1. Introduction

Due to the potential applications in a variety of optics fields, manipulating the propagation of light with sub-wavelength metallic structures has been attracting increasing attention. The most representative example is the discovery of light beaming effect from a subwavelength aperture surrounded by periodic corrugations on the exit side of a metallic film [1-6], which shows a unique path toward achieving photonic miniaturization without the usual scaling limitations in the subwavelength regime. Arrays consisting with narrow slits in the metallic film have been developed to realize the functions of beam focusing and collimating, and the related papers [7, 8] provided novel methods to realize one dimensional phase modulations by varying the width and depth of slits; For the purpose of energy transport on subwavelength scale, a new kind of waveguide based on metallic nanoparticle array was advanced and intensively studied [9-13], and both the experimental and numerical results proved the potential to overcome the mismatch between the electronic components and the optical guiding components in the integration of optoelectronics devices.

In this paper, we move a step forward in this field by designing a kind of metallic lens consisting of subwavelength square holes to realize the function of imaging. In section 2, the principle of the metallic lens is presented and the imaging quality of it is examined by the finite difference time domain method (FDTD) for a variety of field of view. In section 3, a theoretical analysis on the improvement of the imaging quality is performed and an optimized imaging structure is presented for verifying the analyzing result.

2. Principle and the design of the structured lens

In the study of extraordinary transmission of subwavelength square holes perforated in metallic film, J. M. Brok found that when the film is thicker than the incident wavelength, the first order guide mode dominates the transmitted energy [14], therefore, guide modes with higher order are not considered in the design. From the Helmholtz equation and the boundary conditions, the electric field in the square hole is derived and expressed as [15]:

Where a is the side length of the hole whose sides are parallel to the x and y axes. x̂ ₀, ŷ ₀ represent the unit vectors of x, y directions, and A₁, A₂ are the magnitudes of the TE₀₁ and TE₁₀ guide modes respectively. For a square hole with side length a and thickness h, the phase retardation of it is expressed as:

Here, k ₀ is the incident wave vector. We can see from Eq. (2) that the phase retardation of subwavelength square holes can be modulated by varying the side length or the depth of the hole and it is independent of the incident angle. Using this special character, we designed a metallic structured convex lens by perforating arrays consisting of differently sized square holes in the metallic films. The metallic structured lens with focal length of 240 µm and aperture of 280 µm is designed for incident wavelength λ_o=10.6µm as shown in Fig. 1. The square holes are perforated in the metallic film with thickness of 1.8λ_o. For the holes with side lengths larger than λ_o, the higher order guide modes will dissipate on the imaging plane as background and ruin the imaging quality due to their different phases from the first order guide mode. Therefore, the side lengths of the holes which are arranged in square latticed array with period 1.75λ_o are set in the range of 0.52λ_o~0.905λ_o, which also ensures that the phase retardations can be modulated in the range of π~3π. The central hole corresponds to a 2π phase retardation and the hole with distance r away from the centre has a phase difference φ(r) from the central hole determined by:

Here m is an integer to ensure φ(r) being in the range of (-π~π) thereby the side length of the holes will be neither less than 0.52λ_o nor more than 0.905λ_o.

 

Fig. 1. Diagram of the metallic subwavelength structured lens consisting of square latticed hole array, with surface parallel to the xy plane; the optical axis is along z axis and the incident light is y-polarized plane wave with wave vector always in xz plane.

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The electric field on the focal plane is obtained by carrying out the following two steps: First, the electric field at the position of 1µm far away from the metallic film is calculated by FDTD; then the electric field on focal plane is calculated by Rayleigh-Sommerfeld diffraction integrals based on the amplitude and the phase distribution obtained from the first calculation data at the position of 1µm far away from the metallic film. In the FDTD simulation, the unit amplitude of the incident electric field is chosen and the input wave is y-polarized plane wave. Considering the differences in the side lengths of the holes generally being larger than 0.1µm, the grids in xy plane are specified to be 0.1µm while in the z direction, since the thickness is invariable for every hole, the grid length is specified to be 0.4 µm. The result shows that the focused spot is 234µm away from the film surface, which is 6µm less than our designed value, corresponding to a relative error of 2.5%. This discrepancy is consistent with the precision of our FDTD calculation and the near-far field transformation.

Figure 2(a) gives the electric field distribution in 600µm×600µm scope of the focal plane for normal incident. The result shows that the focus is on the place (x=0, y=0) where it should locate, and its full-width of half-maximum (FWHM) is 11.2µm which is very close to the radius (10.8µm) of the Airy disk of traditional lens with the same aperture and focal length. As the incident angle increases from zero, the focus shifts away from the center accordingly. Figure 2(b) presents the electric fields on the x axis when the incident angle is 5°, 10°, 15° and 20° respectively, corresponding to the focus positions, i.e. real imaging heights, of 21µm, 42µm, 66µm, 87µm away from the center, and the FWHM of 11.66µm, 12.69µm, 13.90µm and 14.46µm, while the ideal Gaussian imaging heights are 20.5µm, 41.3µm, 62.7µm and 85.2um respectively.

