Abstract

We develop an approach for the fast and accurate determination of geometrical optics solutions to Maxwell’s equations in inhomogeneous 2D media and for TM polarized electric fields. The eikonal equation is solved by the fast marching method. Particular attention is paid to consistently discretizing the scatterers’ boundaries and matching the discretization to that of the computational domain. The ray tracing is performed, in a direct and inverse way, by using a technique introduced in computer graphics for the fast and accurate generation of textured images from vector fields. The transport equation is solved by resorting only to its integral form, the transport of polarization being trivial for the considered geometry and polarization. Numerical results for the plane wave scattering of two perfectly conducting circular cylinders and for a Luneburg lens prove the accuracy of the algorithm. In particular, it is shown how the approach is capable of properly accounting for the multiple scattering occurring between the two metallic cylinders and how inverse ray tracing should be preferred to direct ray tracing in the case of the Luneburg lens.

© 2014 Optical Society of America

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References

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  1. I.S. Kim and J.R. Wolfgang, “A local mesh refinement algorithm for the time domain-finite difference method using Maxwell’s curl equations,” IEEE Trans. Microw. Theory Tech. 38, 812–815 (1990).
    [Crossref]
  2. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1970).
  3. Y.B. Tao, H. Lin, and H.J. Bao, “KD-tree based fast ray tracing for RCS prediction,” Progr. in Electromagn. Res. PIER 81, 329–341 (2008).
    [Crossref]
  4. A.M. Popovici and J.A. Sethian, “3-D imaging using higher order fast marching traveltimes,” Geophysics 67, 604–609 (2002).
    [Crossref]
  5. A. Capozzoli, C. Curcio, A. Di Vico, and A. Liseno, “NUFFT- & GPU-based fast imaging of vegetation,” IEICE Trans. Commun. E94-B, 2092–2103 (2011).
    [Crossref]
  6. A. Capozzoli, C. Curcio, A. Liseno, and G. Toso, “Phase-only synthesis of flat aperiodic reflectarrays,” Progr. Electromagn. Res. 133, 53–89 (2013).
    [Crossref]
  7. A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Fast phase-only synthesis of conformal reflectarrays,” IET Microw., Antennas Prop. 4, 1989–2000 (2010).
    [Crossref]
  8. H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Luneburg and two-shell lens antennas: radiation characteristics and design optimization,” IEEE Trans. Antennas Prop. 49, 60–69 (2001).
    [Crossref]
  9. S.F. Feldman, E.Y. Shu, and J.Y. Gui, “Synthesis of tapers for fiber optic sensors,” US Patent Number 5,290,398, Mar. 1, 1994.
  10. C.N. Davis, P.Y. Peterson, and S.G. Bilén, “Communication through hypersonic or re-entry plasmas,” roc. of the 49th Aerospace Science Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, FL, Jan. 4–7, 2001, 1–13.
  11. J.A. Sethian, Level Set Methods and Fast Marching Methods (Cambridge University Press, 1999).
  12. R. Rouy and A. Tourin, “A viscosity solutions approach to shape-from-shading,” SIAM J. Numer. Anal. 29, 867–884 (1992).
    [Crossref]
  13. N. Rawlinson and M. Sambridge, “Wave front evolution in strongly heterogeneous layered media using the fast marching method,” Geophys. J. Int. 156, 631–647 (2004).
    [Crossref]
  14. D. Stalling and H.-C. Hege, “Fast and resolution independent line integral convolution,” Proc. of the 22nd Annual Conf. on Computer Graphics and Interactive Tech., Los Angeles, CA, Aug. 6–11, 1995.
  15. D.A. Kapp and G.S. Brown, “A new numerical method for rough-surface scattering calculations,” IEEE Trans. Antennas Prop. 44, 711–721 (1996).
    [Crossref]
  16. G.S. Brown, “An inherent limitation of the Luneburg-Kline representation for the current on a conducting body,” IEEE Trans. Antennas Prop. 38, 1889–1892 (1990).
    [Crossref]
  17. R. Kimmel and J.A. Sethian, “Computing geodesic paths on manifolds,” Proc. Natl. Acad. Sci. USA 95, 8431–8435 (1998).
    [Crossref] [PubMed]
  18. W.-K. Jeong and R.T. Whitaker, “A fast iterative method for eikonal equations,” SIAM J. Sci. Comput. 30, 2512–2534 (2008).
    [Crossref]
  19. A. Capozzoli, C. Curcio, A. Liseno, and S. Savarese, “A comparison of Fast Marching, Fast Sweeping and Fast Iterative Methods for the solution of the eikonal equation,” Proc. of the 21st Telecommunications Forum (TELFOR), Belgrade, Serbia, Nov. 26–28, 2013, 685–688.

