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In this work, the circular dichroisms CD of nanorice heterodimers consisting of two parallel arranged nanorices with the same size but different materials are investigated theoretically.

Symmetry-breaking is introduced by using different materials and oblique incidence to achieve strong CD at the vicinity of Fano resonance peaks. A simple quantitative analysis shows that the structure with larger Fano asymmetry factor has stronger CD. The intensity and peak positions of the CD effect can be flexibly tuned in a large range by changing particle size, shape, the inter-particle distance and surroundings.

Furthermore, CD spectra exhibit high sensitivity to ambient medium in visible and near infrared regions. Our results here are beneficial for the design and application of high sensitive CD sensors and other related fields. Optical activity OAwhose origin can be reduced to a different response of a system to right- and left-circularly polarized light, is a fantastic optical phenomenon discovered more than years ago.

It has two manifestation forms, which are optical rotation and circular dichroism CD 1. Based on OA, optical rotator dispersion ORD spectroscopy, CD spectroscopy and Raman optical activity ROA spectroscopy have been developed to significant analysis methods in the study of medicine diagnosis, crystallography, analytical chemistry, molecular biology and life form in universe 234567.

However, most nature molecules only manifest very weak optical activity, which greatly limits their applications. In recent years, chiroplasmonics is a hotspot of current research in plasmonics due to the giant OA in chiral metallic nanostructures which has potential applications in ultra-sensitive sensing. The strong coupling between light and surface plasmons SPswhich are collective oscillations of free electrons in the interface of metal-dielectric, is responsible for the giant OA.

Since the status of SPs in metallic nanostructures is sensitive to the shape, size, material and configuration of structures, it offers a flexible way to tune OA effect in a broad band from ultraviolet to near-infrared.

So far, giant OA due to different mechanisms have been intensively studied in various chiral plasmonic nanostructures, such as chiral metal particles 8910pairs of mutually twisted planar metal patterns 11single-layered metal saw-tooth gratings 12planar chiral metal patterns 131415DNA based self-assembled metal particles 1617helical metal wires 18etc.

In addition to the above listed intrinsic chiral nanostructures, which are chiral in the sense that they cannot be superimposed on their mirror image by using spatial operation rotation, translation, etc. Extrinsic chirality provides more flexible way to overcome the difficulty in fabrication progress of complex chiral structure and shows even stronger CD than the intrinsic ones.

It was firstly observed in metallic nanostructures by N. Zheludev inwhere extrinsic chirality leads to exceptionally large CD in the microwave region Following that, extrinsic chirality induced CD was expanded to visible and near-infrared range 23 Very recently, Lu et al.We develop spread-theoretic tools to determine when finite planes admit coordinatization by fractional semifields, and to find such semifields when they exist.

We use our results to show that such semifields exist for prime powers 3 n whenever n is an odd integer divisible by 5 or 7. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Caliskan C. Cordero M. Fisher J. Note di Math. Hansen T. Hentzel I. Algebra Comput. Jha V. Belgian Math. Johnson N. Taylor and Francis Group, New York Google Scholar.

Jacobson N. Freeman and Company, San Francisco, Kantor W. In: Proceedings of Conference at Pingree Park, pp. Leone A. Simon Stevin 67— Neumann H. Puccio L. Basel 49— Algebra 22— Schneider T. Reine Angew. Download references. Correspondence to Vikram Jha. Reprints and Permissions. Cordero, M. Fractional dimensions in semifields of odd order. Codes Cryptogr. Download citation. Received : 23 April Revised : 17 October In finite geometrythe Fano plane after Gino Fano is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point.

The Fano plane can be constructed via linear algebra as the projective plane over the finite field with two elements. One can similarly define projective planes over any other finite field, with the Fano plane being the smallest.

Using the standard construction of projective spaces via homogeneous coordinatesthe seven points of the Fano plane may be labeled with the seven non-zero ordered triples of binary digits, and This can be done in such a way that for every two points p and qthe third point on line pq has the label formed by adding the labels of p and q modulo 2. In other words, the points of the Fano plane correspond to the non-zero points of the finite vector space of dimension 3 over the finite field of order 2.

Due to this construction, the Fano plane is considered to be a Desarguesian planeeven though the plane is too small to contain a non-degenerate Desargues configuration which requires 10 points and 10 lines. The lines of the Fano plane may also be given homogeneous coordinates, again using non-zero triples of binary digits.

With this system of coordinates, a point is incident to a line if the coordinate for the point and the coordinate for the line have an even number of positions at which they both have nonzero bits: for instance, the point belongs to the linebecause they have nonzero bits at two common positions. In terms of the underlying linear algebra, a point belongs to a line if the inner product of the vectors representing the point and line is zero. A permutation of the seven points of the Fano plane that carries collinear points points on the same line to collinear points in other words, it "preserves collinearity" is called a " collineation ", " automorphism ", or " symmetry " of the plane.

