WO2014195959A1 - Meta-materials based optical conduits - Google Patents

Meta-materials based optical conduits Download PDF

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Publication number
WO2014195959A1
WO2014195959A1 PCT/IN2013/000355 IN2013000355W WO2014195959A1 WO 2014195959 A1 WO2014195959 A1 WO 2014195959A1 IN 2013000355 W IN2013000355 W IN 2013000355W WO 2014195959 A1 WO2014195959 A1 WO 2014195959A1
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Prior art keywords
optical
energy
hollow cylindrical
optical conduit
metamaterial
Prior art date
Application number
PCT/IN2013/000355
Other languages
French (fr)
Inventor
Sunit Tyagi Dr.
Hermanshu Bhatt Dr.
Original Assignee
Insolare Energy Pvt Ltd
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Insolare Energy Pvt Ltd filed Critical Insolare Energy Pvt Ltd
Publication of WO2014195959A1 publication Critical patent/WO2014195959A1/en

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    • GPHYSICS
    • G02OPTICS
    • G02BOPTICAL ELEMENTS, SYSTEMS OR APPARATUS
    • G02B6/00Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
    • G02B6/02Optical fibres with cladding with or without a coating
    • G02B6/032Optical fibres with cladding with or without a coating with non solid core or cladding
    • GPHYSICS
    • G02OPTICS
    • G02BOPTICAL ELEMENTS, SYSTEMS OR APPARATUS
    • G02B6/00Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
    • G02B6/0001Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings specially adapted for lighting devices or systems
    • G02B6/0096Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings specially adapted for lighting devices or systems the lights guides being of the hollow type
    • GPHYSICS
    • G02OPTICS
    • G02BOPTICAL ELEMENTS, SYSTEMS OR APPARATUS
    • G02B6/00Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
    • G02B6/02Optical fibres with cladding with or without a coating
    • G02B6/02295Microstructured optical fibre
    • G02B6/02309Structures extending perpendicularly or at a large angle to the longitudinal axis of the fibre, e.g. photonic band gap along fibre axis

