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Logarithm conformal mapping brings the cloaking effect.

Xu L, Chen H - Sci Rep (2014)

Bottom Line: In this work, we propose two new kinds of logarithm conformal mappings for invisible device designs.For one of the mappings, the refractive index distribution of conformal cloak varies from 0 to 9.839, which is more feasible for future implementation.Numerical simulations by using finite element method are performed to confirm the theoretical analysis.

View Article: PubMed Central - PubMed

Affiliation: College of Physics, Optoelectronics and Energy, Soochow University, Suzhou 215006, the People's Republic of China.

ABSTRACT
Over the past years, invisibility cloaks have been extensively discussed since transformation optics emerges. Generally, the electromagnetic parameters of invisibility cloaks are complicated tensors, yet difficult to realize. As a special method of transformation optics, conformal mapping helps us design invisibility cloak with isotropic materials of a refractive index distribution. However, for all proposed isotropic cloaks, the refractive index range is at such a breadth that challenges current experimental fabrication. In this work, we propose two new kinds of logarithm conformal mappings for invisible device designs. For one of the mappings, the refractive index distribution of conformal cloak varies from 0 to 9.839, which is more feasible for future implementation. Numerical simulations by using finite element method are performed to confirm the theoretical analysis.

No MeSH data available.


Related in: MedlinePlus

The contour plot of /dw/dz/ in the lower sheet of virtual space.(a) The contour plot of /dw/dz/ for the first conformal mapping. (b) The contour plot of /dw/dz/ for the second conformal mapping.
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f3: The contour plot of /dw/dz/ in the lower sheet of virtual space.(a) The contour plot of /dw/dz/ for the first conformal mapping. (b) The contour plot of /dw/dz/ for the second conformal mapping.

Mentions: Let us now look at the first conformal mapping. The width of the second Riemann sheet is 8π, slightly bigger than twice of the length of the branch cut in virtual space. Therefore the radii of the mirrored Maxwell's fish-eye lenses are set as r1 = l1/2 = 5.53697. As shown in Fig. 2(a), if we put two kissing lenses mentioned above in the lower Riemann sheet in virtual space, all rays impinging the branch cut will enter the lower sheet, propagate in closed circular arc trajectories and return to the upper sheet after reflecting twice at the PEC boundaries of Maxwell's fish-eye lenses. In physical space, all rays will propagate around the PEC boundary (the boundary of the white region in Fig. 2(b)) and leave the device as if nothing is there. According to Eqs. (3), (4) and (6), the refractive index distribution from the first conformal mapping is, The detailed calculation can be found in Ref. 1, 14, 15. Before we move on, let us examine schematically why we use two fish-eye lenses instead of one. In Fig. 3(a), the ribbon pattern is the lower sheet in virtual space of the first conformal mapping in Fig. 2(a). The yellow line with two endpoints (O1 and O2) is the branch cut. Two green lines are the boundaries of the lower sheet, which could be glued together if we roll up the sheet into a cylindrical surface. We plot the contour of /dw/dz/ in the lower sheet, which is small near the branch cut, and gradually grow into infinity away from the branch cut. We use uniform red color to represent large /dw/dz/ in regions far from the branch cut. By applying two kissing mirrored Maxwell's fish-eye lenses (two black circles), whose centers are the endpoints of the branch cut (radii of each lens is half length of the branch cut), as the refractive index of the lenses ranges from 1 to 2, the maximum value of the refractive index of the whole cloaking device shows up around points of A1, A2, A3 and A4, which are near the contour line of 9. These points are symmetric to the branch cut. The reason why we set the radii of the lenses half length of the branch cut is that it constrains the region of two lenses to have a lower upper bound of refractive index. However, if we put one Maxwell's fish-eye lens (shown in blue curves) whose center is O1, the maximum value shows up around points of B1 and B2, which are near the contour line of 40. In this case, the radius of the lens is the length of the branch cut. The blue dashed arc outside the sheet should be mapped to the blue solid arc to form a whole Maxwell's fish-eye lens. Therefore, it is obvious that two fish-eye lenses are better than one in cloaking designs.


