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An analytical investigation of near-field plates
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Metamaterials 4 (2010) 104–111
An analytical investigation of near-field plates Mohammadreza F. Imani, Anthony Grbic ∗ Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122, United States Received 23 December 2009; received in revised form 10 March 2010; accepted 10 March 2010 Available online 19 March 2010
Abstract This paper describes an analytical approach to modeling near-field plates, which are non-periodic grating-like structures that can focus electromagnetic waves to subwavelength dimensions. The analysis provides additional insight into the operation and design of such plates that focus cylindrical waves to subwavelength resolutions. Explicit expressions for the current density induced on the plate and its impedance profile are derived. The analytical expressions are validated numerically. © 2010 Elsevier B.V. All rights reserved. PACS: 41.20.Jb; 42.25.Fx; 42.79.DjNear-field; Evanescent waves; Focusing; Diffraction limit; Metamaterials
1. Introduction In the past few years, there has been considerable interest in near-field superlenses that can focus electromagnetic waves to subwavelength resolutions. Following John Pendry’s work on the “perfect lens” [1], various superlenses consisting of slabs with negative material parameters have been developed and experimentally verified at microwave, infrared and optical frequencies [2–5]. More recently, an alternative approach to subwavelength focusing using patterned, grating-like surfaces was introduced and developed [6,7]. The proposed surfaces have been referred to as near-field plates. A near-field plate is a subwavelengthstructured, planar device that can focus electromagnetic radiation from a source to subwavelength resolutions. The plate’s textured surface (modulated reactance) sets up a highly oscillatory electromagnetic field that converges to a prescribed focal plane in the plate’s near field.
∗
Corresponding author. Fax: +1 7346472106. E-mail address: [email protected] (A. Grbic).
1873-1988/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.metmat.2010.03.002
At microwave frequencies, these plates can be realized as non-periodic arrays of reactive elements, while at optical frequencies, nano-fabricated plates consisting of plasmonic and dielectric materials are envisioned [8–10]. Recent experiments at microwave frequencies have verified a near-field plate’s ability to focus electromagnetic waves to subwavelength resolutions [11]. In Ref. [11], an experimental nearfield plate consisting of an array of interdigitated capacitors was shown to focus 1.027 GHz microwave radiation emanating from an S-polarized cylindrical source to a focus with full width at half maximum, FWHM = λ◦ /18, where λ◦ is the free space wavelength. Recently, a similar approach to subwavelength focusing has been pursued using holography-inspired screens and spatially beam-shifted transmission screens [12,13]. Previous works have characterized near-field plates in simulation and in experiment [7,11,14–16], while this paper characterizes them analytically. The analytical investigation that follows provides added insight into the operation and design of near-field plates over the numerical design approach described earlier. Closed-form
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Fig. 1. A schematic showing the near-field plate (z = 0) and focal plane (z = L). A near-field plate is a non-periodically patterned, planar structure that can focus electromagnetic radiation to lines or spots of arbitrary subwavelength dimension.
expressions for the currents excited on the near-field plates, as well as expressions for the plates’ impedance profile are derived. In the analytical treatment, the plates are assumed to be infinite in width. The current density on the plate is found in the spectral domain and then inverse Fourier transformed to obtain its spatial dependence, as well as the plate’s impedance profile. The analytically derived expressions are compared to those computed numerically for electrically wide plates. 2. Near-field plates for S-polarized radiation Fig. 1 depicts the near-field plate configuration considered in this paper. The plate is located along the z = 0 plane (sheet plane) and the focal plane is assumed to be the z = L plane, where L is the focal length. Furthermore, the electromagnetic fields are assumed to be polarized and invariant in the x direction (s-polarized). In the discussion, the electric field along the focal plane will be referred to as the focal pattern, while the electric field along the surface of the plate will be referred to as Etotal . We will investigate a near-field plate that produces a subwavelength focal pattern of the following form, z = L) = jM|Emax |e−q◦ L q◦ Lsinc(q◦ y)ˆx E(y,
(1)
where L = λ◦ /16, q◦ = 10k◦ and k◦ is the free space wavenumber at the operating frequency f◦ = 1 GHz. Also M = 6 is a real constant known as the amplification factor, which expresses focal pattern in terms of
Fig. 2. Spectral and spatial representation of the focal pattern given by Eq. (1). (a) Spectral representation. (b) Spatial representation.
