Femtosecond laser pulse induced filamentation in atmosphere is susceptible to a number of input laser, focusing optics and medium characteristics. Filamentation of fs pulses in atmosphere is an intense propagation regime where the focusing geometry used to focus the fs laser pulses play an important role influencing the filament intensity and the associated supercontinuum. We identified different optical elements used for focusing the fs laser pulses leading to filamentation in air and classified them according to the induced aberrations. To clearly identify the role of aberrations, all the optical elements were taken to have same focal length. The subsequent filament structure and emissions from the filament were correlated with the aberrations induced by optical element revealed stark differences. The onset of the filamentation, its longitudinal intensity and the associated supercontinuum emission (SCE) have varied drastically with the aberrations induced by the focusing optics. A systematic study directed to choose and identify suitable optical elements according to the usage of the fs pulses for a specific filamentation regime is presented.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Filamentation, a long-range propagation of intense femtosecond pulses in atmospheric air demonstrated 15 years ago  has displayed several fascinating phenomena [2–4], paving the way for the development of multiple tools [5,6] and establish ground for various fundamental research and applications. Filamentation occurs as a result of a quasi-static interplay between plasma defocusing and Kerr self-focusing until the energy of the input fs pulse is lost in the numerous absorption and scattering processes. The intense regime of filament propagation is also associated with the generation of broadband supercontinuum emission which spans over multiple octaves depending on the medium of propagation. The evolving applications of filamentation demand tailoring of the fs pulses over near infrared (from Ti: Sapphire) to mid infrared wavelengths as well as the focusing elements that deliver/modulate these fs pulses to a specific application [7–21]. The broad spectral content of the fs laser pulses make them susceptible to spatio-temporal distortions such as group delay dispersion (GDD) and pulse front delay often resulting in pulse broadening. The temporal characteristics of the fs pulses can be modified by tailoring/pre-compensating the chirp. While the spatial phase of the fs laser beam can be tuned using optical elements which range from transmitting optics such as lenses, spatial phase modulators or reflecting optics such as curved mirrors, off-axis parabolic mirrors and adaptive optical elements. Any aberrations induced on the phase front by these focusing optics will in turn modify the propagation characteristics of fs pulse and filamentation . For example, the pulse curvature is positive or negative in the filament depending on the dominating mechanisms i.e. Kerr self-focusing or plasma defocusing.
The longitudinal onset of filamentation is controlled by the focusing optic by either advancing or delaying the self-focusing distance (ZSF) along the propagation direction . Based on the focusing conditions, the geometry used for filamentation can be classified into loose and tight focusing regime. In the loose focusing regime of filamentation, the intensity of the filament is clamped to ∼ 1013 W/cm2 associated with the constant minimal cutoff wavelength (λmin) for the SCE commonly known as standard intensity clamping [1, 7, 24]. While in the tight focusing conditions the intensity of the filament can be an order of magnitude higher, i.e., ∼ 1014 W/cm2 [25,26] with a continuously shifting λmin. The major difference between the two focusing regimes is the evolution of the phase/pulse front. In the loose focusing regime the curvature of the phase front remains planar along the filament due to multiple cycles of Kerr self-focusing and plasma defocusing . While in the tight focusing regime, the phase front undergoes a Gouy phase shift after which the large divergence of the beam switching off the effects of Kerr self-focusing and plasma defocusing . For majority of applications, such as SCE generation, filament assisted micro/nanomachining, THz generation, VUV, high harmonic generation the tight focusing regime is more suitable. These coherent emissions complimented with high intensity and short pulse duration make filamentation an efficient tool in multiple applications but also makes it highly sensitive to multiple factors like input beam characteristics and medium properties which also influence the high non-linear regime of filamentation. In this regime minor distortions in the curvature or pulse profile can bring about stark differences in filament structure and intensity, because of which understanding of filamentation requires a systematic experimental support. Hence, in this paper we present the results on the effect of focusing optic induced aberrations on filamentation and associated SCG under tight focusing conditions.