It consists of different permutations. The automorphism group is made up of 6 conjugacy classes. All cycle structures except the 7-cycle uniquely define a conjugacy class:. The 48 permutations with a complete 7-cycle form two distinct conjugacy classes with 24 elements:.

See Fano plane collineations for a complete list. The Fano plane contains the following numbers of configurations of points and lines of different types. For each type of configuration, the number of copies of configuration multiplied by the number of symmetries of the plane that keep the configuration unchanged is equal tothe size of the entire symmetry group.

The automorphism group of the group Z 2 3 is that of the Fano plane, and has order The Fano plane is a small symmetric block designspecifically a 2- 7,3,1 -design. The points of the design are the points of the plane, and the blocks of the design are the lines of the plane.

As such it is a valuable example in block design theory. The Fano plane is one of the important examples in the structure theory of matroids. Excluding the Fano plane as a minor is necessary to characterize several important classes of matroids, such as regular, graphic, and cographic ones. If you break one line apart into three 2-point lines you obtain the "non-Fano configuration", which can be embedded in the real plane.

It is another important example in matroid theory, as it must be excluded for many theorems to hold.Thank you for visiting nature. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer.

In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. Photonic and plasmonic devices rely on nanoscale control of the local density of optical states LDOS in dielectric and metallic environments.

The tremendous progress in designing and tailoring the electric LDOS of nano-resonators requires an investigation tool that is able to access the detailed features of the optical localized resonant modes with deep-subwavelength spatial resolution. This scenario has motivated the development of different nanoscale imaging techniques.

Using this technique, we investigate the properties of photonic crystal nanocavities, demonstrating that the resonant modes appear as characteristic Fano line shapes, which arise from interference.

Therefore, by monitoring the spatial variation of the Fano line shape, we locally measure the phase modulation of the resonant modes without the need of external heterodyne detection. This novel, deep-subwavelength imaging method allows us to access both the intensity and the phase modulation of localized electric fields.

Finally, this technique could be implemented on any type of platform, being particularly appealing for those based on non-optically active material, such as silicon, glass, polymers, or metals. The development of integrated nanophotonic and nanoplasmonic optical resonators is currently one of the main goals of the nano-optics community. In optical nano-resonators, the electric local density of optical states LDOSwhich assesses the number of available photonic states in a specific spatial region 1234is dominated by a few strongly localized modes that are not spectrally overlapped Note that for well spectrally resolved modes in nanoresonators the electric LDOS corresponds to the electric field intensity of the modes.

The characterization of the electric LDOS at the proper spatial resolution is an essential step toward the realization of high-density photonic integrated circuits and quantum photonic networks 567.

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For example, photon tunneling in coupled photonic crystal nanocavities PCNs is governed by the electric LDOS spatial overlap between neighboring resonators 8. Similarly, the electric LDOS of a localized mode strongly influences the light—matter interaction of a single quantum emitter being directly related to the spontaneous emission rate 9 as well as to the signal amplification 10and the electric LDOS is crucial for the development of efficient sensors 11 Currently, many experimental techniques are available for imaging the optical resonant mode features at nanoscale resolution.

For example, in III—V semiconductor photonic platforms, multiple quantum wells or quantum dots are embedded to serve as internal light sources In systems such as silicon- glass- polymer- or metal-based devices, the modification of the spontaneous emission rate of small emitters such as fluorescent dyes or nitrogen vacancy centers can be monitored by placing the light source at the end of a scanning probe 16 However, photoluminescence techniques require a spectral matching between the emitter and the photonic mode, and especially in case of functionalized probes, they may suffer the phenomena of bleaching and blinking 18 Pure photonic investigation techniques are potentially more powerful due to the direct investigation of a broad spectral range, the possibility to address the distributions of localized modes in air regions or in metal devices, the absence of polarization constraints and the possibility of phase retrieval when used in combination with heterodyne detection in an external interferometer 2122 A particularly good example is represented by near-field surface enhanced light scattering performed with apertureless probes, as recently applied to various contexts 22 The high sensitivity of this method is largely due to the high refractive index material probes used to enhance the scattering.

However, the disadvantage of this light scattering technique is that the investigation of optical micro- and nano-resonators e. Moreover, this technique, as in the case of cathodoluminescence, suffers from being mainly sensitive to the out-of-plane component of the electric LDOS 22and this technique is not suited for TE-like localized modes.

Therefore, a pure optical method that can be applied on high-Q resonators to retrieve the intensity and the phase of the confined electric field is required. Inspired by far-field resonant scattering experiments, which exhibit Fano-like resonances in high-Q PCNs 262728we implement a full photonic method based on either the resonant back scattering RBS or the resonant forward scattering RFS configuration in the near-field regime.

In addition, we use aluminum-coated aperture probes and plasmonic probes to retrieve the spatial modulation of the phase of the localized modes.In this work, the circular dichroisms CD of nanorice heterodimers consisting of two parallel arranged nanorices with the same size but different materials are investigated theoretically.

