Abstract

We describe and provide a systematic procedure for computationally fast propagation of arbitrary vector electromagnetic (EM) fields through an axially symmetric medium. A cylindrical harmonic field propagator is chosen for this purpose and in most cases, this is the best and the obvious choice. Firstly, we describe the cylindrical harmonic decomposition technique in terms of both scalar and vector basis for a given input excitation field. Then we formulate a generalized discrete Fourier-Hankel transform to achieve efficient vector basis decomposition. We allow a slower, pre-computation step, that finds a representation of the axi-symmetric medium as a transfer matrix in a discrete, cylindrical-harmonic basis. We find this matrix from a series of axi-symmetric (2D) finite element simulations (also known as the 2.5D technique). This transfer matrix approach significantly reduces the computational load when the transverse size or range exceeds about 30 wavelengths. This matrix is independent of the input excitation field for a given space-bandwidth product and hence makes it reusable for different excitation fields. We numerically validate the above approaches for different axi-symmetric EM scattering media which include a hemispherical gradient-index Maxwell’s fish-eye lens, a transformation optics designed spherical invisibility cloak, a thin aspheric lens, and a cylindrical perfect lens.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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2016 (1)

2015 (3)

Y. Ou, D. Pardo, and Y. Chen, “Fourier finite element modeling of light emission in waveguides: 2.5-dimensional fem approach,” Opt. Express 23, 30259–30269 (2015).
[Crossref] [PubMed]

Y. Ye, Z. J. Wong, X. Lu, X. Ni, H. Zhu, X. Chen, Y. Wang, and X. Zhang, “Monolayer excitonic laser,” Nat Photon 9, 733 (2015).
[Crossref]

Y. Zhao, J. Wyrick, F. D. Natterer, J. F. Rodriguez-Nieva, C. Lewandowski, K. Watanabe, T. Taniguchi, L. S. Levitov, N. B. Zhitenev, and J. A. Stroscio, “Creating and probing electron whispering-gallery modes in graphene,” Science 348, 672–675 (2015).
[Crossref] [PubMed]

2014 (2)

G. Lipworth, J. Ensworth, K. Seetharam, D. Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer,” Scientific Reports 4, 3642 (2014).
[Crossref] [PubMed]

O. Furxhi, D. L. Marks, and D. J. Brady, “Echelle crossed grating millimeter wave beam scanner,” Opt. Express 22, 16393–16407 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (2)

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337, 549–552 (2012).
[Crossref] [PubMed]

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (1)

2009 (1)

2007 (1)

R. Merlin, “Radiationless electromagnetic interference: Evanescent-Field lenses and perfect focusing,” Science 317, 927–929 (2007).
[Crossref] [PubMed]

2006 (1)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

2000 (2)

J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Physical Review Letters 85, 3966–3969 (2000).
[Crossref] [PubMed]

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An All-Dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[Crossref] [PubMed]

1986 (1)

B. Thomas, G. L. James, and K. J. Greene, “Design of wide-band corrugated conical horns for cassegrain antennas,” Antennas and Propagation, IEEE Transactions on 34, 750–757 (1986).
[Crossref]

1982 (1)

S. Ali, W. Chew, and J. Kong, “Vector hankel transform analysis of annular-ring microstrip antenna,” IEEE Transactions on Antennas and Propagation 30, 637–644 (1982).
[Crossref]

1981 (1)

G. Crone, A. Rudge, and G. Taylor, “Design and performance of airborne radomes: A review,” Communications, Radar and Signal Processing, IEE Proceedings F 128, 451–464 (1981).
[Crossref]

1971 (1)

M. Meeks and J. Ruze, “Evaluation of the haystack antenna and radome,” Antennas and Propagation, IEEE Transactions on 19, 723–728 (1971).
[Crossref]

Ali, S.

S. Ali, W. Chew, and J. Kong, “Vector hankel transform analysis of annular-ring microstrip antenna,” IEEE Transactions on Antennas and Propagation 30, 637–644 (1982).
[Crossref]

Aubry, A.

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337, 549–552 (2012).
[Crossref] [PubMed]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd Edition (Wiley, 2012).

Bowling, D.

Brady, D. J.

Chen, X.

Y. Ye, Z. J. Wong, X. Lu, X. Ni, H. Zhu, X. Chen, Y. Wang, and X. Zhang, “Monolayer excitonic laser,” Nat Photon 9, 733 (2015).
[Crossref]

Chen, Y.

