Abstract

In planar metamaterial lenses, the focal point moves with the frequency. Here it is shown numerically that this movement can be controlled by properly engineering the dimensions of the metamaterial-based phase shifters that constitute the lens. In particular, such lenses can be designed to exhibit unusual chromatic aberration with the focal length increasing, rather than decreasing, with the frequency. It is proposed that such an artificial “reverse” chromatic aberration may optimize the transverse resolution of millimeter wave diagnostics of plasmas and be useful in compensating for the natural “ordinary” chromatic aberration of other components in an optical system. More generally, optimized chromatic aberration will allow for simultaneous focusing of several objects located at different distances and emitting or reflecting at different frequencies.

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References

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  1. H. J. Hartfuss, T. Geist, and M. Hirsch, “Heterodyne methods in millimetre wave plasma diagnostics with applications to ECE, interferometry and reflectometry,” Plasma Phys. Contr. Fusion 39(11), 1693–1769 (1997).
    [CrossRef]
  2. I. Hutchinson, Principles of Plasma Diagnostics (Cambridge Univ. Press, 2002).
  3. N. C. Luhnann, H. Bindslev, H. Park, J. Sanchez, G. Taylor, and C. X. Yu, “Microwave diagnostics,” Fusion Sci. Technol. 53, 335–396 (2008).
  4. P. K. Chattopadhyay, J. K. Anderson, T. M. Biewer, D. Craig, C. B. Forest, R. W. Harvey, and A. P. Smirnov, “Electron Bernstein wave emission from an overdense reversed field pinch plasma,” Phys. Plasmas 9(3), 752–755 (2002).
    [CrossRef]
  5. J. Wesson, Tokamaks (Oxford University Press, 1987).
  6. M. Al-Joumayly and N. Behdad, “Wideband planar microwave lenses using sub-wavelength spatial phase shifters,” IEEE Trans. Antenn. Propag. 59(12), 4542–4552 (2011).
    [CrossRef]
  7. M. Al-Joumayly and N. Behdad, “A generalized method for synthesizing low-profile, band-pass frequency selective surfaces with non-resonant constituting elements,” IEEE Trans. Antenn. Propag. 58(12), 4033–4041 (2010).
    [CrossRef]
  8. N. Behdad, M. Al-Joumayly, and M. Salehi, “A low-profile third-order bandpass frequency selective surface,” IEEE Trans. Antenn. Propag. 57(2), 460–466 (2009).
    [CrossRef]
  9. J. L. Luxon, “A design retrospective of the DIII-D tokamak,” Nucl. Fusion 42(5), 614–633 (2002).
    [CrossRef]
  10. R. F. Ellis, M. E. Austin, and D. Taussig, “New high spatial resolution optics for the DIII-D ECE radiometer,” in Proc. Of the 14th Joint Workshop on Electron Cyclotron Emission and Electron Cyclotron Resonance Heating (EC 14), Santorini, GR, May 9–12, 2006.

2011 (1)

M. Al-Joumayly and N. Behdad, “Wideband planar microwave lenses using sub-wavelength spatial phase shifters,” IEEE Trans. Antenn. Propag. 59(12), 4542–4552 (2011).
[CrossRef]

2010 (1)

M. Al-Joumayly and N. Behdad, “A generalized method for synthesizing low-profile, band-pass frequency selective surfaces with non-resonant constituting elements,” IEEE Trans. Antenn. Propag. 58(12), 4033–4041 (2010).
[CrossRef]

2009 (1)

N. Behdad, M. Al-Joumayly, and M. Salehi, “A low-profile third-order bandpass frequency selective surface,” IEEE Trans. Antenn. Propag. 57(2), 460–466 (2009).
[CrossRef]

2008 (1)

N. C. Luhnann, H. Bindslev, H. Park, J. Sanchez, G. Taylor, and C. X. Yu, “Microwave diagnostics,” Fusion Sci. Technol. 53, 335–396 (2008).

