However, not all effects of lens distortion are negative. In some cases, it can add a unique perspective to your photos, particularly in wide-angle landscape photos or street photography. The bulging effect of barrel distortion can emphasize scale and depth, making your images more dynamic and engaging.

Fang, Y. & Liu, Y. Optimal control over high-order-harmonic ellipticity in two-color cross-linearly-polarized laser fields. Phys. Rev. A 103, 033116. https://doi.org/10.1103/PhysRevA.103.033116 (2021).

Teichmann, S. M., Silva, F., Cousin, S. L., Hemmer, M. & Biegert, J. 0.5-keV Soft X-ray attosecond continua. Nat. Commun. 7, 11493. https://doi.org/10.1038/ncomms11493 (2016).

Lambert, G. et al. Towards enabling femtosecond helicity-dependent spectroscopy with high-harmonic sources. Nat. Commun. 6, 6167. https://doi.org/10.1038/ncomms7167 (2015).

Elliptically polarizedexamples

Kfir, O. et al. Nanoscale magnetic imaging using circularly polarized high-harmonic radiation. Sci. Adv. 3, eaao4641. https://doi.org/10.1126/sciadv.aao4641 (2017).

Remember, in photography, knowing how to manage imperfections can often be as important as seeking perfection. Embrace the artistic possibilities that lens distortion offers, and use it as another tool to create compelling images. The world of photography is as much about control as it is about creativity.

This work was primarily supported (B.G, S.W, A.B) by a grant from the U.S. Department of Energy, Division of Chemical Sciences, Atomic, Molecular and Optical Sciences Program (Award No. DE-SC0001771). We acknowledge the use of computing resources made possible in part via support by a grant from the U.S. National Science Foundation (Grant No. PHY-1734006).

Circularlypolarizedlight

Baykusheva, D., Ahsan, M. S., Lin, N. & Wörner, H. J. Bicircular high-harmonic spectroscopy reveals dynamical symmetries of atoms and molecules. Phys. Rev. Lett. 116, 123001. https://doi.org/10.1103/PhysRevLett.116.123001 (2016).

While it is more difficult to control the phase difference between the x- and y-components of the harmonic radiation, the relative strength of the components is expected to depend on the intensity ratio of the two cross polarized fields, especially when the intensity of the 400 nm field is significantly smaller than the intensity of the 800 nm field. To confirm this expectation, we changed the intensity ratio of the fields by keeping the intensity of the fundamental pulse at \(10^{14}\) W/\(\hbox {cm}^2\) while varying the intensity of the second harmonic field. As examples of the results we show the ratio of the radiation polarized in the \(\hat{\textbf{y}}\)-direction to that polarized in the \(\hat{\textbf{x}}\)-direction, i.e. \(\rho ^2(\Omega )\), for an even harmonic in the plateau (harmonic 18, blue squares) and in the cut-off region (harmonic 24, red circles) in Fig. 3. For hydrogen atom (panel a) we see that the harmonic intensity ratio increases linearly with respect to the 400 nm field intensity. This observation implies that the ratio, more specifically the \(S_y\)-component, depends on the interaction with just a single photon from the weaker \(2\omega\)-field. The slight deviation from the linear trend for the plateau harmonic (harmonic 18, blue squares) at the highest field intensity ratios likely indicates the impact of higher order 400 nm photon processes. We observe that the results for the \(p_x\)-orbital in argon atom (panel c) are quite similar to those found for the hydrogen atom. This can be understood since the orientation of the orbital is along the polarization direction of the stronger fundamental field and, hence, the coupling with the 800 nm photons is much stronger than with the 400 nm field, which is polarized in the orthogonal direction. In contrast, for the \(p_y\)-orbital, oriented in the direction of the weaker \(2\omega\)-field, the results are different. The \(S_y\)-component is much stronger than the \(S_x\)-component and the overall trend of the harmonic intensity ratio is decreasing rather than increasing linearly. This means that here we cannot interpret the interaction of the \(2\omega\)-field as a single photon coupling or being perturbative with respect to the interaction with the 800 nm field. For the full Ar results (panel b) we therefore conclude that the linear trend at the lowest field intensity ratios is likely due to the dominant impact from the contributions of the harmonic radiation from the \(p_x\)-orbital, while at higher ratios the contribution from the \(p_y\)-orbital becomes more significant, leading to the deviation from the linear increase of the harmonic intensity ratio. We may note that indication of the (first order) perturbative impact of the second harmonic field can also be seen in the results presented in Fig. 2. Beyond the cut-off the ellipticity of the even harmonics (panels on the left) varies periodically as a function of the relative phase (or, equivalently, as a function of the carrier-envelope phase of the 400 nm field), indicating the impact of a single photon transition by the weaker second harmonic field. In contrast, the variation is more complex for harmonics before the cut-off. Furthermore, the observation that highest ellipticities are found when the maximum of the \(2\omega\)-field occurs slightly after the respective ionization time driven by the fundamental field may be interpreted as another indication of the weak perturbative impact of the second harmonic field.

