Propagation dynamics of radially polarized pulses in a gas-filled hollow-core fiber

Ding Wang; Lingling Qiao; Ruirui Zhao; Yu Zhao; Yuxin Leng

doi:10.1364/OE.25.003083

1. Introduction

Radially polarized (RP) beams have found many applications such as trapping and manipulating nanoparticles, microscopy, laser processing, etc [1] due to their unique focusing properties. This kind of beam can be generated by a variety of methods, for example coherently combining two linearly polarized (LP) degenerate Laguerre-Gaussian beams or two circularly-polarized phase vortices [2], designing special laser resonator schemes [3,4], using a segmented wave plates [5], liquid crystals [6], holographic elements [7], and so on. Recently, both the pulse duration and peak power of the RP pulses were boosted to a high level of few optical cycles and gigawatt, which leads to relativistic intensities along the longitudinal direction at moderate pulse energies after tight focusing [8]. This light source was then applied to accelerating electron beams and demonstrated high potential in building particle accelerator [9].

The method of generating such pulses is based on gas-filled hollow core fibers (HCF). Nonlinear compression of laser pulses to few, single or even sub optical cycle based on HCF has been a standard method of delivering energetic sources with extreme temporal characteristics [10–12]. In its implementation, the pulse from a commercial Ti:Sapphire laser system is coupled into a gas-filled HCF for spectral broadening, after which the pulse can be temporally compressed with chirped mirrors. The pulses used are usually linearly polarized and sometimes circularly polarized [13]. For LP pulses, the dominant mode in HCF is EH₁₁ mode which has the least waveguide attenuation. The attenuation gaps between EH₁₁ mode and other modes are large enough that the output from HCF is usually of good beam quality. In the work of [8], the LP pulse after HCF spectral broadening is converted from linear to radial polarization by a customized segmented wave plates before temporal compression, thus resulting in an energetic RP few-cycle pulse. As the techniques involved are all well established, and the segmented wave plates can be made to a large size, the peak power can be further pushed to TW (10¹² W), or even PW (10¹⁵ W) level. The success of this method lies in the fact that the bandwidth of the segmented wave plate is broad enough to obtain a pure radial polarization. As an alternative, the approach of spectral broadening of the RP pulse in HCF is disadvantageous for several reasons according to [8]. First, the fundamental mode of RP pulses in HCF is TM₀₁ mode, and the required waveguide core radius will be significantly larger. Second, the broadening factor and efficiency will be reduced. Third, energy in TM₀₁ mode will transfer to other modes like EH₁₁.

While this approach may be disadvantageous, it also has several advantages. First, the bandwidth of the segmented wave plate can be narrow, thus ensuring a higher degree of purity of radial polarization as well as easy fabrication of the wave plate. Second, the spectral broadening is not limited by the bandwidth of the wave plate. Third, radial polarized pulses at non-conventional wavelengths can be generated through extreme nonlinear processes, such as EUV or X-ray pulses [14]. These advantages would make propagating a RP pulse in gas-filled HCF an attractive approach and understanding of the propagation dynamics necessary and beneficial.

In this work, we theoretically study the propagation of RP pulses in a gas-filled HCF to explore the possibility of spectral broadening of the RP pulses. Unlike conventional modeling using leaky modes expansion [15, 16], the work here resolves the pulse field along the transverse dimension, thus capturing the full spatiotemporal propagation dynamics. We also compare the propagation of LP and RP pulses with the same initial pulse energies and durations in the same gas-filled HCF. According to the simulation results, the energy transmission efficiency and spectral broadening of the RP pulse is similar or even better than that of the LP pulse. The spatial uniformity of the RP pulse after HCF is also better, which results in better focusing properties. These results suggest that spectral broadening of the RP pulse in the gas-filled HCF is feasible, which will be a new powerful method of manipulating cylindrical vector beam and opens new possibilities of generating novel light sources such as RP supercontinuum and X-ray attosecond pulse.

The following of the paper is organized as follows. Section 2 describes the theoretical model used in this work. Particularly, the optimal input condition is discussed. Section 3 presents the simulation results of the RP pulses and compares with the LP pulse propagation dynamics. Section 4 discusses the tight focusing characteristics of the output RP pulse. The paper ends with a conclusion in section 5.

