Naturally harmonic vector fields

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In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. A Nature Research Journal. Strong field laser physics has primarily been concerned with controlling beams in time while keeping their spatial profiles invariant. In the case of high harmonic generation, the harmonic beam is the result of the coherent superposition of atomic dipole emissions.

Therefore, fundamental beams can be tailored in space, and their spatial characteristics will be imparted onto the harmonics. Here we produce high harmonics using a space-varying polarized fundamental laser beam, which we refer to as a vector beam. By exploiting the natural evolution of a vector beam as it propagates, we convert the fundamental beam into high harmonic radiation at its focus where the polarization is primarily linear.

This evolution results in circularly polarized high harmonics in the far field. Such beams will be important for ultrafast probing of magnetic materials. Recent advances allow us to shape the spatial mode of a laser beam in phase and polarization at will, point by point, in their transverse plane 1. Such techniques commonly rely on devices that are composed of aligned liquid crystal molecules whose anisotropy can be tuned with electric fields 2.

Despite their utility in the visible and near infrared, such devices cannot be used in the extreme ultraviolet XUV. However, it is possible through high harmonic generation to upconvert the structured fundamental beams to the extreme ultraviolet 345678. Nevertheless, it is much more common to shape the fundamental beam that produces high harmonics in time rather than in space. A typical method for producing isolated attosecond pulses uses a fundamental pulse with time varying polarization 910 This method relies on the fact that conversion from the fundamental to the XUV requires nearly linear polarization Generating circularly polarized XUV beams by mixing counter rotating bicircular field with different colors is another good example of shaping the laser field in the temporal domain.

The field is shaped to have three-fold symmetry and enables the tunneled electron to recombine with its parental ion every one third of the fundamental optical cycle 1314 The field shaping in temporal domain enables efficient generation of circularly polarized high-order harmonics.

Alternatively, the circularly polarized harmonics are also reported by mixing two non-collinear bicircular driving laser beams 16 Reducing the relative angle is helpful to optimize the phase matching in producing XUV radiation.

To further exploit the spatial degree of freedom, the beam-mixing scheme evolves into the direct modulation of spatial modes of laser beams. In this experiment, we use liquid crystal technology 1819 to shape the polarization of a beam in space rather than in time, which we refer to as a vector beam Such a beam naturally evolves upon propagation, with the polarization structure of the beam changing from near to far field 21 By exploiting the natural evolution of the vector beam, we convert the fundamental beam into high harmonic radiation at its focus where its polarization is primarily linear.

This evolution of the generated high harmonic beam results in circularly polarized XUV radiation in the far field. Comparing to the non-colinear beam-mixing schemes, the laser beam can be modulated pixel-by-pixel enabling a complete and more flexible control of its spatial properties.

This opens a field of vectorizing the radiation in both strong infrared and XUV radiation. The liquid crystal plate that we use to shape the driving laser beam is tuned 1923 to give half-wave retardation at the fundamental wavelength. The liquid crystal molecules are aligned in different orientations in each quadrant, as shown in Fig. With this device and a quarter-wave plate, we can generate the beam shown in Fig.

Specifically, we modify the transverse profile of a Gaussian fundamental beam such that its optical properties vary from one quadrant to another. In this case, adjacent quadrants are defined by opposite circular polarizations with different optical phases. This beam then interacts with a gas-phase nonlinear medium, thus creating high harmonics in the XUV where the spatial structure of the original linearly polarized regions is preserved. As with the fundamental, the polarization state of the resulting XUV vector beam will also evolve as the beam propagates from near to far field see Supplementary Figs.

We show that in the far field, this beam is composed of circularly polarized light in each quadrant, with adjacent quadrants having different handedness.In vector calculusa conservative vector field is a vector field that is the gradient of some function.

Path independence of the line integral is equivalent to the vector field being conservative. A conservative vector field is also irrotational ; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected.

Conservative vector fields appear naturally in mechanics : They are vector fields representing forces of physical systems in which energy is conserved. In a two- and three-dimensional space, there is an ambiguity in taking an integral between two points as there are infinitely many paths between the two points—apart from the straight line formed between the two points, one could choose a curved path of greater length as shown in the figure. Therefore, in general, the value of the integral depends on the path taken.

To visualize this, imagine two people climbing a cliff; one decides to scale the cliff by going vertically up it, and the second decides to walk along a winding path that is longer in length than the height of the cliff, but at only a small angle to the horizontal. Although the two hikers have taken different routes to get up to the top of the cliff, at the top, they will have both gained the same amount of gravitational potential energy.