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Lens distortion, at its core, is a deviation from the 'ideal' image – typically, one that is geometrically perfect. Before we can analyze the intricate aspects of managing lens distortion, it's key to establish a clear understanding of what it actually is.

Schafer, K. J., Yang, B., DiMauro, L. F. & Kulander, K. C. Above threshold ionization beyond the high harmonic cutoff. Phys. Rev. Lett. 70, 1599–1602. https://doi.org/10.1103/PhysRevLett.70.1599 (1993).

As mentioned above, an integral part to enable the generation of elliptically polarized high harmonics is to achieve comparable signal components along the polarization directions of the two laser pulses, in particular a match of the minima in the spectral component of the stronger (fundamental) field and the maxima in the spectral component of the weaker (second harmonic) field. Besides controlling the intensity ratio of the two pulses, this means that the duration over which the two pulses interact with the atom has to be kept short. For a short interaction times the harmonic lines are rather broad, limiting the contrast between maxima and minima in each harmonic component, while for long interaction times the harmonic lines tend to get narrower enhancing the difference between the extrema in the spectra. This is shown in the insets of Fig. 4, which show the harmonic components (blue: x-component, red: y-component) of the 24th harmonic from hydrogen atom. In the left panel the laser pulse duration is very short, the lines are broad, the two components are well separated, and, hence, the polarization of the harmonic is almost linear. At a duration of 20 optical cycles (middle panel) the lines are narrower and more pronounced. This leads to an increase of the maxima and a decrease of the minima in the two components. At this intensity ratio and this duration the two components overlap well for the 24th harmonic, leading to the high ellipticity. At the longest pulse duration considered (right panel) the maximum of the y-component and the minimum of the x-component are even more pronounced, therefore the overlap of the components for the 24th harmonic is less than at intermediate pulse duration and, thus, the ellipticity of the harmonic is lower. The main results in Fig. 4 confirm the expectation that for each of the atoms the ellipticity of the even harmonics can be enhanced significantly by finding the optimum pulse duration. Indeed, we see that very high ellipticities can be obtained, which is in agreement with the experimental observations16. We note that the effective joint interaction time of the two pulses can also be controlled by keeping the pulse duration constant but delaying the pulses to each other and that the optimal pulse duration will depend, at least, on the intensity ratio of the two pulses, and the kind of target atom. These two factors impact the relative strength of the two components of the harmonic signal, which is relevant for the ellipticity of the harmonics.

The theoretical and numerical methods used to obtain the HHG spectra are based on numerical solution of the time-dependent Schrödinger equation. The Hamiltonian for a single-active electron atom in an external electric field is given by (in velocity gauge and dipole approximation, Hartree atomic units are used, \(e = m = \hslash = 1\)),

Enabling elliptically polarized high-order harmonics overcomes a historical limitation in the generation of this highly nonlinear process in atomic, molecular and optical physics with applications in other branches. Here, we shed new light on a controversy between experimental observations and theoretical predictions on the possibility to generate harmonics with large ellipticity using two bichromatic laser pulses which are linearly polarized in orthogonal directions. Results of numerical calculations confirm the previous experimental data that in short laser pulses even harmonics with large ellipticity can be obtained for the interaction of such cross-polarized laser pulses with atoms initially in a s- or p-state, while odd harmonics have low ellipticity. The amount of the ellipticity can be controlled via the relative carrier-envelope phase of the pulses, their intensity ratio and the duration of the pulses.