2002 (2)

P. K. Chattopadhyay, J. K. Anderson, T. M. Biewer, D. Craig, C. B. Forest, R. W. Harvey, and A. P. Smirnov, “Electron Bernstein wave emission from an overdense reversed field pinch plasma,” Phys. Plasmas 9(3), 752–755 (2002).
[CrossRef]

J. L. Luxon, “A design retrospective of the DIII-D tokamak,” Nucl. Fusion 42(5), 614–633 (2002).
[CrossRef]

1997 (1)

H. J. Hartfuss, T. Geist, and M. Hirsch, “Heterodyne methods in millimetre wave plasma diagnostics with applications to ECE, interferometry and reflectometry,” Plasma Phys. Contr. Fusion 39(11), 1693–1769 (1997).
[CrossRef]

Al-Joumayly, M.

M. Al-Joumayly and N. Behdad, “Wideband planar microwave lenses using sub-wavelength spatial phase shifters,” IEEE Trans. Antenn. Propag. 59(12), 4542–4552 (2011).
[CrossRef]

M. Al-Joumayly and N. Behdad, “A generalized method for synthesizing low-profile, band-pass frequency selective surfaces with non-resonant constituting elements,” IEEE Trans. Antenn. Propag. 58(12), 4033–4041 (2010).
[CrossRef]

N. Behdad, M. Al-Joumayly, and M. Salehi, “A low-profile third-order bandpass frequency selective surface,” IEEE Trans. Antenn. Propag. 57(2), 460–466 (2009).
[CrossRef]

Anderson, J. K.

P. K. Chattopadhyay, J. K. Anderson, T. M. Biewer, D. Craig, C. B. Forest, R. W. Harvey, and A. P. Smirnov, “Electron Bernstein wave emission from an overdense reversed field pinch plasma,” Phys. Plasmas 9(3), 752–755 (2002).
[CrossRef]

Behdad, N.

M. Al-Joumayly and N. Behdad, “Wideband planar microwave lenses using sub-wavelength spatial phase shifters,” IEEE Trans. Antenn. Propag. 59(12), 4542–4552 (2011).
[CrossRef]

M. Al-Joumayly and N. Behdad, “A generalized method for synthesizing low-profile, band-pass frequency selective surfaces with non-resonant constituting elements,” IEEE Trans. Antenn. Propag. 58(12), 4033–4041 (2010).
[CrossRef]

N. Behdad, M. Al-Joumayly, and M. Salehi, “A low-profile third-order bandpass frequency selective surface,” IEEE Trans. Antenn. Propag. 57(2), 460–466 (2009).
[CrossRef]

Biewer, T. M.

P. K. Chattopadhyay, J. K. Anderson, T. M. Biewer, D. Craig, C. B. Forest, R. W. Harvey, and A. P. Smirnov, “Electron Bernstein wave emission from an overdense reversed field pinch plasma,” Phys. Plasmas 9(3), 752–755 (2002).
[CrossRef]

Bindslev, H.

N. C. Luhnann, H. Bindslev, H. Park, J. Sanchez, G. Taylor, and C. X. Yu, “Microwave diagnostics,” Fusion Sci. Technol. 53, 335–396 (2008).

Chattopadhyay, P. K.

P. K. Chattopadhyay, J. K. Anderson, T. M. Biewer, D. Craig, C. B. Forest, R. W. Harvey, and A. P. Smirnov, “Electron Bernstein wave emission from an overdense reversed field pinch plasma,” Phys. Plasmas 9(3), 752–755 (2002).
[CrossRef]

Craig, D.

P. K. Chattopadhyay, J. K. Anderson, T. M. Biewer, D. Craig, C. B. Forest, R. W. Harvey, and A. P. Smirnov, “Electron Bernstein wave emission from an overdense reversed field pinch plasma,” Phys. Plasmas 9(3), 752–755 (2002).
[CrossRef]

Forest, C. B.

P. K. Chattopadhyay, J. K. Anderson, T. M. Biewer, D. Craig, C. B. Forest, R. W. Harvey, and A. P. Smirnov, “Electron Bernstein wave emission from an overdense reversed field pinch plasma,” Phys. Plasmas 9(3), 752–755 (2002).
[CrossRef]

Geist, T.

