Abstract

We discuss the numerical simulation of a graded refractive index (GRIN) lens with a general anisotropic medium and compare it with an equivalent nongraded positive-index-of-refraction-material (PIM) lens with an isotropic medium. To evaluate lens performance, we developed a modified eikonal equation valid for the most general form of an anisotropic or chiral medium. Our approach is more comprehensive than previous work in this area and is obtained from the dispersion relation of Maxwell’s equations in the eikonal approximation. The software developed for the numerical integration of the modified eikonal equation is described. Subsequently, a full finite-difference time-dependent simulation was performed to verify the validity of the eikonal calculations. The performance of the GRIN lens is found to be improved over the equivalent PIM one. The GRIN lens is also five to ten times lighter than the equivalent PIM. A GRIN lens operating at 15 GHz is now under fabrication at Boeing, and the experimental results of this lens will be reported in a forthcoming paper.

© 2006 Optical Society of America

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References

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  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and µ," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  2. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  3. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
    [CrossRef] [PubMed]
  4. D. Shurig and D. R. Smith, "Negative index lens aberrations," Phys. Rev. E 70, 065601(R) (2004).
    [CrossRef]
  5. C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Taniliean, and D. C. Vier, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
    [CrossRef]
  6. P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108 (2005).
    [CrossRef]
  7. D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
    [CrossRef]
  8. R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, "Simulation and testing of a graded negative index of refraction lens," Appl. Phys. Lett. 87, 091114 (2005).
    [CrossRef]
  9. E. W. Marchand, Gradient Index Optics (Academic, 1978).
  10. E. Langenbach, "Raytracing in gradient index optics," Proc. SPIE 1780, 486-490 (1993).
  11. M. Born and E. Wolfe, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  12. V. A. De Lorenci, R. Klippert, and D. H. Teodoro, "Birefringence in nonlinear anisotropic dielectric media," Phys. Rev. D 70, 124035 (2004).
    [CrossRef]
  13. M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, 1965).
  14. D. R. Smith, P. Kolinko, and D. Schurig, "Negative refraction in indefinite media," J. Opt. Soc. Am. B 21, 1032-1043 (2004).
    [CrossRef]
  15. See Ref. , Sec. 3.1.1.
  16. P. R. Garabedian, Partial Differential Equations (Wiley, 1964).
  17. D. Hanselman and B. Littlefield, Mastering MATLAB 6 (Prentice Hall, 2001).
    [PubMed]
  18. Computer Simulation Technology of America.
  19. R. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, "Microwave focusing and beam collimation using negative index of refraction lenses," Special Issue on Metamaterials, IEE Proc. H. Microwaves, Antennas Propag., submitted for publication.
  20. S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media/Cambridge U. Press, 1996).
  21. See, for example, M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), Sec. 3.8.3 and 3.12, Example 6.
  22. See Ref. , Chap. 5 and 9.
  23. See Ref. , Table 9.2.
  24. See Ref. , Sec. 9.2.3, Eq. (20).
  25. D. Schurig, Duke University, Durham, N. C. 27707 (personal communication, 2005).

2005

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108 (2005).
[CrossRef]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, "Simulation and testing of a graded negative index of refraction lens," Appl. Phys. Lett. 87, 091114 (2005).
[CrossRef]

2004

V. A. De Lorenci, R. Klippert, and D. H. Teodoro, "Birefringence in nonlinear anisotropic dielectric media," Phys. Rev. D 70, 124035 (2004).
[CrossRef]

D. R. Smith, P. Kolinko, and D. Schurig, "Negative refraction in indefinite media," J. Opt. Soc. Am. B 21, 1032-1043 (2004).
[CrossRef]

D. Shurig and D. R. Smith, "Negative index lens aberrations," Phys. Rev. E 70, 065601(R) (2004).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Taniliean, and D. C. Vier, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

2003

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

2001

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

1968

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and µ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Abramowitz, M.