The questions that are addressed in this paper are: Do the aberrations induced by focusing optics affect onset of filamentation, filament induced breakdown spectral emissions. the on-axis intensities of the filament, plasma electron densities and the supercontinuum emission? The optical elements were chosen such that they have the same focal length and were characterized according to the pulse to phase front delay, chirp, spherical and chromatic aberration induced. The effect of the aberrations on the filament length is studied by longitudinal imaging. The variation of filament intensity and electron density for each optical element was estimated using filament induced breakdown spectra (FIBS) of air. The Supercontinuum from the filament was also collected. The effect of dispersion induced by the shape and material of the optical element was investigated by varying the chirp of the input pulse before the optical element by varying the separation between the compressor gratings in the laser amplifier. The correlation of these measurements with optical element classification shines ample light on the underlying and competing factors during filamentation, considering all the elements have common focusing geometry.
2. Experimental details
Collimated 800 nm, 45 fs pulses from a Ti:Sapphire laser (Coherent Libra operated at 1kHz repetition rate) were tightly focused in air using five different focusing optics all having a focal length of 10 cm in vacuum conditions. The input beam diameter was 8 ± 0.1 mm (1/e2) before the focusing element, leading to an f/12.5 focusing geometry. The focusing optics chosen are off-axis parabolic mirror (OAPM), Achromatic doublet (AD), low GVD thin plano-convex lens (thin lens), uncorrected plano-convex lens (PCX) and a bi-convex lens (BCX). The alignment of the optical components including OAPM was carefully checked both at low intensity from the symmetry of the focal spot size and at high pulse energy from the circular symmetry of supercontinuum emission. Longitudinal images of complete filaments formed by individual pulses in a single frame were monitored in the focal region using a focusing lens coupled to a triggerable CCD camera (GRAS-20, 1/1.8” format 1600 × 1200 pixel camera with C-mount CCD recess, M/s. Ophir Optronics Solution Ltd.). These images show filaments of various diameter and intensities, extending radially over a 60 μm FWHM diameter spot. After the interaction region, the conical emission from the filament is directed on to a 600 lines per mm grating. The first order diffracted spectrum projected on to a teflon sheet was collected by a silica optical fiber (to isolate the pump) coupled to a USB-CCD spectrometer (Maya, Ocean Optics). SCE spectra were collected over an integration time of 30 ms to reduce the noise from pulse-to-pulse fluctuations.
The experiments were performed with linearly polarized (LP) pulses with energy kept at 2 ± 0.05 mJ per pulse corresponding to a peak power of ∼ 44.5 GW. The critical power for self-focusing PCr is taken as 3 GW denoting a reference value calculated with the nonlinear index coefficient n2 = 3.2×10−19 cm2/W. The P/PCr = 14.8 represents that the measurements were performed above the threshold for self-focusing. This is to ensure that all the phenomena associated with the self-focusing, such as plasma formation, filament induced breakdown of air and supercontinuum occur simultaneously. This helps us bring out the role of focusing element induced aberrations more clearly. The longitudinally varying emissions from the filament induced breakdown (FIB) of air is collected using an optical fiber mounted on a translation stage. The fiber is coupled to a spectrograph (Kymera 328i, M/S. ANDOR) with an intensified 2D sCMOS detector (ISTAR-SCMOS-18U-E3, M/s. ANDOR) with a 2 ns temporal resolution. The gate width of intensified detector is kept at 100 ns at a fixed gain for all the measurements presented in the study. For emissions over 320–400 nm, grating with 1200 lines per mm blazed at 300 nm is used in the spectrograph, while the emissions over 750–900 nm are collected using a grating with 600 lines per mm blazed at 750 nm. The experimental schematic is shown in Fig. 1. Wavelength calibration of the spectrograph was done using a mercury argon lamp and intensity calibration was done using a deuterium-halogen lamp. The emission from FIB of air is used to estimate the longitudinally varying filament intensity  and the electron density of the plasma formed .
