Tight focusing of radially- or azimuthally-polarized electromagnetic waves becomes attractive because of the strong field generation in the longitudinal direction. In this paper, we investigate the strength of longitudinal electric field when a radially-polarized femtosecond PW laser pulse is tightly focused by a parabolic surface. From the calculation using the vector diffraction approach, it has been shown that the highest strength of 2.2 × 1013 V/cm can be reached for the longitudinal field with a radially-polarized 11.2-fs, 11.2-J uniform-beam-profile laser pulse. The difference in the strength of longitudinal field with different beam profile and the spectrum of a laser pulse has been also carefully examined. The propagation of a laser spot has been simulated under an extremely-tight-focusing condition (0.25 in terms of f-number) and an achievable field strength for a standing longitudinal field has been examined by colliding two radially-polarized fs PW-level laser pulses.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Tight focusing is of great interest in optical and laser science. It produces a very tiny focal spot and intense light field in the focal plane . However, the electromagnetic field distribution becomes more complicated under the tight focusing condition. Due to the depolarized field component, the focal spot under the tight focusing condition undergoes deviation from a typical intensity distribution predicted by the scalar diffraction theory. The vector diffraction theory should be applied to precisely predict electric and magnetic field distributions for all polarization components under the tight focusing condition [2–5]. Recently, integral formulae based on the Sttraton-Chu integrals have been developed for describing electric and magnetic field distributions for all polarization components tightly focused by a parabolic surface and those have been applied to calculate intensity distributions of all polarization components of a linearly-polarized femtosecond petawatt laser pulse [4,5]. According to the study, the intensity of a longitudinal (or axial) polarization component, which oscillates in the propagation direction, increases to a comparable level of an incident polarization as the f-number of a focusing optics decreases.
The radially- or azimuthally-polarized beam is an example of vector beams on higher-order Poincare sphere and its focusing properties have been well described elsewhere . Interesting features appear when a radially-polarized or azimuthally-polarized light is tightly focused (i.e., f-number < 1 or NA > 0.5). Depending on the incident polarization state, strong longitudinal electric or magnetic field component is formed at the center of focal spot under the tight focusing condition. Because of the strong longitudinal field formed at the center, much attention has been paid to the radially-polarized laser beams for potential applications in microscopy , material processing , and particle acceleration  as well. In the field of laser-driven particle acceleration, several groups have investigated the effect of a longitudinal electric field obtained with linear polarization in the charged-particle acceleration [9–12] and even studied the possibility of utilizing a strong longitudinal field with radial polarization for direct electron acceleration in theoretical and experimental manners [13,14].
As the femtosecond (fs) petawatt (PW) laser system [15–17] becomes a typical tool for accelerating charged particles, it is interesting to understand how strong field strength can be obtained for the longitudinal field when a radially-polarized, fs, PW laser pulse is tightly focused. In this paper, we present the strength of longitudinal electric field that can be achievable when a radially-polarized, fs, 1-PW laser pulse is tightly focused by a parabolic mirror. The integral formulae for radially-polarized and azimuthally-polarized fs laser pulses, based on Stratton-Chu’s and Varga-Török’s integrals [2,3], have been derived for describing electric and magnetic fields tightly-focused by a parabolic surface. Because a fs laser pulse has a broad spectrum, the field strength for a given wavelength in the laser spectrum should be known for the accurate assessment of field strength of a tightly-focused fs laser spot. In our research, instead of using a conventional definition of intensity of , the electric field strengths for all wavelength components are directly calculated from a laser spectrum and pulse energy, and then those strengths are used to assess the strength of a tightly-focused electromagnetic field in the focal region.
Because the spectral phase and wavefront aberration of a fs, PW laser pulse distorts a field distribution in temporal and spatial domains, the effect of the spectral phase and wavefront aberration on the field distribution has been investigated. For the analysis, an optimized spectral phase and a corrected wavefront map taken from a fs, PW laser pulse have been used in the calculation. This analysis provides a more realistic field strength and distribution of a focal spot. Then, the propagation of a radially-polarized, 18 fs, 1 PW laser pulse has been simulated in the vicinity of focal plane. Finally, as an example of multiple colliding laser pulses (MCLP) scheme [18,19], the field strength and distribution of a standing longitudinal field have been investigated when two (propagating and counter-propagating) radially-polarized, fs, 0.5-PW laser pulse are tightly-focused and collide at the focal plane. Having all information on field distributions and the propagation property of radially-polarized light pulses near the focal point will give us an opportunity to investigate the electron motion under an intense and unconventionally-polarized light pulse.