Stremoukhov, S. et al. Origin of ellipticity of high-order harmonics generated by a two-color laser field in the cross-polarized configuration. Phys. Rev. A 94, 013855. https://doi.org/10.1103/PhysRevA.94.013855 (2016).

Ellipticity of the 24th harmonic from hydrogen atom (blue squares) and argon atom (red circles) as a function of the duration of the laser pulses (in cycles of the fundamental field). Other laser parameters are the same as in Fig. 1. The insets show the x- (blue lines) and the y-components of the harmonic signal from hydrogen atom at three pulse durations, (a) 7 cycles, (b) 20 cycles, and (c) 40 cycles.

Harada, Y., Haraguchi, E., Kaneshima, K. & Sekikawa, T. Circular dichroism in high-order harmonic generation from chiral molecules. Phys. Rev. A 98, 021401. https://doi.org/10.1103/PhysRevA.98.021401 (2018).

McPherson, A. et al. Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases. JOSA B 4, 595–601. https://doi.org/10.1364/JOSAB.4.000595 (1987).

Extreme ultraviolet (XUV) and soft x-ray radiation are important light tools for studying the dynamics of fundamental processes in atoms, molecules and materials on ultrashort time scales (see recent reviews1, 2). High harmonic generation (HHG) provides the opportunity to produce such light in a tabletop experimental setting. It is a highly nonlinear process which results from a gas (or solid) interacting with an intense laser field3, 4. On the microscopic (atomic) level the three-step model gives an intuitive picture of the process5,6,7: the intense electric field ionizes an electron from the atom, the electron then propagates in the oscillating field gathering energy, returns to the core and upon recombination with the core emits its energy in an attosecond burst of radiation. In a pulse of several cycles the process repeats every half cycle of the field, giving rise to an attosecond pulse train of high harmonic light. In the frequency domain the light spectrum consists of a series of peaks, or harmonics, at integer multiples of the fundamental frequency reaching up to the soft x-ray regime8,9,10. For a long time experiments generated and applied linearly polarized harmonics, only recently several methods have been proposed and demonstrated to overcome this restriction11,12,13,14,15,16,17,18,19,20. Since then the ability of generating elliptically polarized ultrashort light pulses in the XUV and soft x-ray wavelength regime has enabled various applications with respect to spin-orbit interaction21, molecular symmetries and chirality22, 23, circular dichroism12, 24, and magnetic interactions25.

Pincushion distortion is often seen in telephoto lenses. Here, the image appears pinched at the center, and straight lines near the edges of the frame curve inward.

Many photo editing software programs, like Adobe Lightroom and Photoshop, have lens correction tools that can automatically correct distortion based on the lens used.

Shafir, D., Mairesse, Y., Villeneuve, D. M., Corkum, P. B. & Dudovich, N. Atomic wavefunctions probed through strong-field light-matter interaction. Nat. Phys. 5, 412–416. https://doi.org/10.1038/nphys1251 (2009).

Kyle DeGuzman graduated from San Diego State University with a Bachelor of Science in Television, Film, & New Media. He currently resides in Denver, Colorado spending his time writing, filmmaking, and traveling.

For the numerical calculations the wavefunction is expanded in a basis of spherical harmonics for the angular dimensions and 8th order B-splines in the radial dimension. This approach follows the strategy as outlined in34, 35. The 300 B-splines nodes are placed such that the spacing between nodes is quadratic near the origin then becomes constant at a chosen radius (here, 30 a.u.). Both the maximum orbital angular momentum and maximum magnetic quantum number are fixed to 30. The wavefunction is propagated in a box of 300 a.u. in size with exterior complex scaling being applied to the last 30 a.u., where the radial coordinate is rotated into the complex plane by an angle \(\eta =\pi /4\), to avoid reflections off the walls36. The wavefunction is propagated in time with the Crank-Nicolson method using a time step of 0.1 a.u. The high harmonic signal, \({\textbf{S}}(\omega )\), is calculated by Fourier transforming the dipole acceleration \({\textbf{a}}(t)\),

Lens distortion can significantly affect your photography, particularly if you're shooting architecture or any subject with straight lines. The curving effect of barrel or pincushion distortion can make buildings appear to lean or warp, which can be visually distracting.