2013 (1)

A. Capozzoli, C. Curcio, A. Liseno, and G. Toso, “Phase-only synthesis of flat aperiodic reflectarrays,” Progr. Electromagn. Res. 133, 53–89 (2013).
[Crossref]

2011 (1)

A. Capozzoli, C. Curcio, A. Di Vico, and A. Liseno, “NUFFT- & GPU-based fast imaging of vegetation,” IEICE Trans. Commun. E94-B, 2092–2103 (2011).
[Crossref]

2010 (1)

A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Fast phase-only synthesis of conformal reflectarrays,” IET Microw., Antennas Prop. 4, 1989–2000 (2010).
[Crossref]

2008 (2)

Y.B. Tao, H. Lin, and H.J. Bao, “KD-tree based fast ray tracing for RCS prediction,” Progr. in Electromagn. Res. PIER 81, 329–341 (2008).
[Crossref]

W.-K. Jeong and R.T. Whitaker, “A fast iterative method for eikonal equations,” SIAM J. Sci. Comput. 30, 2512–2534 (2008).
[Crossref]

2004 (1)

N. Rawlinson and M. Sambridge, “Wave front evolution in strongly heterogeneous layered media using the fast marching method,” Geophys. J. Int. 156, 631–647 (2004).
[Crossref]

2002 (1)

A.M. Popovici and J.A. Sethian, “3-D imaging using higher order fast marching traveltimes,” Geophysics 67, 604–609 (2002).
[Crossref]

2001 (1)

H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Luneburg and two-shell lens antennas: radiation characteristics and design optimization,” IEEE Trans. Antennas Prop. 49, 60–69 (2001).
[Crossref]

1998 (1)

R. Kimmel and J.A. Sethian, “Computing geodesic paths on manifolds,” Proc. Natl. Acad. Sci. USA 95, 8431–8435 (1998).
[Crossref] [PubMed]

1996 (1)

D.A. Kapp and G.S. Brown, “A new numerical method for rough-surface scattering calculations,” IEEE Trans. Antennas Prop. 44, 711–721 (1996).
[Crossref]

1992 (1)

R. Rouy and A. Tourin, “A viscosity solutions approach to shape-from-shading,” SIAM J. Numer. Anal. 29, 867–884 (1992).
[Crossref]

1990 (2)

I.S. Kim and J.R. Wolfgang, “A local mesh refinement algorithm for the time domain-finite difference method using Maxwell’s curl equations,” IEEE Trans. Microw. Theory Tech. 38, 812–815 (1990).
[Crossref]

G.S. Brown, “An inherent limitation of the Luneburg-Kline representation for the current on a conducting body,” IEEE Trans. Antennas Prop. 38, 1889–1892 (1990).
[Crossref]

Bao, H.J.

Y.B. Tao, H. Lin, and H.J. Bao, “KD-tree based fast ray tracing for RCS prediction,” Progr. in Electromagn. Res. PIER 81, 329–341 (2008).
[Crossref]

Bilén, S.G.

C.N. Davis, P.Y. Peterson, and S.G. Bilén, “Communication through hypersonic or re-entry plasmas,” roc. of the 49th Aerospace Science Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, FL, Jan. 4–7, 2001, 1–13.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1970).

Brown, G.S.

D.A. Kapp and G.S. Brown, “A new numerical method for rough-surface scattering calculations,” IEEE Trans. Antennas Prop. 44, 711–721 (1996).
[Crossref]

G.S. Brown, “An inherent limitation of the Luneburg-Kline representation for the current on a conducting body,” IEEE Trans. Antennas Prop. 38, 1889–1892 (1990).
[Crossref]

Capozzoli, A.