2. Electric and magnetic fields tightly-focused by a parabolic mirror
2.1 Monochromatic wave case
2.1.1 Radially-polarized electromagnetic wave
The radially-polarized electric field can be generated when a linearly-polarized field passes through an ideal radial polarizer. A generalized Jones matrix (RP(δλ,ϕ)) for the ideal radial polarizer including an arbitrary phase retardation (δλ) at a wavelength is given by,20–22]. In order to calculate electric and magnetic fields in the focal region, let us assume that an x-polarized electromagnetic field () passes through the radial polarizer from the + z-direction. Then, the electric field after radial polarizer changes its polarization state as follows:Eqs. (2) and (3) are focused and the field distributions for all polarization components in the focal region should be calculated with these fields through the vector diffraction theory.
The same mathematical procedure used in [3,4] has been taken for deriving integral expressions to describe field distributions of all polarization components for a radially-polarized light. The use of a parabolic mirror is assumed and the slowly-varying amplitude approximation is applied. After tedious and straightforward calculation, integral expressions for all polarization components of electric fields in an observation position (xp, yp, zp) near the focal point can be obtained as follows:Eqs. (5) - (7). The ratio between the radius and the focal length determines the f-number of a focusing optics. In a similar way, one obtains integral expression for magnetic fields in an observation position (xp, yp, zp) near the focal point as follows:Eqs. (4) and (8) with Eqs. (5) – (7) and Eqs. (9) – (11) over a solid angle explicitly provides electric and magnetic field distributions near the focal point under normal and tight focusing conditions. As can be seen in Eq. (11), under the perfect phase retardation condition, no longitudinal magnetic field component is formed with the radially-polarized light. Thus, it is concluded that only a longitudinal electric field (Ez) can be formed along the center of the focal spot and the longitudinal electric field becomes dominant when a radially-polarized electric field is tightly focused, as will be seen later.
2.1.2 Azimuthally-polarized electromagnetic wave
An azimuthally-polarized light can be generated by rotating the radial polarizer by π/2 against the polarization of the incident x-polarized light. The rotation angle of ϕ in Eq. (1) can be replaced with ϕ + π/2. Then, a generalized Jones’ matrix for an azimuthal polarizer, AP(δλ,ϕ), can be expressed as:
With the same procedure taken as before, we obtain the integrands, αi(θS,ϕS) and βi(θS,ϕS), under the perfect phase retardation condition as follows:Eq. (17), no longitudinal electric field component is formed for the azimuthally-polarized case under the perfect phase matching condition. Only a strong magnetic field (Hz) can be generated at the center of focus along the propagation direction. The explicit expressions of integrands, αi(θS,ϕS) and βi(θS,ϕS), for general case can be found in Appendix.
2.2 Tightly-focused, radially-polarized femtosecond laser pulse
2.2.1 Electric field strength calculated from spectrum and total energy
For a monochromatic plane wave delivering a power of P, the electric field strength of an incident wave is calculated by the relation of . Here, ε0 is the permittivity, c the speed of light, and A the transverse area of an incident laser beam. Figure 1 shows a schematic diagram for focusing a radially-polarized femtosecond laser pulse with a spherical mirror. A fs laser pulse contains a broad spectrum of the order of nm in spectral bandwidth. Thus, the electric field strength for a given wavelength of λ in the laser spectrum should be calculated from a measured laser spectrum () and total energy (TE). The total energy under the laser spectrum can be expressed as
Here, means the beam profile with the definition of . In the spectrum measurement, a certain amount of exposure time, τexp, that satisfies the relation of is required to resolve two adjacent wavelengths of λ1 and λ2. Then, the exposure time becomes
Because the electromagnetic energy density at a wavelength λ is given by , the energy delivered by a wavelength λ is given by and the total energy by . By using Eqs. (21) and (22), we finally obtain the expression for the electric field strength at a specific wavelength in a fs laser pulse as follows:Eq. (23) into Eqs. (4) and (8), the electric field strength and energy density in the focal region can be directly calculated for a wavelength in the fs laser pulse.