However, due to the inherent imperfections in lens design and the curved nature of the lens, light rays often converge at different points, causing image distortion. These distortions can manifest as barrel distortion, where straight lines appear curved outward, or pincushion distortion, where straight lines appear curved inward.

Lens distortion is a common characteristic of all types of camera lenses to varying extents. There are several effective techniques you can employ to minimize its impact and achieve optimal image quality. These include careful lens selection, proper shooting techniques, and post-processing adjustments.

Circular polarization

Bordo, E. et al. Interlocked attosecond pulse trains in slightly bi-elliptical high harmonic generation. J. Phys. Photonics 2, 034005. https://doi.org/10.1088/2515-7647/ab8a5c (2020).

In Fig. 1 we display the results of numerical calculations, based on the solution of the time-dependent Schrödinger equation (see “Methods”), for microscopic high-harmonic spectra obtained for the interaction of (a) a hydrogen atom and (b) an argon atom with cross-polarized laser pulses. For the argon atom the calculations were performed in the single-active electron approximation, using a single-active-electron potential fitted to a density functional theory calculation33. The two laser pulses were centered at 800 nm (\(\omega\)-field, polarized in \({\hat{\textbf{x}}}\)-direction) and at 400 nm (\(2\omega\)-field, polarized in \({\hat{\textbf{y}}}\)-direction). Like in the experiment16, we used a high peak intensity \(I_{\omega } = 10^{14}\) W/\(\hbox {cm}^2\) for the field at the fundamental wavelength, while the second harmonic pulse intensity was an order of magnitude lower, \(I_{2\omega } = 10^{13}\) W/\(\hbox {cm}^2\). Both pulses had the same duration with the peak of the envelopes coinciding and the relative carrier-to-envelope phases as given in the Figure caption.

Eichmann, H. et al. Polarization-dependent high-order two-color mixing. Phys. Rev. A 51, R3414–R3417. https://doi.org/10.1103/PhysRevA.51.R3414 (1995).

The science behind lens distortion lies in how a lens works and the path of light as it interacts with the lens. When light enters the lens, it undergoes a process called refraction, where its direction changes due to the change in medium.

Elliptically polarizedlight

where \({\textbf{p}}\) is the kinetic momentum operator of the electron, \({\textbf{A}}(t)\) is the vector potential, and V(r) is the single-active electron atomic potential, which is fitted to a density functional theory calculation33. The vector potential of the orthogonal two-color field is

Fleischer, A., Kfir, O., Diskin, T., Sidorenko, P. & Cohen, O. Spin angular momentum and tunable polarization in high-harmonic generation. Nat. Photonics 8, 543–549. https://doi.org/10.1038/nphoton.2014.108 (2014).

Young, L. et al. Roadmap of ultrafast x-ray atomic and molecular physics. J. Phys. B At. Mol. Opt. Phys. 51, 032003. https://doi.org/10.1088/1361-6455/aa9735 (2018).

We’re in a golden age of TV writing and development. More and more people are flocking to the small screen to find daily entertainment. So how can you break put from the pack and get your idea onto the small screen? We’re here to help.

Hernández-García, C. et al. Schemes for generation of isolated attosecond pulses of pure circular polarization. Phys. Rev. A 93, 043855. https://doi.org/10.1103/PhysRevA.93.043855 (2016).

With this work we intend to provide new insights into the feasibility of generating elliptically polarized high harmonics at and beyond the cut-off with cross-polarized bichromatic laser pulses. Using ab-initio microscopic solutions of the time-dependent Schrödinger equation we show that indeed highly elliptically polarized harmonics from a s-valence orbital in atomic hydrogen and from a p-orbital in atomic argon can be generated. This confirms the experimental observations, including the result that the ellipticities of the generated even harmonics are in general larger than those of the odd harmonics. We relate the observed ellipticity to the parameters of the fields, which provides an intuitive picture of the underlying physical process, and address differences in the results depending on the initial orbital shape. Finally, we show that restricting the effective interaction time of both pulses, e.g. by shortening the pulses, is essential for achieving high ellipticities.