See, for example, M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), Sec. 3.8.3 and 3.12, Example 6.

Born, M.

M. Born and E. Wolfe, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

De Lorenci, V. A.

V. A. De Lorenci, R. Klippert, and D. H. Teodoro, "Birefringence in nonlinear anisotropic dielectric media," Phys. Rev. D 70, 124035 (2004).
[CrossRef]

Garabedian, P. R.

P. R. Garabedian, Partial Differential Equations (Wiley, 1964).

Greegor, R.

R. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, "Microwave focusing and beam collimation using negative index of refraction lenses," Special Issue on Metamaterials, IEE Proc. H. Microwaves, Antennas Propag., submitted for publication.

Greegor, R. B.

R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, "Simulation and testing of a graded negative index of refraction lens," Appl. Phys. Lett. 87, 091114 (2005).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Taniliean, and D. C. Vier, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Hanselman, D.

D. Hanselman and B. Littlefield, Mastering MATLAB 6 (Prentice Hall, 2001).
[PubMed]

Kay, I. W.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, 1965).

Kline, M.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, 1965).

Klippert, R.

V. A. De Lorenci, R. Klippert, and D. H. Teodoro, "Birefringence in nonlinear anisotropic dielectric media," Phys. Rev. D 70, 124035 (2004).
[CrossRef]

Kolinko, P.

Koltenbah, B. E.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Langenbach, E.

E. Langenbach, "Raytracing in gradient index optics," Proc. SPIE 1780, 486-490 (1993).

Li, K.

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Taniliean, and D. C. Vier, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Littlefield, B.

D. Hanselman and B. Littlefield, Mastering MATLAB 6 (Prentice Hall, 2001).
[PubMed]

Lu, W. T.

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108 (2005).
[CrossRef]

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, 1978).

Mock, J. J.

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

Nielsen, J. A.

R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, "Simulation and testing of a graded negative index of refraction lens," Appl. Phys. Lett. 87, 091114 (2005).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Taniliean, and D. C. Vier, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

R. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, "Microwave focusing and beam collimation using negative index of refraction lenses," Special Issue on Metamaterials, IEE Proc. H. Microwaves, Antennas Propag., submitted for publication.

Parazzoli, C. G.

R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, "Simulation and testing of a graded negative index of refraction lens," Appl. Phys. Lett. 87, 091114 (2005).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Taniliean, and D. C. Vier, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

R. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, "Microwave focusing and beam collimation using negative index of refraction lenses," Special Issue on Metamaterials, IEE Proc. H. Microwaves, Antennas Propag., submitted for publication.

Parimi, P. V.

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108 (2005).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

R. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, "Microwave focusing and beam collimation using negative index of refraction lenses," Special Issue on Metamaterials, IEE Proc. H. Microwaves, Antennas Propag., submitted for publication.

Schurig, D.

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

D. R. Smith, P. Kolinko, and D. Schurig, "Negative refraction in indefinite media," J. Opt. Soc. Am. B 21, 1032-1043 (2004).
[CrossRef]

D. Schurig, Duke University, Durham, N. C. 27707 (personal communication, 2005).

R. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, "Microwave focusing and beam collimation using negative index of refraction lenses," Special Issue on Metamaterials, IEE Proc. H. Microwaves, Antennas Propag., submitted for publication.

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shurig, D.

D. Shurig and D. R. Smith, "Negative index lens aberrations," Phys. Rev. E 70, 065601(R) (2004).
[CrossRef]

Smith, D. R.

R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, "Simulation and testing of a graded negative index of refraction lens," Appl. Phys. Lett. 87, 091114 (2005).
[CrossRef]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

D. R. Smith, P. Kolinko, and D. Schurig, "Negative refraction in indefinite media," J. Opt. Soc. Am. B 21, 1032-1043 (2004).
[CrossRef]

D. Shurig and D. R. Smith, "Negative index lens aberrations," Phys. Rev. E 70, 065601(R) (2004).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

R. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, "Microwave focusing and beam collimation using negative index of refraction lenses," Special Issue on Metamaterials, IEE Proc. H. Microwaves, Antennas Propag., submitted for publication.