The filament intensity was estimated from the ratio of fluorescence line intensity of two different ionized nitrogen species (at 337 nm and 391 nm) which are populated by varying orders of intensity . The line centred at 391 nm is the fluorescence transition of the nitrogen ions ionized by direct ionization of inner valence electrons. Whereas, the intensity of the 337 nm line is directly proportional to the total number of nitrogen ions i.e. both species formed by direct ionization of inner valence electrons and normal ionization of least bound electron. The population of these ionized species is directly proportional to the intensity at different orders of non-linearity and proportionality. Hence the ratio of emission line intensity at 391 nm and 337 nm provides a direct correlation to filament intensity in line with earlier reports . The electron density of the plasma was calculated from the line width of the Stark broadened oxygen line at 844.66 nm which is fitted to Lorentzian line shape . The width of the emission line FWHM (Δλ) is therefore related to electron density (Ne) as Ne = (Δλ × 1016)/2ω with ω is the impact width having a value of 0.05140 nm at 844.66 nm . Supercontinuum emission (SCE) from the filaments was also acquired over varying chirp. The input pulse was profiled using a single shot autocorrelator (SSA) before the entry to the lens, for the varying chirp. The magnitude of the chirp (C) is related to the measured pulse duration through the Eq. (1), where τ0 is the FWHM of the pulse profile at zero chirp, τin is the FWHM of the chirped pulse. The sign of the chirp is determined by the phase distribution of the different frequencies. If the slope of the phase vs frequency graph is positive, the chirp is positive and vice versa.1 summarizes the optical properties of the material and physical dimensions of the focusing optics used in the study. Other than the OAPM, all other focusing elements are transmissive optics, which introduce dispersion which in turn modifies the pulse duration. The OAPM used is a 90° off-axis parabolic mirror with an effective focal length of 100 mm.
3. Results and discussion
3.1. Effect of aberrations of focusing optic on the fs pulses
The aberrations induced by a spherical focusing element such as a lens to a fs pulse are spatially varying pulse to phase front delay (PPD), chirp and extension of the focal volume due to longitudinal spherical aberration (LSA) and chromaticity/dispersion of the lens medium. The PPD is due to the first order dispersion of the lens and the chirp is due to the second order dispersion i.e., Group Delay Dispersion (GDD) in the lens. The PPD and chirp for the focusing optics used were estimated by taking into account the optical element dimensions and medium properties as shown in Table 1. Five different focusing elements with the same focal length were used in the study: a) Off Axis Parabolic mirror (OAPM) b) Dispersion corrected thin plano-convex lens for fs pulses (Thin lens) c) Thick BK-7 Bi-Convex lens (BCX) d) Thick BK-7 Plano-Convex lens (PCX) e) Achromatic doublet (AD). The optics were chosen such that they had varying degrees of aberrations and dispersion. The OAPM is the ideal focusing element with no apparent aberration, some possible degree of cylindrical asymmetry (arising only due to optic element alignment errors) and no dispersion. The AD, despite possessing minimal (zero) spherical and chromatic aberrations, induces good amount of overall GDD and PPD that induces a curvature in the pulse front which is opposite to that of the other optics. The thin lens (designed for femtosecond pulses) has low group delay dispersion (GDD) and PPD, but still has a radially varying dispersion component. The thick lens (Plano-convex and Bi-Convex lens) induce both radially varying and overall GDD, PPD along with a high degree of spherical and chromatic aberration as well as high GDD due to the thickness. It is to be noted that although the radii of curvature for the thick lenses are different, the length of propagation in the lens according to the radial position are the same, because of which we find that the main difference between the two thick lenses is the longitudinal spherical aberration (LSA).
The first order dispersion manifests in the form of a temporal shift in the pulse peak causing a delay between the pulse front and phase front because of which we have an added curvature on the pulse front but not on the phase front (pulse to phase front delay (PPD)). The second order dispersion or GDD manifests as chirp in the pulse causing a broadening of the pulse. These temporal distortions vary according to the radial position on the lens. The input power of the pulse is also kept constant, to explicitly understand the role of spatial and temporal aberrations induced by the focusing optics.
The extent of chirp or PPD induced due to the GDD and dispersion over the length of the propagation in the medium (L(r)) was calculated using equations (2), (3) and (4) as a function of the radial position (r) on the lens . Where d is the thickness of the lens at the geometric center (along the optic axis), R1 and R2 are the radii of curvature of the lens τ0 is the initial pulse duration, Δτ(r) is the radially varying PPD, n is the refractive index, dn/dλ is the first order dispersion factor, C(r) is the radially varying chirp, k″ is the second order dispersion factor, λ is the central wavelength and c is speed of light. The formulae used for estimation are30,31], which is characterized by the longitudinal shift (LSA) of the rays from center of the lens to the periphery of the lens at the focus. SA is primarily defined according to the quality factor of the lens which is defined by the radii of curvature of the lens (R1, R2). It is also dependent on the numerical aperture and refractive index of the medium. In our experiment, numerical aperture is fixed, and refractive index variation was considered according to the medium of the optical element used. The LSA of the Achromat doublet is 1 mm, Bi-Convex lens is 1.7 mm, Plano-Convex thick lens and thin lens are 1.2 mm.