2.2.2 Pulse formation and propagation
Because the strength of incident electric field is a function of wavelength and the strengths of electric and magnetic fields [Ei(P) and Hi(P) in Eqs. (4) and (8)] are dependent on the wavelength (λ) or frequency (ω), Ei(P) and Hi(P) can be written as Ei(ω:P) and Hi(ω:P). Temporal shapes for electric field having a spectral bandwidth can be calculated by taking the Fourier transform of Ei(ω:P) with spectral profile and phase. Numerically, the Fourier transform can be completed by coherently superposing monochromatic electric fields with given laser spectrum and spectral phase, as described in [4,23]. The coherent superposition method showed a good agreement in predicting a temporal profile for the linearly-polarized case. In our numerical calculations, monochromatic electric fields calculated in 1-nm (spectral resolution) steps are superposed to provide information on the spatio-temporal distribution. Finally, superposing all electric fields in the focal region provides information on resultant electric field distribution, energy density, and total energy. The total energy in the focal region was calculated for the validity of the method. The calculation of total energy was performed with a linearly-polarized, transform-limited, 11.2-fs, 11.2-J laser pulse having an uniform beam profile and Gaussian spectrum. Under a normal focusing condition (f-number of 4), more than 98% of the incident laser pulse energy was confined in a calculation box having a dimension of around the focal spot. Thus, it can be concluded that the method used for the direct calculation of energy density and field strength of a fs laser pulse is valid within an accuracy of >98%.
Multiplying by a phase factor, , introduces the traveling property to the monochromatic electric and magnetic fields in the z-direction. The superposition of travelling monochromatic electromagnetic waves with given spectrum and spectral phase instantly yields the propagation property of a laser pulse. Therefore, the superposition of travelling monochromatic electric fields calculated in the focal region also shows the propagation of a focal spot. It is assumed, in our calculation, that an electromagnetic wave diffracted from the parabolic surface reaches at the focal point at t = 0. Thus, t can be interpreted as the time delay. This means that the electric field calculated at t = −5 fs represents electric field distribution formed at 5 fs earlier than it arrives at the focal point. The propagation of a tightly focused spot can be visualized by calculating electric fields from negative to positive time delays. The calculation of electric and magnetic fields in our method does not assume a weak longitudinal field. It only assumes a slowly-varying amplitude to assure the field amplitude to remain the same in a short distance of calculation region and provides explicit description on the electromagnetic field and its propagation under the tight focusing condition.
3. Numerical results
An x-polarized electromagnetic field is assumed as an incident field. It is also assumed that the x-polarized electromagnetic wave changes its polarization into the radial or azimuthal polarization state without changing the beam profile and power. The electromagnetic wave comes from the + z-direction, is reflected by an on-axis parabolic surface, and propagates to the -z-direction. A detailed configuration can be found elsewhere . The input field is digitized for the numerical calculation with grids of 20 x180 pixels in θS and ϕS domains. Thus, in general, any kind of input beam profile such as Gaussian and Laguerre-Gaussian (LG) can be used in the calculation. Even though the diffraction limit (DL = 1.22 × λ × f-number) and the Rayleigh range (RL = π × DL2/λ) are mostly chosen to determine the calculation volume, in principle there is no restriction in determining the calculation volume and resolution.
Figures 2 and 3 show the square of electric field (|E|2) calculated in the focal plane for linearly-, radially-, and azimuthally-polarized electromagnetic waves under two different f-number conditions (f-numbers of 3 and 0.25). The calculations have been performed with monochromatic and uniform plane wave having a wavelength of 800 nm. The intensity distributions calculated with Eqs. (4) and (8) represent general characteristics of linearly-, radially-, and azimuthally-polarized focused light fields. Under the normal focusing condition (f-number > 1), other polarization component than the incident one is not significant. Thus, typical shapes, such as circular and annular, are observed in the focal plane. Interesting features appear under the tight focusing condition (f-number < 1). As the f-number decreases, the longitudinal polarization components become strong to be comparable or to exceed the strength of transverse field component. Because the azimuthal polarization does not have a longitudinal component even after focusing, the shape of focal spot for the azimuthal polarization remains the same irrespective of the focusing condition. As for the magneticfield, situation is a little bit different but the results are almost similar. The azimuthally-polarized field produces a strong magnetic field in the longitudinal direction [as shown in Eq. (20)] while the radially-polarized field does not [as shown in Eq. (11)]. This can be easily understood by considering the oscillation of electric and magnetic fields. Only the radial component generates the longitudinal field when it is tightly focused.