Chew, W.

S. Ali, W. Chew, and J. Kong, “Vector hankel transform analysis of annular-ring microstrip antenna,” IEEE Transactions on Antennas and Propagation 30, 637–644 (1982).
[Crossref]

Chilkoti, A.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

Ciraci, C.

C. Ciraci, Y. Urzhumov, and D. R. Smith, “Far-field analysis of axially symmetric three-dimensional directional cloaks,” Opt. Express 21, 9397–9406 (2013).
[Crossref] [PubMed]

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

Cossairt, O. S.

Crone, G.

G. Crone, A. Rudge, and G. Taylor, “Design and performance of airborne radomes: A review,” Communications, Radar and Signal Processing, IEE Proceedings F 128, 451–464 (1981).
[Crossref]

Ensworth, J.

G. Lipworth, J. Ensworth, K. Seetharam, D. Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer,” Scientific Reports 4, 3642 (2014).
[Crossref] [PubMed]

Fan, S.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An All-Dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[Crossref] [PubMed]

Feng, S.

Fernández-Dominguez, A. I.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

Fink, Y.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An All-Dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[Crossref] [PubMed]

Furxhi, O.

Greene, K. J.

B. Thomas, G. L. James, and K. J. Greene, “Design of wide-band corrugated conical horns for cassegrain antennas,” Antennas and Propagation, IEEE Transactions on 34, 750–757 (1986).
[Crossref]

Halterman, K.

Hill, R. T.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

Huang, D.

G. Lipworth, J. Ensworth, K. Seetharam, D. Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer,” Scientific Reports 4, 3642 (2014).
[Crossref] [PubMed]

Ibanescu, M.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An All-Dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[Crossref] [PubMed]

James, G. L.

B. Thomas, G. L. James, and K. J. Greene, “Design of wide-band corrugated conical horns for cassegrain antennas,” Antennas and Propagation, IEEE Transactions on 34, 750–757 (1986).
[Crossref]

Joannopoulos, J. D.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An All-Dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[Crossref] [PubMed]

Johnson, C. C.

C. C. Johnson, Field and Wave Electrodynamics (McGraw-Hill, 1965).

Kong, J.

S. Ali, W. Chew, and J. Kong, “Vector hankel transform analysis of annular-ring microstrip antenna,” IEEE Transactions on Antennas and Propagation 30, 637–644 (1982).
[Crossref]

Lee, J. S.

G. Lipworth, J. Ensworth, K. Seetharam, D. Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer,” Scientific Reports 4, 3642 (2014).
[Crossref] [PubMed]

Levitov, L. S.

Y. Zhao, J. Wyrick, F. D. Natterer, J. F. Rodriguez-Nieva, C. Lewandowski, K. Watanabe, T. Taniguchi, L. S. Levitov, N. B. Zhitenev, and J. A. Stroscio, “Creating and probing electron whispering-gallery modes in graphene,” Science 348, 672–675 (2015).
[Crossref] [PubMed]

Lewandowski, C.

Y. Zhao, J. Wyrick, F. D. Natterer, J. F. Rodriguez-Nieva, C. Lewandowski, K. Watanabe, T. Taniguchi, L. S. Levitov, N. B. Zhitenev, and J. A. Stroscio, “Creating and probing electron whispering-gallery modes in graphene,” Science 348, 672–675 (2015).
[Crossref] [PubMed]

Lipworth, G.

G. Lipworth, J. Ensworth, K. Seetharam, D. Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer,” Scientific Reports 4, 3642 (2014).
[Crossref] [PubMed]

Lu, X.

Y. Ye, Z. J. Wong, X. Lu, X. Ni, H. Zhu, X. Chen, Y. Wang, and X. Zhang, “Monolayer excitonic laser,” Nat Photon 9, 733 (2015).
[Crossref]

Maier, S. A.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337, 549–552 (2012).
[Crossref] [PubMed]

Marks, D. L.

Meeks, M.

M. Meeks and J. Ruze, “Evaluation of the haystack antenna and radome,” Antennas and Propagation, IEEE Transactions on 19, 723–728 (1971).
[Crossref]

Merlin, R.

R. Merlin, “Radiationless electromagnetic interference: Evanescent-Field lenses and perfect focusing,” Science 317, 927–929 (2007).
[Crossref] [PubMed]

Miau, D.

Mittra, R.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, Volume 4 of IEEE Press Series on Electromagnetic Wave Theory (Wiley, 1998).