Sridar, S.

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108 (2005).
[CrossRef]

Starr, A. F.

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

Stegun, I.

See, for example, M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), Sec. 3.8.3 and 3.12, Example 6.

Tanielian, M.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Tanielian, M. H.

R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, "Simulation and testing of a graded negative index of refraction lens," Appl. Phys. Lett. 87, 091114 (2005).
[CrossRef]

R. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, "Microwave focusing and beam collimation using negative index of refraction lenses," Special Issue on Metamaterials, IEE Proc. H. Microwaves, Antennas Propag., submitted for publication.

Taniliean, M. H.

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Taniliean, and D. C. Vier, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

Teodoro, D. H.

V. A. De Lorenci, R. Klippert, and D. H. Teodoro, "Birefringence in nonlinear anisotropic dielectric media," Phys. Rev. D 70, 124035 (2004).
[CrossRef]

Thompson, M. A.

R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, "Simulation and testing of a graded negative index of refraction lens," Appl. Phys. Lett. 87, 091114 (2005).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Taniliean, and D. C. Vier, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

R. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, "Microwave focusing and beam collimation using negative index of refraction lenses," Special Issue on Metamaterials, IEE Proc. H. Microwaves, Antennas Propag., submitted for publication.

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and µ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Vetter, A. M.

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Taniliean, and D. C. Vier, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

Vier, D. C.

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Taniliean, and D. C. Vier, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

R. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, "Microwave focusing and beam collimation using negative index of refraction lenses," Special Issue on Metamaterials, IEE Proc. H. Microwaves, Antennas Propag., submitted for publication.

Vodo, P.

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108 (2005).
[CrossRef]

Wolfe, E.

M. Born and E. Wolfe, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Wolfram, S.

S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media/Cambridge U. Press, 1996).

Appl. Phys. Lett.

R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, "Simulation and testing of a graded negative index of refraction lens," Appl. Phys. Lett. 87, 091114 (2005).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Taniliean, and D. C. Vier, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108 (2005).
[CrossRef]

J. Opt. Soc. Am. B

Phys. Rev. D

V. A. De Lorenci, R. Klippert, and D. H. Teodoro, "Birefringence in nonlinear anisotropic dielectric media," Phys. Rev. D 70, 124035 (2004).
[CrossRef]

Phys. Rev. E

D. Shurig and D. R. Smith, "Negative index lens aberrations," Phys. Rev. E 70, 065601(R) (2004).
[CrossRef]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

Phys. Rev. Lett.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Science

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and µ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other

E. W. Marchand, Gradient Index Optics (Academic, 1978).

E. Langenbach, "Raytracing in gradient index optics," Proc. SPIE 1780, 486-490 (1993).

M. Born and E. Wolfe, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, 1965).

See Ref. , Sec. 3.1.1.

P. R. Garabedian, Partial Differential Equations (Wiley, 1964).

D. Hanselman and B. Littlefield, Mastering MATLAB 6 (Prentice Hall, 2001).
[PubMed]

Computer Simulation Technology of America.

R. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, "Microwave focusing and beam collimation using negative index of refraction lenses," Special Issue on Metamaterials, IEE Proc. H. Microwaves, Antennas Propag., submitted for publication.

S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media/Cambridge U. Press, 1996).

See, for example, M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), Sec. 3.8.3 and 3.12, Example 6.

See Ref. , Chap. 5 and 9.

See Ref. , Table 9.2.

See Ref. , Sec. 9.2.3, Eq. (20).