A. Capozzoli, C. Curcio, A. Liseno, and G. Toso, “Phase-only synthesis of flat aperiodic reflectarrays,” Progr. Electromagn. Res. 133, 53–89 (2013).
[Crossref]

A. Capozzoli, C. Curcio, A. Di Vico, and A. Liseno, “NUFFT- & GPU-based fast imaging of vegetation,” IEICE Trans. Commun. E94-B, 2092–2103 (2011).
[Crossref]

A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Fast phase-only synthesis of conformal reflectarrays,” IET Microw., Antennas Prop. 4, 1989–2000 (2010).
[Crossref]

A. Capozzoli, C. Curcio, A. Liseno, and S. Savarese, “A comparison of Fast Marching, Fast Sweeping and Fast Iterative Methods for the solution of the eikonal equation,” Proc. of the 21st Telecommunications Forum (TELFOR), Belgrade, Serbia, Nov. 26–28, 2013, 685–688.

Curcio, C.

A. Capozzoli, C. Curcio, A. Liseno, and G. Toso, “Phase-only synthesis of flat aperiodic reflectarrays,” Progr. Electromagn. Res. 133, 53–89 (2013).
[Crossref]

A. Capozzoli, C. Curcio, A. Di Vico, and A. Liseno, “NUFFT- & GPU-based fast imaging of vegetation,” IEICE Trans. Commun. E94-B, 2092–2103 (2011).
[Crossref]

A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Fast phase-only synthesis of conformal reflectarrays,” IET Microw., Antennas Prop. 4, 1989–2000 (2010).
[Crossref]

A. Capozzoli, C. Curcio, A. Liseno, and S. Savarese, “A comparison of Fast Marching, Fast Sweeping and Fast Iterative Methods for the solution of the eikonal equation,” Proc. of the 21st Telecommunications Forum (TELFOR), Belgrade, Serbia, Nov. 26–28, 2013, 685–688.

D’Elia, G.

A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Fast phase-only synthesis of conformal reflectarrays,” IET Microw., Antennas Prop. 4, 1989–2000 (2010).
[Crossref]

Davis, C.N.

C.N. Davis, P.Y. Peterson, and S.G. Bilén, “Communication through hypersonic or re-entry plasmas,” roc. of the 49th Aerospace Science Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, FL, Jan. 4–7, 2001, 1–13.

Di Vico, A.

A. Capozzoli, C. Curcio, A. Di Vico, and A. Liseno, “NUFFT- & GPU-based fast imaging of vegetation,” IEICE Trans. Commun. E94-B, 2092–2103 (2011).
[Crossref]

Feldman, S.F.

S.F. Feldman, E.Y. Shu, and J.Y. Gui, “Synthesis of tapers for fiber optic sensors,” US Patent Number 5,290,398, Mar. 1, 1994.

Gui, J.Y.

S.F. Feldman, E.Y. Shu, and J.Y. Gui, “Synthesis of tapers for fiber optic sensors,” US Patent Number 5,290,398, Mar. 1, 1994.

Hege, H.-C.

D. Stalling and H.-C. Hege, “Fast and resolution independent line integral convolution,” Proc. of the 22nd Annual Conf. on Computer Graphics and Interactive Tech., Los Angeles, CA, Aug. 6–11, 1995.

Jeong, W.-K.

W.-K. Jeong and R.T. Whitaker, “A fast iterative method for eikonal equations,” SIAM J. Sci. Comput. 30, 2512–2534 (2008).
[Crossref]

Kapp, D.A.

D.A. Kapp and G.S. Brown, “A new numerical method for rough-surface scattering calculations,” IEEE Trans. Antennas Prop. 44, 711–721 (1996).
[Crossref]

Kim, I.S.

I.S. Kim and J.R. Wolfgang, “A local mesh refinement algorithm for the time domain-finite difference method using Maxwell’s curl equations,” IEEE Trans. Microw. Theory Tech. 38, 812–815 (1990).
[Crossref]

Kimmel, R.

R. Kimmel and J.A. Sethian, “Computing geodesic paths on manifolds,” Proc. Natl. Acad. Sci. USA 95, 8431–8435 (1998).
[Crossref] [PubMed]

Lin, H.

Y.B. Tao, H. Lin, and H.J. Bao, “KD-tree based fast ray tracing for RCS prediction,” Progr. in Electromagn. Res. PIER 81, 329–341 (2008).
[Crossref]

Liseno, A.