This is because a gravitational field is conservative. As an example of a non-conservative field, imagine pushing a box from one end of a room to another. Pushing the box in a straight line across the room requires noticeably less work against friction than along a curved path covering a greater distance.

Escher's painting Ascending and Descending illustrates a non-conservative vector field, impossibly made to appear to be the gradient of the varying height above ground as one moves along the staircase. It is rotational in that one can keep getting higher or keep getting lower while going around in circles. It is non-conservative in that one can return to one's starting point while ascending more than one descends or vice versa.

On a real staircase, the height above the ground is a scalar potential field: If one returns to the same place, one goes upward exactly as much as one goes downward. Its gradient would be a conservative vector field and is irrotational. The situation depicted in the painting is impossible. The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field.

This holds as a consequence of the chain rule and the fundamental theorem of calculus. For this reason, such vector fields are sometimes referred to as curl-free vector fields or curl-less vector fields.

They are also referred to as longitudinal vector fields. In a simply connected open region, an irrotational vector field has the path-independence property. This can be seen by noting that in such a region, an irrotational vector field is conservative, and conservative vector fields have the path-independence property.

The result can also be proved directly by using Stokes' theorem. In a simply connected open region, any vector field that has the path-independence property must also be irrotational.

The vorticity of an irrotational field is zero everywhere. Kelvin's circulation theorem states that a fluid that is irrotational in an inviscid flow will remain irrotational. This result can be derived from the vorticity transport equationobtained by taking the curl of the Navier-Stokes Equations. For a two-dimensional field, the vorticity acts as a measure of the local rotation of fluid elements.

Note that the vorticity does not imply anything about the global behavior of a fluid. It is possible for a fluid traveling in a straight line to have vorticity, and it is possible for a fluid that moves in a circle to be irrotational. The most prominent examples of conservative forces are the gravitational force and the electric force associated to an electrostatic field.We find all the left-invariant harmonic unit vector fields on the oscillator groups.

Besides, we determine the associated harmonic maps from the oscillator group into its unit tangent bundle equipped with the associated Sasaki metric. Moreover, we investigate the stability and instability of harmonic unit vector fields on compact quotients of four dimensional oscillator group G 1 1. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Boeckx, L. Vanhecke : Harmonic and minimal vector fields on tangent and unit tangent bundles.

Vanhecke : Harmonic and minimal radial vector fields. Acta Math. Google Scholar. Boucetta, A. Medina : Solutions of the Yang-Baxter equations on quadratic Lie groups: the case of oscillator groups. Calvaruso : Harmonicity of vector fields on four-dimensional generalized symmetric spaces.

Gadea, J. Gil-Medrano : Relationship between volume and energy of vector fields. Vanhecke : Examples of minimal unit vector fields. Global Anal. Vanhecke : Minimal and harmonic characteristic vector fields on three-dimensional contact metric manifolds. Vanhecke : Energy and volume of unit vector fields on three-dimensional Riemannian manifolds. Levichev : Chronogeometry of an electromagnetic wave given by a bi-invariant metric on the oscillator group. Russian original. Tohoku Math. In French.

Milnor : Curvatures of left invariant metrics on Lie groups. Onda : Examples of algebraic Ricci solitons in the pseudo-Riemannian case. Tsukada, L. Vanhecke : Minimality and harmonicity for Hopf vector fields.

Vanhecke, J. Unione Mat. B, Artic. Wiegmink : Total bending of vector fields on Riemannian manifolds. Download references. Correspondence to Ju Tan.

Reprints and Permissions. Xu, N. Invariant harmonic unit vector fields on the oscillator groups. Czech Math J 69, — Download citation.An excellent reference for anyone needing to examine properties of harmonic vector fields to help them solve research problems. It emphasizes a combination of theoretical development with practical applications for a solid treatment of the subject useful to those new to research using differential geometric methods in extensive detail.

Chapter Eight. Lorentz Geometry and Harmonic Vector Fields 8. A Few Notions of Lorentz Geometry.

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naturally harmonic vector fields

Search for books, journals or webpages All Pages Books Journals. However, due to transit disruptions in some geographies, deliveries may be delayed. View on ScienceDirect. Authors: Sorin Dragomir Domenico Perrone. Hardcover ISBN: Imprint: Elsevier. Published Date: 26th October Page Count: For regional delivery times, please check When will I receive my book? Sorry, this product is currently out of stock. Flexible - Read on multiple operating systems and devices.Thank you for visiting nature.

