Abstract

In this work the least squares method is used to reduce anisotropy in transformation optics technique. To apply the least squares method a power series is added on the coordinate transformation functions. The series coefficients were calculated to reduce the deviations in Cauchy-Riemann equations, which, when satisfied, result in both conformal transformations and isotropic media. We also present a mathematical treatment for the special case of transformation optics to design waveguides. To demonstrate the proposed technique a waveguide with a 30° of bend and with a 50% of increase in its output width was designed. The results show that our technique is simultaneously straightforward to be implement and effective in reducing the anisotropy of the transformation for an extremely low value close to zero.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. G. E. Shilov, Elementary Real and Complex Analysis (MIT, 1973).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
    [CrossRef] [PubMed]
  11. H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010).
  12. Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. 61(12), 5910–5922 (2013).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  17. D. E. Blair, Inversion Theory and Conformal Mapping (American Mathematical Society, 2000).

2014 (1)

C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys. 16(2), 023030 (2014).
[CrossRef]

2013 (2)

Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. 61(12), 5910–5922 (2013).
[CrossRef]

D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express 21(12), 14223–14243 (2013).
[CrossRef] [PubMed]

2010 (5)

2009 (2)

N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

2008 (1)

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

2006 (1)

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

1953 (1)

L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math. 3(1), 1–58 (1953).
[CrossRef]

Ahlfors, L.

L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math. 3(1), 1–58 (1953).
[CrossRef]

Chang, Z.

Cui, T. J.

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010).

Gabrielli, L. H.

García-Meca, C.

C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys. 16(2), 023030 (2014).
[CrossRef]

Hao, Y.

Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. 61(12), 5910–5922 (2013).
[CrossRef]

Hu, G.

Hu, J.

Jiang, Z. H.

Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. 61(12), 5910–5922 (2013).
[CrossRef]

Johnson, S. G.

Kundtz, N.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[CrossRef] [PubMed]

Landy, N. I.

Leonhardt, U.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Li, J.

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Lipson, M.

Liu, D.

Ma, H. F.

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010).

Ma, Y. G.

Martí, J.

C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys. 16(2), 023030 (2014).
[CrossRef]

Martínez, A.

C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys. 16(2), 023030 (2014).
[CrossRef]

Ong, C. K.

Ortuño, R.

C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys. 16(2), 023030 (2014).
[CrossRef]

Padilla, W. J.

Pendry, J. B.

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

Poitras, C. B.

Quevedo-Teruel, O.

Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. 61(12), 5910–5922 (2013).
[CrossRef]

Smith, D. R.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[CrossRef] [PubMed]

Spadoti, D. H.

Tang, W.

Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. 61(12), 5910–5922 (2013).
[CrossRef]

Turpin, J. P.

Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. 61(12), 5910–5922 (2013).
[CrossRef]

Wang, N.

Werner, D. H.

Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. 61(12), 5910–5922 (2013).
[CrossRef]

Wu, Q.

Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. 61(12), 5910–5922 (2013).
[CrossRef]

Zhou, X.

IEEE Trans. Antenn. Propag. (1)

Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. 61(12), 5910–5922 (2013).
[CrossRef]

J. Anal. Math. (1)

L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math. 3(1), 1–58 (1953).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nat. Commun. (1)

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010).

Nat. Mater. (1)

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[CrossRef] [PubMed]

New J. Phys. (1)

C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys. 16(2), 023030 (2014).
[CrossRef]

Opt. Express (4)

Phys. Rev. Lett. (1)

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Prog. Opt. (1)

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

Science (1)

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Other (4)

K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton University, 2008).

G. E. Shilov, Elementary Real and Complex Analysis (MIT, 1973).

W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflection less transformation media in an isotropic and homogenous background,” arXiv:0806.3231 (2008).

D. E. Blair, Inversion Theory and Conformal Mapping (American Mathematical Society, 2000).

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Figures (4)

Fig. 1
Fig. 1

General transformation applied to a y-directed waveguide. A LSM optimizes the transformation with sample points chosen iteratively.

Fig. 2
Fig. 2

Algorithm used to choose the sampling points for the LSM.

Fig. 3
Fig. 3

Simple example of coordinate transformation. (a) Non-transformed media. (b) Transformed media with no anisotropy reduction (case 1). (c) Transformed media with anisotropy reduction (case 7).

Fig. 4
Fig. 4

Numerical simulations results. (a) Original medium. Only the core and cladding regions are transformed, not the entire simulation domain. (b) Electric filed distribution over the original transformed medium, without anisotropy reduction (case 1). The simulated medium is isotropic, therefore any anisotropy in the transformation results in propagation distortions, as demonstrated by the cross-section plot (line plot) of the device output. (c) Same as (b) but with anisotropy reduction (case 7). The beam propagates through the structure without distortions. (d) Refractive index profile for case 1. (e) Refractive index profile for case 7.

Tables (1)

Tables Icon

Table 1 Conditions. Peak and Mean Anisotropy. Maximum and Minimum Refractive Index

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

ε ij =ε g g ij
μ ij =μ g g ij
K= tr( J T J) 2det(J)
n 2 = n 2 tr( g 2 g ij )
x = f x (x,y)+b(x,y) Σ i=0 p Σ j=0 q A i j x i y j
y = f y ( x,y )+b( x,y ) Σ i=0 p Σ j=0 q B ij x i y j
b( x,y ) x | bnd =0 or b( x,y ) y | bnd =0
tg( θ )= y y x y = f y y f x y
tg( φ )= y x x x = f y x f x x
E( A,B )= i=0 k [ ( x x y y ) 2 + ( x y + y x ) 2 ] i
tg( θ )tg( φ )1
x =( x o +ecosφx )( y yf )+x( 1 y yf )
y =( y 0 +esinφx )( y yf )
x =( 4+1.5 3 2 x )( y 10 )+x( 1 y 10 )+sin( πy/10 ) i=0 q j=0 p A ij x i y j
y =( 91.5 1 2 x )( y 10 )+sin( πy/10 ) i=0 q j=0 p B ij x i y j

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