A. Capozzoli, C. Curcio, A. Liseno, and G. Toso, “Phase-only synthesis of flat aperiodic reflectarrays,” Progr. Electromagn. Res. 133, 53–89 (2013).
[Crossref]

A. Capozzoli, C. Curcio, A. Di Vico, and A. Liseno, “NUFFT- & GPU-based fast imaging of vegetation,” IEICE Trans. Commun. E94-B, 2092–2103 (2011).
[Crossref]

A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Fast phase-only synthesis of conformal reflectarrays,” IET Microw., Antennas Prop. 4, 1989–2000 (2010).
[Crossref]

A. Capozzoli, C. Curcio, A. Liseno, and S. Savarese, “A comparison of Fast Marching, Fast Sweeping and Fast Iterative Methods for the solution of the eikonal equation,” Proc. of the 21st Telecommunications Forum (TELFOR), Belgrade, Serbia, Nov. 26–28, 2013, 685–688.

Mosallaei, H.

H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Luneburg and two-shell lens antennas: radiation characteristics and design optimization,” IEEE Trans. Antennas Prop. 49, 60–69 (2001).
[Crossref]

Peterson, P.Y.

C.N. Davis, P.Y. Peterson, and S.G. Bilén, “Communication through hypersonic or re-entry plasmas,” roc. of the 49th Aerospace Science Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, FL, Jan. 4–7, 2001, 1–13.

Popovici, A.M.

A.M. Popovici and J.A. Sethian, “3-D imaging using higher order fast marching traveltimes,” Geophysics 67, 604–609 (2002).
[Crossref]

Rahmat-Samii, Y.

H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Luneburg and two-shell lens antennas: radiation characteristics and design optimization,” IEEE Trans. Antennas Prop. 49, 60–69 (2001).
[Crossref]

Rawlinson, N.

N. Rawlinson and M. Sambridge, “Wave front evolution in strongly heterogeneous layered media using the fast marching method,” Geophys. J. Int. 156, 631–647 (2004).
[Crossref]

Rouy, R.

R. Rouy and A. Tourin, “A viscosity solutions approach to shape-from-shading,” SIAM J. Numer. Anal. 29, 867–884 (1992).
[Crossref]

Sambridge, M.

N. Rawlinson and M. Sambridge, “Wave front evolution in strongly heterogeneous layered media using the fast marching method,” Geophys. J. Int. 156, 631–647 (2004).
[Crossref]

Savarese, S.

A. Capozzoli, C. Curcio, A. Liseno, and S. Savarese, “A comparison of Fast Marching, Fast Sweeping and Fast Iterative Methods for the solution of the eikonal equation,” Proc. of the 21st Telecommunications Forum (TELFOR), Belgrade, Serbia, Nov. 26–28, 2013, 685–688.

Sethian, J.A.

A.M. Popovici and J.A. Sethian, “3-D imaging using higher order fast marching traveltimes,” Geophysics 67, 604–609 (2002).
[Crossref]

R. Kimmel and J.A. Sethian, “Computing geodesic paths on manifolds,” Proc. Natl. Acad. Sci. USA 95, 8431–8435 (1998).
[Crossref] [PubMed]

J.A. Sethian, Level Set Methods and Fast Marching Methods (Cambridge University Press, 1999).

Shu, E.Y.

S.F. Feldman, E.Y. Shu, and J.Y. Gui, “Synthesis of tapers for fiber optic sensors,” US Patent Number 5,290,398, Mar. 1, 1994.

Stalling, D.

D. Stalling and H.-C. Hege, “Fast and resolution independent line integral convolution,” Proc. of the 22nd Annual Conf. on Computer Graphics and Interactive Tech., Los Angeles, CA, Aug. 6–11, 1995.

Tao, Y.B.

Y.B. Tao, H. Lin, and H.J. Bao, “KD-tree based fast ray tracing for RCS prediction,” Progr. in Electromagn. Res. PIER 81, 329–341 (2008).
[Crossref]

Toso, G.

A. Capozzoli, C. Curcio, A. Liseno, and G. Toso, “Phase-only synthesis of flat aperiodic reflectarrays,” Progr. Electromagn. Res. 133, 53–89 (2013).
[Crossref]

Tourin, A.

R. Rouy and A. Tourin, “A viscosity solutions approach to shape-from-shading,” SIAM J. Numer. Anal. 29, 867–884 (1992).
[Crossref]

Whitaker, R.T.