Mock, J. J.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

Natterer, F. D.

Y. Zhao, J. Wyrick, F. D. Natterer, J. F. Rodriguez-Nieva, C. Lewandowski, K. Watanabe, T. Taniguchi, L. S. Levitov, N. B. Zhitenev, and J. A. Stroscio, “Creating and probing electron whispering-gallery modes in graphene,” Science 348, 672–675 (2015).
[Crossref] [PubMed]

Nayar, S. K.

Ni, X.

Y. Ye, Z. J. Wong, X. Lu, X. Ni, H. Zhu, X. Chen, Y. Wang, and X. Zhang, “Monolayer excitonic laser,” Nat Photon 9, 733 (2015).
[Crossref]

Nomura, T.

G. Lipworth, J. Ensworth, K. Seetharam, D. Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer,” Scientific Reports 4, 3642 (2014).
[Crossref] [PubMed]

Ou, Y.

Overfelt, P. L.

Pardo, D.

Pendry, J. B.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337, 549–552 (2012).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Physical Review Letters 85, 3966–3969 (2000).
[Crossref] [PubMed]

Peterson, A. F.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, Volume 4 of IEEE Press Series on Electromagnetic Wave Theory (Wiley, 1998).

Ray, S. L.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, Volume 4 of IEEE Press Series on Electromagnetic Wave Theory (Wiley, 1998).

Reynolds, M. S.

G. Lipworth, J. Ensworth, K. Seetharam, D. Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer,” Scientific Reports 4, 3642 (2014).
[Crossref] [PubMed]

Rodriguez-Nieva, J. F.

Y. Zhao, J. Wyrick, F. D. Natterer, J. F. Rodriguez-Nieva, C. Lewandowski, K. Watanabe, T. Taniguchi, L. S. Levitov, N. B. Zhitenev, and J. A. Stroscio, “Creating and probing electron whispering-gallery modes in graphene,” Science 348, 672–675 (2015).
[Crossref] [PubMed]

Rudge, A.

G. Crone, A. Rudge, and G. Taylor, “Design and performance of airborne radomes: A review,” Communications, Radar and Signal Processing, IEE Proceedings F 128, 451–464 (1981).
[Crossref]

Ruze, J.

M. Meeks and J. Ruze, “Evaluation of the haystack antenna and radome,” Antennas and Propagation, IEEE Transactions on 19, 723–728 (1971).
[Crossref]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Schmalenberg, P.

G. Lipworth, J. Ensworth, K. Seetharam, D. Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer,” Scientific Reports 4, 3642 (2014).
[Crossref] [PubMed]

Schurig, D.

Seetharam, K.

G. Lipworth, J. Ensworth, K. Seetharam, D. Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer,” Scientific Reports 4, 3642 (2014).
[Crossref] [PubMed]

Smith, D. R.

G. Lipworth, J. Ensworth, K. Seetharam, D. Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer,” Scientific Reports 4, 3642 (2014).
[Crossref] [PubMed]

C. Ciraci, Y. Urzhumov, and D. R. Smith, “Far-field analysis of axially symmetric three-dimensional directional cloaks,” Opt. Express 21, 9397–9406 (2013).
[Crossref] [PubMed]

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337, 549–552 (2012).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

Song, R.

Stroscio, J. A.

Y. Zhao, J. Wyrick, F. D. Natterer, J. F. Rodriguez-Nieva, C. Lewandowski, K. Watanabe, T. Taniguchi, L. S. Levitov, N. B. Zhitenev, and J. A. Stroscio, “Creating and probing electron whispering-gallery modes in graphene,” Science 348, 672–675 (2015).
[Crossref] [PubMed]

Taniguchi, T.

Y. Zhao, J. Wyrick, F. D. Natterer, J. F. Rodriguez-Nieva, C. Lewandowski, K. Watanabe, T. Taniguchi, L. S. Levitov, N. B. Zhitenev, and J. A. Stroscio, “Creating and probing electron whispering-gallery modes in graphene,” Science 348, 672–675 (2015).
[Crossref] [PubMed]

Taylor, G.

G. Crone, A. Rudge, and G. Taylor, “Design and performance of airborne radomes: A review,” Communications, Radar and Signal Processing, IEE Proceedings F 128, 451–464 (1981).
[Crossref]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Thomas, B.