D. Schurig, Duke University, Durham, N. C. 27707 (personal communication, 2005).

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Figures (10)

Fig. 1
Fig. 1

Integration domain for (a) planar interfaces Ω 12 and Ω 23 and (b) spherical interfaces. The radius of curvature in (b) is positive (negative) for a convex (concave) surface, where r 12 < 0 ( r 23 > 0 ) , and light travels from left to right. The characteristics in the free-space regions R 1 and R 3 are coincident with the optical rays; however, this is not the case in the GRIN lens, region R 2 .

Fig. 2
Fig. 2

Characteristic lines computed on (a) a PIM convex–plano lens using an aperture of 3 cm, (b) the same lens with a 12 cm aperture, and (c) a GRIN lens with an aperture of 12 cm. Note the tighter focus on the GRIN compared with the PIM of the same aperture size.

Fig. 3
Fig. 3

Evolution of the eikonal surface through the (a) PIM and (b) GRIN calculations as described in Figs. 2b, 2c, respectively. Eikonal surfaces are shown at the following positions: (1) launch pad (violet), (2) entrance to the lens (blue), (3) exit of the lens (red), and (4) further toward the focus (green). These surfaces correspond to the following x 3 positions: 5 , 1 , 0, 5 cm, respectively. For (a) the blue surface (2) partially intersects the interior of the PIM lens owing to its shape. Comparing (2) and (3) in (b), we see that the optical paths diminish through the GRIN lens owing to its negative phase velocity.

Fig. 4
Fig. 4

Plots of the eikonal value versus x 3 for selected characteristic lines for the PIM (solid curve) and GRIN (dashed curve) as described in Figs. 3a, 3b, respectively. Owing to the shape of the PIM lens, the lines refract at different positions. This is not the case with the GRIN, which also shows a sharper focus.

Fig. 5
Fig. 5

Plots of (a) spherical aberration coefficient A 040 and (b) normalized intensity i versus pupil radius R for PIM (solid curve), isotropic GRIN (dashed curve), and anisotropic GRIN (dotted curve).

Fig. 6
Fig. 6

Electric field amplitude of a 12 cm circular aperture illuminated by a plane wave traveling from left to right: (a) longitudinal plane through the aperture center and (b) far-field line plot.

Fig. 7
Fig. 7

Electric field amplitude in the longitudinal plane from a PIM lens simulation at 15 GHz.

Fig. 8
Fig. 8

Index-of-refraction radial distribution for a GRIN lens at 15 GHz. (a) Radial distribution, (b) rings of constant n.

Fig. 9
Fig. 9

Electric field amplitude from FDTD simulation results of the 12 cm aperture GRIN lens.

Fig. 10
Fig. 10

Normalized intensity at the Gaussian focus from eikonal (solid and dashed curves) and FDTD calculations (dotted and dashed–dotted curves) for the PIM and GRIN lenses, respectively, as a function of the pupil radius.

Equations (88)