Definitions

  • the present invention is related to energy systems. More particularly the present invention is related to energy system employing meta-materials based optical conduits for energy transfer.
  • optical conduits are used for the transfer of energy from one point to a second point.
  • Examples of such applications include telecommunication systems, semiconductor arrays, energy systems and the like.
  • conventional optical fibers have several disadvantages such as high transmission losses and bend losses.
  • Metamaterials are periodically repeating, synthetic composite structures that are specifically engineered to circumvent inconvenient bulk material properties.
  • Photonic bandgap crystals such as synthetic opals are a subset example of metamaterials.
  • the exceptional characteristics and response functions of metamaterials are not observed in the individual constituent materials of the composite, and these phenomena arise as a direct result of the periodic inclusion of functional materials such as metals, semiconductors or polymers embedded within the composite.
  • an optical conduit for the transmission of optical energy comprises a hollow cylindrical pipe configured to guide optical energy from a first end to a second end.
  • a metamaterial is disposed on an inner wall of the hollow cylindrical tube and the metamaterial is configured to exhibit a negative refractive index.
  • the optical conduit further comprises a core formed in the center of the hollow cylindrical pipe and configured to confine the optical signals with the hollow cylindrical pipe.
  • a light energy transmission system for transmitting optical energy.
  • the system comprises a light energy source configured to generate light energy and a flexible optical conduit configured to transmit the light energy.
  • the optical conduit comprises a hollow cylindrical pipe configured to guide optical energy from a first end to a second end.
  • a metamaterial is disposed on an inner wall of the hollow cylindrical tube wherein the metamaterial is configured to exhibit a negative refractive index.
  • the system further comprises a core formed in the center of the hollow cylindrical pipe and configured to confine the optical signals within the hollow cylindrical.
  • FIG. 1 is an embodiment of a system for transmitting optical energy implemented according to one aspect of the present technique
  • FIG 2A is a cross sectional view of an embodiment of an optical conduit implemented according to aspects of the present technique
  • FIG. 2B is a perspective view of an embodiment of an optical conduit implemented according to aspects of the present technique
  • FIG. 3 and FIG. 4 are sectional views of an embodiment of an optical conduit illustrating a manner in which optical energy is transmitted;
  • FIG. 5 is an isometric view of the metamaterial a helical arrangement of two types of materials implemented according to aspects of the present technique.
  • the present disclosure discusses an embodiment of a meta-material based light pipe also known as optical conduit.
  • the optical conduit is used for the transmission of optical energy and it includes a hollow cylindrical pipe configured to guide optical energy from a first end to a second end that has walls comprising optical meta-material s.
  • the meta-materials are disposed on the inner walls of the optical conduit along with cladding material.
  • an energy transmission system 10 for transmitting of energy is provided.
  • energy refers in general to electromagnetic energy and specifically to solar energy.
  • the system includes an energy source 12 configured to generate electromagnetic energy.
  • the energy source refers to the sun and the electromagnetic energy refers to solar energy.
  • Energy collectors 14 are configured to collect solar energy and are placed in an energy collection area. Examples of energy collectors include optical reflectors, solar panels, etc.
  • Energy converter 18 is configured to convert the collected solar energy to an alternate , form- of .energy such as thermal energy, electrical energy, etc. In one embodiment, the energy converter converts solar energy to corresponding electrical energy.
  • the energy collector 14 is coupled to energy converter using optical conduit 18.
  • the optical conduit 18 is fabricated using metamaterials and is engineered to have a negative refractive index.
  • the metamaterial is designed for broadband visible spectrum optical properties using a metastructure made of regular meta-atoms whose resonant interaction with light energy gives the desired optical properties. Examples of the metamaterials used include split ring resonators, plasmonic materials, dielectric materials, etc.
  • the optical conduit is configured to guide the solar energy from the energy collector to the energy converter. In one embodiment, the optical conduit is flexible.
  • Power converter 20 is configured to condition the electrical energy generated by the energy converter 18 to a standard form, such as, 220V at 50 Hz output power. Examples of conversion include but are not limited to DC to AC to DC, DC to AC or just changing the voltage or frequency or the combinations thereof.
  • the output of the power converter 20 is filtered through a filter 22 for removing unwanted harmonics and it is then fed to different devices shown by reference numeral 24, 26 and 28.
  • the optical conduit is fabricated using metamaterials and is engineered to have a negative refractive index.