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Harmonic form

In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. A Nature Research Journal. Strong field laser physics has primarily been concerned with controlling beams in time while keeping their spatial profiles invariant. In the case of high harmonic generation, the harmonic beam is the result of the coherent superposition of atomic dipole emissions.

Therefore, fundamental beams can be tailored in space, and their spatial characteristics will be imparted onto the harmonics. Here we produce high harmonics using a space-varying polarized fundamental laser beam, which we refer to as a vector beam. By exploiting the natural evolution of a vector beam as it propagates, we convert the fundamental beam into high harmonic radiation at its focus where the polarization is primarily linear.

This evolution results in circularly polarized high harmonics in the far field. Such beams will be important for ultrafast probing of magnetic materials. Recent advances allow us to shape the spatial mode of a laser beam in phase and polarization at will, point by point, in their transverse plane 1. Such techniques commonly rely on devices that are composed of aligned liquid crystal molecules whose anisotropy can be tuned with electric fields 2.

Despite their utility in the visible and near infrared, such devices cannot be used in the extreme ultraviolet XUV. However, it is possible through high harmonic generation to upconvert the structured fundamental beams to the extreme ultraviolet 345678.

Nevertheless, it is much more common to shape the fundamental beam that produces high harmonics in time rather than in space. A typical method for producing isolated attosecond pulses uses a fundamental pulse with time varying polarization 910 This method relies on the fact that conversion from the fundamental to the XUV requires nearly linear polarization Generating circularly polarized XUV beams by mixing counter rotating bicircular field with different colors is another good example of shaping the laser field in the temporal domain.

The field is shaped to have three-fold symmetry and enables the tunneled electron to recombine with its parental ion every one third of the fundamental optical cycle 1314 The field shaping in temporal domain enables efficient generation of circularly polarized high-order harmonics.

Alternatively, the circularly polarized harmonics are also reported by mixing two non-collinear bicircular driving laser beams 16 Reducing the relative angle is helpful to optimize the phase matching in producing XUV radiation. To further exploit the spatial degree of freedom, the beam-mixing scheme evolves into the direct modulation of spatial modes of laser beams.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. Question: Is there relation between two above definitions? Please give a simple example. Theorem 1.

Theorem 2. There are several good sources for study of this notion of harmonicity of unit vector fields. Dragomir and Domenico Perrone Elsevier,but there are also articles that you may find useful: For example, see the survey article Volume, energy and generalized energy of unit vector fields on Berger spheres.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Relation between harmonic vector field and harmonic 1-form Ask Question. Asked 4 years, 2 months ago.

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naturally harmonic vector fields

Update: I find some theorem in this topic: Theorem 1. YCor G 2, 3 3 gold badges 15 15 silver badges 42 42 bronze badges. What is "the energy function" here?

But the restriction to unit vector fields is a bit unusual, and doesn't give the same Euler-Lagrange equations, I imagine. It would appear to me that Definition 1 is a very special definition, largely unrelated to Definition 2. You can of course talk about harmonic vector fields i.

Active Oldest Votes. Robert Bryant Robert Bryant Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown.It follows from the Hodge theorem that this mapping is an isomorphism. In particular, harmonic functions, i. A parallel theory of harmonic forms exists for Hermitian manifolds cf.

Laplace—Beltrami equation. In such a case. In particular. The study of harmonic functions and forms on Riemann surfaces originates with B. Riemann, whose existence theorems were fully proved at the beginning of the 20th century.

The theory of harmonic forms on compact Riemannian manifolds was first presented by W. Hodge [1]. Various generalizations of the theory of harmonic forms were subsequently given.

Vectorizing the spatial structure of high-harmonic radiation from gas

Differential form. Harmonic forms are a powerful tool in the study of the cohomology of real and complex manifolds and of cohomology spaces of discrete groups. Harmonic forms can be used to establish a connection between the curvature of a compact Riemannian manifold and the triviality of some of its cohomology groups [6][7]. Similar connections have also been obtained in complex analytic geometry [4][5] and in the theory of discrete transformation groups [8].

Log in. Namespaces Page Discussion. Views View View source History. Jump to: navigationsearch. References [1] W. Hodge, "The theory and application of harmonic integrals"Cambridge Univ.

naturally harmonic vector fields

Press [2] G. Schwartz, "Equaciones diferenciales parciales elipticas"Univ. Colombia [3b] L. Schwartz, "Variedades analiticas complejas"Univ. Colombia [4] R. Wells jr. Chern, "Complex manifolds without potential theory"Springer [6] S.


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