Now, the field strength for radially-polarized, fs, 1-PW laser pulses has been carefully investigated in the focal region. Figure 4 shows the dependence of field strength on polarization and focusing condition. In this case, the field strength is calculated with 11.2-fs, 11.2-J, uniform-beam profile laser pulses having a Gaussian spectrum. It is assumed that a linearly-polarized uniform beam laser pulse pass through a radial polarizer and is focused immediately after the polarizer. Uniform and radially-polarized beam profile can be used under the assumption of a very short distance between a waveplate and focusing optics. The Gaussian spectrum centered at 800 nm is generated to support a 11.2-fs laser pulse. Linear and radial polarizations are considered for comparison. As shown in Fig. 4, the highest field strength is observed at the f-number of 0.25, which means peripheral beam is incident on the focus at the right angle of π/2 (i.e., θmin = π/2). The longitudinal electric field of the radially-polarized laser pulse showed the highest field strength of 2.2 × 1013 V/cm. The calculation result implies that the longitudinal field strength of over 4.8 × 1013 V/cm (corresponding to ~3 × 1024 W/cm2 in terms of intensity) is possible with a currently-existing 5-PW laser  under the extreme tight focusing condition of 0.25.
Despite the same pulse duration, detailed temporal profiles between two different laser spectrums can be slightly different. And, although the generation of radially-polarized uniform beam laser pulse was tried by just placing a segmented waveplate in the beam path , the Laguerre-Gaussian (LG01) beam profile is a more realistic beam distribution for the radial polarization. Thus, three different laser spectrums [shown in Fig. 5(a)] and two different beam profiles (uniform and LG01) have been considered in order to examine the dependence of field strength on the laser spectrum and beam profile. The LS1 and LS2 in Fig. 5(a) are laser spectrums taken from a PW Ti:sapphire laser system . The transform- limited pulse profile for each spectrum is shown in Fig. 5(b). As shown in Fig. 5(b), laser spectrums, LS1 and LS2, support 11.7-fs (at FWHM) and 18.1-fs laser pulses with different tails, respectively. The strengths of longitudinal fields as well as transverse ones are calculated with these laser spectrums. In the calculation, the peak power of a laser pulse is set to 1-PW by assigning energies of 11.2 J, 11.7 J, and 18.1 J to Gaussian, LS1, and LS2 laser spectrums, respectively.
Figures 6(a) and 6(b) show the summary of field strengths for radially-polarized uniform and LG01 laser pulses. The focusing condition is 0.25 in terms of f-number. The field strength for the plane wave case is shown in the left for comparison. The uniform beam profile provides a higher field strength in the longitudinal direction because the outer part of a beam profile contributes more to the formation of longitudinal electric field in the focal region. The highest field strength for the radially-polarized LG01 beam is 1.63 × 1013 V/cm with Gaussian spectrum. The laser pulse having a Gaussian spectrum shows the closest strength to that of plane wave. The difference between two strengths is less than 1.5%. As shown in Fig. 6, the difference becomes larger with realistic laser spectrums, yielding differences of 2.2% and 5.2% for LS1 and LS2 spectrums, respectively. The difference is related to the energy spread in time. The Gaussian pulse profile shows the best confinement in time, showing the highest field strength. When the time (t) is normalized to the time (tm) which means the first intensity minimum in time, the pulse profile from LS2 showed a 5% wider profile in the central region than that from LS1. The different energy spread in time might cause the change in the field strength.
The fs, PW laser pulse is not ideal in temporal and spatial domains. It is temporally and spatially distorted by the spectral phase and wavefront aberration. Thus, the effect of the spectral phase and wavefront aberration should be considered in evaluating the field strength. In the following analysis, the reduction in the field strength expressed in the energy density is evaluated with a spectral phase and wavefront aberration contained in a fs, PW laser pulse. Figures 7(a) and 7(b) are the spectral phase and wavefront aberration used in the calculation. The 18.1-fs, 18.1-J uniform laser pulse is chosen for the analysis. The peak-to-valley is 0.66 μm for the wavefront aberration and the Strehl ratio calculated with the wavefront was about 0.6. The spectral phase was well-optimized in the spectral range and does not much distort the temporal profile. Figures 7(c), 7(d), and 7(e) show energy density maps under the f-number of 0.25 with spectral phase and wavefront aberration. In this case, although the most of laser energy is confined within 0.77 μm (~1λ) in the x-direction and 0.7 μm (~0.88λ) in the y-direction, the reduction in energy density due to the wavefront aberration is observed. The peak energy density for the longitudinal field reaches at 1.1 × 1013 J/cm3, which is ~60% of an ideal case showing a good agreement with the Strehl ratio.