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

× H + i k o ϵ E = 0 ,
× E i k o μ H = 0 ,
( ϵ E ) = 0 ,
( μ H ) = 0 .
E = e exp [ i k o ζ ( r ) ] , H = h exp [ i k o ζ ( r ) ] ,
( 1 e ) ( e x i ) k o i = 1 , 2 , 3 ,
ζ × h + ϵ e = 0 ,
ζ × e μ h = 0 ,
ζ e = 0 ,
ζ h = 0 .
A I st X = 0 .
( ζ ) 2 = μ ϵ = n 2 .
A I st = [ ϵ 0 0 0 ζ 3 ζ 2 0 ϵ 0 ζ 3 0 ζ 1 0 0 ϵ ζ 2 ζ 1 0 0 ζ 3 ζ 2 μ 0 0 ζ 3 0 ζ 1 0 μ 0 ζ 2 ζ 1 0 0 0 μ ] .
ϵ μ ( ζ 1 2 + ζ 2 2 + ζ 3 2 ϵ μ ) 2 = 0 .
I = c 8 π Re [ e × h * ] .
I = c 8 π μ ( e e * ) ζ .
d d s ( n d r d s ) = n .
ϵ E ϵ E , μ H μ H .
ζ × h + ϵ e = 0 ,
ζ × e μ h = 0 ,
ζ e = 0 ,
ζ h = 0 .
A Anist X = 0 .
A Anist = [ ϵ 1 0 0 0 ζ 3 ζ 2 0 ϵ 2 0 ζ 3 0 ζ 1 0 0 ϵ 3 ζ 2 ζ 1 0 0 ζ 3 ζ 2 μ 1 0 0 ζ 3 0 ζ 1 0 μ 2 0 ζ 2 ζ 1 0 0 0 μ 3 ] ,
ϵ 1 ϵ 2 ϵ 3 μ 1 μ 2 μ 3 ϵ 1 μ 1 ζ 1 4 ϵ 2 μ 2 ζ 2 4 ϵ 3 μ 3 ζ 3 4 + ϵ 1 μ 1 ( ϵ 2 μ 3 + ϵ 3 μ 2 ) ζ 1 2 + ϵ 2 μ 2 ( ϵ 1 μ 3 + ϵ 3 μ 1 ) ζ 2 2 + ϵ 3 μ 3 ( ϵ 1 μ 2 + ϵ 2 μ 1 ) ζ 3 2 ( ϵ 1 μ 2 + ϵ 2 μ 1 ) ζ 1 2 ζ 2 2 ( ϵ 1 μ 3 + ϵ 3 μ 1 ) ζ 1 2 ζ 3 2 ( ϵ 2 μ 3 + ϵ 3 μ 2 ) ζ 2 2 ζ 3 2 = 0 .
ϵ μ ( ζ 1 2 + ζ 2 2 + ζ 3 2 ϵ μ ) 2 = 0 ,
F [ ϵ i ( x ) , μ i ( x ) , ζ i ( x ) ] = 0 i = 1 , 2 , 3 .
d x i d s = F ζ i , i = 1 , 2 , 3 ,
d ζ d s = ζ 1 F ζ 1 + ζ 2 F ζ 2 + ζ 3 F ζ 3 ,
d ζ i d s = ( F x i + ζ i F ζ ) , i = 1 , 2 , 3 .
d x i d s = 2 ζ i i = 1 , 2 , 3 ,
d ζ d s = 2 ( ζ 1 2 + ζ 2 2 + ζ 3 2 ) ,
d ζ i d s = μ ( x ) d ϵ ( x ) d x i + ϵ ( x ) d μ ( x ) d x i i = 1 , 2 , 3 .
ζ θ = i = 1 3 ζ i x i ( s = 0 , θ , ϕ ) θ , ζ ϕ = i = 1 3 ζ i x i ( s = 0 , θ , ϕ ) ϕ ,
F { ϵ i [ x ( s = 0 , θ , ϕ ) ] , μ i [ x ( s = 0 , θ , ϕ ) ] , ζ i [ x ( s = 0 , θ , ϕ ) ] } = 0 .
k o ζ In = k o ζ Rfl = k o ζ Trn
( ζ 1 ) In = ( ζ 1 ) Trn , ( ζ 2 ) In = ( ζ 2 ) Trn .