  • the manner in which the optical conduit is implemented is described in further detail below.
  • FIG. 2A and 2B illustrates a cross sectional view and perspective view of an optical conduit implemented according to aspects of the present technique.
  • the optical conduit comprises a hollow cylindrical pipe with a core formed in the center and configured to guide optical energy.
  • the optical conduit has an inner wall lined with metamaterials 34.
  • the metamaterial 34 is configured to exhibit a negative refractive index and a high reflectance.
  • negative index metamaterials or negative index materials are artificial structures where the refractive index has a negative value over a frequency range.
  • the index of refraction of the metamaterial is engineered to be negative in specific direction of energy transfer.
  • the metamaterials 34 include alternating structures of plasmonic materials and dielectric materials.
  • the metamaterial 34 includes either distributed or lumped resonant sub units such as split ring resonators.
  • split ring resonators may comprise an inner square with a split on one side embedded in an outer square with a split on the other side to produce the desired magnetic response. The use of SRR creates the necessary strong magnetic coupling or response.
  • the optical conduit is made of rolled tube of metamaterial.
  • the optical conduit is configured to guide the light energy over long distances and around curves with minimal losses.
  • the energy is transmitted for distances over 50 kilometerss.
  • the core 32 is formed in the center of the hollow cylindrical pipe and configured to confine the electromagnetic energy with the hollow cylindrical pipe.
  • the core is filled with low density gases.
  • the optical conduit comprises a vacuum core.
  • Cladding layer 36 is used as an outer layer encapsulating the hollow cylindrical piper and is configured to confine the electromagnetic energy within the core.
  • the cladding layer is formed using one or more layers of material of lower refractive index.
  • the cladding layer is in contact with metamaterial layer. Further, the refractive index of the cladding layer is higher than that of the metamaterial layer. The high enough refractive index contrast ensures that the propagation is prohibited in any direction within a characteristic range of frequencies.
  • the cladding layer 36 ensures that the electromagnetic energy is confined within the optical conduit by the principle of total internal reflection at the boundaries of the cladding layer and the metamaterial layer.
  • FIG. 3 is a cross sectional view of optical conduit illustrating a manner in which energy is transmitted from a point of entry to a point of exit according to aspects of the present technique. As shown, energy enters the conduit 40 at point of entry 42 and leaves the conduit at point of exit 44. [0027] The traversal path taken by the electromagnetic energy inside the optical conduit is illustrated generally by reference numeral 46.
  • the metamaterial 34 is engineered to refract the electromagnetic energy into the core 32.
  • the refraction angle is adjusted to be lesser than the incident angle. By adjusting the angle of refraction, energy losses are minimized.
  • the metamaterial layer comprises alternating repetitive structures of dielectric materials 52 and plasmonic materials 54.
  • the effect of repeating structure ensures that the interference is constructed in well- defined propagation directions, which leads to scattering and complete reflection.
  • Plasmonic metals exploit surface plasmons, which are produced from the interaction of light with metal-dielectric materials.
  • the incident light couples with the surface plasmons to create self-sustaining, propagating waves known as surface plasmon polaritons. Once launched, the surface plasmon polaritons ripple along the metal- dielectric interface and do not stray from the narrow path.
  • Plasmonic materials generally present minimum losses and high conductivity. Examples include silver and gold particles. Examples of dielectric materials include silicon nitride, silicon dioxide, glass and other wide gap materials.
  • the dielectric material and the plasmonic material are arranged in a cylindrical or helical form like that of a spring.
  • the cylindrical symmetry of the arrangement provides the necessary properties required for providing the right optical properties.
  • FIG. 5 is an isometric view of the metamaterial 60 showing the helical arrangement of two types of materials represented by reference numeral 62 and 64 respectively.
  • the helical arrangement of plasmonic and dielectric material provides the necessary nonlinear response required for creating right amount of resonance at optical frequencies and results in negative refractive index to give the required reflection characteristics.
  • the arrangement is a repetition of sub-wavelength pattern of plasmonic metal and dielectrics, the combination of these provides the desired optical response.
  • the material is arranged in a spiral ring around the circumference of the pipe.
  • the optical conduit provides an efficiency , of about 99% to 99.99%.
  • the use of vacuum or gas minimizes the loss of energy due to interaction of light with material.
  • the negative index of refraction at the edges of this volume refracts or reflects the light back serving to confine and contain the light.
  • the optical conduit is a low loss flexible waveguide, able to take light across long distances with minimal energy loss or degradation in energy spectrum.

