Abstract

The concept of super-resolution refers to various methods for improving the angular resolution of an optical imaging system beyond the classical diffraction limit. In optical microscopy, several techniques have been successfully developed with the aim of narrowing the central lobe of the illumination point spread function. In astronomy, however, no similar techniques can be used. A feasible method to design antennas and telescopes with angular resolution better than the diffraction limit consists of using variable transmittance pupils. In particular, discrete binary phase masks (0 or π) with finite phase-jump positions, known as Toraldo pupils (TPs), have the advantage of being easy to fabricate but offer relatively little flexibility in terms of achieving specific trade-offs between design parameters, such as the angular width of the main lobe and the intensity of sidelobes. In this paper, we show that a complex transmittance filter (equivalent to a continuous TP, i.e., consisting of infinitely narrow concentric rings) can achieve more easily the desired trade-off between design parameters. We also show how the super-resolution effect can be generated with both amplitude- and phase-only masks and confirm the expected performance with electromagnetic numerical simulations in the microwave range.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

2016 (1)

L. Olmi, P. Bolli, L. Cresci, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Super-resolution with Toraldo pupils: analysis with electromagnetic numerical simulations,” Proc. SPIE 9906, 99065Y (2016).
[Crossref]

2015 (1)

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

2013 (1)

2012 (1)

2010 (1)

L. N. Hazra and N. Reza, “Optimal design of Toraldo super resolving filters,” Proc. SPIE 7787, 77870D (2010).
[Crossref]

2008 (1)

L. Liu and G. Wang, “Designing superresolution optical pupil filter with constrained global optimization algorithm,” Optik 119, 481–484 (2008).
[Crossref]

2004 (3)

V. F. Canales, D. M. de Juana, and M. P. Cagigal, “Superresolution in compensated telescopes,” Opt. Lett. 29, 935–937 (2004).
[Crossref]

A. Ranfagni, D. Mugnai, and R. Ruggeri, “Beyond the diffraction limit: super-resolving pupils,” J. Appl. Phys. 95, 2217–2222 (2004).
[Crossref]

M. P. Cagigal, V. F. Canales, and J. E. Oti, “Design of continuous superresolving masks for ground-based telescopes,” Publ. Astron. Soc. Pac. 116, 965–970 (2004).
[Crossref]

2003 (2)

2002 (1)

2000 (1)

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197, 219–223 (2000).
[Crossref]

1998 (1)

M. Bersanelli, E. Mattaini, E. Santambrogio, A. Simonetto, S. Cirant, F. Gandini, C. Sozzi, N. Mandolesi, and F. Villa, “A low-sidelobe, high frequency corrugated feed horn for CMB observations,” Exp. Astron. 8, 231–238 (1998).
[Crossref]

1997 (1)

1952 (1)

G. Toraldo di Francia, “Nuove pupille superrisolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

Bersanelli, M.

M. Bersanelli, E. Mattaini, E. Santambrogio, A. Simonetto, S. Cirant, F. Gandini, C. Sozzi, N. Mandolesi, and F. Villa, “A low-sidelobe, high frequency corrugated feed horn for CMB observations,” Exp. Astron. 8, 231–238 (1998).
[Crossref]

Bolli, P.

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

L. Olmi, P. Bolli, L. Cresci, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Super-resolution with Toraldo pupils: analysis with electromagnetic numerical simulations,” Proc. SPIE 9906, 99065Y (2016).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Bryant, G. W.

Caballero, M. T.

Cagigal, M. P.

Canales, V. F.

Cirant, S.

M. Bersanelli, E. Mattaini, E. Santambrogio, A. Simonetto, S. Cirant, F. Gandini, C. Sozzi, N. Mandolesi, and F. Villa, “A low-sidelobe, high frequency corrugated feed horn for CMB observations,” Exp. Astron. 8, 231–238 (1998).
[Crossref]

Cresci, L.

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

L. Olmi, P. Bolli, L. Cresci, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Super-resolution with Toraldo pupils: analysis with electromagnetic numerical simulations,” Proc. SPIE 9906, 99065Y (2016).
[Crossref]

D’Agostino, F.

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

de Juana, D. M.

Gandini, F.

M. Bersanelli, E. Mattaini, E. Santambrogio, A. Simonetto, S. Cirant, F. Gandini, C. Sozzi, N. Mandolesi, and F. Villa, “A low-sidelobe, high frequency corrugated feed horn for CMB observations,” Exp. Astron. 8, 231–238 (1998).
[Crossref]

Gao, P.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

Haynes, V.

Hazra, L. N.