B. Thomas, G. L. James, and K. J. Greene, “Design of wide-band corrugated conical horns for cassegrain antennas,” Antennas and Propagation, IEEE Transactions on 34, 750–757 (1986).
[Crossref]

Thomas, E. L.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An All-Dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[Crossref] [PubMed]

Urzhumov, Y.

G. Lipworth, J. Ensworth, K. Seetharam, D. Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer,” Scientific Reports 4, 3642 (2014).
[Crossref] [PubMed]

C. Ciraci, Y. Urzhumov, and D. R. Smith, “Far-field analysis of axially symmetric three-dimensional directional cloaks,” Opt. Express 21, 9397–9406 (2013).
[Crossref] [PubMed]

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

Venkatesh, S.

Viswanathan, N.

Wang, Y.

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Figures (10)

Fig. 1
Fig. 1 Transfer Matrix T in its sparse, block-diagonal form. The m = 0th block is highlighted in magenta.
Fig. 2
Fig. 2 Shows the total number of band-limited modes; S as a function of space-bandwidth product R × B along with a quadratic fit.
Fig. 3
Fig. 3 Hemispherical Maxwell’s fish-eye gradient index lens schematic for 2.5 D technique. Blue outline box indicates the simulated domain , z, ϕ = 0).
Fig. 4
Fig. 4 Shows the comparison between 2.5D technique and complete 3D simulations. (a). Electric field norm on xz plane (y = 0) using 2.5 D technique (Only fields pre and post the scattering medium are computed using the transfer matrix. Hence the fields inside the lens in (a) are blank). (b). Electric field norm on xz plane (y = 0) computed using conventional 3D approach in COMSOL. (c). Electric field norm at the focal plane of the lens. The white dashed line indicates the focal line. (d). Offset Gaussian beam with a beam waist=2λ propagating along +z direction is the source excitation field in both 2.5D and 3D simulations. (e). Normalized absolute error between 2.5D technique and 3D simulations at the focal plane. Electric fields are in the units of V/m.
Fig. 5
Fig. 5 Spherical invisibility cloak schematic for 2.5 D technique. Blue outline box indicates the simulated domain.
Fig. 6
Fig. 6 Shows the real part of the field (in V/m) due to incident offset Gaussian vector beam on xz plane (y = 0) inside and outside the spherical cloak simulated using 2.5D technique. The subfigures below show the individual component of the material property tensors required inside the cloaking region.
Fig. 7
Fig. 7 Aspheric thin plano-convex lens schematic for 2.5 D technique. Blue outline box indicates the simulated domain.
Fig. 8
Fig. 8 (a). Numerically computed fields norm of electric field pre- and post-lens (in V/m). (b). Comparison between the output Gaussian beam radius as a function of z calculated numerically (black solid) and analytically (dashed red).
Fig. 9
Fig. 9 Perfect Lens schematic for 2.5D technique. Blue outline box shows the simulated domain.
Fig. 10
Fig. 10 (a) Shows the source plane electric field norm (in V/m) of the chosen arbitrary excitation source at z = 0 plane. (b) Shows the difference map between the image and the object plane electric field norms (in V/m).

Tables (6)

Tables Icon

Table 1 Composite indices l, i, and α constructed for an example case: R = 1, B = 2π, Bs = 1.1B and δ = 0.5. Highlighted (grey colored rows) in l index indicate that the harmonics have space-bandwidth product less than or equal to RB. Highlighted (grey colored rows) in α index indicate that the spatial coordinate is within the domain radius R.

Tables Icon

Table 2 Comparison of Computational Performance between complete 3D simulation and Cylindrical Harmonic Mode Propagation technique (2.5D technique) for a domain radius R = 8λ.

Tables Icon

Table 3 Comparison of computational load between Plane wave propagator and Cylindrical harmonic propagator using the same 2D axi-symmetric solver (2.5D technique) for a domain radius R = 8λ.

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Table 4 Parity Condition for TM modes.

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Table 5 Parity Condition for TE modes.