2. Theoretical model

We assume cylindrical symmetry of the propagation in the simulations. The mathematical model is based on the generalized unidirectional pulse propagation equations with lossy boundary conditions (gUPPE-b) [17–19]:

\partial_{z} E (z, r, ω) = i(\sqrt{\hat{L}} - \frac{ω}{v_{g}}) E (z, r, ω) + \frac{ω}{2 β c^{2}} [i ω \frac{P (z, r, ω)}{ε_{0}} - \frac{j (z, r, ω)}{ε_{0}}],

where

E (z, r, ω)

is the pulse in frequency domain; linear operator

\hat{L} = ω^{2} ε (ω) / c^{2} + Δ_{⊥}

describes both the dispersion and diffraction effects;

v_{g}

is the velocity of the moving frame;

β = ω n (ω) / c

is the frequency-dependent wavenumber;

P (z, r, ω)

and

j (z, r, ω)

are the nonlinear polarizations related to bounded and free electrons respectively;

ω

,

c

and

ε_{0}

are angular frequency, light speed and vacuum permittivity respectively. For cubic Kerr effect, the nonlinear polarization in time domain is

P / ε_{0} = 2 n_{0} n_{2} I E

which does not include harmonic generation.

I

is the pulse intensity envelope. The gas ionization effect is modeled by

j (t) / ε_{0} = c n_{0} η W (I) U_{i} (ρ_{nt} - ρ) E / I

, where

W (I)

is ionization rate calculated according to Perelomov, Popov and Terent’ev (PPT) model [20];

U_{i}

is the ionization potential;

ρ

is electron density;

ρ_{nt}

is gas density. The plasma effect is modeled by

j (ω) / ε_{0} = \frac{τ_{c} (1 + i ω τ_{c})}{1 + ω^{2} τ_{c}^{2}} \frac{e^{2}}{ε_{0} m_{e}} F F T [ρ E]

, where e,

m_{e}

,

τ_{c}

are electron charge, mass and collision time, respectively.

F F T [\dots]

is the Fourier transform. Assuming the ionized electrons are born at rest, the density

ρ

evolves as

\frac{\partial ρ}{\partial t} = W (I) (ρ_{nt} - ρ) + \frac{σ ρ I}{U_{i}},

where

σ

is the impact ionization cross section.

There are two boundary conditions for the propagation of RP pulse in HCF. First, the field on the central axis is null, or $E_{axis} = 0$ [21]. Second, the field at the intersection of gas and fiber clad is $E_{intersction} = (4 E_{1} - E_{2}) / (3 - 2 i k_{clad} Δ r / n_{clad}^{2})$ , where $E_{1}$ and $E_{2}$ are fields at $Δ r$ and $2 \cdot Δ r$ away from the boundary of the hollow core [17]; $k_{clad}$ and $n_{clad}$ are the wavenumber and linear refractive index of the fiber clad respectively. The envelop of the initial pulse takes the following form,

E (z = 0, t, r) = E_{0} f (r) \exp {- \frac{r^{2}}{w_{0}^{2}}} \exp {- \frac{t^{2}}{2 t^{2}_{0}}},

where

f (r) = 1

indicates the pulse is Gaussian in spatial domain;

f (r) = r

corresponds to the profile of RP pulse [1]. Figure 1(a) shows the spatial intensity profiles of LP and RP pulses with the same energy and duration, where

a

is the inner radius of HCF and

κ = w_{0} / a

. The peak intensity is normalized to that of the LP pulse. It can be seen that the peak intensities of RP pulses are lower than the LP pulse at the chosen beam sizes. For the LP Gaussian pulse, the optimal coupling condition for the HCF is

w_{0} \approx 0.65 a

, i.e., 98% pulse energy is coupled to the fundamental mode EH₁₁ for efficient transmission. To determine the optimal coupling condition for RP pulse, one can follow the same way of reasoning. The fundamental mode of the RP beam is TM₀₁ mode [22]. The dependence of coupling efficiency from free space into TM₀₁ mode on the beam size can be calculated according to

η_{cp} = \frac{{\int_{0}^{1} r^{2} \exp {- \frac{r^{2}}{κ^{2}}} V_{TM01} (r) d r}^{2}}{\int_{0}^{1} r V_{TM01}^{2} (r) d r \cdot \int_{0}^{\infty} r^{3} \exp {- \frac{2 r^{2}}{κ^{2}}} d r},

where

V_{TM01} (r) = J_{1} (u_{1} r)

.

J_{1} (\cdot)

is Bessel function of the first kind with order 1 and

u_{1}

is its first zero point. The coupling efficiency is shown in Fig. 1(b). When

w_{0} \approx 0.565 a

, the efficiency is about 0.97. However, with this beam size the intensity at the intersection of gas and clad is a little high as shown in black squares in Fig. 1(a). This may cause damage to the fiber clad when the input pulse is energetic. The intensity profile near the fiber clad approaches that of the optimal LP pulse when

w_{0} = 0.5 a

(shown in red crosses in Fig. 1(a)), but the coupling efficiency is a little lower at about 0.94. This beam size can be chosen as the initial condition.