The propagation property of radially-polarized fs PW pulses is shown in Fig. 8. The laser pulse propagates from -z- to + z-direction and the x-z plane is taken as the cross-section. The energy densities of a femtosecond focal spot are calculated at various time delays from −20 fs to 0 fs. Figure 8 shows snap shots of energy densities for a focal spot taken at moments of −16.6 fs, −8.3 fs, and 0 fs. According to the wave optics theory, the field distribution of a monochromatic wave in the vicinity of focal plane is determined by the Rayleigh range related to the f-number of a focusing optics. It means that a shorter length in field distribution can be obtained with a shorter-focal-length optics. However, it is different for the fs laser pulse. Two parameters (f-number and pulse duration) interplay in determining the field distribution. In case of a larger Rayleigh range (f-number >> 1) than a pulse length (cτ) given by the speed of light (c) and the pulse duration (τ), the field distribution spreads in a distance corresponding to the pulse length in the z-direction. The field distribution propagates through the Rayleigh range and remains almost the same within the range. Under a tight focusing condition (f-number << 1), the Rayleigh range becomes comparable to or even shorter than the pulse length. In this case, the field distribution is determined by the Rayleigh range and the field can be tightly confined in a shorter region than the pulse length (see Fig. 8). Thefield strength varies in time, according to the pulse temporal profile. It is worth noting that, although the strongest longitudinal field is formed at the center of a spot, the Poynting vector in the propagation direction is zero, which means no electromagnetic energy flow occurs along the spot center.
Now, it is interesting to investigate how strong standing longitudinal field can be formed when two (propagating and counter-propagating) pulses collide at the focus. We assume that two radially-polarized, LG01, 18.1-fs, 0.5-PW laser pulses are reflected by two OAPs and collide at the focus [see Fig. 9(a)]. The same spectral phase and wavefront map as in Fig. 7(b) were used in the calculation. The counter-propagating field are calculated with Eqs. (4) and (8), but with an opposite propagation direction. Because of the space for the mirror to reflect laser pulse to OAPs, the use of OAPs having a f-number of 0.6 is assumed in the calculation. Figure 9(b) show the energy density and electric field distributions in the x-z plane for thestanding wave formed by colliding. With no phase difference between two pulse, the highest field strengths of 7.7 × 1012 V/cm and 5.9 × 1012 V/cm are obtained for transverse and longitudinal components, respectively. When two pulses collide with a phase difference of π, the peaks shift to λ/4 in space and the highest strengths were almost the same. Although the highest strength for the longitudinal field reaches only to 5.9 × 1012 V/cm, it is worthy to note that a strong and standing longitudinal field can be generated by colliding two radially-polarized fs, 0.5-PW laser pulses. This is only an example generating a strong standing light field with the longitudinal polarization. Other colliding pulse schemes might be considered for efficient generation of gamma-ray and electron-positron pair production.
The field strength of a longitudinal field that can be obtained by tightly-focusing a radially-polarized fs PW laser pulse has been investigated. The dependence of field strength on the spatial and temporal profiles is examined. The propagation of a tightly-focused femtosecond spot has been simulated under an extremely-tight-focusing condition of 0.25 and an achievable field strength for a standing longitudinal field has been examined when two radially-polarized fs 0.5 PW laser pulse collide at the focus. The approach taken in this research provides a general way to calculate the theoretical prediction for field strength and intensity with experimentally measured spatial (beam profile and wavefront) and temporal (spectrum and spectral phase) data and total energy, and also presents theoretical limit for strengths of transverse and longitudinal fields. Because the longitudinal electric field components play an important role in understanding the electron, it is expected that the result of this paper will be helpful by providing precise electric and magnetic field distributions in understanding electron motions under strong and unconventional electromagnetic field.
For a radially-polarized wave having an arbitrary phase delay between fast and slow axes, the integrands, αi(θS,ϕS), are given by
High Field Initiative (CZ.02.1.01/0.0/0.0/15_003/0000449); Advanced research using high intensity laser produced photons and particles (CZ.02.1.01/0.0/0.0/16_019/0000789).
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