W.-K. Jeong and R.T. Whitaker, “A fast iterative method for eikonal equations,” SIAM J. Sci. Comput. 30, 2512–2534 (2008).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1970).

Wolfgang, J.R.

I.S. Kim and J.R. Wolfgang, “A local mesh refinement algorithm for the time domain-finite difference method using Maxwell’s curl equations,” IEEE Trans. Microw. Theory Tech. 38, 812–815 (1990).
[Crossref]

Geophys. J. Int. (1)

N. Rawlinson and M. Sambridge, “Wave front evolution in strongly heterogeneous layered media using the fast marching method,” Geophys. J. Int. 156, 631–647 (2004).
[Crossref]

Geophysics (1)

A.M. Popovici and J.A. Sethian, “3-D imaging using higher order fast marching traveltimes,” Geophysics 67, 604–609 (2002).
[Crossref]

IEEE Trans. Antennas Prop. (3)

H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Luneburg and two-shell lens antennas: radiation characteristics and design optimization,” IEEE Trans. Antennas Prop. 49, 60–69 (2001).
[Crossref]

D.A. Kapp and G.S. Brown, “A new numerical method for rough-surface scattering calculations,” IEEE Trans. Antennas Prop. 44, 711–721 (1996).
[Crossref]

G.S. Brown, “An inherent limitation of the Luneburg-Kline representation for the current on a conducting body,” IEEE Trans. Antennas Prop. 38, 1889–1892 (1990).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

I.S. Kim and J.R. Wolfgang, “A local mesh refinement algorithm for the time domain-finite difference method using Maxwell’s curl equations,” IEEE Trans. Microw. Theory Tech. 38, 812–815 (1990).
[Crossref]

IEICE Trans. Commun. (1)

A. Capozzoli, C. Curcio, A. Di Vico, and A. Liseno, “NUFFT- & GPU-based fast imaging of vegetation,” IEICE Trans. Commun. E94-B, 2092–2103 (2011).
[Crossref]

IET Microw., Antennas Prop. (1)

A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Fast phase-only synthesis of conformal reflectarrays,” IET Microw., Antennas Prop. 4, 1989–2000 (2010).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

R. Kimmel and J.A. Sethian, “Computing geodesic paths on manifolds,” Proc. Natl. Acad. Sci. USA 95, 8431–8435 (1998).
[Crossref] [PubMed]

Progr. Electromagn. Res. (1)

A. Capozzoli, C. Curcio, A. Liseno, and G. Toso, “Phase-only synthesis of flat aperiodic reflectarrays,” Progr. Electromagn. Res. 133, 53–89 (2013).
[Crossref]

Progr. in Electromagn. Res. (1)

Y.B. Tao, H. Lin, and H.J. Bao, “KD-tree based fast ray tracing for RCS prediction,” Progr. in Electromagn. Res. PIER 81, 329–341 (2008).
[Crossref]

SIAM J. Numer. Anal. (1)

R. Rouy and A. Tourin, “A viscosity solutions approach to shape-from-shading,” SIAM J. Numer. Anal. 29, 867–884 (1992).
[Crossref]

SIAM J. Sci. Comput. (1)

W.-K. Jeong and R.T. Whitaker, “A fast iterative method for eikonal equations,” SIAM J. Sci. Comput. 30, 2512–2534 (2008).
[Crossref]

Other (6)

A. Capozzoli, C. Curcio, A. Liseno, and S. Savarese, “A comparison of Fast Marching, Fast Sweeping and Fast Iterative Methods for the solution of the eikonal equation,” Proc. of the 21st Telecommunications Forum (TELFOR), Belgrade, Serbia, Nov. 26–28, 2013, 685–688.

D. Stalling and H.-C. Hege, “Fast and resolution independent line integral convolution,” Proc. of the 22nd Annual Conf. on Computer Graphics and Interactive Tech., Los Angeles, CA, Aug. 6–11, 1995.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1970).

S.F. Feldman, E.Y. Shu, and J.Y. Gui, “Synthesis of tapers for fiber optic sensors,” US Patent Number 5,290,398, Mar. 1, 1994.

C.N. Davis, P.Y. Peterson, and S.G. Bilén, “Communication through hypersonic or re-entry plasmas,” roc. of the 49th Aerospace Science Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, FL, Jan. 4–7, 2001, 1–13.