It should be noted that for the aberrations that have been mentioned, the curvature of the beam discussed is the curvature of the pulse front. The curvature of the phase front is different from that of the pulse front. The spherical aberration acts on both the phase front and the pulse front, while PPD acts only on the pulse front. Aberrations on phase front add to the pulse front, but not vice-versa. These aberrations are summarized in Table 2. These aberrations influence not only the curvature of the pulse but spatial and temporal distribution of the intensity. The longitudinal spherical aberration (LSA) was characterized by the shift in the focus of the rays from the periphery of the optical element to the centre of the optical element. The PPD and chirp much before the initiation of filamentation processes was simulated (Fig. 2).
3.2. Filament characterization
The onset of filamentation due to the aberrations is captured by imaging the visible emissions of the filaments with reference to the focal plane of the focusing optic at 100 mm. Figure 3 shows the filament images captured under similar experimental conditions. This clearly indicates that the onset of filament is explicitly dependent on the aberrations. We find that although there is considerable extension of the focus of the order of 100 μm due to the large spectral content and chromaticity of the optical element when compared to the Rayleigh range, it is found to be much smaller than the visible extent of the plasma column of the filament itself which is of the order of a few millimeters.
The observations from the experiment are shown in Fig. 3. We find a large contrast in the filament structure from the multiple optical elements. The observable structural changes can be distinctly classified by two contrasting filament shapes and dimensions. Firstly, the filament with long tail and intense end generated from the OAPM and achromat doublet that are ∼ 3 mm long visible portion. Second class of filaments are ∼ 1 mm long visible portion with most intense part at the centre, generated from the Thick lenses and thin lens. The filament structure generated by OAPM has a very characteristic comet like shape with a tail (∼3mm) and with the brightest part at the end/trailing edge of the filament. The total length of the filament is ∼3.5 mm. The SCG extent with OAPM was found to be maximum of the optics used. The filament generated using Achromat doublet, has a structure similar to that of the filament generated using OAPM with a length of ∼ 3mm.
For the filament generated using a thin lens made of fused silica, Bi-Convex (BCX) and thick Plano-Convex (PCX) lenses made of BK-7, most intense part is in the center of the filament with ∼1mm long visible portion. The self-emission images indicate multiple filament formation which could be due to multiple breakdown or due to the dynamic interplay of Kerr self-focusing and plasma defocusing . Contrary to intuition considering the large amount of temporal and spatial aberrations, the filaments observed here were found to have the maximum intensity and electron density among all the optics including the OAPM. The filaments are the widest for these lenses. The main difference between the PCX and BCX is the onset of filamentation, onset of filament from BCX was observed to overshoot other filaments by ∼7 mm.
Taking into account the first type of filaments, the AD has only SA, but filament structure does not differ from that of the OAPM generated filament. The Achromat also induces large overall chirp and PPD, with minute radial variation with opposite curvature when compared to other transmitting optics. In the case of filaments generated using lenses (thin lens, BCX, PCX) the radially varying PPD plays a significant role in the early termination of the filament, leading to 1 mm long visible portion of filaments. Onset of the filament generated from the BCX (LSA∼1.7 mm) was found to overshoot that of the filament from thin lens and PCX (LSA∼1.2 mm) by 7 mm (7 times filament length). This is because of the added curvature on the phase front and pulse front due to SA, which adds an opposing curvature to the curvature caused by Kerr self-focusing and radially varying PPD, thereby extending the onset of filamentation. This demonstrates the influence of SA on filament onset and formation. We can say that spherical aberration influences the onset of filamentation and radially varying PPD influences the collapse of filamentation.
The emissions due to the filament induced breakdown of air are mainly collected over two spectral regions of 320–400 nm and 750–900 nm as shown in Fig. 4. The spectra shown in Fig. 4 are corrected for Bremsstrahlung background. These two spectral regions predominantly represent the emissions from N2 and O2 respectively, which are the major constituents of the molecular mixture of air. Moreover, the ratio of the spectral emissions of N2 over the 320–400 nm region helps us estimate the intensity of the filament .
