupled into a gas-filled hollow-core fiber (HCF) for spectral broadening which is simulated through integration of the propagation Eq. (1) in the manuscript. After that the pulse is collimated and reflected upon several chirped mirrors for chirp compensation. As a result the pulse is compressed. The chirped mirrors are carefully designed to induce fixed second order chirp in the pulse per reflection. Therefore, the pulse compression process can be simulated by simply adding a phase term on the pulse as follows:Eq. (3) describe is the pulse after propagation in the HCF and before reflection on the chirped mirrors and the pulse after reflection upon chirped mirrors which induce the total group delay dispersion (GDD) of , and is the central angular frequency.
There are two boundary conditions for RP pulses transmitting through the HCF. First, the field is zero or null on the central axis of the HCF, or ; second, the field at the interface of the fiber core and fiber clad is , where and are the fields at Δr and 2⋅Δr away from the boundary of the hollow core , respectively, and and are the wavenumber and refractive index of the fiber clad, respectively. The envelope of the initial input RP pulses can be described by the following expression:22]. For the input pulse, the coupling efficiency must be a significant parameter with the same energy and duration, where a is the inner radius of the HCF and v = . For Gaussian LP pulses, the optimal coupling condition is v = 0.65, when 98% of the pulse energy is coupled to the fundamental mode EH11 for efficient transmission. According to , for RP pulses, the coupling efficiency can reach up to 97% when v = 0.565, however, this may cause damage to the fiber clad when the input pulse energy is high because of the higher intensity at the interface of the fiber core and fiber clad. Though the coupling efficiency decreases to 94% for v = 0.5, the intensity at the interface of the fiber core and fiber clad in this condition is relatively lower. Hence the transmission loss reaches a minimum after pulse propagation through the 1-m-long HCF when v = 0.5. In this study, the initial coupling condition of RP pulses in the HCF inlet is set to v = 0.5.
3. Results and discussion
The numerical values given in this section correspond to the initial Fourier-transform-limited pulse is 40 fs (FWHM) centered at 800 nm, the input energy of RP pulses are ranging from 5.0 mJ to 7.0 mJ, the length and inner diameter of the HCF filled with neon gas are 1 m and 500 μm, respectively, and the pressure within the HCF is 650 mbar. First, we compare the energy transmissions of different input pulse energies in Fig. 1. It is evident that the energy transmission decreases with increasing the input pulse energy. For instance, for pulse energy is 5.0 mJ, the energy transmission is up to 85%, and for pulse energy is 7.0 mJ, the energy transmission is dropping to 67%. Moreover, the spatio-temporal distribution of pulses are closer to the fiber core which cause less losses and the boundary of the HCF which cause more losses along with the pulses propagation in HCF are the main reason why the lines are not smooth in Fig. 1, the descriptions of the problem are also shown in .
Figure 2 shows the spatio-temporal distribution of the compressed pulses with proper GDD compensation for input energies of 5.0–7.0 mJ. The compressed results from Fig. 2 indicate that the pulse duration can be compressed to 9.0–7.0 fs (FWHM) for the initial input energy of 5.0–7.0 mJ, respectively. The GDD compensation for input energies of 5.0, 5.5, 6.0, 6.5, and 7.0 mJ are –69.5, –56.5, –44.3, –40, and –32.5 fs2. As shown in Fig. 2, the spatio-temporal distribution of the compressed pulses is slightly nonuniform with higher input energies. We can see that more sidelobes of compressed pulses appear with the increase in initial input energy in Figs. 2 and 3, it is clear that the intensity or amount of sidelobes of the compressed pulses are greater than the others when the initial input energy is 7.0 mJ, especially as the intensity of the sidelobe is close to 20%. In other words, the RP pulses are suitable to compress when the initial input energy is less than 6.5 mJ.