J.A. Sethian, Level Set Methods and Fast Marching Methods (Cambridge University Press, 1999).

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Figures (19)

Fig. 1
Fig. 1 Flow-chart illustrating the main steps of the approach.
Fig. 2
Fig. 2 Geometry of the problem.
Fig. 3
Fig. 3 Flow-chart illustrating the main steps of the FMM.
Fig. 4
Fig. 4 Illustrating the narrowband and the upwind and downwind nodes for the FMM.
Fig. 5
Fig. 5 The 2D Cartesian stencil.
Fig. 6
Fig. 6 Matching the curve Γ to the Cartesian computational grid by a local triangular mesh. Bullets: Cartesian grid. Triangles: backing nodes.
Fig. 7
Fig. 7 Mesh triangles sharing one vertex point P.
Fig. 8
Fig. 8 Illustrating the eikonal update for triangular meshes.
Fig. 9
Fig. 9 Choice of a backing node to divide one obtuse triangle into two acute triangles.
Fig. 10
Fig. 10 Case of two scattering cylinders: Amplitude (in dB) of the total field obtained by the set up algorithm when second-order reciprocal interaction are considered.
Fig. 11
Fig. 11 Case of two scattering cylinders: Amplitude (in dB) of the total field obtained by FEKO.
Fig. 12
Fig. 12 Case of two scattering cylinders: Cuts of the amplitude (in dB) of the total field obtained by the set up algorithm (red dashed line) and by FEKO (blue dashed line) along a portion of the y = 0.9m-axis. Second-order reciprocal interactions have been considered.
Fig. 13
Fig. 13 Case of two scattering cylinders: Amplitude (in dB) of the total field obtained by the set up algorithm when third-order reciprocal interaction are considered.
Fig. 14
Fig. 14 Case of two scattering cylinders: Cuts of the amplitude (in dB) of the total field obtained by the set up algorithm (red dashed line) and by FEKO (blue dashed line) along a portion of the y = 0.9m-axis. Third-order reciprocal interactions have been considered.
Fig. 15
Fig. 15 Case of Luneburg lens: Field phase arising from the determination of the eikonal.
Fig. 16
Fig. 16 Case of Luneburg lens: Directly traced rays (blue line); lens boundary (red line).
Fig. 17
Fig. 17 Case of Luneburg lens: Normalized field amplitude for the case of direct ray tracing.
Fig. 18
Fig. 18 Case of Luneburg lens: Inversely traced rays (blue line); lens boundary (red line).
Fig. 19
Fig. 19 Case of Luneburg lens: Normalized field amplitude for the case of inverse ray tracing.

Equations (11)

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E s ( ρ _ ) E 0 ( ρ _ ) e j k 0 L ( ρ _ ) ,
{ | _ L ( ρ _ ) | 2 = n 2 ( ρ _ ) in Ω L ( ρ _ ) = L 0 ( ρ _ ) on Γ ,
{ _ ( I ( ρ _ ) n ) _ L ( ρ _ ) + I ( ρ _ ) n 2 L ( ρ _ ) = 0 in Ω I ( ρ _ ) = I 0 ( ρ _ ) on Γ
[ max { D i j x ( L ) , D i j + x ( L ) , 0 } 2 + max { D i j y ( L ) , D i j + y ( L ) , 0 } 2 ] 1 / 2 = n i j ,
D i j x ( L ) = L i j L i + 1 j h D i j + x ( L ) = L i + 1 j L i j h .
switch i j x ( L ) = { D 2 i j x ( L ) if L i 2 j L i 1 j D i j x ( L ) otherwise switch i j + x ( L ) = { D 2 i j + x ( L ) if L i + 2 j L i + 1 j D i j + x ( L ) otherwise ,
D 2 i j x ( L ) = 3 L i j 4 L i 1 j + L i 2 j 2 Δ x
D 2 i j + x ( L ) = 3 L i j + 4 L i + 1 j L i + 2 j 2 Δ x ,
( a 2 + b 2 2 a b cos θ ) t 2 + 2 b Δ u ( a cos θ b ) t + b 2 ( Δ u 2 n 2 a 2 sin 2 θ ) = 0 ,
a cos θ < b ( t Δ L ) t < a cos θ
d d s ( x ( s ) , y ( s ) ) = _ L ( x ( s ) , y ( s ) )

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