In this paper, we numerically demonstrate the advantage of utilizing continuous amplitude and phase modulation in super-oscillation focusing lens design. Numerical results show that compared with simple binary amplitude modulation, continuous amplitude and phase modulation can greatly improve the super-oscillation focusing performance by increasing the central lobe intensity and the ratio of its energy to the total energy, reducing the sidelobe intensity, and substantially extending the field of view. Our study also reveals the role of phase distribution in reducing the spatial frequency bandwidth of the super-oscillation optical field on the focal plane. Based on continuous amplitude and binary phase modulation, a lens was designed with double layer metal slit array for wavelength of 4.6 µm. COMSOL is used to carry out the 2D simulation. The lens focal length is 40.18λ and the focal spot FWHM is 0.308λ. Two largest sidelobes are located right next to the central lobe with intensity about 40% of the central lobe intensity. Except for the two sidelobes, other sidelobes have intensity less than 25% of the central lobe intensity, which leads to a clear field of view on the whole focal plane.
© 2014 Optical Society of America
Recently, there has been a growing interest in the design and fabrication of far field subwavelength focusing devices for applications including optical lithography and microscopy. Super-oscillation is one promising way to realize subwavelength focusing in far field . Design of lens transmission function is the key to realize super-oscillation focusing. According to the phase transformation of ideal thin lens, continuous phase modulation was used to design micro-lenses, where the phase distribution was realized with subwavelength structure, such as metallic pillars , subwavelength square waveguides , circular waveguide , and slits [5–7]. The depth-to-width ratio is a great challenge in the fabrication of phase modulation based microlens, where large phase delay requires high depth-to-width ratio in subwavelength scale. Especially, a tight focus requires large phase modulation range, and therefore a huge depth-to-width ratio, which is formidable for state-of-art fabrication techniques. Binary amplitude modulation is another candidate for subwavelength focusing, where slit is commonly used as a binary amplitude mask for multi-slit interference [8–10]. Resolution of tens of nanometers has been demonstrated with a microscopy system illuminated with subwavelength hot spot created by a super-oscillation focusing lens based on binary amplitude mask . Binary amplitude and binary phase modulation was proposed in the lens design by Qiu . A theoretical design of super-oscillation focusing lens was proposed based on continuous modulation of both phase and amplitude . However, the central lobe energy is negligible compared with the huge sidelobe several wavelengths from the central lobe, and it requires a huge phase tuning range up to hundreds of π. To our knowledge, in all reported designs, there is always a huge sidelobe in the nearby region of the hot spot, which limits the application of super-oscillation focusing. Recently, we have reported a way to realize continuous amplitude and phase modulation with double layer metallic holes , where an example of subwavelength focusing lens based on continuous amplitude modulation was designed with this double layer metallic holes array. In this paper, we demonstrate the advantages of using continuous amplitude and phase modulation in improving super-oscillation focusing. The importance of the phase distribution on the focal plane is also revealed in the formation of super-oscillation focusing. A lens based on continuous amplitude and binary phase was designed with double layer metallic slits and its super-oscillation subwavelength focusing ability is demonstrated with COMSOL simulation.
2. Theoretical consideration
Phase plays a very important role in achieving optical super-oscillation, which we will discuss more in detail later. Generally, optical field can be expressed by its electrical field, or, where andare amplitude and phase respectively. According to Helmholtz equation, amplitude and phase satisfy the following equations,
The equations show clear couple between phase and amplitude. As shown later, in a super-oscillation field, phase and amplitude have to satisfy certain spatial distribution, which allows subwavelength focusing beyond the diffraction limit. From this point of view, the lens transmission function is the key to generate such phase and amplitude distribution on the focal plane to achieve subwavelength super-oscillation focusing. Continuous phase and amplitude modulation gives more freedoms in lens design and more room for improving lens performance. Especially, in cases where large phase change is challenging, continuous amplitude modulation becomes quite important.
3. Amplitude and phase modulation in super-oscillation focusing
Figure 1 shows the transmission function of a lens based on slit array, where only the right half of the lens is plotted because of symmetry. The lens consists of 2N slits, with constant period T and width W. Each slit may have different amplitude transmittance Ai and phase delay φi. For given focal length f0 and focal spot full width at half maximum (FWHM), genetic algorithm  is employed to find the optimized values of Ai and φi for each slit to achieve the best focusing performance. The gene is in the format of binary number. The gene of the ith slit consists of M bits with first N1 bits for amplitude and the rest N2 = M-N1 bits for phase. For example, N1 = 4, N2 = 2, and the gene is 101110, then the amplitude and phase are given by Ai = (1 × 23 + 0 × 22 + 1 × 21 + 1 × 2°) × Amax/(24-1) and φi = (1 × 21 + 0 × 2°) × φmax/(22-1), where Amax and φmax are the maximum amplitude transmittance and the maximum phase delay. The gene of each slit is generated in a random way iteratively  until the difference between diffraction pattern of the designed lens and the target diffraction pattern does not change any more.