( ζ 3 ) Trn = ± ϵ μ ( ζ 1 ) In 2 ( ζ 2 ) In 2 .
( ζ 3 ) Trn = 1 ( ζ 1 ) In 2 ( ζ 2 ) In 2 .
f ( x ) = C 0 + C 2 ( x 1 2 + x 2 2 ) + C 4 [ ( x 1 2 + x 2 2 ) ] 2 + C x 3 ( x 3 x 3 , Min ) .
ϵ i i ( x ) = μ i i ( x ) = 1.1 0.0501 ( x 1 2 + x 2 2 ) + 0.0001 [ ( x 1 2 + x 2 2 ) ] 2 ,
A 040 = 2 π ϵ 04 0 1 R 0 4 ( ρ ) ρ d ρ 0 2 π ζ ( ρ , θ ) cos ( 4 θ ) d θ .
i = 1 π 2 0 1 0 2 π exp ( i k 0 ζ ) ρ d ρ d θ 2 .
F = ϵ 1 ϵ 2 ϵ 3 μ 1 μ 2 μ 3 i = 1 3 ϵ i μ i ζ i 4 + i , j , k = 1 3 δ i j k ϵ i μ i ϵ j μ k ζ i 2 i , j = 1 3 δ i j ϵ i μ j ζ i 2 ζ j 2 = 0 .
d x i d s = F ζ i = 4 ϵ i μ i ζ i 3 + 2 j , k = 1 3 δ i j k ϵ i μ i ϵ j μ k ζ i 2 j = 1 3 δ i j ϵ i μ i ζ i ζ j 2 i = 1 , 2 , 3 ,
d ζ d s = i = 1 3 ζ i F ζ i = 4 i = 1 3 ϵ i μ i ζ i 4 + 2 i , j , k = 1 3 δ i j k ϵ i μ i ϵ j μ k ζ i 2 2 i , j = 1 3 δ i j ϵ i μ j ζ i 2 ζ j 2 .
d ζ n d s = ( F x n + ζ n F ζ ) = F x n = + ( ϵ 1 ϵ 2 ϵ 3 μ 1 μ 2 μ 3 ) x n + i = 1 3 ( ϵ i μ i ) x n ζ i 4 i , j , k = 1 3 δ i j k ( ϵ i μ i ϵ j μ k ) x n ζ i 2 + i , j = 1 3 δ i j ( ϵ i μ j ) x n ζ i 2 ζ j 2 = 0 n = 1 , 2 , 3 .
ζ 3 = ± { 1 2 ϵ 3 μ 3 [ ( ϵ 1 μ 2 + ϵ 2 μ 1 ) ϵ 3 μ 3 ( ϵ 3 μ 1 + ϵ 1 μ 3 ) ζ 1 2 ( ϵ 3 μ 2 + ϵ 2 μ 3 ) ζ 2 2 ± Γ 1 ] } 1 2 ,
Γ 1 = { [ ( ϵ 3 μ 1 ϵ 1 μ 3 ) 2 ζ 1 2 + ( ϵ 3 μ 2 ϵ 2 μ 3 ) 2 ζ 2 2 ] 2 + ϵ 3 μ 3 ( ϵ 1 μ 2 ϵ 2 μ 1 ) Γ 2 } 1 2 ,
Γ 2 = ϵ 3 μ 3 ( ϵ 1 μ 2 ϵ 2 μ 1 ) + 2 ( ϵ 3 μ 1 ϵ 1 μ 3 ) 2 ζ 1 2 2 ( ϵ 3 μ 2 ϵ 2 μ 3 ) 2 ζ 2 2 .
P = ( x 1 , x 2 , x 3 ) = [ r sin ( θ ) cos ( ϕ ) , r sin ( θ ) sin ( ϕ ) , x 3 C + r cos ( θ ) ] ,
ζ In = R ( θ , ϕ ) ζ In ,
( ζ 1 In ζ 2 In ζ 3 In ) = ( ζ θ  ∕ r ζ ϕ  ∕ r sin θ ζ 3 In ) = [ cos θ cos ϕ cos θ sin ϕ sin θ sin ϕ cos ϕ 0 sin θ cos ϕ sin θ sin ϕ cos θ ] ( ζ 1 In ζ 2 In ζ 3 In ) .