the maximum of the incident wave at the surface of the plate: |Emax | = |Einc (y = 0)|. These values for L, q◦ , f◦ , and M are assumed throughout the paper. As explained in Ref. [7], the imaginary number j ensures that the plate’s surface impedance is primarily reactive. A near-field plate that produces a sinc focal pattern is chosen since such a plate was numerically and experimentally investigated in Refs. [11,14]. For the assumed parameters, the sinc focal pattern and its spectrum are shown in Fig. 2. The constant q◦ represents the maximum transverse wavenumber ky = q◦ that contributes to the focal pattern, and as a result determines its nullto-null beamwidth (BWNN ): BWNN = 2π/q◦ = λ◦ /10. Finally, the plate is assumed to be electrically thin in the z direction (thickness λ◦ ) so that current densities in the direction normal to the plate can be neglected, and the plate can be modeled as a sheet with surface impedance ηsheet (y) [17]. The first step to designing a near-field plate involves finding the electric field, Etotal (y), that is needed at the surface of the plate to produce the focal pattern [7]. By back-propagating the focal pattern given in Eq. (1), Etotal (y) can be easily obtained through an inverse
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Fourier transform (IFT): total (ky )} total (y) = F−1 {E E = jM|Emax |F−1 {πLe(−q◦ +jkz )L (ky , q◦ )}ˆx (2) where the symbol F−1 denotes the IFT with respect to transverse wavenumber ky ; (ky , q◦ ) is a square pulse in the spectral domain with amplitude equal to one extending from ky = −q◦ to ky = q◦ ; and kz is the wavenumber in the z direction defined as: ⎧ ⎨ k◦ 2 − k y 2 : ky 2 < k◦2 kz = (3) ⎩ −j ky 2 − k◦ 2 : ky 2 > k◦2 . Since the configuration is two-dimensional and the electric fields and current densities only have x components, the scalar form of these quantities with a subscript x will be used. Figs. 3 and 4 show the spectral Etotal (ky ) and spatial Etotal (y) computed in this manner. It should be noted that the numerically computed values for Etotal (y) that are used here are different from those in Refs. [7,11,14], where an approximate analytical expression for Etotal (y) was used. The second design step involves finding the current density Jx (y) on the plate that produces Etotal (y). The current density can be computed by solving the following integral equation [7]: k◦ η W/2 (2) Jx (y )H0 (k◦ |y − y |)dy Einc (y) − 4 −W/2 = Etotal (y)
(4)
which represents the boundary condition at the plate’s surface. The function Einc (y) denotes the electric field (2) incident on the plate from an external source, H0 is the zeroth order Hankel function of the second kind, η = 120π is the free space wave impedance, and W is the width of the near-field plate. Substituting Eq. (2) into Eq. (4) yields an integral equation which can be solved to find Jx (y). Once Jx (y) is known, ηsheet (y) can be computed by simply taking the ratio of Etotal (y) to Jx (y): ηsheet (y) =
Etotal (y) . Jx (y)
Fig. 3. Spectral representation of the electric field at the surface of the plate. The solid line represents Etotal (ky ) given by Eq. (2). The dashed line is the spectral domain representation of the approximate Etotal (y) given by Eq. (16). The ripples are due to the truncation of the numerical Fourier transform (Gibb’s phenomena). (a) Real part (b) Imaginary part.
(5)
In order to find Jx (y) and ηsheet (y), Eq. (4) was solved numerically in Refs. [7,11,14]. In this paper, we solve Eq. (4) analytically, to obtain approximate closed-form expressions for both Jx (y) and ηsheet (y). In the analytical
treatment, we assume that the plate is infinitely wide, in order to simplify the integral on the left hand side of Eq. (4) to a convolution: k◦ η ∞ (2) Einc (y) − Jx (y )H0 (k◦ |y − y |)dy 4 −∞ = jM|Emax |F−1 {πLe(−q◦ +jkz )L (ky , q◦ )}.
(6)
Since the convolution becomes a multiplication in the spectral domain, an expression for Jx (ky ) can be found in the spectral domain and then inverse Fourier transformed to obtain approximate expressions for Jx (y) and ηsheet (y). The Fourier transform properties of Hankel and Bessel functions as well as their integral representations [18] are utilized in deriving the approximate expressions. Although we consider a near-field plate under a specific excitation (i.e. cylindrical wave) the analytical approach presented here can be applied to near-field
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formed exactly to obtain its spectral domain counterpart: −
k◦ η Jx (ky ) k◦ ηI e(−jkz d) − 2 kz 2 kz = jπM|Emax |Le(−q◦ +jkz )L (ky , q◦ )
(8)
By rearranging Eq. (8), the following expression for the spectral domain Jx (ky ) can be obtained: Jx (ky ) = Jxinc (ky ) + Jxfoc (ky ) = −Ie(−jkz d) −
j2πLM|Emax |e(−q◦ +jkz )L kz (ky , q◦ ) (9) k◦ η
Eq. (9) reveals that the current density on the nearfield plate consists of two parts. The first term, referred to as Jxinc (ky ), cancels the incident cylindrical wave in the region z > 0, while the second term, referred to as Jxfoc (ky ), forms the desired focal pattern. The first term, Jxinc , can be inverse Fourier transformed directly: jk◦ dI (2) H1 (k◦ d 2 + y2 ) Jxinc (y) = (10) 2 d 2 + y2
Fig. 4. Spatial representation of the electric field at the surface of the plate. The solid line represents Etotal (y) given by Eq. (2) while the dashes and dots represent the analytical expression given by Eq. (16). (a) Real part. (b) Imaginary part.