To show the advantages of continuous modulation of phase and amplitude, six different lenses were designed at wavelength λ for normal incident plane wave with s-polarization. The plane wave intensity is 1. All lenses have the same focal length f0 = 40λ, lens radius R = 380λ, and FWHM = 0.34λ, which is smaller than the diffraction limit 0.503λ and the super-oscillation criterion 0.382λ. For comparison, all lenses were designed with the same geometric size (T = 0.948λ, W = 0.79λ, and R = 380λ), but with different types of amplitude and phase modulation, that is, binary amplitude (BA), continuous amplitude (CA), binary amplitude and binary phase (BABP), binary amplitude and continuous phase (BACP), continuous amplitude and binary phase (CABP), and continuous amplitude and continuous phase (CACP). Here, continuous phase modulation has 256 values (corresponding to N2 = 8) with equal interval of φmax/255, and continuous amplitude modulation has 16 values (corresponding to N1 = 4) with equal interval of Amax/15. Although more values are desirable for focusing performance, limited but enough numbers of values are more feasible for lens fabrication.
The optical intensity distribution on the focal plane was obtained for all six lenses with angular spectrum approach. The corresponding normalized intensity distribution is plotted against x/λ in Fig. 2(a)–2(f), where x/λ is in log scale. The detailed information on focusing performance is listed in Table 1, including the central lobe intensity Ic, the ratio of the maximum sidelobe intensity to the central lobe intensity Rsmax, the field of view for Rsmax less than 25%, and the ratio of central lobe energy to the total transmitted energy RE. As required, the central lobe FWHM is 0.34λ at focal length 40 λ for all six lenses. On the focal plane, there are sidelobes distributed on the two sides of the central lobe. For the lens based on BA modulation, the focal central lobe intensity Ic is 1.44 and RE is 0.37%. As shown in Fig. 2(a), there are obvious large sidelobes outside the area beyond x = 15.8λ. The maximum sidelobe intensity is about 59% of the central lobe intensity Ic at x = 91.8λ. The field of view with sidelobe intensity less than 0.25Ic is [-22λ, 22λ]. Figure 2(b) gives the normalized intensity distribution of the lens based on CA modulation, which shows a clear drop in the sidelobe intensity and an increase in the field of view. The maximum sidelobe intensity is 0.41Ic at x = 79.8λ, and the field of view with sidelobe intensity less than 0.25Ic is [-61λ, 61λ], three times larger than that of the lens based on BA modulation. The improvement indicates that CA modulation helps to suppress the sidelobe intensity and increase the field of view. The lens also show increased Ic = 3.65 and RE = 0.5%. The intensity generated by the lens based on BABP modulation is illustrated in Fig. 2(c). The phase π leads to a clear improvement in the focusing performance. The maximum sidelobe intensity is 30% of Ic, which is only half that of the lens based on BA modulation. The field of view with sidelobe intensity less than 0.25Ic is [-37λ, 37λ], which is larger than that of the lens based on BA modulation, but smaller than that of the lens based on CA modulation, due to the spike appearing at x = 37.89λ. In most of the area, this lens has smaller sidelobes than the previous two lenses. As given in Table 1, the lens also shows an improved focusing ability with enhanced Ic = 5.8 and RE = 0.82%. Figure 2(d) illustrates the result of the lens based on CABP modulation. On the focal plane, the intensity distribution has no obvious sidelobes. The central lobe intensity Ic is 8.58, and Rsmax is only 25% with a clear field of view on the whole focal plane. The ratio of the energy in the central lobe also increases to 1.2%. Figure 2(e) shows the intensity distribution of the lens based on BACP modulation. Again, the result shows an increase in the central lobe intensity and further suppression of sidelobes. As listed in Table 1, Ic is 11.28, Rsmax is only 25% and RE is 1.7%. As demonstrated in Fig. 2(f), when CACP modulation is adopted, the central lobe intensity reaches Ic = 13.23 and the optical energy in the central lobe is 2.06% of the total transmitted energy. Compared with the case of BA modulation, in CACP modulation, the central lobe intensity increases by 9.2 times and RE increases by 5.56 times. In all later three cases, that is, CABP, BACP and CACP, there is a clear field of view on the whole focal plane with Rsmax less than 25%. Especially in the cases of BACP and CACP, there are only six large sidelobes located in the nearby region around the central lobe, and the largest sidelobe intensity has Rsmax = 25%. In rest of the focal plane, the sidelobe intensity is less than 10% that of the central lobe. Unlike most reported super-oscillation lenses which has large sidelobe outside of the central lobe area; here, we have a clear field of view in the whole focal plane.