Spectral lines of N2 have more longitudinal extent than the visible images captured for all the focusing optics. The stark broadened line width of O2 emission at 845 nm from the first ionized species of neutral oxygen molecule, is used to estimate the plasma electron density within the filament. The longitudinal intensity of the filament and plasma electron density were estimated from the FIBS measured along the propagation of the filament (Fig. 5 & 6). The intensity and electron density estimation shown in Figs. 5 and 6 show that the maximum filament intensity is for the plano-convex lens and the biconvex, followed by the thin lens then by the OAPM and AD, with the AD having the least intensity.
There is a stark contrast in plasma properties for the two different filament classes which were classified earlier, we observe that the electron density for the long filaments i.e. from achromat and OAPM is much lower and is measurable only at the most intense part of the filament unlike the shorter filaments where electron density is measurable over the entire visible part of the filament. This shows that that filament intensity and electron density increased due to early termination of the filament. Comparison of visible filament images, intensity and electron density plots show that the visibly brightest part of filament does not coincide with peak electron or most intense part of the filament.
3.3. Supercontinuum emission and effect of Input pulse chirp variation
The supercontinuum emission (SCE) is due to self-phase modulation of the fs laser pulse  which primarily depends on the rate of change of pulse intensity in a given medium . The rate of change of pulse intensity in the case of filamentation of air causes a broadening of the pulse spectrum which manifests as SCE. The SCE shows significant variation among the filaments from the different optical elements. The constant frequency upshift bounding the supercontinuum spectrum of an ultrashort laser pulse undergoing filamentation is used as a measure of conversion efficiency of fs pulse to SCE. This can be obtained from the minimum wavelength obtained from the SCE spectrum, λmin. Figure 7 compares the SCE from different focusing elements at “zero” chirp of the pulse, collected under similar experimental conditions of input pulse energy, polarization, and focusing geometry to explicitly understand the role of aberrations. The OAPM generates the best SCE in terms of spectral amplitude and the extent of broadening (λmin ∼ 550 nm) as it does not induce any GDD and the filament does not terminate early due to the absence of radially varying PPD. As the AD induces maximum GDD, the spectrum is limited to λmin ∼ 660 nm. The other lenses generate higher filament intensity and as the extent of broadening of supercontinuum is proportional to the filament intensity, we observe a larger broadening (characterized by λmin varying over 580–610 nm), when compared to that of AD.
The aberrations of the optical elements are known to modify the propagation characteristics of a fs pulse. The filamentation and the associated SCE are due to the dynamic balance of Kerr self-focusing and plasma defocusing that explicitly depend on the propagation distance. Along with aberrations the dispersion characteristics of the optical element play an important role in determining the peak intensities of the focusing fs pulse. The radially varying first and second order dispersion modifies the pulse width of the focused fs pulse. The material dispersion can be compensated by pre-chirping the pulse temporally. But the spatial variation of dispersion can only be compensated by using specialized optical elements such as deformable mirrors or spatial light modulators. To this effect, we varied the only parameter in our control, which is chirp associated with the input pulse to tailor the process of filamentation that generates a bright SCE.
From Fig. 3 and 7, it is clear that the onset of the filamentation and the associated SCE explicitly depend on the aberrations. Just before the focusing optic the beam was directed into a single shot auto correlator (SSA) and the output compression of the regenerative amplifier was adjusted such that a transform limited pulse was obtained for the focusing optic. Any further chirp induced on the pulse for the current experiments was obtained by varying the grating separation of the output compressor. The chirp induces a quadratic phase difference between frequencies and thereby increasing the temporal duration. We can safely assume that although the induced chirp is quadratic in nature, the variation of the temporal duration is symmetric for negative and positive chirp in this range of grating separation (Supplement 1 ). Figure 8 shows the SCE spectrum and the corresponding filament while varying the chirp of the input pulse for each focusing optic. The role of overall and radially varying GDD induced by the optics manifesting as pulse broadening can be observed by varying the chirp.
We find that some of the optics generate bright filaments when chirped rather than at zero chirp. However, spectral study at these regions does not show increase in light intensity. Brighter visible filaments need not be essentially the most intense filaments because the visible emissions are associated with the first order ionization of air. While the second order ionization emission is in the UV, that are used to estimate the intensity of the filaments. The OAPM has shown the best SCE at zero chirp and has highest filament intensity (Fig5). While the visible image of the filament is not the brightest at C=0. The maximum SCE spread for an achromat is observed at a significantly shifted value of chirp ∼2. The filament becomes almost invisible at this point, again showing that brightness of filament is not most intense part of filament. The breakdown spectra in the 320–400 nm range (Fig. 4) for these bright filaments is dominated by a bremsstrahlung and the reduction of the ionized fluorescence line at 391nm. The maximum SCE spread is slightly shifted (chirp value ∼1) for the thick lenses also.