plates designed for arbitrary incident waves and desired focal patterns, provided that the Fourier transformation of the incident and desired focal patterns are known. 3. Focusing with infinitely wide near-field plates: analytical formulation A near-field plate is considered that can focus a cylindrical wave to a subwavelength focal pattern given by Eq. (1) [11]. The source of the cylindrical wave will be an x-directed electric line source with current I located at (y = 0, z = −d). Throughout this section, it will be assumed that I = 1 mA and d = λ◦ /16. The electric field produced by the line source at the plate’s surface is [19]: k◦ ηI (2) H0 (k◦ y2 + d 2 ) (7) Einc (y) = − 4 Substituting Einc (y) into the integral equation Eq. (6) yields an integral equation that can be Fourier trans-
The real and imaginary parts of the IFT of Jxfoc (ky ) are found separately. The IFT of the imaginary part of foc Jx (ky ) is found exactly, by transforming the propagating spectrum (|ky | < k◦ ) of Jxfoc (ky ), and retaining its imaginary part: πML|Emax |k◦ e−q◦ L Im{Jxfoc (y)} = − . 2η (11) y 2 − L2 2 2 2 2 J0 (k◦ y + L ) + 2 J2 (k◦ y + L ) y + L2 where J0 and J2 denote Bessel functions of the first kind. The real part of Jxfoc (y) can be derived approximately. Given that the second term of Eq. (9) increases exponentially as a function of ky for |ky | k◦ , we apply the following quasi-static approximation, k◦2 − ky2 ≈ −j|ky |. (12) Under this assumption, the real part of Jxfoc (ky ) simplifies to: Re{Jxfoc (ky )} ≈
−2πLM|Emax |e(−q◦ +|ky |)L |ky |(ky , q◦ ) . k◦ η
(13)
The quasi-static approximation is valid when q◦ k◦ , in other words for near-field plates that focus electromagnetic waves to extremely subwavelength resolutions. It should be noted that the growing nature for
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4. Focusing with finite near-field plates: Re{Jxfoc (ky )} is expected, since it results from restoranalytical vs. numerical ing/amplifying the evanescent electric field from the focal plane (z = L) to the sheet plane (z = 0) (see In the previous section, we derived the current density Fig. 3(b)). Inverse Fourier transforming Eq. (13), leads and impedance for an infinitely wide near-field plate that to the following approximate expression for the real part foc focuses a cylindrical of Jx (y),
2ML|Emax | L2 − y 2 2Ly foc Re{Jx (y)} ≈ − cos(q◦ y) yq◦ − 2 sin(q◦ y) (14) Lq◦ − 2 k◦ η(L2 + y2 ) L + y2 L + y2 By combining Eqs. (11) and (14), the following expression for Jx (y) is obtained: jk◦ dI jπM|Emax |Lk◦ e−q◦ L y 2 − L2 (2) 2 2 2 2 2 2 . J0 (k◦ y + L ) + 2 H1 (k◦ d +y )− J2 (k◦ y + L ) Jx (y)= 2η y + L2 2 d 2 +y2 (15)
2M|Emax |L L2 − y 2 2Ly − Lq◦ − 2 cos(q◦ y) + yq◦ − 2 sin(q◦ y) . k◦ η(L2 + y2 ) L + y2 L + y2 The accuracy of Eq. (15) can be verified by comparing its numerical Fourier transform, to the exact expression wave to a subwavelength focus. In this section, we comfor Jx (ky ) given by Eq. (9). The two plots are compared in pare the analytical results to numerical results for a Fig. 5 and show close agreement. The ripples at the sharp finite plate that is W = 10λ◦ wide (electrically wide). transitions are due to the truncation of the numerical The Method of Moments (MoM) is used to simulate Fourier transform (Gibb’s phenomena). In order to have the finite-width plate. To be precise, the point matching a completely analytical expression for ηsheet , we must method is employed with a discretization of λ◦ /40. The find a closed-form expression for Etotal (y). In Ref. [7], current densities obtained analytically and numerically an analytical expression for Etotal (y) was derived which are depicted in Fig. 6. They match up rather well, we will use here with a slight modification:
sin(q◦ y) Etotal (y) = jM|Emax |L L cos(q◦Ly)+y +2πq◦ e(−q◦ L) (cos(k◦ L) − 1)sinc(q◦ y) 2 +y2 (16) 2 −q◦ L max | 2 + y2 ). J (k L − πL e √ k2◦ M|E 1 ◦ 2 2
L +y
The first term is the same as the formula used in Ref. [7], and can be derived under the quasi-static approximation Eq. (12). The second term is a modification factor which is included to match the approximate expression for Etotal (ky ) with its exact value at ky = 0. The third term is the real part of Etotal (y) resulting from the propagating spectrum (|ky | < k◦ ). The analytical expressions for Etotal (ky ) and Etotal (y) are compared to the numerically computed ones in Figs. 3 and 4, and show close agreement. Now that closed-form expressions for the current density and Etotal on the infinitely wide plate are known, the plate’s sheet impedance can be found by substituting Eqs. (15) and (16) into Eq. (5). Figs. 4 and 6 reveal the fact that the imaginary part of the current density Jx (y) and the real part of Etotal (y) are much smaller than their real and imaginary counterparts, respectively. This fact explains why only the reactive part of sheet impedance Im{ηsheet (y)} is typically used in the implementation of practical near-field plates [11].