Fig. 1 (a) Intensity profiles at the fiber inlet for optimal Gaussian distribution (solid blue line) and RP pulses with different beam sizes. (b) Coupling efficiency of RP pulses to TM₀₁ mode versus different beam sizes.

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On the other hand, according to the linear propagation characteristics of LP pulse the spatial distribution will converge to the fundamental mode after long enough propagation. To find out what transverse distribution the RP pulse will converge to and help to determine the optimal coupling conditions, Eq. (1) is integrated without nonlinear source terms. Figure 2(a) shows the spatial intensity distributions obtained with different conditions for RP pulses (shown in blue-dot and red-square lines). The black-circle line indicates the profile of initial pulse with $w_{0} = 0.5 a$ ; the green-cross line is the TM₀₁ mode. It can be seen that the stable transverse profile of RP pulse in HCF deviates a lot from the fundamental TM₀₁ mode. This is quite different from the situations of LP Gaussian pulse in the HCF for which the EH₁₁ mode is the fundamental mode. To the knowledge of the authors, the stable transverse intensity profile in HCF for RP pulse does not correspond to any known eigenmodes in terms of Bessel functions. From Fig. 2(a) it can be seen that the modal field shape is singular at r = 0 with a large radial derivative. This is due to the fact that we only impose the boundary condition $E_{axis} = 0$ at r = 0 without requiring $\partial_{r} E = 0$ at r = 0 because the field is not continuous at r = 0 along any direction. This modal field also emerges under other initial conditions as long as the same boundary conditions are imposed.

Fig. 2 (a) Transverse intensity profiles for stable linear propagation of RP input pulses in HCF with 250 $μ m$ (blue-dotted line) and 500 $μ m$ (red-square line) inner diameters, input RP pulse with $κ$ equals 0.5 (black-circle line) and TM₀₁ mode (green-cross line). (b) Waveguide attenuation of RP input pulses with different initial beam sizes in a 250 $μ m$ HCF versus propagation distance.

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Figure 2(b) shows the pulse energy evolution of the RP pulses with different initial beam sizes during linear propagation in a 250 $μ m$ HCF. The beam size with $w_{0} = 0.5 a$ has the least attenuation. Therefore, we set $w_{0} = 0.5 a$ for all the RP pulse propagation in the following. It should be pointed out here that the transmission efficiency of LP pulse under the same conditions is about 87%, higher than the RP pulse. If we take into account the coupling efficiency, the total energy efficiencies of linear transmission for LP and RP pulses are 85% and 73%, respectively. The loss of RP pulse is higher because the transverse energy distribution is closer to the hollow core boundary which is the only path of attenuation.

It should be noted that in the following we use “LP scheme” to denote the configuration where the linearly polarized Gaussian pulse is first spectrally broadened in a gas-filled HCF, and then calibrated, temporally compressed and converted to radially polarized pulse, and tightly focused; “RP scheme” denotes the configuration where a radially polarized pulse is first spectrally broadened in a gas-filled HCF, and then calibrated and temporally compressed, and tightly focused.

3. Simulation results and discussions

Having established the theoretical model, we present the results of the numerical studies. The simulation parameters mimic the experimental conditions in [8]. The initial Fourier-transform-limited pulse is 1.5mJ/40fs FWHM centered at 800nm. The 1m long HCF has a diameter of 500 $μ m$ and is filled with argon. The pressure is gradient with 0.2mbar at the inlet and 689.3mbar at the outlet. To better understand the propagation characteristics of RP pulses, we also simulate the LP pulses under the same conditions for comparison. Figure 1 shows the output results of LP pulse. The spectrum in Fig. 1(a) covers from about 650nm to 950nm, which is similar to the experimental result in [8]. It can be seen that the spectrum has a little spatial chirp in the short wavelength range. This is due to the fact that the short wavelength portion usually comes from nonlinearity such as self-steepening and plasma effects that induce asymmetry in the spectrum. The output pulse can be temporally compressed with proper chirp compensation. Figure 1(b) shows the temporal power profiles of the compressed pulse versus different chirp compensations. Between −30 fs² to −70 fs², the main peak of the pulse can be compressed to about 6 fs FWHM. It should be noted that we do not consider the chirp induced by the propagation after the HCF and the chamber window. Therefore, the optimal chirp compensation is different from that in the experiment. Figure 1(c) shows the spatiotemporal distribution of the compressed with GDD of −50 fs². Due to the spatial chirp in the spectrum, the temporal intensity profile of the compressed pulse also shows a little ununiformity. The blue-solid line and red dash-dotted line in Fig. 1(d) indicate the corresponding on-axis temporal intensity and phase profiles, respectively; the green-dash line is the temporal power profile. Although the pulse is compressed to few-cycle level, there still exists high order chirp. The discrepancy between the on-axis intensity and temporal power profiles also reflects the spatial ununiformity.