H. J. Hartfuss, T. Geist, and M. Hirsch, “Heterodyne methods in millimetre wave plasma diagnostics with applications to ECE, interferometry and reflectometry,” Plasma Phys. Contr. Fusion 39(11), 1693–1769 (1997).
[CrossRef]

Hartfuss, H. J.

H. J. Hartfuss, T. Geist, and M. Hirsch, “Heterodyne methods in millimetre wave plasma diagnostics with applications to ECE, interferometry and reflectometry,” Plasma Phys. Contr. Fusion 39(11), 1693–1769 (1997).
[CrossRef]

Harvey, R. W.

P. K. Chattopadhyay, J. K. Anderson, T. M. Biewer, D. Craig, C. B. Forest, R. W. Harvey, and A. P. Smirnov, “Electron Bernstein wave emission from an overdense reversed field pinch plasma,” Phys. Plasmas 9(3), 752–755 (2002).
[CrossRef]

Hirsch, M.

H. J. Hartfuss, T. Geist, and M. Hirsch, “Heterodyne methods in millimetre wave plasma diagnostics with applications to ECE, interferometry and reflectometry,” Plasma Phys. Contr. Fusion 39(11), 1693–1769 (1997).
[CrossRef]

Luhnann, N. C.

N. C. Luhnann, H. Bindslev, H. Park, J. Sanchez, G. Taylor, and C. X. Yu, “Microwave diagnostics,” Fusion Sci. Technol. 53, 335–396 (2008).

Luxon, J. L.

J. L. Luxon, “A design retrospective of the DIII-D tokamak,” Nucl. Fusion 42(5), 614–633 (2002).
[CrossRef]

Park, H.

N. C. Luhnann, H. Bindslev, H. Park, J. Sanchez, G. Taylor, and C. X. Yu, “Microwave diagnostics,” Fusion Sci. Technol. 53, 335–396 (2008).

Salehi, M.

N. Behdad, M. Al-Joumayly, and M. Salehi, “A low-profile third-order bandpass frequency selective surface,” IEEE Trans. Antenn. Propag. 57(2), 460–466 (2009).
[CrossRef]

Sanchez, J.

N. C. Luhnann, H. Bindslev, H. Park, J. Sanchez, G. Taylor, and C. X. Yu, “Microwave diagnostics,” Fusion Sci. Technol. 53, 335–396 (2008).

Smirnov, A. P.

P. K. Chattopadhyay, J. K. Anderson, T. M. Biewer, D. Craig, C. B. Forest, R. W. Harvey, and A. P. Smirnov, “Electron Bernstein wave emission from an overdense reversed field pinch plasma,” Phys. Plasmas 9(3), 752–755 (2002).
[CrossRef]

Taylor, G.

N. C. Luhnann, H. Bindslev, H. Park, J. Sanchez, G. Taylor, and C. X. Yu, “Microwave diagnostics,” Fusion Sci. Technol. 53, 335–396 (2008).

Yu, C. X.

N. C. Luhnann, H. Bindslev, H. Park, J. Sanchez, G. Taylor, and C. X. Yu, “Microwave diagnostics,” Fusion Sci. Technol. 53, 335–396 (2008).

Fusion Sci. Technol. (1)

N. C. Luhnann, H. Bindslev, H. Park, J. Sanchez, G. Taylor, and C. X. Yu, “Microwave diagnostics,” Fusion Sci. Technol. 53, 335–396 (2008).