other than the slight differences near the edges of the finite plate that result from edge diffraction. It should be noted that the analytical expression for Re{Jx (y)} has been derived based on the quasi-static assumption (Eq. 12), which is valid for q◦ k◦ . For larger focal spots, or equivalently smaller values of q◦ , it becomes less valid. As a rule of thumb, the quasi-static assumption holds for q◦ > 3k◦ . The focal patterns, produced by both the analytical and numerical current densities, were computed using the two-dimensional free space Green’s function [19], Ex (y, z = L)
k◦ ηI (2) H0 (k◦ y2 + (d + L)2 ) =− 4 k◦ η W/2 (2) − Jx (y )H0 (k◦ (y − y )2 + L2 )dy 4 −W/2 (17)
M.F. Imani, A. Grbic / Metamaterials 4 (2010) 104–111
Fig. 5. The (a) real and (b) imaginary parts of the spectral representation of the current density, Jx (ky ). The solid line represents Eq. (9) and the dashed line represents the numerical Fourier transform of Eq. (15). (a) Real part. (b) Imaginary part.
and are plotted in Fig. 7. The focal patterns produced by the analytically derived current density show close agreement with those computed using the MoM. Furthermore, 2D plots of the vertical electric field computed using the analytically derived and numerically computed current densities (Jx ) are shown in Fig. 8. The incident electric field excites a highly oscillatory field on the surface of the plate (z = 0). This highly oscillatory field forms a subwavelength focus along the focal plane and diverges rapidly beyond the focal plane (z > L). Reflection from the plate is evident in the region z < 0. Finally, since one of the main goals was to find an expression for the plate’s sheet impedance, the reactive sheet impedance resulting from both methods is plotted in Fig. 9. Impedance values for a 4λ◦ plate are shown since it is simpler to implement than a plate that is 10λ◦ wide. Given that only the reactive part of the sheet impedance was used in the implementation of practical near-field plates [11,14], the reactive part is only shown. The impedances show good agreement over the majority of the plate with the exception of the plate’s edges.
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Fig. 6. The (a) real and (b) imaginary parts of the current density Jx (y). The solid line represents the analytically derived current density given by Eq. (15). The dots and dashes represent the numerically (MoM) computed current density. The current density is only shown on half of the plate since it is symmetric about y = 0. (a) Real part. (b) Imaginary part.
Fig. 7. Close-up view of the magnitude of electric field at the focal plane. The solid line represents the electrical field at the focal plane produced by the analytically derived current density given by Eq. (15). The dashed line represents the electric field at the focal plane produced by the numerically computed (MoM) current density. The dotted line represents the ideal focal pattern given by Eq. (1).
5. Conclusion In this paper, we have analytically modeled and characterized near-field plates. Expressions for the induced
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current density and surface impedance of infinitely wide plates were found. Further, it was shown that the expressions obtained for infinitely wide plates can be used to approximate the characteristics of finite-width plates. In fact, we have shown that the current density on a nearfield plate can be split into two parts. One part cancels the incident wave on the focus-side of the plate, while the other part produces the desired subwavelength focal pattern. The paper also mathematically demonstrated why only the reactive part of a near-plate’s sheet impedance can be used in their design. Acknowledgments The authors would like to acknowledge discussions with T.B.A. Senior. This work was supported by a NSF Faculty Early Career Development Award (ECCS - 0747623), a Presidential Early Career Award for Scientist and Engineers: (FA9550-09-1-0696), and through the Multidisciplinary University Research Initiative Program (FA9550-06-01-0279). References
Fig. 8. 2D plots of the vertical electric field (in dB) surrounding the 10λ◦ near-field plate. The electric field is computed for the current densities obtained (a) numerically (MoM) and (b) analytically. (a) MoM. (b) Analytical.
Fig. 9. Reactive sheet impedance, Im{ηsheet (y)}. The solid line represents the analytically derived Im{ηsheet (y)} using Eqs. (15) and (16) in Eq. (5). The dots represents numerically obtained Im{ηsheet (y)} using MoM. The surface impedance is only shown for half of the plate since it is symmetric about y = 0.
[1] J.B. Pendry, Negative refraction makes a perfect lens, Physical Review Letters 85 (2000) 3966–3969. [2] A. Grbic, G.V. Eleftheriades, Overcoming the diffraction limit with a planar left-handed transmission-line lens, Physical Review Letters 92 (2004) 117403. [3] N. Fang, H. Lee, C. Sun, X. Zhang, Sub-diffraction-limited optical imaging with a silver superlens, Science 308 (2005) 534– 537. [4] D. Melville, R. Blaikie, Super-resolution imaging through a planar silver layer, Optics Express 13 (2005) 2127–2134. [5] F. Mesa, M.J. Freire, R. Marques, J.D. Baena, Three- dimensional superresolution in metamaterial slab lenses: experiment and theory, Physical Review B 72 (2005) 235117. [6] R. Merlin, Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing, Science 317 (2007) 927–929. [7] A. Grbic, R. Merlin, Near-field focusing plates and their design, IEEE Transactions on Antennas and Propagation 56 (2008) 3159–3165. [8] R. Gordon, Proposal for superfocusing at visible wavelengths using radiationless interference of a plasmonic array, Physical Review Letters 102 (2009) 207402. [9] I. Tsukerman, Superfocusing by nanoshells, Optics Letters 34 (2009) 1057–1059. [10] L. Scorrano, F. Bilotti, L. Vegni, Design of a meta-screen for near-zone field focalization at optical frequencies, Microwave and Optical Technology Letters 51 (2009) 2718–2721. [11] A. Grbic, L. Jiang, R. Merlin, Near-field plates: subdiffraction focusing with patterned surfaces, Science 320 (2008) 511– 513. [12] G.V. Eleftheriades, A.M.H. Wong, Holography-inspired screens for subwavelength focusing in the near field, IEEE Microwave and Wireless Component Letters 18 (2008) 236–238.