Barrel distortion is common in wide-angle lenses and is characterized by the image appearing to bulge outwards from the center. In barrel distortion, straight lines near the edges of the frame appear curved outward.

Ratio of radiation polarized in the \(\hat{\textbf{y}}\)-direction to radiation polarized in the \(\hat{\textbf{x}}\)-direction emitted from (a) hydrogen atom and (b) argon atom as a function of the intensity ratio of the two fields. The intensity of the fundamental field was kept fixed at \(10^{14}\) W/\(\hbox {cm}^2\), while that of the \(2\omega\)-field was varied. Separately shown are the results from the (c) \(p_x\)- and (d) \(p_y\)-orbital in argon. Blue squares represent the results for a plateau harmonic (harmonic 18) and red squares those for a cut-off harmonic (harmonic 24). The dotted lines indicate a linear increase and are shown for the sake of comparison. Other laser parameters are the same as in Fig. 1.

Components of HHG spectra and ellipticity for the interaction of (a) a hydrogen atom and (b) an argon atom with an intense laser pulse. In both cases the driving laser intensities are \(I_{800} = 10^{14}\) W/\(\hbox {cm}^2\) and \(I_{400} = 10^{13}\) W/\(\hbox {cm}^2\) with a duration of 26.8 fs, which corresponds to 10 (20) cycles of the fundamental (\(2\omega\)) field. The carrier envelope phase (CEP) of the 800 nm component is \(\phi _{\textrm{800}}=0\) while the CEP of the 400 nm component is \(\phi ^{(\mathrm H)}_{\textrm{400}}=-\frac{\pi }{4}\) for hydrogen and \(\phi ^{(\mathrm Ar)}_{\textrm{400}}=-\frac{5\pi }{8}\) for argon. Solid vertical lines indicate the ionization threshold and the cutoff due to the stronger 800 nm pulse.

Now that we've explored lens distortion, its types, and how to mitigate its effects, it's time to look deeper into the world of lenses. In our next article, we break down how the camera lens works with light and capturing images.

Ghomashi, B., Walker, S. & Becker, A. Enabling elliptically polarized high harmonic generation with short cross polarized laser pulses. Sci Rep 13, 12843 (2023). https://doi.org/10.1038/s41598-023-39814-y

Shafir, D. et al. Probing the symmetry of atomic wavefunctions from the point of view of strong field-driven electrons. New J. Phys. 12, 073032. https://doi.org/10.1088/1367-2630/12/7/073032 (2010).

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

First, we show in Fig. 2 results of additional calculations for hydrogen (panels in upper row) and argon (panels in lower row) in which the carrier waves of the two cross-polarized fields are offset to each other while the envelope remains unchanged. The relative phase of the two fields, \(\phi _{\textrm{rel}} = \phi _{800}-\phi _{400}/2\), is varied over one cycle of the fundamental field. Shown are the ellipticities of the (a) even and (b) odd harmonics as function of the relative phase of the \(2\omega\)-field measured with respect to the peak of the fundamental field. The results show a few distinct features: First, large values of ellipticity are found only for even harmonics, while the odd harmonics are close to linearly polarized independent of the relative phase of the two fields. Next, the ellipticity for the even harmonics varies largely as a function of the relative phase. While for the harmonics near and beyond the cut-off one sees a smooth variation with two extrema having opposite signs, there is no order in the pattern of the plateau harmonics. This indicates that the generation of elliptically polarized harmonics around and after the cut-off can be more easily observed and controlled which may explain why in the experimental report16 ellipticities for cut-off harmonics have been reported. Furthermore, we expect that the inherent physical picture behind the results may be more evident at the highest harmonics. Within the three-step model of high harmonic generation ionization and recombination times for electrons leading to the generation of the harmonics can be identified. According to this picture the relative phase, at which maximum values of ellipticities for the cut-off harmonics are found, indicates that the maximum of the \(2\omega\)-field oscillation has to occur slightly after the respective ionization time.