ζ Ref = R 1 ( θ , ϕ ) ζ Ref ,
( ζ 1 Ref ζ 2 Ref ζ 3 Ref ) = [ cos θ cos ϕ sin ϕ sin θ cos ϕ cos θ sin ϕ cos ϕ sin θ sin ϕ sin θ 0 cos θ ] ( ζ θ  ∕ r ζ ϕ  ∕ r sin θ ζ 3 Ref ) .
F = ϵ μ + ( ζ θ  ∕ r ) 2 + ( ζ ϕ  ∕ r sin θ ) 2 + ζ 3 Ref 2 = 0 .
ζ 1 Ref = cos θ cos ϕ ( ζ θ  ∕ r ) sin ϕ ( ζ ϕ  ∕ r sin θ ) + sin θ cos ϕ ζ 3 Ref ,
ζ 2 Ref = cos θ sin ϕ ( ζ θ  ∕ r ) + cos ϕ ( ζ ϕ  ∕ r sin θ ) + sin θ sin ϕ ζ 3 Ref ,
ζ 3 Ref = sin θ ( ζ θ  ∕ r ) + cos θ ζ 3 Ref ,
ζ 3 Ref = ± [ ϵ μ ( ζ θ  ∕ r ) 2 ( ζ ϕ  ∕ r sin θ ) 2 ] 1 2 .
sin α In = [ 1 ( Δ x 1 sin θ cos ϕ + Δ x 2 sin θ sin ϕ + Δ x 3 cos θ ) 2 ζ 2 ] 1 2 ,
sin α Ref = 1 ϵ μ [ 1 ( Δ x 1 sin θ cos ϕ + Δ x 2 sin θ sin ϕ + Δ x 3 cos θ ) 2 ζ 2 ] 1 2 .
det [ ϵ 11 ϵ 12 ϵ 13 0 ζ 3 Ref ζ 2 In ϵ 12 ϵ 22 ϵ 23 ζ 3 Ref 0 ζ 1 In ϵ 13 ϵ 23 ϵ 33 ζ 2 In ζ 1 In 0 0 ζ 3 Ref ζ 2 In μ 11 μ 12 μ 13 ζ 3 Ref 0 ζ 1 In μ 12 μ 22 μ 23 ζ 2 In ζ 1 In 0 μ 13 μ 23 μ 33 ] = 0 .
C 4 ζ 3 Ref 4 + C 3 ζ 3 Ref 3 + C 2 ζ 3 Ref 2 + C 1 ζ 3 Ref + C 0 = 0 .
C 4 = ϵ 33 μ 33 , C 3 = 2 ( ϵ 33 μ 13 + ϵ 13 μ 33 ) ζ 1 In 2 ( ϵ 33 μ 23 + ϵ 23 μ 33 ) ζ 2 In ,
C 2 = β 211 ζ 1 In 2 β 212 ζ 2 In 2 2 β 22 ζ 1 In ζ 2 In + β 23 ,
C 1 = 2 β 111 ζ 1 In 3 2 β 112 ζ 2 In 3 2 β 121 ζ 1 In 2 ζ 2 In 2 β 122 ζ 1 In ζ 2 In 2 + 2 β 131 ζ 1 In + 2 β 132 ζ 2 In ,
C 0 = ϵ 11 μ 11 ζ 1 In 4 ϵ 22 μ 22 ζ 2 In 4 2 β 011 ζ 1 In 3 ζ 2 In 2 β 012 ζ 1 In ζ 2 In 3 β 02 ζ 1 In 2 ζ 2 In 2 + β 031 ζ 1 In 2 + β 032 ζ 2 In 2 + 2 β 04 ζ 1 In ζ 2 In ,
β 21 i = ϵ 33 μ i i + ϵ i i μ 33 + 4 ϵ i 3 μ i 3 , β 22 = ϵ 33 μ 12 + ϵ 12 μ 33 + 2 ϵ 13 μ 23 + 2 ϵ 23 μ 13 ,
β 23 = ( ϵ 33 ϵ 11 ϵ 13 2 ) ( μ 33 μ 22 μ 23 2 ) + ( ϵ 33 ϵ 22 ϵ 23 2 ) ( μ 33 μ 11 μ 13 2 ) 2 ( ϵ 33 ϵ 12 ϵ 13 ϵ 23 ) ( μ 33 μ 12 μ 13 μ 23 ) ,