M.F. Imani, A. Grbic / Metamaterials 4 (2010) 104–111 [13] L. Markley, A.M.H. Wong, Y. Wang, G.V. Eleftheriades, Spatially shifted beam approach to subwavelength focusing, Physical Review Letters 101 (2008) 113901. [14] A. Grbic, L. Jiang, R. Merlin, Near-field focusing plates, in: IEEE MTT-S International Microwave Syposium WE1A-02 Atlanta, GA, 2008, pp. 209–212. [15] M.F. Imani, A. Grbic, Near-field focusing with a corrugated surface, IEEE Antenna and Wirless Propagation Letters 8 (2009) 421–424. [16] M.F. Imani, A. Grbic, Tailoring near-field patterns with concentrically corrugated plates, Applied Physics Letters 95 (2009) 111107.
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[17] T.B.A. Senior, J.L. Volakis, Approximate boundary conditions in electromagnetics, The Institution of Electrical Engineers, 1995. [18] M. Abramovitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, 1970. [19] C.A. Balanis, Advanced Engineering Electromagnetics, JohnWiley and Sons, 1989.
Metamaterials 4 (2010) 104–111
An analytical investigation of near-field plates Mohammadreza F. Imani, Anthony Grbic ∗ Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122, United States Received 23 December 2009; received in revised form 10 March 2010; accepted 10 March 2010 Available online 19 March 2010
Abstract This paper describes an analytical approach to modeling near-field plates, which are non-periodic grating-like structures that can focus electromagnetic waves to subwavelength dimensions. The analysis provides additional insight into the operation and design of such plates that focus cylindrical waves to subwavelength resolutions. Explicit expressions for the current density induced on the plate and its impedance profile are derived. The analytical expressions are validated numerically. © 2010 Elsevier B.V. All rights reserved. PACS: 41.20.Jb; 42.25.Fx; 42.79.DjNear-field; Evanescent waves; Focusing; Diffraction limit; Metamaterials
1. Introduction In the past few years, there has been considerable interest in near-field superlenses that can focus electromagnetic waves to subwavelength resolutions. Following John Pendry’s work on the “perfect lens” [1], various superlenses consisting of slabs with negative material parameters have been developed and experimentally verified at microwave, infrared and optical frequencies [2–5]. More recently, an alternative approach to subwavelength focusing using patterned, grating-like surfaces was introduced and developed [6,7]. The proposed surfaces have been referred to as near-field plates. A near-field plate is a subwavelengthstructured, planar device that can focus electromagnetic radiation from a source to subwavelength resolutions. The plate’s textured surface (modulated reactance) sets up a highly oscillatory electromagnetic field that converges to a prescribed focal plane in the plate’s near field.
∗
Corresponding author. Fax: +1 7346472106. E-mail address: [email protected] (A. Grbic).
1873-1988/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.metmat.2010.03.002
At microwave frequencies, these plates can be realized as non-periodic arrays of reactive elements, while at optical frequencies, nano-fabricated plates consisting of plasmonic and dielectric materials are envisioned [8–10]. Recent experiments at microwave frequencies have verified a near-field plate’s ability to focus electromagnetic waves to subwavelength resolutions [11]. In Ref. [11], an experimental nearfield plate consisting of an array of interdigitated capacitors was shown to focus 1.027 GHz microwave radiation emanating from an S-polarized cylindrical source to a focus with full width at half maximum, FWHM = λ◦ /18, where λ◦ is the free space wavelength. Recently, a similar approach to subwavelength focusing has been pursued using holography-inspired screens and spatially beam-shifted transmission screens [12,13]. Previous works have characterized near-field plates in simulation and in experiment [7,11,14–16], while this paper characterizes them analytically. The analytical investigation that follows provides added insight into the operation and design of near-field plates over the numerical design approach described earlier. Closed-form
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Fig. 1. A schematic showing the near-field plate (z = 0) and focal plane (z = L). A near-field plate is a non-periodically patterned, planar structure that can focus electromagnetic radiation to lines or spots of arbitrary subwavelength dimension.
expressions for the currents excited on the near-field plates, as well as expressions for the plates’ impedance profile are derived. In the analytical treatment, the plates are assumed to be infinite in width. The current density on the plate is found in the spectral domain and then inverse Fourier transformed to obtain its spatial dependence, as well as the plate’s impedance profile. The analytically derived expressions are compared to those computed numerically for electrically wide plates. 2. Near-field plates for S-polarized radiation Fig. 1 depicts the near-field plate configuration considered in this paper. The plate is located along the z = 0 plane (sheet plane) and the focal plane is assumed to be the z = L plane, where L is the focal length. Furthermore, the electromagnetic fields are assumed to be polarized and invariant in the x direction (s-polarized). In the discussion, the electric field along the focal plane will be referred to as the focal pattern, while the electric field along the surface of the plate will be referred to as Etotal . We will investigate a near-field plate that produces a subwavelength focal pattern of the following form, z = L) = jM|Emax |e−q◦ L q◦ Lsinc(q◦ y)ˆx E(y,
(1)
where L = λ◦ /16, q◦ = 10k◦ and k◦ is the free space wavenumber at the operating frequency f◦ = 1 GHz. Also M = 6 is a real constant known as the amplification factor, which expresses focal pattern in terms of
Fig. 2. Spectral and spatial representation of the focal pattern given by Eq. (1). (a) Spectral representation. (b) Spatial representation.