The blue dot-dash line and red solid line in Fig. 3 indicate the temporal power profile and the profile of peak intensity, respectively. As the initial input energy increases, the difference between temporal power profile and the profile of peak intensity increases. The discrepancy between the temporal power profile and the profile of peak intensity correlates the spatial nonuniformity. In summary, the phenomenon of nonuniformity of spatio-temporal distribution is influenced by stronger nonlinear effects such as self-focusing effects and plasma defocusing effects.
Figures 4 and 5 present the output power spectra and spatiospectral distributions of the output spectrum after the RP pulse propagates through the HCF. From Fig. 4, it is evident that the modulation and blue-shift are enhanced in the RP pulse spectrum when the input pulse energy increases. We can see that the output pulses have the wider spectrum with increasing the input energies. From Fig. 5, as the input pulse energy increases and the peak intensity of the RP pulses increases, more nonuniform spatio-spectral distribution for the output RP pulse, we can declare that energetic input energies which are equal to higher peak power lead to stronger self-focusing effects and then give rise to nonuniform spatio-spectral distributions for RP pulses propagation in HCF.
Figures 6 and 7 displays the normalized spatial intensity profiles of the input pulse at the entrance of the HCF and output pulse through the 1-m-long and neon-filled HCF when the initial input energy is 5–7 mJ. It is observed that the spot diameter of the output pulse is smaller than that of the input pulse by comparing Fig. 6(a) with Figs. 7(a)–(e). Moreover, the higher input energy causes stronger self-focusing effects and a greater spatial chirp, the spot diameter decreases, and the spatial intensity profile of the output pulse is closer to the central axis of the HCF. From Fig. 7(f), the spatial intensity profile of the LP output pulse after the HCF indicates that the LP pulse with an input energy of 2 mJ has a far smaller spot diameter than the RP pulse.
The dependences on the propagation length of the maximum peak intensity and peak plasma density on the cross section of HCF are shown in Figs. 8(a) and 8(b). From Fig. 8(a), it is clear that the maximum peak intensity of RP pulses is lower than the maximum peak intensity of LP pulses with pulses propagation in HCF. It is shown that the transverse distribution of RP pulses is annular and more scattered than LP pulses. From Fig. 8(b), the maximum peak plasma density of LP pulses along with propagation in HCF is higher than RP pulses because of the maximum peak intensity of LP pulses is higher than the maximum peak intensity of RP pulses.
Comparing the pulse propagation by HCF and filamentation , the common feature is that in both cases the pulses are confined in a smaller and relatively constant transverse area during propagation. This leads to the nonlinear effect enhancement, such as spectral broadening which is desirable for pulse compression. For the pulse propagation in HCF, the pulse beam is confined in a constant waveguide during propagation. For the pulse filamentation, it is the dynamical balance of self-focusing effect and plasma defocusing effect that limits the beam in a small transverse area . In HCF, the spectral broadening is mainly due to self phase modulation (SPM) effect. As a result, the extra chirp compensation is needed to compress the spectrum broadened pulse generally . In filamentation, the plasma defocusing effect also plays an important role. Therefore, with proper conditions, the pulse self-compression is possible. Then, both methods can be used to compress pulses efficiently.
4. Conclusions and discussion
The simulation results show that a compressed pulse reaching down to 7 fs and up to 5 mJ could be generated by the HCF filled with neon when the input laser source pulse was radially polarized with a transmission efficiency of more than 67%, and the spatio-temporal intensity distributions are more uniform for lower input pulse energies. Compared with LP pulses, the advantages of the dynamic propagation and compression of RP pulses in HCF are that the RP pulse possesses annular distributions and a smaller peak power with the same input energy. The use of a gas-filled HCF provides a new method for manipulating cylindrical vector pulses. It also opens new possibilities of generating novel RP sources at non-conventional wavelengths which have applications in fields from strong field laser physics to nanophotonics.
National Natural Science Foundation of China (NSFC) (Grant No. 61205208), Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB1603), and International S and T Cooperation Program of China (Grant No. 2016YFE0119300).
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