We noticed that with our present designed parameters, there are eight weaker diffraction spots around the main focus [Fig. 2(a)], and the secondary diffractive spots shift in the same direction with the focus as the incident angle increases. When the incident angle is 20°, it can be seen from Fig. 2(b) that the intensity of secondary diffractive spot exceeds the main focus spot. On the one hand, the secondary diffractive spots in the focal plane represent fake spatial frequency information of the object for imaging, and they will mix up with the real spatial frequency information sometimes thereby ruin the imaging quality. On the other hand, they take away part of the transmitted energy, and the energy loss of higher spatial frequency information is more than the lower spatial frequency information, which greatly decreases the contrast of the imaging.

 

Fig. 2. The electric fields on the focal plane (z=234 µm) of the metallic lens with hole array period equal to 1.75λ_o. (a) the electric field in the 600µm×600µm region on the focal plane with normal incident. (b) Electric field on the line y=0 and x from -200µm to 200 µm when the incident angle is 5°, 10°, 15°, 20° respectively; peaks in x<0 are the secondary maximum spots and in x>0 are the focuses. The maximum electric field ratios of the focus to the secondary diffraction maximum are 1.64, 1.52, 1.10, and 0.856 for the incident angle of 5°, 10°, 15°, 20° respectively.

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3. Optimization for the structured lens

In order to improve the imaging performance by optimizing the parameters of the structured lens, the physical origin of the secondary diffractive spots are investigated firstly. Since it is hard to get the explicit expression of the electric field on focal plane from the Rayleigh-Sommerfeld diffraction integral, Fresnel diffraction integral approximation [16] is used to perform the theoretical analysis.

For the metallic structure with N holes, by neglecting the diversity of side lengths of the holes, the complex amplitude of transmitted electric field for the incident angle θ can be approximately expressed as:

(x_n, y_n) is the central location of the nth hole, a is the side length of a single hole, and rect(x, y) is 2D rectangle function. By applying Fresnel diffraction integrals, the electric amplitude E_f(x′, y′) on the focal plane can be obtained:

In Eq. (5), (u, v) corresponds to a bright spot in the focal plane if they make the values of x_n(u − sin θ/λ)+y_nv be integers for all (x_n, y_n). There is a diffraction maximum at u₀=sinθ/λ, v₀=0, and it corresponds to the designed focus. For square hole arrays with period of ^Δ, combining with the properties of the sinc function, the location (u′, v′) of the secondary diffractive maximum can be obtained with the condition of:

From the analysis we know that the generation of the secondary diffractive spots is due to the discrete property of output wave front. However, if the secondary diffractive spots are very weak comparing to the focus in brightness, their influence on imaging contrast can be neglected. According to the traditional definition, we take the natural logarithm e as a standard for the ratio of maximum electric field of the focus to that of the secondary diffractive spots. By substituting (u₀, v₀) and (u′, v′) into Eq. (5), the relationship between the ratio, the array period and incident angle can be obtained as depicted in Fig. 3. The isogram of natural algorithm is also plotted. It can be seen from this figure that for a given incident angle, the contrast can be improved by diminishing the value of |^Δ/a|; Since the side length a is determined by Eqs. (2) and (3), an effective way to optimizing the metallic structured lens is to increase the number of holes in per unit area.

 

Fig. 3. The maximum electric field amplitude ratio of main focus to the secondary diffraction maximum as a function of the incident angle and array period. During our calculation, the holes’ side length a is set to be 0.8λ₀,

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As comparison, we changed the period of the holes from 1.75λ_o to λ_o. The optimized metallic lens with the same aperture and focal length is designed. Fig. 4(a) and Fig. 4(b) present the electric field distribution on the focal plane obtained in the same way as we did for the unoptimized one, namely, the near fields are calculated by using the same FDTD simulation parameters and the electric fields on focal plane are calculated by Rayleigh-Sommerfeld diffraction integrals. The results indicate that both the imaging height and the FWHM of the focus keep stable at different incident angles, while the secondary diffraction maxima have departed further from the center and their maximum amplitudes are reduced dramatically relative to the amplitude of the main focus. The amplitude at the main focus with incident angle of 20° is 2.38 times larger than that of the secondary diffractive maximum, as shown in Fig. 4(b). Obviously the imaging quality in contrast is much improved with the optimized structure.

Comparing the calculated results of these two metallic lenses, we conclude that the imaging quality is greatly improved by increasing the number of the holes per unit area. Concerning the metallic structured lens with the aperture of 280µm and the focal length of 240µm designed in this paper, the view angle range of ±15° is achieved. However, this is not the view angle limit of this structure, since it can be further improved by decreasing the lattice constant between the holes.

 

Fig. 4. The electric fields on the focal plane (z=234 µm) of the optimized metallic lens. (a) The electric field in the 800µm×800µm region on the focal plane with normal incident. (b) Electric field on the line y=0 and x from -300µm to 300 µm; peaks in x<0 are the secondary maximum spots and in x>0 are the focuses. The maximum electric field ratios of the focus to the secondary diffraction maximum are 6.44, 4.80, 3.10, and 2.38 when the incident angle is 5°, 10°, 15°, 20° respectively.

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4. Conclusions

A new kind of metallic sub-wavelength structure is developed for its great potential and advantages in the application of imaging and its imaging ability is testified by means of FDTD calculation and the Rayleigh-Sommerfeld diffraction integrals. The calculation results indicate that both the imaging height and the size of focus show the reasonable values and the resolving power is close to the diffraction-limited lens. Theoretical analysis indicates that the field of view and the imaging contrast can be greatly improved by increasing the number of holes in per unit area and the corresponding positive verification is also given.