the maximum of the incident wave at the surface of the plate: |Emax | = |Einc (y = 0)|. These values for L, q◦ , f◦ , and M are assumed throughout the paper. As explained in Ref. [7], the imaginary number j ensures that the plate’s surface impedance is primarily reactive. A near-field plate that produces a sinc focal pattern is chosen since such a plate was numerically and experimentally investigated in Refs. [11,14]. For the assumed parameters, the sinc focal pattern and its spectrum are shown in Fig. 2. The constant q◦ represents the maximum transverse wavenumber ky = q◦ that contributes to the focal pattern, and as a result determines its nullto-null beamwidth (BWNN ): BWNN = 2π/q◦ = λ◦ /10. Finally, the plate is assumed to be electrically thin in the z direction (thickness λ◦ ) so that current densities in the direction normal to the plate can be neglected, and the plate can be modeled as a sheet with surface impedance ηsheet (y) [17]. The first step to designing a near-field plate involves finding the electric field, Etotal (y), that is needed at the surface of the plate to produce the focal pattern [7]. By back-propagating the focal pattern given in Eq. (1), Etotal (y) can be easily obtained through an inverse
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Fourier transform (IFT): total (ky )} total (y) = F−1 {E E = jM|Emax |F−1 {πLe(−q◦ +jkz )L (ky , q◦ )}ˆx (2) where the symbol F−1 denotes the IFT with respect to transverse wavenumber ky ; (ky , q◦ ) is a square pulse in the spectral domain with amplitude equal to one extending from ky = −q◦ to ky = q◦ ; and kz is the wavenumber in the z direction defined as: ⎧ ⎨ k◦ 2 − k y 2 : ky 2 < k◦2 kz = (3) ⎩ −j ky 2 − k◦ 2 : ky 2 > k◦2 . Since the configuration is two-dimensional and the electric fields and current densities only have x components, the scalar form of these quantities with a subscript x will be used. Figs. 3 and 4 show the spectral Etotal (ky ) and spatial Etotal (y) computed in this manner. It should be noted that the numerically computed values for Etotal (y) that are used here are different from those in Refs. [7,11,14], where an approximate analytical expression for Etotal (y) was used. The second design step involves finding the current density Jx (y) on the plate that produces Etotal (y). The current density can be computed by solving the following integral equation [7]: k◦ η W/2 (2) Jx (y )H0 (k◦ |y − y |)dy Einc (y) − 4 −W/2 = Etotal (y)
(4)
which represents the boundary condition at the plate’s surface. The function Einc (y) denotes the electric field (2) incident on the plate from an external source, H0 is the zeroth order Hankel function of the second kind, η = 120π is the free space wave impedance, and W is the width of the near-field plate. Substituting Eq. (2) into Eq. (4) yields an integral equation which can be solved to find Jx (y). Once Jx (y) is known, ηsheet (y) can be computed by simply taking the ratio of Etotal (y) to Jx (y): ηsheet (y) =
Etotal (y) . Jx (y)
Fig. 3. Spectral representation of the electric field at the surface of the plate. The solid line represents Etotal (ky ) given by Eq. (2). The dashed line is the spectral domain representation of the approximate Etotal (y) given by Eq. (16). The ripples are due to the truncation of the numerical Fourier transform (Gibb’s phenomena). (a) Real part (b) Imaginary part.
(5)
In order to find Jx (y) and ηsheet (y), Eq. (4) was solved numerically in Refs. [7,11,14]. In this paper, we solve Eq. (4) analytically, to obtain approximate closed-form expressions for both Jx (y) and ηsheet (y). In the analytical
treatment, we assume that the plate is infinitely wide, in order to simplify the integral on the left hand side of Eq. (4) to a convolution: k◦ η ∞ (2) Einc (y) − Jx (y )H0 (k◦ |y − y |)dy 4 −∞ = jM|Emax |F−1 {πLe(−q◦ +jkz )L (ky , q◦ )}.
