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The design mentality of an optimal metalens model, based on the electromagnetic susceptibility, a synthesis of subwavelength-thick metasurfaces (MSs) is presented in this paper. First, based on the finite difference method of generalized sheet transition conditions, the surface susceptibility function of the MS with spatial discontinuities can be determined. Then, the paper analyzed the remaining corresponding physical field conditions for the scale of metalens. In order to adapt to the physical limitations encountered in the near-field focusing of the metalens, a standard parabolic phase design is proposed in this paper, and its upsides and downsides of the two-phase processing in different aspects are compared. Using COMSOL software with numerical simulation, it can be seen that the standard design can easily obtain high resolution in the near field, while the focusing effect is more stable when the focal length is small by the parabolic phase design.

The lens is the most basic element of the optical system, and its history can even be traced back to the ancient crystal ball used for divination and “seeing”. Today, as a device that converges or diffuses electromagnetic radiation, it is widely used in various fields such as optical instruments, cameras, security, and vehicles. Traditional lenses realize light convergence and divergence by adjusting the optical path of light with different wavelengths. Common optical instruments always use multiple curved lenses with different thicknesses and materials to focus images. Inevitably, this increases the scale of the instruments’ volume. However, in the face of increasing requirements of imaging with higher standards, it is difficult for traditional lenses to be designed to be thinner and more compact [

A metalens is a planar optical device that uses a combination of artificial sub-wavelength units to perform wave-front manipulation. Different from the traditional lens, the metalens is processed by the increasingly mature nanofabrication technology, the field intensity and phase of the light wave are modulated through the surface nano structure. Therefore, different functions can be realized by setting the properties of each nano-element and its spatial position, without the need for size or curved surface to accumulate appropriate propagation phase as well as complex lens group in a view to achieving electromagnetic wave manipulation [

This article explains in detail the metalens synthesis method based on Huygens principle, and distinguishes the pros and cons (flexibility to Angle and frequency of incident light) of the two focus phase types in the near-field. Finally, it will introduce the improvement of the susceptibility function of metalens synthesis under some design requirements, and expand some ideas for metalens design.

When discussing metasurface, it generally refers to a set of scatters composed of composite media (even thin-film conforming to classical theory). Just like the three-dimensional (volume) metamaterial [

The effective-medium theory is the traditional and most convenient method for modeling metamaterials, yet the simple introduction of bulk parameters to the metamaterial is not accurate enough to the electromagnetic discontinuity of the metasurface boundary [

z ^ × Δ H = j ω P ∥ − z ^ × ∇ ∥ M z , (1a)

Δ E × z ^ = j ω μ M ∥ − ∇ ∥ ( P z ε ) × z ^ . (1b)

For the convenience of closed form solutions, paper makes P z = M z = 0 , P and M are electric and magnetic polarization densities, respectively, then can be expressed as

P = ε χ ¯ ¯ e e E a v + χ ¯ ¯ e m μ ε H a v , (2a)

M = χ ¯ ¯ m m H a v + χ ¯ ¯ m e ε μ E a v . (2b)

Here the solution is expressed in terms of the transverse susceptibility functions, χ ¯ ¯ e e , χ ¯ ¯ e m , χ ¯ ¯ m m , χ ¯ ¯ m e , which represent the electric/magnetic (e/m) transverse polarization responses (first subscript) to transverse electric/magnetic (e/m) field

excitations (second subscript). Ψ a v = [ Ψ t + ( Ψ r + Ψ i ) ] 2 , Ψ = H , E . Substituting

the formula (2) into the formula (1), it can obtain the constraint condition of the susceptibility,

( − Δ H y Δ H x ) = j ω ε ( χ e e x x χ e e x y χ e e y x χ e e y y ) ( E x , a v E y , a v ) + j ω ε μ ( χ e m x x χ e m x y χ e m y x χ e m y y ) ( H x , a v H y , a v ) , (3a)

( Δ E y − Δ E x ) = j ω μ ( χ m m x x χ m m x y χ m m y x χ m m y y ) ( H x , a v H y , a v ) + j ω ε μ ( χ m e x x χ m e x y χ m e y x χ m e y y ) ( E x , a v E y , a v ) . (3b)

To simplify the model, it considers here the design of a single-refraction lens. The MS with no thickness is placed at z = 0, and the light enters from one side ( χ ¯ ¯ e m ≡ χ ¯ ¯ m e = 0 and diagonal χ e e x y ≡ χ e e y x ≡ χ m m x y ≡ χ m m y x = 0 ) [

χ e e x x = − Δ H y j ω ε E x , a v , (4a)

χ e e y y = Δ H x j ω ε E y , a v , (4b)

χ m m x x = Δ E y j ω μ H x , a v , (4c)

χ m m y y = − Δ E x j ω μ H y , a v . (4d)

In order to get a lossless ( k = ω μ ε ) and non-reflective metalens, A t = A i ,