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Algorithm 1 Finding Composite Index l

Equations (50)

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E t ( ρ , ϕ , z 0 ) E t ( ρ , ϕ , z 1 ) or a l m n T b l m n or E z ( ρ , ϕ , z 0 ) , H z ( ρ , ϕ , z 0 ) E z ( ρ , ϕ , z 1 ) , H z ( ρ , ϕ , z 1 )
E z ( ρ , ϕ , z 0 ) = m , n a 1 m n ψ 1 m n ( ρ , ϕ )
η 0 H z ( ρ , ϕ , z 0 ) = m , n a 2 m n ψ 2 m n ( ρ , ϕ )
E z ( ρ , ϕ , z 1 ) = m , n b 1 m n ψ 1 m n ( ρ , ϕ )
η 0 H z ( ρ , ϕ , z 1 ) = m , n b 2 m n ψ 2 m n ( ρ , ϕ )
ψ l m n ( ρ , ϕ ) = C l m n J m ( β l m n ρ ) exp ( j m ϕ )
J m ( β 1 m n R ) = 0 and J m ( β 2 m n R ) = 0
J m ( χ 1 m n ) = 0 and J m ( χ 2 m n ) = 0
C 1 m n 1 = π R J m + 1 ( χ 1 m n )
C 2 m n 1 = π R [ J m 2 ( χ 2 m n ) J m 1 ( χ 2 m n ) J m + 1 ( χ 2 m n ) ] 1 / 2
ψ l m n | ψ l m n = 0 R π π ψ l m n * ( ρ , ϕ ) ψ l m n ( ρ , ϕ ) d ϕ ρ d ρ = δ m m δ n n
a l m n = ψ l m n | E z l = 1 η 0 H z l = 2 = 0 R π π ψ l m n * ( ρ , ϕ ) { E z ( ρ , ϕ , z 0 ) l = 1 η 0 H z ( ρ , ϕ , z 0 ) l = 2 } d ϕ ρ d ρ
E t = E E z z ^
E t ( ρ , ϕ , z 0 ) = l , m , n a l m n ϒ l m n Ψ l m n ( ρ , ϕ )
E t ( ρ , ϕ , z 1 ) = l , m , n b l m n ϒ l m n Ψ l m n ( ρ , ϕ )
ϒ l m n = j ω 2 0 μ 0 β l m n 2 δ l 1
Ψ 1 m n ( ρ , ϕ ) = C 1 m n [ J m ( β 1 m n ρ ) ρ ^ + j m J m ( β 1 m n ρ ) β 1 m n ρ ϕ ^ ] exp ( j m ϕ )
Ψ 2 m n ( ρ , ϕ ) = C 2 m n [ j m J m ( β 2 m n ρ ) β 2 m n ρ ρ ^ + J m ( β 2 m n ρ ) ϕ ^ ] exp ( j m ϕ )
Ψ l m n | Ψ l m n = 0 R π π Ψ l m n * ( ρ , ϕ ) Ψ l m n ( ρ , ϕ ) d ϕ ρ d ρ = δ l l δ m m δ n n
a l m n ϒ l m n = Ψ l m n | E t = 0 R π π Ψ l m n * ( ρ , ϕ ) E t ( ρ , ϕ , z 0 ) d ϕ ρ d ρ
b l m n = l , m , n T l m n l m n a l m n
T l m n l m n = δ l l δ m m δ n n exp ( j β z l m n ( z 1 z 0 ) ) where , β z l m n = ω 2 μ β l m n 2
M = max { m | χ 1 m 1 B R or χ 2 m 1 BR } N = max { n | χ 10 n B R or χ 21 n BR }
{ l } = { l , m , n | χ l m n BR } { l = 1 , 2 } { M m M 1 } { 1 n N }
J = max { j | χ 1 j 1 B s R or χ 2 j 1 B s R } B s B M K = max { k | χ 10 k B s R or χ 21 k B s R } B s B N
ρ l m k = χ l m k B s and ϕ j = j π j
{ i } = { i , j , k } = { i = 1 , 2 } { J j J 1 } { 1 k K }
x α = α δ and y γ = γ δ
ρ α γ = δ α 2 + γ 2 and ϕ α γ = atan 2 ( γ , α )
{ α } = { i , α , γ | ρ α γ R } { i = 1 , 2 } { A α A } { A γ A }
a = diag ( HE 0 )
b = Ta
E 1 = H ¯ b
E 1 = H ¯ T diag ( HE 0 )