(6)
Since the convolution becomes a multiplication in the spectral domain, an expression for Jx (ky ) can be found in the spectral domain and then inverse Fourier transformed to obtain approximate expressions for Jx (y) and ηsheet (y). The Fourier transform properties of Hankel and Bessel functions as well as their integral representations [18] are utilized in deriving the approximate expressions. Although we consider a near-field plate under a specific excitation (i.e. cylindrical wave) the analytical approach presented here can be applied to near-field
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formed exactly to obtain its spectral domain counterpart: −
k◦ η Jx (ky ) k◦ ηI e(−jkz d) − 2 kz 2 kz = jπM|Emax |Le(−q◦ +jkz )L (ky , q◦ )
(8)
By rearranging Eq. (8), the following expression for the spectral domain Jx (ky ) can be obtained: Jx (ky ) = Jxinc (ky ) + Jxfoc (ky ) = −Ie(−jkz d) −
j2πLM|Emax |e(−q◦ +jkz )L kz (ky , q◦ ) (9) k◦ η
Eq. (9) reveals that the current density on the nearfield plate consists of two parts. The first term, referred to as Jxinc (ky ), cancels the incident cylindrical wave in the region z > 0, while the second term, referred to as Jxfoc (ky ), forms the desired focal pattern. The first term, Jxinc , can be inverse Fourier transformed directly: jk◦ dI (2) H1 (k◦ d 2 + y2 ) Jxinc (y) = (10) 2 d 2 + y2
Fig. 4. Spatial representation of the electric field at the surface of the plate. The solid line represents Etotal (y) given by Eq. (2) while the dashes and dots represent the analytical expression given by Eq. (16). (a) Real part. (b) Imaginary part.
plates designed for arbitrary incident waves and desired focal patterns, provided that the Fourier transformation of the incident and desired focal patterns are known. 3. Focusing with infinitely wide near-field plates: analytical formulation A near-field plate is considered that can focus a cylindrical wave to a subwavelength focal pattern given by Eq. (1) [11]. The source of the cylindrical wave will be an x-directed electric line source with current I located at (y = 0, z = −d). Throughout this section, it will be assumed that I = 1 mA and d = λ◦ /16. The electric field produced by the line source at the plate’s surface is [19]: k◦ ηI (2) H0 (k◦ y2 + d 2 ) (7) Einc (y) = − 4 Substituting Einc (y) into the integral equation Eq. (6) yields an integral equation that can be Fourier trans-
The real and imaginary parts of the IFT of Jxfoc (ky ) are found separately. The IFT of the imaginary part of foc Jx (ky ) is found exactly, by transforming the propagating spectrum (|ky | < k◦ ) of Jxfoc (ky ), and retaining its imaginary part: πML|Emax |k◦ e−q◦ L Im{Jxfoc (y)} = − . 2η (11) y 2 − L2 2 2 2 2 J0 (k◦ y + L ) + 2 J2 (k◦ y + L ) y + L2 where J0 and J2 denote Bessel functions of the first kind. The real part of Jxfoc (y) can be derived approximately. Given that the second term of Eq. (9) increases exponentially as a function of ky for |ky | k◦ , we apply the following quasi-static approximation, k◦2 − ky2 ≈ −j|ky |. (12) Under this assumption, the real part of Jxfoc (ky ) simplifies to: Re{Jxfoc (ky )} ≈
−2πLM|Emax |e(−q◦ +|ky |)L |ky |(ky , q◦ ) . k◦ η
(13)
The quasi-static approximation is valid when q◦ k◦ , in other words for near-field plates that focus electromagnetic waves to extremely subwavelength resolutions. It should be noted that the growing nature for
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4. Focusing with finite near-field plates: Re{Jxfoc (ky )} is expected, since it results from restoranalytical vs. numerical ing/amplifying the evanescent electric field from the focal plane (z = L) to the sheet plane (z = 0) (see In the previous section, we derived the current density Fig. 3(b)). Inverse Fourier transforming Eq. (13), leads and impedance for an infinitely wide near-field plate that to the following approximate expression for the real part foc focuses a cylindrical of Jx (y),
2ML|Emax | L2 − y 2 2Ly foc Re{Jx (y)} ≈ − cos(q◦ y) yq◦ − 2 sin(q◦ y) (14) Lq◦ − 2 k◦ η(L2 + y2 ) L + y2 L + y2 By combining Eqs. (11) and (14), the following expression for Jx (y) is obtained: jk◦ dI jπM|Emax |Lk◦ e−q◦ L y 2 − L2 (2) 2 2 2 2 2 2 . J0 (k◦ y + L ) + 2 H1 (k◦ d +y )− J2 (k◦ y + L ) Jx (y)= 2η y + L2 2 d 2 +y2 (15)
2M|Emax |L L2 − y 2 2Ly − Lq◦ − 2 cos(q◦ y) + yq◦ − 2 sin(q◦ y) . k◦ η(L2 + y2 ) L + y2 L + y2 The accuracy of Eq. (15) can be verified by comparing its numerical Fourier transform, to the exact expression wave to a subwavelength focus. In this section, we comfor Jx (ky ) given by Eq. (9). The two plots are compared in pare the analytical results to numerical results for a Fig. 5 and show close agreement. The ripples at the sharp finite plate that is W = 10λ◦ wide (electrically wide). transitions are due to the truncation of the numerical The Method of Moments (MoM) is used to simulate Fourier transform (Gibb’s phenomena). In order to have the finite-width plate. To be precise, the point matching a completely analytical expression for ηsheet , we must method is employed with a discretization of λ◦ /40. The find a closed-form expression for Etotal (y). In Ref. [7], current densities obtained analytically and numerically an analytical expression for Etotal (y) was derived which are depicted in Fig. 6. They match up rather well, we will use here with a slight modification:
sin(q◦ y) Etotal (y) = jM|Emax |L L cos(q◦Ly)+y +2πq◦ e(−q◦ L) (cos(k◦ L) − 1)sinc(q◦ y) 2 +y2 (16) 2 −q◦ L max | 2 + y2 ). J (k L − πL e √ k2◦ M|E 1 ◦ 2 2
L +y
The first term is the same as the formula used in Ref. [7], and can be derived under the quasi-static approximation Eq. (12). The second term is a modification factor which is included to match the approximate expression for Etotal (ky ) with its exact value at ky = 0. The third term is the real part of Etotal (y) resulting from the propagating spectrum (|ky | < k◦ ). The analytical expressions for Etotal (ky ) and Etotal (y) are compared to the numerically computed ones in Figs. 3 and 4, and show close agreement. Now that closed-form expressions for the current density and Etotal on the infinitely wide plate are known, the plate’s sheet impedance can be found by substituting Eqs. (15) and (16) into Eq. (5). Figs. 4 and 6 reveal the fact that the imaginary part of the current density Jx (y) and the real part of Etotal (y) are much smaller than their real and imaginary counterparts, respectively. This fact explains why only the reactive part of sheet impedance Im{ηsheet (y)} is typically used in the implementation of practical near-field plates [11].