Logarithm conformal mapping brings the cloaking effect.

Xu L, Chen H - Sci Rep (2014)

The contour plot of /dw/dz/ in the lower sheet of virtual space.(a) The contour plot of /dw/dz/ for the first conformal mapping. (b) The contour plot of /dw/dz/ for the second conformal mapping.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4215307&req=5

f3: The contour plot of /dw/dz/ in the lower sheet of virtual space.(a) The contour plot of /dw/dz/ for the first conformal mapping. (b) The contour plot of /dw/dz/ for the second conformal mapping.
Mentions: Let us now look at the first conformal mapping. The width of the second Riemann sheet is 8π, slightly bigger than twice of the length of the branch cut in virtual space. Therefore the radii of the mirrored Maxwell's fish-eye lenses are set as r1 = l1/2 = 5.53697. As shown in Fig. 2(a), if we put two kissing lenses mentioned above in the lower Riemann sheet in virtual space, all rays impinging the branch cut will enter the lower sheet, propagate in closed circular arc trajectories and return to the upper sheet after reflecting twice at the PEC boundaries of Maxwell's fish-eye lenses. In physical space, all rays will propagate around the PEC boundary (the boundary of the white region in Fig. 2(b)) and leave the device as if nothing is there. According to Eqs. (3), (4) and (6), the refractive index distribution from the first conformal mapping is, The detailed calculation can be found in Ref. 1, 14, 15. Before we move on, let us examine schematically why we use two fish-eye lenses instead of one. In Fig. 3(a), the ribbon pattern is the lower sheet in virtual space of the first conformal mapping in Fig. 2(a). The yellow line with two endpoints (O1 and O2) is the branch cut. Two green lines are the boundaries of the lower sheet, which could be glued together if we roll up the sheet into a cylindrical surface. We plot the contour of /dw/dz/ in the lower sheet, which is small near the branch cut, and gradually grow into infinity away from the branch cut. We use uniform red color to represent large /dw/dz/ in regions far from the branch cut. By applying two kissing mirrored Maxwell's fish-eye lenses (two black circles), whose centers are the endpoints of the branch cut (radii of each lens is half length of the branch cut), as the refractive index of the lenses ranges from 1 to 2, the maximum value of the refractive index of the whole cloaking device shows up around points of A1, A2, A3 and A4, which are near the contour line of 9. These points are symmetric to the branch cut. The reason why we set the radii of the lenses half length of the branch cut is that it constrains the region of two lenses to have a lower upper bound of refractive index. However, if we put one Maxwell's fish-eye lens (shown in blue curves) whose center is O1, the maximum value shows up around points of B1 and B2, which are near the contour line of 40. In this case, the radius of the lens is the length of the branch cut. The blue dashed arc outside the sheet should be mapped to the blue solid arc to form a whole Maxwell's fish-eye lens. Therefore, it is obvious that two fish-eye lenses are better than one in cloaking designs.

Bottom Line: In this work, we propose two new kinds of logarithm conformal mappings for invisible device designs.For one of the mappings, the refractive index distribution of conformal cloak varies from 0 to 9.839, which is more feasible for future implementation.Numerical simulations by using finite element method are performed to confirm the theoretical analysis.

View Article: PubMed Central - PubMed

Affiliation: College of Physics, Optoelectronics and Energy, Soochow University, Suzhou 215006, the People's Republic of China.

ABSTRACT
Over the past years, invisibility cloaks have been extensively discussed since transformation optics emerges. Generally, the electromagnetic parameters of invisibility cloaks are complicated tensors, yet difficult to realize. As a special method of transformation optics, conformal mapping helps us design invisibility cloak with isotropic materials of a refractive index distribution. However, for all proposed isotropic cloaks, the refractive index range is at such a breadth that challenges current experimental fabrication. In this work, we propose two new kinds of logarithm conformal mappings for invisible device designs. For one of the mappings, the refractive index distribution of conformal cloak varies from 0 to 9.839, which is more feasible for future implementation. Numerical simulations by using finite element method are performed to confirm the theoretical analysis.

No MeSH data available.


Related in: MedlinePlus