The induced chirp by the element is positive and the chirp value at lowest λmin is slightly positive chirp for Thin lens and PCX. The λmin varies with either increasing or decreasing the input chirp. The lowest λmin at positive chirp is because of the combined effect of induced chirp and the overall PPD. When a chirped pulse experiences PPD, the central frequency of the input pulse shifts to the trailing frequencies which will be higher for positively chirped pulses and vice-versa. Therefore, for a positively chirped pulse we see a decrease in λmin as chirp is increased till a particular value. Beyond which it starts dropping again since the induced chirp also reduces the peak intensity of the pulse. This positive chirp value at least λmin is directly proportional to the overall PPD induced as shown from the observations (Fig. 9).
The evolution of SCE with chirp of the lenses (with filaments of length 1 mm as shown in Fig. 2 & 3) also shows interesting results. Figure 3 showed that the pulse curvature from PPD and SA define the onset of filament and its structure. In this respect the PCX and thin lens have lesser SA compared to BCX, thereby influencing the onset of filamentation. Although the PCX is thicker and made of a different material when compared to the thin lens, the only difference that was established is the overall PPD, which does not influence the curvature. Filament from the thin lens undergoes minimal broadening. As we increase the chirp, the dominating aberration changes for thin lens, whose filament moves forward indicating the gradual cancelling of collapse of filament, making SA the dominating aberration. The intensity estimated here from the breakdown spectra at the extreme chirp was found to not vary much from that of zero chirp. Whereas for PCX, with increase in chirp the dominating aberration is not only PPD but also radially varying GDD, with large pulse broadening at center of the beam, such that the pulse is shorter at the periphery of the beam than at the center. The spatial intensity profile of the beam is an accumulation of the input spatial intensity profile and the intensity reduction due to pulse broadening after propagating through the lens. This shifts the intensity of the pulse towards the edges of the beam, further enhances the collapse of the filament. This is observed at large chirp for AD too. The BCX on the other hand also induces the same amount of GDD but we see that the filament behavior with varying chirp is completely different from that of the PCX. This is because although the pulse is broadened, the induced SA which is much larger than that of the PCX and keeps the peak intensity in the center of the beam making SA the dominating aberration. The images of the filaments from PCX and BCX over chirp show that they do not move in space, but the dominating mechanisms restricting their movements are different. In the case of PCX it is the termination of filamentation due to radially varying pulse broadening and PPD, but for BCX it is predominantly SA. For the thin lens we find that at zero chirp the dominating mechanism is radially varying PPD but SA becomes more significant as we increase the chirp, which is resulting in the forwards movement of the filament with increasing chirp.
The influence of different aberrations on the filament and the associated SCE was studied using different reflecting and transmitting optics of same focal geometry under similar experimental conditions of input pulse energy, polarization and pulse duration. The quality of the filaments generated was studied in terms of its onset, length, intensity and the plasma electron density (measured via spectral emissions). The SCE associated with the filament is also characterized for each focusing optic in terms of the spectral amplitude and the minimum wavelength. We have seen that the variation of input spatial (SA) and temporal aberrations (PPD) has affected a large extent of the filament characteristics like Onset of filamentation and visible length with variation of SA and radially varying PPD. The OAPM and AD were observed to produce longer filaments with uniform filament intensity. While the PCX, BCX and thin lens have produced a shorter and intense filament. The best SCE is observed using OAPM with least aberrations, while the BCX which has significant PPD, highest LSA and chromatic aberration has resulted in a comparable SCE with a λmin ∼ 580 nm. We have successfully correlated the various optical elements and aberrations to the filament and SCE characteristics to arrive at the appropriate usage of optical elements, for instance where we have counter intuitively shown that maximum confinement of energy was achieved with PCX without loss in peak intensity.
Defence Research and Development Organisation (ERIP/ER/1501138/M/01/319/D(R&D)).
Samuel Anurag Nalam thanks CSIR, India for Research Fellowship.
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
SeeSupplement 1. for supporting content.
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