L. N. Hazra and N. Reza, “Optimal design of Toraldo super resolving filters,” Proc. SPIE 7787, 77870D (2010).
[Crossref]

Juskaitis, R.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197, 219–223 (2000).
[Crossref]

Kim, H.

Liu, L.

L. Liu and G. Wang, “Designing superresolution optical pupil filter with constrained global optimization algorithm,” Optik 119, 481–484 (2008).
[Crossref]

Luo, X.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

Maffei, B.

Mandolesi, N.

M. Bersanelli, E. Mattaini, E. Santambrogio, A. Simonetto, S. Cirant, F. Gandini, C. Sozzi, N. Mandolesi, and F. Villa, “A low-sidelobe, high frequency corrugated feed horn for CMB observations,” Exp. Astron. 8, 231–238 (1998).
[Crossref]

Markwardt, C. B.

C. B. Markwardt, Astronomical Data Analysis Software and Systems XVIII, D. A. Bohlender, D. Durand, and P. Dowler, eds., Astronomical Society of the Pacific Conference Series (Astronomical Society of the Pacific, 2009), Vol. 411, p. 251.

Martinez-Corral, M.

Mattaini, E.

M. Bersanelli, E. Mattaini, E. Santambrogio, A. Simonetto, S. Cirant, F. Gandini, C. Sozzi, N. Mandolesi, and F. Villa, “A low-sidelobe, high frequency corrugated feed horn for CMB observations,” Exp. Astron. 8, 231–238 (1998).
[Crossref]

Migliozzi, M.

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

Morris, G. M.

Mugnai, D.

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

L. Olmi, P. Bolli, L. Cresci, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Super-resolution with Toraldo pupils: analysis with electromagnetic numerical simulations,” Proc. SPIE 9906, 99065Y (2016).
[Crossref]

A. Ranfagni, D. Mugnai, and R. Ruggeri, “Beyond the diffraction limit: super-resolving pupils,” J. Appl. Phys. 95, 2217–2222 (2004).
[Crossref]

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A 311, 77–81 (2003).
[Crossref]

Natale, E.

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

L. Olmi, P. Bolli, L. Cresci, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Super-resolution with Toraldo pupils: analysis with electromagnetic numerical simulations,” Proc. SPIE 9906, 99065Y (2016).
[Crossref]

Neil, M. A. A.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197, 219–223 (2000).
[Crossref]

Nesti, R.

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

L. Olmi, P. Bolli, L. Cresci, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Super-resolution with Toraldo pupils: analysis with electromagnetic numerical simulations,” Proc. SPIE 9906, 99065Y (2016).
[Crossref]

Ng, M. W.

Olmi, L.

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

L. Olmi, P. Bolli, L. Cresci, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Super-resolution with Toraldo pupils: analysis with electromagnetic numerical simulations,” Proc. SPIE 9906, 99065Y (2016).
[Crossref]

Oti, J. E.

M. P. Cagigal, V. F. Canales, and J. E. Oti, “Design of continuous superresolving masks for ground-based telescopes,” Publ. Astron. Soc. Pac. 116, 965–970 (2004).
[Crossref]

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607–609 (2003).
[Crossref]

Ozturk, F.

Panella, D.

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

L. Olmi, P. Bolli, L. Cresci, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Super-resolution with Toraldo pupils: analysis with electromagnetic numerical simulations,” Proc. SPIE 9906, 99065Y (2016).
[Crossref]

Pisano, G.

Pu, M.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

Ranfagni, A.

A. Ranfagni, D. Mugnai, and R. Ruggeri, “Beyond the diffraction limit: super-resolving pupils,” J. Appl. Phys. 95, 2217–2222 (2004).
[Crossref]

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A 311, 77–81 (2003).
[Crossref]

Reza, N.

L. N. Hazra and N. Reza, “Optimal design of Toraldo super resolving filters,” Proc. SPIE 7787, 77870D (2010).
[Crossref]

Ruggeri, R.

A. Ranfagni, D. Mugnai, and R. Ruggeri, “Beyond the diffraction limit: super-resolving pupils,” J. Appl. Phys. 95, 2217–2222 (2004).
[Crossref]

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A 311, 77–81 (2003).
[Crossref]

Sales, T. R. M.

Santambrogio, E.

M. Bersanelli, E. Mattaini, E. Santambrogio, A. Simonetto, S. Cirant, F. Gandini, C. Sozzi, N. Mandolesi, and F. Villa, “A low-sidelobe, high frequency corrugated feed horn for CMB observations,” Exp. Astron. 8, 231–238 (1998).
[Crossref]

Simonetto, A.