{ E r = 0 H r = 0 . For Huygens type MS, the transmitted electromagnetic wave is { E t = E i e j φ t H t = H i e j φ t , where the φ t is retardation imparted by MS, the transmission coefficient is T x = E x t / E x i = e j φ t , in TM mode,

Δ E x = A i ( e j φ t − 1 ) , (5a)

Δ H y = A i η ( e j φ t − 1 ) , (5b)

E x , a v = A i 2 ( e j φ t + 1 ) , (5c)

H y , a v = A i 2 η ( e j φ t + 1 ) . (5d)

the same in TE mode. Substituting Equation (5) into Equation (4), it can finally obtain the electric/magnetic polarization coefficient

χ = χ e e x x = χ m m y y = χ m m x x = χ e e y y = 2 j k ( T x − 1 T x + 1 ) .

Nanfang Yu and Capasso et al. have produced a variety of superior superlenses with excellent performance [

In general, for a metalens with a given focal length f, the phase shift of transmitted light at an arbitrary point m on the profile needs to be adjusted Δ φ m (

Δ φ m = k ⋅ m Q ¯ = 2 π λ ( ( y m 2 + f 2 ) − f ) (6)

Here, the λ is the wavelength of electromagnetic waves in free space, y m is the distance of the scattering element from the center of the MS. The transmitted light passing through point m on the MS will form a circular equal phase surface with the adjacent scattered light at point Q following the Huygens principle, and form the focus at point F. Hence, the control function χ about the e/m susceptibility on the MS can be obtained.

For the convenient, it is assumed that the plane wave (100 GHz) is irradiated on the surface through vacuum from the apture at 2λ on the left, and then focused at point F in vacuum on the right after modulation. In order to ensure the focusing effect, the aperture size is limited by the scale of the focal length. In the study of near field focusing, the incident width of 8λ is set for convenience, moreover, the MS thickness d is selected as λ/10 to satisfy the subwavelength and facilitate manufacturing (as long as d maintains the subwavelength, the permittivity and permeability can be effectively approximated to ε r = 1 + χ E S / d , μ r = 1 + χ M S / d , respectively) [

Although such design of Huygens metalens has the natural advantage of super-resolution in near-field focusing [

To this end, it optimizes the metalens based on Huygens’ principle and use the isophased surface for the second time. It is assumed that the transmitted light first forms a parabolic wave front, and then forms a circular wave front, the plane wave incident to MS also coherently focuses at the point F. For convenience’s sake, a special situation is selected (see

Δ φ ′ m = k ⋅ 2 m L ¯ = 2 π λ ⋅ y m 2 2 f (7)

Paraboloidal phase processing (PPP) requires a larger phase difference than the general focusing phase model, which results in smaller amplitude on the wave front and smaller focal spot energy. Of course, since the phase shift of adjacent elements is larger in this design, it also avoids the problem of insufficient phase shift of adjacent elements, which is easy to occur in the fabrication of MS elements [

Based on the design method of metalens introduced above, this article uses COMSOL multiphysics simulation software to analyze its electromagnetic characteristics in detail. Just write the equivalent dielectric property ε r , μ r into an equal-thickness medium layer of sufficient length. Three sets of Huygens metalenses with different focal lengths ( f 1 = 3 λ , f 2 = λ , f 3 = 0.5 λ ) are given below. Since the uniform incident light width of 8λ is set, the equivalent numerical aperture is N A 1 = 0.8 , N A 2 = 0.97 , N A 3 = 0.99 , respectively, as shown in

In order to improve the design, it uses the new parabolic phase design introduced above to compare with the old design of the metalens at a specific focal length in the near field. It can be seen from

the two, the parabolic peak intensity is about half of the former (nearly double phase shift), but the contrast color legend shows that there are very few areas above the half peak and no interference peaks at all. In addition, the new phase design super lens with a focal length of 0.5λ has a half-peak width of 0.42λ. Although the resolution has been reduced, it is still a good performance. Next, paper analyzes the adaptability of the metalens to the deflection angle of the incident light (see

focus can adapt to a wider frequency range of focus shift, and the focal spot energy does not change much compared with the reference frequency, which is obviously better than the old design.

In this paper, the e/m susceptibility control method applies GSTC theory to design and analyze the near-field transmission and focusing characteristics of lossless subwavelength Huygens hypersurface from a two-dimensional perspective, and provides a method for solving the near-field physical limitation of the Huygens metalens. Huygens metalens can achieve a good focusing effect, and the new phase design brings more theoretical advantages (such as diameter size, incident Angle, etc.) to the near field focusing. In addition, people can also try different expressions of e/m susceptibility to adapt to different design requirements, metalens has a huge application potential.

The author declares no conflicts of interest regarding the publication of this paper.

Fu, Z.G. (2021) Near-Field Focusing With Subwavelength Thickness Metalenses via Electromagnetic Susceptibility Models. Optics and Photonics Journal, 11, 197-209. https://doi.org/10.4236/opj.2021.117014