Ellipticity of the even (panels on the left) and odd (panels on the right) harmonics as a function of harmonic order and relative phase \(\phi _{\textrm{rel}} = \phi _{800}-\frac{\phi _{400}}{2}\) for (a, b) hydrogen atom and (c, d) argon atom. The dashed line indicates the harmonic cut-off, associated with the stronger 800 nm field. Other laser parameters are the same as in Fig. 1.

where \(\sigma =0.125\) is a dimensionless quantity. The results show that the even harmonics near, at and past the cut off have a significant amount of ellipticity while the odd harmonics are in general close to being linearly polarized. This result is in qualitative agreement with the experimental observations16. From Eq. (1) we notice that a large ellipticity can be obtained by matching the phase difference of the two components to \(\pi /2\) and the ratio of the two components, \(\rho (\Omega )\), close to 1. As we can see from the results in Fig. 1 the signal at each of the harmonics is broad and the minima for the x-component match up rather well with the maxima of the y-components of the generated radiation at the frequencies of the even harmonics with high ellipticity, especially near and at the cut-off. In contrast, we observe a large difference between the respective maxima (in \(|S_x|^2\)) and minima (in \(|S_y|^2\)) at the frequencies of the odd harmonics and, hence, those harmonics are close to being linearly polarized. Indeed, we can generalize this observation: If the intensity of the second harmonic field is sufficiently weaker than that of the fundamental pulse, we can expect that the component of the high harmonic generation in the direction of the \(2\omega\)-field is weaker than the component in the direction of the \(\omega\)-field. Therefore, the requirement of matching the intensities of the two components for generating high-order harmonics with a high ellipticity is more likely to be achieved for the even than for the odd harmonics. In order to address these points further, we will discuss in the remainder of this work which parameters of the two fields impact the degree of ellipticity.

Popmintchev, T. et al. Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers. Science 336, 1287–1291. https://doi.org/10.1126/science.1218497 (2012).

Ideally, all the light rays should converge at a single point on the sensor, resulting in a sharp and undistorted image. To understand this visually, check out our breakdown on types of camera lenses.

Mustache distortion, also known as complex distortion, is a combination of barrel and pincushion distortion. The image appears to warp in a wavy pattern, with lines curving outward near the center and inward at the edges.

Schoenlein, R. et al. Recent advances in ultrafast X-ray sources. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 377, 20180384. https://doi.org/10.1098/rsta.2018.0384 (2019).

Ferray, M. et al. Multiple-harmonic conversion of 1064 nm radiation in rare gases. J. Phys. B At. Mol. Opt. Phys. 21, L31. https://doi.org/10.1088/0953-4075/21/3/001 (1988).

While lens distortion can be a unique quality you’d like to keep within your photography, it can also be one that you want to know how to mitigate as much as possible.

Ferré, A. et al. A table-top ultrashort light source in the extreme ultraviolet for circular dichroism experiments. Nat. Photonics 9, 93–98. https://doi.org/10.1038/nphoton.2014.314 (2015).

B.G., S.W. and A.B. conceived the study and developed the theoretical framework. B.G. wrote the code and performed the numerical calculations. B.G., S.W., and A.B. analyzed and interpreted the results and wrote the manuscript.

Dorney, K. M. et al. Controlling the polarization and vortex charge of attosecond high-harmonic beams via simultaneous spin-orbit momentum conservation. Nat. Photonics 13, 123–130. https://doi.org/10.1038/s41566-018-0304-3 (2019).

Understanding lens distortion can actually help you improve the quality of your images and give you more control over the final result.

Composition is also a factor. Position your subject in the center of the frame where distortion is usually least noticeable. For example, in portrait photography, frame your subject in the center of the frame for the least amount of lens distortion on your subject's face.