β 11 i = ϵ i i μ i 3 + ϵ i 3 μ i i , β 12 i = ϵ i i μ j 3 + ϵ j 3 μ i i + 2 ϵ i j μ i 3 + 2 ϵ i 3 μ i j ,
β 13 i = ( ϵ 33 ϵ i i ϵ 13 2 ) ( μ j j μ i 3 μ i j μ j 3 ) + ( ϵ j j ϵ i 3 ϵ i j ϵ j 3 ) ( μ 33 μ i i μ i 3 2 ) + ( ϵ 33 ϵ i j ϵ i 3 ϵ j 3 ) ( μ i i μ j 3 μ i j μ i 3 ) + ( ϵ i i ϵ j 3 ϵ i j ϵ i 3 ) ( μ 33 μ i j μ i 3 μ j 3 ) ,
β 01 i = ϵ i i μ i j + ϵ i j μ i i , β 02 = ϵ 11 μ 22 + ϵ 22 μ 11 + 4 ϵ 12 μ 12 ,
β 03 i = ( ϵ 33 ϵ i i ϵ i 3 2 ) ( μ i i μ j j μ i j 2 ) + ( ϵ i i ϵ j j ϵ i j 2 ) ( μ 33 μ i i μ i 3 2 ) 2 ( ϵ i i ϵ j 3 ϵ i j ϵ i 3 ) ( μ i i μ j 3 μ i j μ i 3 ) ,
β 04 = ( ϵ 11 ϵ 22 ϵ 12 2 ) ( μ 33 μ 12 μ 13 μ 23 ) + ( ϵ 33 ϵ 12 ϵ 13 ϵ 23 ) ( μ 22 μ 11 μ 12 2 ) + ( ϵ 11 ϵ 23 ϵ 12 ϵ 23 ) ( μ 22 μ 13 μ 12 μ 23 ) + ( ϵ 22 ϵ 13 ϵ 12 ϵ 23 ) ( μ 11 μ 23 μ 12 μ 13 ) ,
i = 1 , 2 , j i 3 .
Spherical Abreration = 1 2 A 040 ( 6 ρ 4 6 ρ 2 + 1 ) ,
Coma = A 031 ( 3 ρ 3 2 ρ ) cos ( θ ) ,
Astigmatism = A 022 ρ 2 [ 2 cos 2 ( θ ) 1 ] ,
Curvature of field = 1 2 A 120 ( 2 ρ 2 1 ) ,
Distortion = A 111 ρ cos ( θ ) .
i = 1 π 2 0 1 0 2 π exp ( i k 0 ζ ) ρ d ρ d θ 2 .
ζ ( ρ , θ ) = Σ n , m ϵ n m A ln m R n m ( ρ ) cos ( m θ ) .
2 l + m + n = 4 , ϵ n m = { 1 2 m = 0 , n 1 1 otherwise } .
1 0 R n l ( ρ ) R n l ( ρ ) ρ d ρ = δ n n 2 ( n + 1 ) , 0 2 π cos ( m θ ) cos ( m θ ) d θ = π δ m m m > 0 ,
0 1 R n m ( ρ ) ρ d ρ 0 2 π ζ ( ρ , θ ) cos ( m θ ) d θ = Σ n , m ϵ n m A ln m 0 1 R n m ( ρ ) R n m ( ρ ) ρ d ρ 0 2 π cos ( m θ ) cos ( m θ ) d θ .
0 1 R n m ( ρ ) ρ d ρ 0 2 π ζ ( ρ , θ ) cos ( m θ ) d θ = Σ n , m ϵ n m A ln m δ n n 2 ( n + 1 ) π δ m m = π 2 ( n + 1 ) ϵ n m A l n m ,
A l m n = 2 ( n + 1 ) π ϵ n m 0 1 R n m ( ρ ) ρ d ρ 0 2 π ζ ( ρ , θ ) cos ( m θ ) d θ .

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