E 0 = E il 0 = E i j k l m n 0 b = b l = b l m n H = H li = H l m n i j k T = T l l = T l m n l m n H ¯ = H ¯ α l = H ¯ i α γ l m n a = a l = a l m n E 1 = E α 1 = E i α γ 1
Input Samples & Forward DFHT Inverse DFHT & Output Samples E i j k l m n 0 = δ i l { E z ( ρ 1 m k , ϕ j , z 0 ) l = 1 η H z ( ρ 2 m k , ϕ j , z 0 ) l = 2 H ¯ i α γ l m n = δ i l ψ l m n ( ρ α γ , ϕ α γ ) H l m n i j k = δ i l 2 π 2 R 2 J B s 2 C l m k 2 ψ l m n * ( ρ l m k , ϕ j ) E i α γ 1 = { E z ( ρ α γ , ϕ α γ , z 1 ) i = 1 η H z ( ρ α γ , ϕ α γ , z 1 ) i = 2
Input Samples & Forward DFHT Inverse DFHT & Output Samples E i j k l m n 0 = E t ( ρ l m k , ϕ j , z 0 ) e ^ i H ¯ i α γ l m n = Ψ l m n ( ρ α γ , ϕ α γ ) e ^ i H l m n i j k = 2 π 2 R 2 J B s 2 C l m k 2 Ψ l m n * ( ρ l m k , ϕ j ) e ^ i E i α γ 1 = E t ( ρ α γ , ϕ α γ , z 1 ) e ^ i
E ( ρ , ϕ , z ) = E ˜ ( ρ , z ) exp ( j m ϕ )
[ j m ρ ϕ ^ ] × [ 1 μ ( j m ρ ϕ ^ ) × E ˜ ] β 2 E ˜ = 0
E l m n bg ( ρ , z ) = [ ϒ l m n Ψ l m n ( ρ , 0 ) + δ l 1 ψ 1 m n ( ρ , 0 ) z ^ ] exp ( j β z l m n ( z z 0 ) )
T l m n l m n = b l m n l m n = 0 R π π ψ l m n * ( ρ , ϕ ) exp ( j m ϕ ) f l l m n d ϕ ρ d ρ = 2 π δ m m C l m n 0 R J m ( β l m n ρ ) f l l m n ρ d ρ
f l l m n ( ρ ) = δ l l ψ l m n ( ρ , 0 ) exp ( j β z l m n ( z 1 z 0 ) ) + { E z sc ( ρ , z 1 ) l = 1 η H z sc ( ρ , z 1 ) l = 2
T l m n l m n = k = 1 K H l m n m k F k l l m n H l m n m k = δ m m 4 π 2 R 2 B s 2 C l m n C l m k 2 J m ( β l m n ρ l m k ) F k l l m n = f l l m n ( ρ l m k )
E z ( ρ , ϕ , z 0 ) or η 0 H z ( ρ , ϕ , z 0 ) = E 0 sin θ i exp ( j β 0 z 0 cos θ i ) m = j m J m ( β 0 ρ sin θ i ) exp ( j m ϕ )
= μ = [ r r r θ r ϕ θ r θ θ θ ϕ ϕ r ϕ θ ϕ ϕ ] = [ b b a ( r a ) 2 r 2 0 0 0 b b a 0 0 0 b b a ]
[ ρ ρ ρ ϕ ρ z ϕ ρ ϕ ϕ ϕ z z ρ z ϕ z z ] = [ 1 2 ( r r + θ θ + ( θ θ r r ) cos 2 θ ) 0 ( r r θ θ ) cos θ sin θ 0 ϕ ϕ 0 ( r r θ θ ) cos θ sin θ 0 r r cos 2 θ + θ θ sin 2 θ ]
0 R J m ( χ 1 m n ρ ) J m ( χ 1 m n ρ ) ρ d ρ = R 2 2 J m + 1 2 ( χ 1 m n ) δ n n
0 R J m ( χ 2 m n ρ ) J m ( χ 2 m n ρ ) ρ d ρ = R 2 2 [ J m 2 ( χ 2 m n ) J m 1 ( χ 2 m n ) J m + 1 ( χ 2 m n ) ] δ n n
π π exp ( j m ϕ ) exp ( j m ϕ ) d ϕ = 2 π δ m m
[ b ] = [ # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # ] 17 × 17 [ a 2 , 4 , 1 a 2 , 3 , 1 a 1 , 2 , 1 a 2 , 2 , 1 a 1 , 1 , 1 a 2 , 1 , 1 a 2 , 1 , 2 a 1 , 0 , 1 a 1 , 0 , 2 a 2 , 0 , 1 a 2 , 0 , 2 a 1 , 1 , 1 a 2 , 1 , 1 a 2 , 1 , 2 a 1 , 2 , 1 a 2 , 2 , 1 a 2 , 3 , 1 ] 17 × 1

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