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  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Optics & Photonics (AREA)
  • Optical Modulation, Optical Deflection, Nonlinear Optics, Optical Demodulation, Optical Logic Elements (AREA)

Abstract

An optical conduit (30) for the transmission of optical energy is provided. The optical conduit (30) comprises a hollow cylindrical pipe (36) configured to guide optical energy from a first end to a second end. A metamaterial (34) is disposed on inner wall of the hollow cylindrical tube (36) and the metamaterial (34) is configured to exhibit a negative refractive index. The optical conduit (30) further comprises a core (32) formed in the center of hollow cylindrical pipe (36) and configured to confine the optical signals with the hollow cylindrical pipe (36).

Description

META-MATERIALS BASED OPTICAL CONDUITS
BACKGROUND
[0001] The present invention is related to energy systems. More particularly the present invention is related to energy system employing meta-materials based optical conduits for energy transfer.
[0002] In many applications, optical conduits are used for the transfer of energy from one point to a second point. Examples of such applications include telecommunication systems, semiconductor arrays, energy systems and the like. However, conventional optical fibers have several disadvantages such as high transmission losses and bend losses.
[0003] In recent times, the use of metamaterials in the fabrication of optical fibers has been contemplated. Metamaterials are periodically repeating, synthetic composite structures that are specifically engineered to circumvent inconvenient bulk material properties. Photonic bandgap crystals such as synthetic opals are a subset example of metamaterials. The exceptional characteristics and response functions of metamaterials are not observed in the individual constituent materials of the composite, and these phenomena arise as a direct result of the periodic inclusion of functional materials such as metals, semiconductors or polymers embedded within the composite.
[0004J The use of metamaterials in optical conduits can significantly increase the efficiency of transmission and reduce bend losses. However, one of the challenges is the engineering of metamaterials with the correct composition of structures to ensure maximum efficiency.
[0005] Therefore there is a need for metametrial based optical: conduits that are capable of transferring energy over large distances with minimum energy losses. SUMMARY
[0006] Briefly, according to one embodiment of the invention, an optical conduit for the transmission of optical energy is provided. The optical conduit comprises a hollow cylindrical pipe configured to guide optical energy from a first end to a second end. A metamaterial is disposed on an inner wall of the hollow cylindrical tube and the metamaterial is configured to exhibit a negative refractive index. The optical conduit further comprises a core formed in the center of the hollow cylindrical pipe and configured to confine the optical signals with the hollow cylindrical pipe.
[0007] In another embodiment, a light energy transmission system for transmitting optical energy is provided. The system comprises a light energy source configured to generate light energy and a flexible optical conduit configured to transmit the light energy. The optical conduit comprises a hollow cylindrical pipe configured to guide optical energy from a first end to a second end. A metamaterial is disposed on an inner wall of the hollow cylindrical tube wherein the metamaterial is configured to exhibit a negative refractive index. The system further comprises a core formed in the center of the hollow cylindrical pipe and configured to confine the optical signals within the hollow cylindrical.
DRAWINGS
[0008] FIG. 1 is an embodiment of a system for transmitting optical energy implemented according to one aspect of the present technique;
[0009] FIG 2A is a cross sectional view of an embodiment of an optical conduit implemented according to aspects of the present technique;
[0010] FIG. 2B is a perspective view of an embodiment of an optical conduit implemented according to aspects of the present technique; [0011] FIG. 3 and FIG. 4 are sectional views of an embodiment of an optical conduit illustrating a manner in which optical energy is transmitted; and
[0012] FIG. 5 is an isometric view of the metamaterial a helical arrangement of two types of materials implemented according to aspects of the present technique.
DETAILED DESCRIPTION
[0013] The present disclosure discusses an embodiment of a meta-material based light pipe also known as optical conduit. The optical conduit is used for the transmission of optical energy and it includes a hollow cylindrical pipe configured to guide optical energy from a first end to a second end that has walls comprising optical meta-material s. In one embodiment, the meta-materials are disposed on the inner walls of the optical conduit along with cladding material.
[0014] As illustrated in FIG. 1, an energy transmission system 10 for transmitting of energy is provided. As used herein, energy refers in general to electromagnetic energy and specifically to solar energy. The system includes an energy source 12 configured to generate electromagnetic energy. For the purpose of the description only, the energy source refers to the sun and the electromagnetic energy refers to solar energy.
[0015] Energy collectors 14 are configured to collect solar energy and are placed in an energy collection area. Examples of energy collectors include optical reflectors, solar panels, etc. Energy converter 18 is configured to convert the collected solar energy to an alternate, form- of .energy such as thermal energy, electrical energy, etc. In one embodiment, the energy converter converts solar energy to corresponding electrical energy.
[0016] The energy collector 14 is coupled to energy converter using optical conduit 18. The optical conduit 18 is fabricated using metamaterials and is engineered to have a negative refractive index. In one embodiment, the metamaterial is designed for broadband visible spectrum optical properties using a metastructure made of regular meta-atoms whose resonant interaction with light energy gives the desired optical properties. Examples of the metamaterials used include split ring resonators, plasmonic materials, dielectric materials, etc. The optical conduit is configured to guide the solar energy from the energy collector to the energy converter. In one embodiment, the optical conduit is flexible.
[0017] Power converter 20 is configured to condition the electrical energy generated by the energy converter 18 to a standard form, such as, 220V at 50 Hz output power. Examples of conversion include but are not limited to DC to AC to DC, DC to AC or just changing the voltage or frequency or the combinations thereof. The output of the power converter 20 is filtered through a filter 22 for removing unwanted harmonics and it is then fed to different devices shown by reference numeral 24, 26 and 28.
[0018] As described above, the optical conduit is fabricated using metamaterials and is engineered to have a negative refractive index. The manner in which the optical conduit is implemented is described in further detail below.