To find out how continuous amplitude and phase modulation improve the focusing performance, in Fig. 3, the focal plane phase distribution is plotted for ideal lens (blue), the lens based on BA modulation (black), and two lenses based on CACP modulation (red for phase range 0~π, and green for phase range 0~10π), respectively. All lenses have the same focal length of 40λ. In Fig. 3, it is noticed that, compared with the lens based on BA modulation, the two lenses based on CACP have phase distribution much closer to that of the ideal lens on the focal plane. This is the reason why the two lenses have better focusing performance, such as higher central lobe intensity and larger energy ratio in the central lobe, as shown in Table 1. Because of the same reason, for the two lenses based on CACP modulation, the one with larger phase modulation range, that is 0~10π, has better focusing performance with Ic = 16.2 and RE = 2.6%.
According to above discussion, continuous modulation of both phase and amplitude can improve focusing ability of the super-oscillation lens, including increase in the central lobe intensity, the ratio of central lobe energy to the total transmitted energy and the field of view, and reduction in Rsmax. Actually the reason lies behind the fact that, phase and amplitude are coupled to each other, and delicate phase and amplitude spatial distribution on the lens is the key to generate super-oscillation optical field on the focal plane. In the ideal case, it is expected to have a better performance if arbitrary values of amplitude and phase are available in the lens design.
To see how the amplitude and phase spatial distribution help to create optical super-oscillation, a super-oscillation focusing lens was designed based on continuous amplitude and binary phase (0, π) modulation, and its optical field on the focal plane is plotted in Fig. 4. The focal length is 40λ, the numerical aperture is 0.986, the central lobe intensity is 52, the energy ratio in the central lobe is about 6.78%, and the largest sidelobe intensity is less than 30% of the central lobe intensity. The corresponding intensity (solid) and phase (dashed) distribution is plotted within the range of [-5λ, 5λ] in Fig. 4(a). The optical intensity I(x) (solid line) has FWHM of 0.34λ (diffraction limit 0.507λ). The corresponding phase φ(x) (dashed line) shows rapid change at valleys of the intensity I(x). Especially, at the two zero points of the central lobe, the phase shift is about 3.128 rad., close to π, which indicates a phase inverse at the intensity zero points. According to the definition of super-oscillation, a rapid phase change leads to a high local spatial frequency with value larger than the cut-off frequency 1/λ. To see a clear physical picture of this phase shift, the real (dots) and imaginary (dashed) parts of the optical field E(x) is plotted along with its field amplitude A(x) (solid) in Fig. 4(b). It is interesting to note that both real and imaginary parts of the electrical field oscillate two times slower than the amplitude and intensity. Since Re[E(x)], Im[E(x)], A(x) and φ(x) have following relations, or Re[E(x)] = A(x)cos[φ(x)] and Im[E(x)] = A(x)sin[φ(x)], this indicates that phase φ(x) helps to reduce the spatial frequency of the optical field. In the inset of Fig. 4(b), the intensity distribution is plotted for the right part of the focal plane.
Figures 5(a) and 5(b) illustrate the Fourier spectrum of the super-oscillation optical field amplitude A(x) and the Fourier spectrum of the complex field A(x)exp[jφ(x)] itself respectively, where fco is the spatial cut-off frequency determined by optical wavelength, i.e. fco = 1/λ. It is found that the Fourier component of amplitude A(x) is nonzero far beyond the cut-off frequency. However, the Fourier component of the optical field A(x)exp[jφ(x)] shows a clear drop right at fco. This is the direct evidence that phase distribution φ(x) helps to confine the optical electrical field within the cut-off frequency, when the amplitude A(x) has higher frequency component far beyond fco in a super-oscillation optical field.
4. Design of super-oscillation lens based on continuous amplitude and binary phase modulation
As discussed above, phase plays a very important role in the super-oscillation focusing, it is important to include the phase distribution in the lens design. In the following, we give an example of a super-oscillation focusing lens based on CABP modulation, which is easier to be realized using subwavelength structure compared with one based on CACP. Firstly, the lens transmission function is designed to achieve desired focal plane intensity distribution, and then double layer metal slits (DLMS)  are used to realize the designed transmission function. Finally, a 2D DLMS lens model is constructed in COMSOL and FDTD simulation is conducted to investigate the lens performance.