Elliptical polarization equation derivation

Elliptically polarizedformula

Following an initial proposal in 199526, most of the techniques for the generation of elliptically polarized high harmonics are based on the application of bichromatic, typically \(\omega - 2\omega\), fields in various focusing configurations11,12,13,14,15,16,17,18,19,20. Perhaps, one of the simplest configurations to implement is the use of two linearly polarized laser pulses at frequencies \(\omega\) and \(2\omega\) in a geometry in which the polarization vectors are orthogonal to each other. Yet, generation of high harmonics in this configuration entails an important open question. Due to the reflection symmetry in this configuration, it has been predicted that only linearly polarized harmonics are generated, independent of whether the active valence electron in the target atom is initially in a s- or in a p-state27, 28. For the interpretation it is assumed that the driving field is periodic, envelope effects are negligible and that the pulse is long. The analysis then shows that all the odd harmonics will be polarized along the axis parallel to the fundamental field and the even harmonics are polarized along the perpendicular axis. Even relaxing the long-pulse assumption, it was considered that as long as the active orbital is spherically symmetric the generated harmonics can only be linearly polarized in the direction along the recollision angle of the electron trajectory associated with that harmonic. However, in 2015 an experimental study reported the generation of bright highly elliptically polarized high harmonics with cross-polarized bichromatic laser pulses16, most prominent at and beyond the HHG cut-off. Very few theoretical studies have addressed this controversy between experimental observation and theoretical prediction so far29,30,31. For example, it has been shown that the assumption that each of the attosecond bursts of high harmonic generation is linearly polarized27, 28 is inaccurate32. However, an overarching picture explaining the experimental observations has not emerged to the best of our knowledge.

Lens distortion, in simple terms, is a deviation from the perfect image caused by the optical design of a lens. This deviation results in the image being distorted in a way that differs from the actual scene. The amount and type of distortion depend on the lens design, with wide-angle lenses typically exhibiting more distortion than telephoto lenses. But this depends on the manufacturer and when it was made.

There are three primary types of lens distortions. Understanding and recognizing these distortions is essential for photographers and videographers. The three types of lens distortion are barrel distortion, pincushion distortion, and mustache distortion.

We have also obtained the ellipticities of the generated harmonics (open squares), by defining the ellipticity at a given frequency \(\Omega\) as32:

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Fleischer, A., Bordo, E., Kfir, O., Sidorenko, P. & Cohen, O. Polarization-fan high-order harmonics. J. Phys. B At. Mol. Opt. Phys. 50, 034001. https://doi.org/10.1088/1361-6455/aa507c (2017).

Johnson, A. S. et al. High-flux soft x-ray harmonic generation from ionization-shaped few-cycle laser pulses. Sci. Adv. 4, eaar3761. https://doi.org/10.1126/sciadv.aar3761 (2018).

Mahieu, B. et al. Control of ellipticity in high-order harmonic generation driven by two linearly polarized fields. Phys. Rev. A 97, 043857. https://doi.org/10.1103/PhysRevA.97.043857 (2018).

Azoury, D. et al. Interferometric attosecond lock-in measurement of extreme-ultraviolet circular dichroism. Nat. Photonics 13, 198–204. https://doi.org/10.1038/s41566-019-0350-5 (2019).

Corkum, P. B. Plasma perspective on strong field multiphoton ionization. Phys. Rev. Lett. 71, 1994–1997. https://doi.org/10.1103/PhysRevLett.71.1994 (1993).

where \(\rho (\Omega )=\left| S_y(\Omega )/S_x(\Omega )\right|\) and \(S_{x,y}\) are the amplitudes of the harmonics in the two orthogonal directions. We then obtained the ellipticity at a given harmonic order as an average over the range of one harmonic using a Gaussian filter:

Elliptically polarizedlight equation

Each panel of the Fig. 1 consists of two graphs: in the upper graph shown are the intensities of the components of the generated high harmonic fields in the \({\hat{\textbf{x}}}\)-direction (blue line) and in the \({\hat{\textbf{y}}}\)-direction (red line) as a function of multiples of the fundamental frequency. According to our calculations and in agreement with theoretical predictions the component in the direction perpendicular to the polarization plane (i.e., the \({\hat{\textbf{z}}}\)-direction) is negligible and therefore not shown. As it is expected, the HHG intensities in the polarization direction (i.e., \({\hat{\textbf{x}}}\)-direction) of the fundamental driving field, display maxima at odd harmonics, while those in the orthogonal direction have maxima at the even harmonics. This trend is most clearly visible for harmonics below the ionization threshold and near and past the cut-off, which are indicated by the vertical lines in the Figure.