[0019] FIG. 2A and 2B illustrates a cross sectional view and perspective view of an optical conduit implemented according to aspects of the present technique. The optical conduit comprises a hollow cylindrical pipe with a core formed in the center and configured to guide optical energy.
[0020] The optical conduit has an inner wall lined with metamaterials 34. The metamaterial 34 is configured to exhibit a negative refractive index and a high reflectance. In general, negative index metamaterials or negative index materials (NIM) are artificial structures where the refractive index has a negative value over a frequency range. In one embodiment, the index of refraction of the metamaterial is engineered to be negative in specific direction of energy transfer.
[0021] In one embodiment, the metamaterials 34 include alternating structures of plasmonic materials and dielectric materials. In another embodiment, the metamaterial 34 includes either distributed or lumped resonant sub units such as split ring resonators. For example, split ring resonators (SRR) may comprise an inner square with a split on one side embedded in an outer square with a split on the other side to produce the desired magnetic response. The use of SRR creates the necessary strong magnetic coupling or response.
[0022] In another embodiment, the optical conduit is made of rolled tube of metamaterial. The optical conduit is configured to guide the light energy over long distances and around curves with minimal losses. In one embodiment, the energy is transmitted for distances over 50 Kilometers.
[0023] The core 32 is formed in the center of the hollow cylindrical pipe and configured to confine the electromagnetic energy with the hollow cylindrical pipe. In one embodiment, the core is filled with low density gases. In a further embodiment, the optical conduit comprises a vacuum core.
[0024] Cladding layer 36 is used as an outer layer encapsulating the hollow cylindrical piper and is configured to confine the electromagnetic energy within the core. In one embodiment, the cladding layer is formed using one or more layers of material of lower refractive index.
[0025] In one embodiment, the cladding layer is in contact with metamaterial layer. Further, the refractive index of the cladding layer is higher than that of the metamaterial layer. The high enough refractive index contrast ensures that the propagation is prohibited in any direction within a characteristic range of frequencies. The cladding layer 36 ensures that the electromagnetic energy is confined within the optical conduit by the principle of total internal reflection at the boundaries of the cladding layer and the metamaterial layer.
[0026] FIG. 3 is a cross sectional view of optical conduit illustrating a manner in which energy is transmitted from a point of entry to a point of exit according to aspects of the present technique. As shown, energy enters the conduit 40 at point of entry 42 and leaves the conduit at point of exit 44. [0027] The traversal path taken by the electromagnetic energy inside the optical conduit is illustrated generally by reference numeral 46. In one embodiment, the metamaterial 34 is engineered to refract the electromagnetic energy into the core 32. In one embodiment, the refraction angle is adjusted to be lesser than the incident angle. By adjusting the angle of refraction, energy losses are minimized.
[0028] Referring now to FIG. 4, an example layer of metamaterial layer used in an optical conduit is illustrated. In one embodiment, the metamaterial layer comprises alternating repetitive structures of dielectric materials 52 and plasmonic materials 54. The effect of repeating structure ensures that the interference is constructed in well- defined propagation directions, which leads to scattering and complete reflection.
[0029] Plasmonic metals exploit surface plasmons, which are produced from the interaction of light with metal-dielectric materials. The incident light couples with the surface plasmons to create self-sustaining, propagating waves known as surface plasmon polaritons. Once launched, the surface plasmon polaritons ripple along the metal- dielectric interface and do not stray from the narrow path.
[0030] Plasmonic materials generally present minimum losses and high conductivity. Examples include silver and gold particles. Examples of dielectric materials include silicon nitride, silicon dioxide, glass and other wide gap materials.
[0031] In one embodiment, the dielectric material and the plasmonic material are arranged in a cylindrical or helical form like that of a spring. The cylindrical symmetry of the arrangement provides the necessary properties required for providing the right optical properties.
[0032] FIG. 5 is an isometric view of the metamaterial 60 showing the helical arrangement of two types of materials represented by reference numeral 62 and 64 respectively. The helical arrangement of plasmonic and dielectric material provides the necessary nonlinear response required for creating right amount of resonance at optical frequencies and results in negative refractive index to give the required reflection characteristics. The arrangement is a repetition of sub-wavelength pattern of plasmonic metal and dielectrics, the combination of these provides the desired optical response. The material is arranged in a spiral ring around the circumference of the pipe.
[0033] Thus, by using metamaterials engineered to have very high reflectance and negative refractive index in the optical conduit, energy losses are substantially reduced and the light can be transmitted efficiently. In one embodiment, the optical conduit provides an efficiency , of about 99% to 99.99%. The use of vacuum or gas minimizes the loss of energy due to interaction of light with material. The negative index of refraction at the edges of this volume refracts or reflects the light back serving to confine and contain the light. Further, the optical conduit is a low loss flexible waveguide, able to take light across long distances with minimal energy loss or degradation in energy spectrum.
[0034] For example, as an aid to understanding, the following appended claims may contain usage of the introductory phrases "at least one" and "one or more" to introduce claim recitations. However, the use of such phrases should not be construed to imply that the introduction of a claim recitation by the indefinite articles "a" or "an" limits any particular claim containing such introduced claim recitation to embodiments containing only one such recitation, even when the same claim includes the introductory phrases "one or more" or "at least one" and indefinite articles such as "a" or "an" (e.g., "a" and/or "an" should be interpreted to mean "at least one" or "one or more"); the same holds true for the use of definite articles used to introduce claim recitations, in addition, even if a specific number of an introduced claim recitation is explicitly recited, those skilled in the art will recognize that such recitation should be interpreted to mean at least the recited number (e.g., the bare recitation of "two recitations," without other modifiers, means at least two recitations, or two or more recitations).
[0035] While only certain features of several embodiments have been illustrated and described herein, many modifications and changes will occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the invention.