M. Bersanelli, E. Mattaini, E. Santambrogio, A. Simonetto, S. Cirant, F. Gandini, C. Sozzi, N. Mandolesi, and F. Villa, “A low-sidelobe, high frequency corrugated feed horn for CMB observations,” Exp. Astron. 8, 231–238 (1998).
[Crossref]

Sozzi, C.

M. Bersanelli, E. Mattaini, E. Santambrogio, A. Simonetto, S. Cirant, F. Gandini, C. Sozzi, N. Mandolesi, and F. Villa, “A low-sidelobe, high frequency corrugated feed horn for CMB observations,” Exp. Astron. 8, 231–238 (1998).
[Crossref]

Stefani, L.

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

L. Olmi, P. Bolli, L. Cresci, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Super-resolution with Toraldo pupils: analysis with electromagnetic numerical simulations,” Proc. SPIE 9906, 99065Y (2016).
[Crossref]

Stelzer, E. H. K.

Stranick, S. J.

Swoger, J.

Tang, D.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

Toraldo di Francia, G.

G. Toraldo di Francia, “Nuove pupille superrisolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

G. Toraldo di Francia, La diffrazione della luce (Einaudi, 1958).

Villa, F.

M. Bersanelli, E. Mattaini, E. Santambrogio, A. Simonetto, S. Cirant, F. Gandini, C. Sozzi, N. Mandolesi, and F. Villa, “A low-sidelobe, high frequency corrugated feed horn for CMB observations,” Exp. Astron. 8, 231–238 (1998).
[Crossref]

Wang, C.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

Wang, G.

L. Liu and G. Wang, “Designing superresolution optical pupil filter with constrained global optimization algorithm,” Optik 119, 481–484 (2008).
[Crossref]

Wang, J.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

Wang, Y.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

Wilson, T.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197, 219–223 (2000).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Yan, W.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

Zhang, Y.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

Zhao, Z.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

Appl. Opt. (1)

Atti Fond. Giorgio Ronchi (1)

G. Toraldo di Francia, “Nuove pupille superrisolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

Exp. Astron. (2)

M. Bersanelli, E. Mattaini, E. Santambrogio, A. Simonetto, S. Cirant, F. Gandini, C. Sozzi, N. Mandolesi, and F. Villa, “A low-sidelobe, high frequency corrugated feed horn for CMB observations,” Exp. Astron. 8, 231–238 (1998).
[Crossref]

L. Olmi, P. Bolli, L. Cresci, F. D’Agostino, M. Migliozzi, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications,” Exp. Astron. 43, 285–309 (2017).
[Crossref]

J. Appl. Phys. (1)

A. Ranfagni, D. Mugnai, and R. Ruggeri, “Beyond the diffraction limit: super-resolving pupils,” J. Appl. Phys. 95, 2217–2222 (2004).
[Crossref]

J. Microsc. (1)

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197, 219–223 (2000).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Optik (1)

L. Liu and G. Wang, “Designing superresolution optical pupil filter with constrained global optimization algorithm,” Optik 119, 481–484 (2008).
[Crossref]

Phys. Lett. A (1)

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A 311, 77–81 (2003).
[Crossref]

Proc. SPIE (2)

L. N. Hazra and N. Reza, “Optimal design of Toraldo super resolving filters,” Proc. SPIE 7787, 77870D (2010).
[Crossref]

L. Olmi, P. Bolli, L. Cresci, D. Mugnai, E. Natale, R. Nesti, D. Panella, and L. Stefani, “Super-resolution with Toraldo pupils: analysis with electromagnetic numerical simulations,” Proc. SPIE 9906, 99065Y (2016).
[Crossref]

Publ. Astron. Soc. Pac. (1)

M. P. Cagigal, V. F. Canales, and J. E. Oti, “Design of continuous superresolving masks for ground-based telescopes,” Publ. Astron. Soc. Pac. 116, 965–970 (2004).
[Crossref]

Sci. Rep. (1)

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2015).
[Crossref]

Other (5)

C. B. Markwardt, Astronomical Data Analysis Software and Systems XVIII, D. A. Bohlender, D. Durand, and P. Dowler, eds., Astronomical Society of the Pacific Conference Series (Astronomical Society of the Pacific, 2009), Vol. 411, p. 251.

http://www.exelisvis.com/ProductsServices/IDL.aspx .

http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html .