The lens is designed at wavelength λ = 4.6 µm for s-polarized plane wave. In the design, sixteen values in [0, 1] are used for amplitude transmittance, and two values of 0 and π/2 are used for phase delay. The designed focal length is f0 = 40λ, and the numerical aperture N.A. is 0.994. Therefore, the diffraction limit is 0.503λ and the super-oscillation criterion is 0.38λ. The detailed parameters of the lens are listed in Table 2. The optimized lens amplitude transmittance and phase distributions, Ai(x) and φi(x), are plotted in Figs. 6(a) and 6(b), respectively. There is a clear variation in the amplitude transmittance along the x axis, especially in the region (0, 30λ). The phase delay shows a non-uniform distribution on the lens surface. In the designed diffraction pattern, the central lobe intensity is 33.2 times of that of incident plane wave and its FWHM is 0.34λ, which is smaller than the super-oscillation criterion. As shown in the normalized diffraction pattern (solid) in Fig. 7, the maximum sidelobe is next to the central lobe and its intensity is only 25% of the central lobe intensity. The intensity distribution also shows a clear field of view on the whole focal plane.
According to the amplitude and phase distribution given in Fig. 6, a metal (Aluminum) lens consisting of 730 double layer metal slits (DLMS)  was designed. The unit structure of the lens is illustrated in the inset of Fig. 7, where w1 and w2 are the width of the top and bottom slits, t1 and t2 are the thickness of the top and bottom layers, and T is the unit size. On the lens surface, for given pairs of Ai(x) and φi(x), values of w1, w2, t1, and t2 are determined according to the far field amplitude and phase of the periodical slit array . Here, t1 is 200 nm, t2 is 1.5 µm, T is 4 µm, and w2 is 2.5 µm for phase 0 and 3.5 µm for phase π/2. Continuous amplitude value is realized by varying the top slit width w1. The relation between the effective amplitude transmittance is plotted against w1 for 0 phase delay (solid squares) and π/2 phase delay (open circles) respectively in the inset of Fig. 7. Due to enhanced throughput caused by surface plasmon , some values of the effective amplitude transmittance are larger than 1.
COMSOL was used to conduct 2D simulation for this double layer metal slit lens. A hot spot was found at 40.18λ, which is a little bit larger than the designed focal length of 40λ. The normalized intensity distribution (circles) on the focal plane was plotted in Fig. 7. The hot spot FWHM is 0.308λ, smaller than the designed FWHM of 0.34λ. The central lobe intensity is 14, less than the designed intensity of 32.8. The intensity of sidelobes near the central lobe is about 40% of the central lobe intensity, which is larger than the designed value of 25%. The detailed information on the focusing field can be found in Table 2. The difference between the designed diffraction pattern and the DLMS lens diffraction pattern obtained with COMSOL is believed to be caused by the difference between DLMS amplitude and phase distribution derived from the far field pattern of periodical DLMS structure  and the exact DLMS amplitude and phase distribution on the output surface of the lens. For future fabrication, electron beam lithography or focusing ion beam milling can be used to fabricate such DLMS lens.
In conclusion, phase distribution plays a very important role in the formation of super-oscillation optical field, because it reduces the bandwidth of the complex optical field and keeps the spatial frequency component of the optical field within the spatial cut-off frequency. Numerical results show that compared with simple binary amplitude modulation, continuous amplitude and phase modulation can greatly improve the super-oscillation focusing performance by increasing the central lobe intensity and the ratio of its energy to the total energy, reducing the sidelobe intensity, and substantially extending the field of view. Based on continuous amplitude and binary phase modulation, a lens transmittance function was designed to realize super-oscillation focusing. According to the transmittance function, a lens was designed based on DLMS structures. 2D FTDT simulation shows a focal spot with FWHM of 0.308λ at focal length of 40.18λ. There are four largest sidelobes next to the central lobe. Their intensity is about 40% of the central lobe intensity. On the focal plane, except for these four sidelobes, other sidelobes have intensity less than 25% of the central lobe intensity. This leads to a clear field of view on the whole focal plane. Our approach can be further improved by taking into account the detailed field distribution of DLMS during the theoretical design stage to minimize the difference between the designed diffraction pattern and the one generated by DLMS lens.
Authors would like to acknowledge the financial support from China National Key Basic Research and Development Program under Grant No. 2013CBA01700. This work is also partially supported by the China National Science Foundation under Grant No. 61077057 and 61177093.
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