other than the slight differences near the edges of the finite plate that result from edge diffraction. It should be noted that the analytical expression for Re{Jx (y)} has been derived based on the quasi-static assumption (Eq. 12), which is valid for q◦ k◦ . For larger focal spots, or equivalently smaller values of q◦ , it becomes less valid. As a rule of thumb, the quasi-static assumption holds for q◦ > 3k◦ . The focal patterns, produced by both the analytical and numerical current densities, were computed using the two-dimensional free space Green’s function [19], Ex (y, z = L)
k◦ ηI (2) H0 (k◦ y2 + (d + L)2 ) =− 4 k◦ η W/2 (2) − Jx (y )H0 (k◦ (y − y )2 + L2 )dy 4 −W/2 (17)
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Fig. 5. The (a) real and (b) imaginary parts of the spectral representation of the current density, Jx (ky ). The solid line represents Eq. (9) and the dashed line represents the numerical Fourier transform of Eq. (15). (a) Real part. (b) Imaginary part.
and are plotted in Fig. 7. The focal patterns produced by the analytically derived current density show close agreement with those computed using the MoM. Furthermore, 2D plots of the vertical electric field computed using the analytically derived and numerically computed current densities (Jx ) are shown in Fig. 8. The incident electric field excites a highly oscillatory field on the surface of the plate (z = 0). This highly oscillatory field forms a subwavelength focus along the focal plane and diverges rapidly beyond the focal plane (z > L). Reflection from the plate is evident in the region z < 0. Finally, since one of the main goals was to find an expression for the plate’s sheet impedance, the reactive sheet impedance resulting from both methods is plotted in Fig. 9. Impedance values for a 4λ◦ plate are shown since it is simpler to implement than a plate that is 10λ◦ wide. Given that only the reactive part of the sheet impedance was used in the implementation of practical near-field plates [11,14], the reactive part is only shown. The impedances show good agreement over the majority of the plate with the exception of the plate’s edges.
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Fig. 6. The (a) real and (b) imaginary parts of the current density Jx (y). The solid line represents the analytically derived current density given by Eq. (15). The dots and dashes represent the numerically (MoM) computed current density. The current density is only shown on half of the plate since it is symmetric about y = 0. (a) Real part. (b) Imaginary part.
Fig. 7. Close-up view of the magnitude of electric field at the focal plane. The solid line represents the electrical field at the focal plane produced by the analytically derived current density given by Eq. (15). The dashed line represents the electric field at the focal plane produced by the numerically computed (MoM) current density. The dotted line represents the ideal focal pattern given by Eq. (1).
5. Conclusion In this paper, we have analytically modeled and characterized near-field plates. Expressions for the induced
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current density and surface impedance of infinitely wide plates were found. Further, it was shown that the expressions obtained for infinitely wide plates can be used to approximate the characteristics of finite-width plates. In fact, we have shown that the current density on a nearfield plate can be split into two parts. One part cancels the incident wave on the focus-side of the plate, while the other part produces the desired subwavelength focal pattern. The paper also mathematically demonstrated why only the reactive part of a near-plate’s sheet impedance can be used in their design. Acknowledgments The authors would like to acknowledge discussions with T.B.A. Senior. This work was supported by a NSF Faculty Early Career Development Award (ECCS - 0747623), a Presidential Early Career Award for Scientist and Engineers: (FA9550-09-1-0696), and through the Multidisciplinary University Research Initiative Program (FA9550-06-01-0279). References
Fig. 8. 2D plots of the vertical electric field (in dB) surrounding the 10λ◦ near-field plate. The electric field is computed for the current densities obtained (a) numerically (MoM) and (b) analytically. (a) MoM. (b) Analytical.
Fig. 9. Reactive sheet impedance, Im{ηsheet (y)}. The solid line represents the analytically derived Im{ηsheet (y)} using Eqs. (15) and (16) in Eq. (5). The dots represents numerically obtained Im{ηsheet (y)} using MoM. The surface impedance is only shown for half of the plate since it is symmetric about y = 0.
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