For easy comparison, the layout of Fig. 3 is the same as that of Fig. 4. For the RP pulse with the same energy coupled into the same HCF, the output results are shown in Fig. 4. It can be seen that while the spectral broadening, chirp compensation and the duration of the compressed pulse are similar between RP and LP pulses, the spatial distributions for RP pulse are more uniform than those of LP pulse, and temporal intensities of the compressed RP pulse are cleaner than LP pulse. Figure 5 further reveals the difference between LP and RP pulses. Figure 5(a) shows the energy transmission of LP (red-circle line) and RP (blue-dotted line) pulses along propagation. The RP pulse has more than 10% transmission efficiency than the LP pulse after 1m propagation. Before 40cm where the gas pressure is relatively low, the transmission efficiencies of LP and RP pulses are almost the same; after 40cm, the LP pulse is more and more attenuated than the RP pulse with the increase of gas density. The higher transmission efficiency of RP pulse originates from two reasons. First, the RP pulse has lower peak intensity than the LP pulse. Therefore, loss due to ionization is lower for RP pulse. The high peak intensity also induces more blue-shift in the spectrum of LP pulse as shown in Fig. 5(b). Second, the transverse energy distribution of RP pulse (shown in Fig. 5(d)) is almost as far away from the hollow core boundary as the LP pulse (shown in Fig. 5(c)). Since the waveguide loss is through the boundary in HCF, this loss for RP pulse is close to that of LP pulse. As a result, the energy efficiency of RP pulse is higher than the LP pulse during nonlinear propagation in HCF, unlike the linear propagation.

Fig. 3 Propagation results of 1.5mJ LP pulse in 500 $μ m$ HCF filled with pressure-gradient gases. (a) Spatiospectral distribution of output spectrum; (b) temporal power profiles of compressed pulses versus compensated GDDs; (c) spatiotemporal distribution of compressed pulse with GDD −50fs²; (d) spectral intensity and phase profiles of the compressed pulse. The blue solid line and red dot-dash line indicate the on-axis profiles of intensity and phase; the green dashed line indicates the temporal power profile.

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Fig. 4 Propagation results of 1.5mJ RP pulse with $κ$ equals 0.5 in 500 $μ m$ HCF filled with pressure-gradient gases. (a) Spatiospectral distribution of output spectrum; (b) temporal power profiles of compressed pulses versus compensated GDDs; (c) spatiotemporal distribution of compressed pulse with GDD −50fs²; (d) spectral intensity and phase profiles of the compressed pulse. The blue solid line and red dot-dash line indicate the profiles of peak intensity and phase; the green dashed line indicates the temporal power profile.

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Fig. 5 Comparison between RP and LP schemes. (a) Energy transmissions of LP (red-circle line) and RP (blue-dotted line) pulses along propagation in HCF; (b) output power spectra of LP (red-dashed line) and RP (blue-solid line) pulses; transverse energy distributions of (c) LP and (d) RP pulses along propagation.

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With the above comparison, it suggests that in order to generate a few-cycle energetic RP pulse spectral broadening the RP pulse in a gas-filled HCF is advantageous to the method in [8]. However, there are still several problems that need to be considered, such as the effect of self-focusing. In fact, the transverse profile of the above output RP pulse is different from the linear profile in Fig. 2(a): the intensity peak is closer to the axis, and its width is also narrower. This should be due to the self-focusing effect. If the initial energy is increased to 4.0 mJ, this trend is more obvious (not shown here) and spatiotemporal splitting occurs, which puts an upper limit on the initial energy.