IEEE Trans. Antenn. Propag. (3)

M. Al-Joumayly and N. Behdad, “Wideband planar microwave lenses using sub-wavelength spatial phase shifters,” IEEE Trans. Antenn. Propag. 59(12), 4542–4552 (2011).
[CrossRef]

M. Al-Joumayly and N. Behdad, “A generalized method for synthesizing low-profile, band-pass frequency selective surfaces with non-resonant constituting elements,” IEEE Trans. Antenn. Propag. 58(12), 4033–4041 (2010).
[CrossRef]

N. Behdad, M. Al-Joumayly, and M. Salehi, “A low-profile third-order bandpass frequency selective surface,” IEEE Trans. Antenn. Propag. 57(2), 460–466 (2009).
[CrossRef]

Nucl. Fusion (1)

J. L. Luxon, “A design retrospective of the DIII-D tokamak,” Nucl. Fusion 42(5), 614–633 (2002).
[CrossRef]

Phys. Plasmas (1)

P. K. Chattopadhyay, J. K. Anderson, T. M. Biewer, D. Craig, C. B. Forest, R. W. Harvey, and A. P. Smirnov, “Electron Bernstein wave emission from an overdense reversed field pinch plasma,” Phys. Plasmas 9(3), 752–755 (2002).
[CrossRef]

Plasma Phys. Contr. Fusion (1)

H. J. Hartfuss, T. Geist, and M. Hirsch, “Heterodyne methods in millimetre wave plasma diagnostics with applications to ECE, interferometry and reflectometry,” Plasma Phys. Contr. Fusion 39(11), 1693–1769 (1997).
[CrossRef]

Other (3)

I. Hutchinson, Principles of Plasma Diagnostics (Cambridge Univ. Press, 2002).

J. Wesson, Tokamaks (Oxford University Press, 1987).

R. F. Ellis, M. E. Austin, and D. Taussig, “New high spatial resolution optics for the DIII-D ECE radiometer,” in Proc. Of the 14th Joint Workshop on Electron Cyclotron Emission and Electron Cyclotron Resonance Heating (EC 14), Santorini, GR, May 9–12, 2006.

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Figures (5)

Fig. 1
Fig. 1

Topology of a third-order bandpass MEFSS. (a) Exploded view of overall lens design. (b) Top views of a unit cell capacitive and inductive layers. (c) Equivalent circuit model valid for normal incidence.

Fig. 2
Fig. 2

The phase delay profile of a typical glass lens is a smooth function of radial distance (a). This is lost in discretized lens systems (b) but regained in the limit of small aperture features (c).

Fig. 3
Fig. 3

Parameter sweeps highlighting the behavior of a 3rd order MEFSS when varying (a) g1 (b) g2 or (c) w, all the rest remaining fixed.

Fig. 4
Fig. 4

3D iso-phase contour plot over g1, g2, and w for f = 10GHz. Note that for every desired ϕ, parameter solution g1, g2, w is not unique for a given f. However, constraining ϕ at multiple f reduces the choices of g1, g2, and w.

Fig. 5
Fig. 5

Phase response of the numerically optimized zones (solid lines) and their corresponding desired phase responses (dashed lines).

Tables (1)

Tables Icon

Table 1 Optimal Parameter Dimensions by Zone

Equations (11)

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C 1 = ε 0 ε eff 2D π ln[ 1 sin( π g 1 2D ) ]
L= μ 0 μ eff D 2π ln[ 1 sin( πw 2D ) ]
T= 2 A+ B Z 0 +C Z 0 +D
( A B C D )=( 1 0 jω C 1 1 )( cosβh jZsinβh j Z sinβh cosβh )( 1 0 1 jωL 1 )× ( cosβh jZsinβh j Z sinβh cosβh )( 1 0 jω C 2 1 )( cosβh jZsinβh j Z sinβh cosβh )× ( 1 0 1 jωL 1 )( cosβh jZsinβh j Z sinβh cosβh )( 1 0 jω C 1 1 )
A=2 [ ( Δ ϕ max k +l ) 2 l 2 ] 1 2
ϕ 0 =k( ( A 2 ) 2 + l 2 l )+ ϕ
ϕ m =k( ( A 2 ) 2 + l 2 l )+ ϕ k( r m 2 + l 2 l )
Δϕ= ϕ 0 ϕ 6 =k( r 6 2 + l 2 l )
Δϕ=218°
Δϕ=81°
G i = f=8,8.5,...,12GHz [ ϕ f ( w, g 1 , g 2 ) ϕ f (i) ] 2

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