Claims

CLAIMS:
1. An optical conduit for the transmission of optical energy, the optical conduit comprising;
a hollow cylindrical pipe configured to guide optical energy from a first end to a second end, wherein a metamaterial is disposed on an inner wall of the hollow cylindrical tube, and wherein the metamaterial is configured to exhibit a negative refractive index; and
a core formed in the center of the hollow cylindrical pipe and configured to confine the optical signals with the hollow cylindrical pipe.
2. The optical conduit of claim 1, wherein the core comprises vacuum.
3. The optical conduit of claim 1, wherein the core comprises low density gases.
4. The optical conduit of claim 1, wherein the metamaterial is configured to alter the direction of the optical energy incident thereon.
5. The optical conduit of claim 1, wherein the hollow cylindrical tube is flexible.
6. The optical conduit of claim 1, wherein the optical energy is transmitted for distances more than about 50 KM.
7. The optical conduit of claim 1 , wherein in the metamaterial is formed using plasmonic material and dieletric material.
8. The optical conduit of claim 7, wherein the plasmonic material alternates with the dielectric material.
9. The optical conduit of claim 7, wherein the plasmonic material and the dielectric material are arranged in a helical pattern.
10. The optical conduit of claim 1, further comprising a cladding layer disposed on an outer wall of the hollow cylindrical pipe and configured to confine the optical energy within the core.
11. An light energy transmission system for transmitting optical energy, the system comprising;
a light energy source configured to generate light energy; and;
a flexible optical conduit configured to transmit the light energy, the optical conduit comprising:
a hollow cylindrical pipe configured to guide optical energy from a first end to a second end, wherein a metamaterial is disposed on an inner wall of the hollow cylindrical tube, and wherein the metamaterial is configured to exhibit a negative refractive index; and
a core formed in the center of the hollow cylindrical pipe and configured to confine the optical signals within the hollow cylindrical.
12. The system of claim 1 1 wherein the metamaterial is configured to alter the direction of the optical energy incident thereon.
13. The optical conduit of claim 10, wherein in the metamaterial is formed using plasmonic material and dieletric material.
14. The optical conduit of claim 7, wherein the plasmonic material and the dielectric material are arranged in a helical pattern.
15. The optical conduit of claim 7, wherein the plasmonic material alternates with the dielectric material.
PCT/IN2013/000355 2013-06-04 2013-06-04 Meta-materials based optical conduits WO2014195959A1 (en)

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