4. Focusing properties

The wide interest in RP pulses lies in the focusing characteristics of such beams and their large longitudinal components. The focal field distribution of a RP beam can be calculated according to [23]

\begin{array}{l} E_{r} (r, z) = \frac{1}{λ} \int_{0}^{θ_{max}} | P (θ) | \sqrt{\cos θ} \sin θ \cos θ J_{1} (2 π \frac{r}{λ} \sin θ) \exp (i 2 π \frac{z}{λ} \cos θ + i ϕ (θ)) d θ, \\ E_{z} (r, z) = i \frac{1}{λ} \int_{0}^{θ_{max}} | P (θ) | \sqrt{\cos θ} \sin^{2} θ J_{0} (2 π \frac{r}{λ} \sin θ) \exp (i 2 π \frac{z}{λ} \cos θ + i ϕ (θ)) d θ, \end{array}

where

E_{r}

and

E_{z}

are the transverse and longitudinal components respectively; the azimuthal component is 0 due to symmetry. Note that the above integral is for single wavelength

λ

.

r

and

z

here are the transverse and longitudinal coordinates near the focus;

θ_{max}

is the maximal angle determined by the NA of the focal lens;

J_{n} (\cdot)

is the Bessel function of the first kind with order n;

| P (θ) | \exp (i ϕ (θ))

is the transverse field before focusing.

Figure 6 shows the tight focusing characteristics of the RP pulse after HCF propagation, 3m free diffraction and calibration with 3m focal lens. The focusing objective has an NA of 0.8. The first to third rows are intensity profiles for 700nm, 800nm and 900nm components, respectively. The first to third columns are ${| E_{z} |}^{2}$ , ${| E_{r} |}^{2}$ and ${| E_{z} |}^{2}$ + ${| E_{r} |}^{2}$ , respectively. It can be seen that the longitudinal components are strongest on the propagation axis, and the transverse components leave a gap where the longitudinal components are intense. This separation is well suited to many applications such as electron acceleration. The peak intensity of ${| E_{z} |}^{2}$ is about the same level as that of ${| E_{r} |}^{2}$ , which is different from the plane wave in [23] where the peak intensity of ${| E_{z} |}^{2}$ is nearly two times that of ${| E_{r} |}^{2}$ . This is due to the spatial variation of the field. As for the LP pulse, the peak value of ${| E_{z} |}^{2}$ is smaller than the ${| E_{r} |}^{2}$ component at some wavelengths (not shown here). It can be seen in Fig. 6 that the overall distributions of ${| E_{z} |}^{2}$ component for different wavelengths deviate a lot from each other. This will affect the spatiotemporal distributions of the focused field at different locations.

Fig. 6 Tight focusing characteristics of RP pulse after HCF propagation followed by 3m free diffraction and calibration. The focusing objective has an NA of 0.8. The first to third rows are intensity profiles for 700nm, 800nm and 900nm, respectively. The first to third columns are ${| E_{z} |}^{2}$ , ${| E_{r} |}^{2}$ and ${| E_{z} |}^{2}$ + ${| E_{r} |}^{2}$ , respectively. The radially polarized pulse is generated by the RP scheme.

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Figure 7 shows the spatiotemporal intensity distributions of ${| E_{z} |}^{2}$ and ${| E_{r} |}^{2}$ components of the RP pulse at five locations near the focus. Although the profiles change a little at different locations, they preserve localized shape over a least 2 $μ m$ distance. About 16 percent of total energy is in ${| E_{z} |}^{2}$ component, i.e. 206 $μ J$ ; the highest peak intensity of ${| E_{z} |}^{2}$ component is estimated to be more than $9 \times 10^{18}$ W/cm². On the other hand, the spatiotemporal distributions for the LP scheme show much irregularity (not shown here) as expected, which will lower the application performance.

Fig. 7 Spatiotemporal intensity profiles near the focus after tight focusing with an NA = 0.8 objective. The first row indicates the intensity profiles of ${| E_{z} |}^{2}$ components; the second is ${| E_{r} |}^{2}$ components. The intensities are normalized to the peak values of corresponding ${| E_{z} |}^{2}$ components. The radially polarized pulse is generated by the RP scheme.

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5. Conclusions

In conclusion, we numerically studied the propagation dynamics of RP pulses in a gas-filled HCF. It is found that the stable transverse mode of RP pulse in HCF is not TM₀₁ mode, nor any Bessel modes. Compared with the LP scheme, the RP scheme has higher transmission efficiency, more uniform spectral broadening, and cleaner temporal profile after compression and better focusing properties after highly nonlinear propagation under the same conditions. These results suggest that energetic few-cycle RP pulses can be generated more efficiently with the RP scheme than the LP scheme. The use of gas-filled HCF provides a new method of 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.

Funding

This work is supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11604351, 11204328, 61521093, 61078037, 11127901, 11134010, and 61205208) and the National Basic Research Program of China (Grant No. 2011CB808101).

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Propagation dynamics of radially polarized pulses in a gas-filled hollow-core fiber

Abstract

1. Introduction

2. Theoretical model

3. Simulation results and discussions

4. Focusing properties

5. Conclusions

Funding

References and links

Cited By

Figures (7)

Equations (5)

Optics Express