Symmetry-breaking is introduced by using different materials and oblique incidence to achieve strong CD at the vicinity of Fano resonance peaks. A simple quantitative analysis shows that the structure with larger Fano asymmetry factor has stronger CD. The intensity and peak positions of the CD effect can be flexibly tuned in a large range by changing particle size, shape, the inter-particle distance and surroundings.

Furthermore, CD spectra exhibit high sensitivity to ambient medium in visible and near infrared regions. Our results here are beneficial for the design and application of high sensitive CD sensors and other related fields.

Optical activity OAwhose origin can be reduced to a different response of a system to right- and left-circularly polarized light, is a fantastic optical phenomenon discovered more than years ago. It has two manifestation forms, which are optical rotation and circular dichroism CD 1. Based on OA, optical rotator dispersion ORD spectroscopy, CD spectroscopy and Raman optical activity ROA spectroscopy have been developed to significant analysis methods in the study of medicine diagnosis, crystallography, analytical chemistry, molecular biology and life form in universe 234567.

However, most nature molecules only manifest very weak optical activity, which greatly limits their applications. In recent years, chiroplasmonics is a hotspot of current research in plasmonics due to the giant OA in chiral metallic nanostructures which has potential applications in ultra-sensitive sensing. The strong coupling between light and surface plasmons SPswhich are collective oscillations of free electrons in the interface of metal-dielectric, is responsible for the giant OA.

Since the status of SPs in metallic nanostructures is sensitive to the shape, size, material and configuration of structures, it offers a flexible way to tune OA effect in a broad band from ultraviolet to near-infrared. So far, giant OA due to different mechanisms have been intensively studied in various chiral plasmonic nanostructures, such as chiral metal particles 8910pairs of mutually twisted planar metal patterns 11single-layered metal saw-tooth gratings 12planar chiral metal patterns 131415DNA based self-assembled metal particles 1617helical metal wires 18etc.

In addition to the above listed intrinsic chiral nanostructures, which are chiral in the sense that they cannot be superimposed on their mirror image by using spatial operation rotation, translation, etc. Extrinsic chirality provides more flexible way to overcome the difficulty in fabrication progress of complex chiral structure and shows even stronger CD than the intrinsic ones. It was firstly observed in metallic nanostructures by N.

Zheludev inwhere extrinsic chirality leads to exceptionally large CD in the microwave region Following that, extrinsic chirality induced CD was expanded to visible and near-infrared range 23 Very recently, Lu et al.

Kato et al.It has been conjectured that all non-desarguesian projective planes contain a Fano subplane. We will show that there is an embedded Fano subplane in the Figueroa plane of order q 3 for q any prime power. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Batten L. Brown, J. In: Proceedings of the 3rd Congress of Geometry, pp Brown J.

Simon Stevin 12— Caliskan, C. Cherowitzo W. Dempwolff U. Basel 43— Utilitas Math, Winnepeg Figueroa R. Fisher, J. Hall M. Hering, C. In: Jungnickel, D. Springer, Berlin Johnson N.

### Fano subplanes in finite Figueroa planes

Note Mat. Moorhouse G. Neumann H. Download references. Correspondence to Bryan Petrak. Reprints and Permissions. Petrak, B. Fano subplanes in finite Figueroa planes. Download citation. Received : 15 November Revised : 11 January Published : 08 June Issue Date : December Search SpringerLink Search.

Abstract It has been conjectured that all non-desarguesian projective planes contain a Fano subplane. References 1 Batten L. Utilitas Math, Winnepeg 10 Figueroa R.

Springer, Berlin 15 Johnson N.In mathematicsprojective geometry is the study of geometric properties that are invariant with respect to projective transformations.

This means that, compared to elementary Euclidean geometryprojective geometry has a different setting, projective spaceand a selective set of basic geometric concepts.

The basic intuitions are that projective space has more points than Euclidean spacefor a given dimension, and that geometric transformations are permitted that transform the extra points called " points at infinity " to Euclidean points, and vice-versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations the affine transformations.

The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometrybecause angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinityonce the concept is translated into projective geometry's terms.

Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century.

This included the theory of complex projective spacethe coordinates used homogeneous coordinates being complex numbers. Several major types of more abstract mathematics including invariant theorythe Italian school of algebraic geometryand Felix Klein 's Erlangen programme resulting in the study of the classical groups were based on projective geometry.

It was also a subject with many practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry. The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry the study of projective varieties and projective differential geometry the study of differential invariants of the projective transformations.

Projective geometry is an elementary non- metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines.

That there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art.

The simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" i. Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. For example, the different conic sections are all equivalent in complex projective geometry, and some theorems about circles can be considered as special cases of these general theorems.

During the early 19th century the work of Jean-Victor PonceletLazare Carnot and others established projective geometry as an independent field of mathematics. After much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations.

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