Kfir, O. et al. Generation of bright phase-matched circularly-polarized extreme ultraviolet high harmonics. Nat. Photonics 9, 99–105. https://doi.org/10.1038/nphoton.2014.293 (2015).

Bachau, H., Cormier, E., Decleva, P., Hansen, J. E. & Martín, F. Applications of B -splines in atomic and molecular physics. Rep. Prog. Phys. 64, 1815–1943. https://doi.org/10.1088/0034-4885/64/12/205 (2001).

Lens distortion is an inherent characteristic of camera lenses that can significantly affect your photography. While it can be detrimental in certain contexts, such as architectural photography, it can also add a unique perspective to images when used intentionally.

Argenti, L. & Moccia, R. K-matrix method with B-splines: σnℓ, βn and resonances in He photoionization below N = 4 threshold. J. Phys. B At. Mol. Opt. Phys. 39, 2773–2790. https://doi.org/10.1088/0953-4075/39/12/012 (2006).

Hickstein, D. D. et al. Non-collinear generation of angularly isolated circularly polarized high harmonics. Nat. Photonics 9, 743–750. https://doi.org/10.1038/nphoton.2015.181 (2015).

Reiff, R., Joyce, T., Jaroń-Becker, A. & Becker, A. Single-active electron calculations of high-order harmonic generation from valence shells in atoms for quantitative comparison with TDDFT calculations. J. Phys. Commun. 4, 065011. https://doi.org/10.1088/2399-6528/ab9a68 (2020).

Fan, T. et al. Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism. Proc. Natl. Acad. Sci. 112, 14206–14211. https://doi.org/10.1073/pnas.1519666112 (2015).

Huang, P.-C. et al. Polarization control of isolated high-harmonic pulses. Nat. Photonics 12, 349–354. https://doi.org/10.1038/s41566-018-0145-0 (2018).

Photography is more than just clicking the shutter. It’s about truly understanding your camera and the lenses you use as tools. One aspect that often gets overlooked is lens distortion. It’s a common occurrence, yet many photographers are unaware of its impact on their photos.

Baykusheva, D. & Wörner, H. J. Chiral discrimination through bielliptical high-harmonic spectroscopy. Phys. Rev. X 8, 031060. https://doi.org/10.1103/PhysRevX.8.031060 (2018).

Elliptical polarization example

McCurdy, C. W. & Martín, F. Implementation of exterior complex scaling in B-splines to solve atomic and molecular collision problems. J. Phys. B At. Mol. Opt. Phys. 37, 917–936. https://doi.org/10.1088/0953-4075/37/4/017 (2004).

Different focal lengths exhibit different levels of distortion. Wide-angle lenses tend to have more distortion, so for subjects with straight lines, consider using a telephoto or standard lens.

Lewenstein, M., Balcou, P., Ivanov, M. Y., L’Huillier, A. & Corkum, P. B. Theory of high-harmonic generation by low-frequency laser fields. Phys. Rev. A 49, 2117–2132. https://doi.org/10.1103/PhysRevA.49.2117 (1994).

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where \(\omega\) is the central frequency of the fundamental field, N is the number of cycles of the fundamental field, \(\phi _{800/400}\) is the carrier envelope phase, \(A_{800/400}\) is the amplitude of the x/y component.

In summary, by presenting and interpreting results of numerical calculations for high-harmonic spectra in hydrogen and argon atoms interacting with cross-polarized laser pulses we shed new light on a controversy between experimental observations and theoretical predictions concerning the generation of elliptically polarized high harmonics in this set-up. Our results confirm the experimental observations that, for a short laser pulse, near the cut-off large ellipticities for the even harmonics are generated while those for the odd harmonics remain small. In the long pulse limit our results agree with theoretical predictions that the harmonics are linearly polarized for a periodic cross-polarized field. The analysis of the results provides a picture in which the second harmonic acts as a perturbation after the time of birth of the electron in the continuum by the fundamental field. Furthermore, it is shown that the amount of ellipticity can be controlled via the relative phase between the two pulses, their intensity ratio and the pulse durations. Enabling the generation of elliptically polarized harmonics in this rather simple set-up extends the pool of techniques for helicity-dependent high harmonic generation which have a large application range in dynamical studies on ultrashort time scales.