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

G. Toraldo di Francia, La diffrazione della luce (Einaudi, 1958).

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Figures (7)

Fig. 1.
Fig. 1. Top panel. Diffraction pattern at ν = 20    GHz by a 3-coronae TP, as given by the square of Eq. (1) for n = 3 (black solid line), and that of an open pupil of equal diameter (blue solid line). The red dashed line, almost exactly overlapping with the black solid line, represents the diffracted amplitude as obtained by Eq. (8) using the input illumination shown in the bottom panel. All curves are normalized with respect to the on-axis value. Bottom panel. Pupil illumination for a 3-coronae TP, with a 9 cm diameter, as required by Eq. (1). The “*” symbols represent the relative magnitudes of the coefficients k 1 , k 2 , and k 3 , while the dashed green line represents the second-order polynomial best fit to the three points. The black solid line represents the ideal entrance pupil illumination, as given by Eq. (1), where the amplitude of the illumination is uniform over each separate corona. A negative amplitude in the middle corona indicates a phase inversion.
Fig. 2.
Fig. 2. Illustration of the notation used in determining Eq. (9). P is a point near the focus, F, f is the focal length, and R is the pupil radius. w represents the radial coordinate in the focal plane.
Fig. 3.
Fig. 3. Top panel. Diffraction pattern at ν = 20    GHz by a continuous TP. The red dashed line represents the desired intensity in the FF, as obtained by using a sinc 2 function. The black solid line represents the best-fit diffraction pattern, obtained using Eq. (8) and solving for the aperture illumination, A ( r ) . The blue solid line is the same as in Fig. 1, and all curves are normalized with respect to the on-axis value. Bottom panel. Pupil illumination of the continuous TP, A ( r ) , as obtained from a best-fit procedure applied to Eq. (8). As before, a negative amplitude indicates a phase inversion.
Fig. 4.
Fig. 4. Same as Fig. 3, but using as aperture illumination, A ( r ) , the linear combination of a Gaussian function and a polynomial.
Fig. 5.
Fig. 5. Same as Fig. 3, but using a phase-only aperture illumination, i.e., A ^ ( r ) = exp [ j ϕ ( r ) ] . In this case, the phase function, ϕ ( r ) , shown in the bottom panel, was a simple sine function.
Fig. 6.
Fig. 6. Top panel. Parameters used in the description of the phase-only pupil. Bottom panel. The phase illumination converted into a phase mask at ν = 20    GHz . The mask shows the required thickness variations in a dielectric material with a refraction index 1.5 .
Fig. 7.
Fig. 7. Top panel. FEKO 3D model to implement the phase-only pupil. The dielectric material composing the pupil has a refraction index n = 1.5 , and the pupil aperture is realized in an infinite ground plane. The arrows below the pupil show the direction of incidence of a plane wave (blue arrow), and also the direction of the linear polarization (red arrow). Bottom panel. FF results from the FEKO simulation at ν = 20    GHz , normalized to the on-axis value.

Equations (13)

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A tot ( x ) = i = 0 n 1 k i + 1 x [ ρ i + 1 J 1 ( ρ i + 1 x ) ρ i J 1 ( ρ i x ) ] = i = 0 n 1 F ( x ; ρ , k ) ,
ρ i + 1 = r i + Δ r R = ρ i + Δ r R .
F ( x ; ρ , k ) F ( x ; r , Δ r , k ) = k ( r + Δ r ) x × [ r + Δ r R J 1 ( r + Δ r R x ) r R J 1 ( r R x ) ] .
J 1 ( r + Δ r R x ) = J 1 ( z + Δ z ) J 1 ( z ) + Δ z J 1 ,
J 1 ( z ) = J 0 ( z ) 1 z J 1 ( z ) ,
J 1 ( z + Δ z ) J 1 ( z ) ( 1 Δ z z ) + Δ z J 0 ( z ) .
F ( x ; r , Δ r , k ) k ( r + Δ r ) z 2 ( r R ) 2 × [ z Δ z J 0 ( z ) + ( Δ z ) 2 ( J 0 ( z ) J 1 ( z ) z ) ] ,
F ( x ; r , Δ r , k ) k ( r + Δ r ) R 2 J 0 ( r R x ) r Δ r ,
k ( r + Δ r ) k ( r ) + Δ r k ( r ) .
A tot ( x ) = 2 π λ 2 0 R A ( r ) J 0 ( r R x ) r d r = 2 π R 2 λ 2 0 1 A ( ρ ) J 0 ( ρ x ) ρ d ρ ,
A tot ( v ) 0 1 A ( ρ ) J 0 ( v ρ ) ρ d ρ ,
Δ l ( r ) = { n s ( r ) + n o [ s max s ( r ) ] } n o s max = ( n n o ) s ( r ) ,
s ( r ) = s max ϕ ( r ) 2 π ,

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