Controllable design of super-oscillatory planar lenses for sub-diffraction-limit optical needles

Jinshuai Diao; Weizheng Yuan; Yiting Yu; Yechuan Zhu; Yan Wu

doi:10.1364/OE.24.001924

1. Introduction

Sub-diffraction-limit optical needles have attracted much attention in the scientific community for their novel properties and potential applications in many fields such as optical beam lithography [1], super-resolution imaging [2], particle acceleration [3], and high-density optical data storage [4]. In recent years, considerable research has been reported both theoretically and experimentally on the realization of optical needles [5–12 ]. The involved methods can be roughly divided into four categories. First, under the illumination of a radially polarized Bessel-Gaussian beam, a longitudinally polarized optical needle with sub-diffraction-limit beam size along the optical axis was achieved with a lens of high numerical aperture (NA) and a binary optical element [5,6 ]. Second, a reflection paraboloidal or spherical mirror system was established to create a Gaussian optical needle with a long depth of focus (DOF) illuminated by the radially polarized beam [7,8 ]. Third, when two radially polarized beams were directed to a 4Pi focusing system, a high-purity optical needle with an arbitrarily chosen DOF was created by reversing the field pattern [9]. At last, a binary mask was used more recently to produce the optical needles under the illumination of different polarized beams [10–12 ].

Recently, it was demonstrated that a sub-diffraction-limit hotspot could be formed through tailoring the constructive interference of a large number of waves diffracted from a binary annular mask that was called the super-oscillatory lens (SOL), and the imaging experiments were performed by a modified conventional microscope with the SOL showing the resolution beyond the diffraction limit [2]. The realized sub-diffraction-limit hot spot was proved to be relevant to the phenomenon that the band-limited functions can locally oscillate much faster than their highest frequency of Fourier components, which is called the super-oscillation [2,11,13,14 ]. Later, the same group proposed that an optical needle could be obtained by converting the central area of the SOL into an opaque region to form a shadow, and changing the diameter of the blocking region without varying the rest of SOL, a reasonable compromise of the optical needle’s DOF and intensity was achieved [10]. The same method was also considered to achieve a longer DOF of the optical needle [11]. However, it is difficult to achieve a precisely controllable high-quality optical needle with this method. For example, the intensity distribution of the optical needle was not uniform but increasing, and the light energy contained in the optical needle was relatively low compared with the surroundings on the optical axis. Moreover, the focal length and DOF of the needle cannot be arbitrarily controlled by just changing the diameter of the blocking area. As we know, a controllable high-quality sub-diffraction-limit optical needle is influenced by many factors such as the focal length, DOF, the full-width at half-maximum (FWHM), and the light uniformity of the needle. Considering all these factors, we present a multi-objective and multi-constraint optimization model compromising the above-mentioned factors to achieve a compromised high-quality optical needle. The optimizing procedure of the model is designed using the Matlab programming language based on the genetic algorithm (GA) and the fast Hankel transform algorithm, as well as taking into account the computing efficiency. The achieved results are in good agreement with those calculated using the vectorial Rayleigh-Sommerfeld integral. In addition, the multi-objective and multi-constraint optimization model can be applied to different polarized beams, e.g. the linearly polarized beam, the circularly polarized beam and the radially polarized beam.

2. Design and optimization

2.1 Multi-objective and multi-constraint modeling

For an adequately thin binary annular mask, the electric field immediately behind the mask plane can be approximated by multiplying the illuminated electric field in the mask plane and the complex transmittance function through the mask [2,15,16 ]. If the electric field immediately behind the mask plane is known, the electric field at an arbitrary location away from the mask can be accurately derived by the angular spectrum theory.

Assume that a linearly polarized beam (electric field polarized along the x direction) normally illuminates the binary mask and propagates along the + z direction, as shown in Fig. 1 . Under the cylindrical coordinate system, according to the vectorial angular spectrum theory [15,16 ], the electric field components of any point P(r, φ, z) at the observation plane (z > 0) can be expressed as

{\begin{cases} E_{x} (r, z) = \int_{0}^{\infty} A_{0} (l) \exp [j 2 π q (l) z] J_{0} (2 π l r) 2 π l d l \\ E_{y} (r, z) = 0 \\ E_{z} (r, φ, z) = - j \cos φ \int_{0}^{\infty} \frac{l}{q (l)} A_{0} (l) \exp [j 2 π q (l) z] J_{1} (2 π l r) 2 π l d l \end{cases}

with

q (l) = {\begin{cases} {(1 / λ^{2} - l^{2})}^{1 / 2}, \overset{}{} \overset{}{} \overset{}{} l \leq 1 / λ \\ j {(l^{2} - 1 / λ^{2})}^{1 / 2}, \overset{}{} \overset{}{} l > 1 / λ \end{cases}

and

A_{0} (l) = \int_{0}^{\infty} t (r) g (r) J_{0} (2 π l r) 2 π r d r

where, J_n(·) denotes the n^th-order Bessel function of the first kind; λ = λ ₀/n with λ ₀ being the illuminating wavelength and n being the refractive index of the output medium; l is the spatial frequency component; t(r) is the transmittance function of SOL; g(r) denotes the amplitude distribution of the incident plane wave. E_x and A₀ are expressed as the 0 ^th-order Hankel transform and E_z means the 1 ^st-order Hankel transform. The y-component of the electric field is null, which is determined by the polarization of the incident light. The transversely and longitudinally polarized electric energy densities are calculated by

{| E_{x} (r, z) |}^{2}

and

{| E_{z} (r, φ, z) |}^{2}

, respectively; thus, the total electric energy density is

I (r, φ, z) = {| E_{x} (r, z) |}^{2} + {| E_{z} (r, φ, z) |}^{2} .

In a high-NA microscopic imaging system, the transversely polarized electric field component is dominant, while the longitudinally one is always strongly attenuated in the image plane due to the polarization filtering of this imaging system [17]. Therefore, we do not consider the longitudinal component and the total electric energy density is approximated by

I (r, z) = {| E_{x} (r, z) |}^{2}

, which demonstrated an acceptable agreement with the experimental results [2,10–12 ].

Fig. 1 Schematic diagram of sub-diffraction-limit focusing by a super-oscillatory lens.

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In order to characterize a required sub-diffraction-limit optical needle, the key point is to find a solution to make a compromise between the focal length, DOF, FWHM, and the light uniformity of the needle, as shown in Fig. 2 . Here, a multi-objective and multi-constraint optimization model is established as

M i n i m i z e {\begin{cases} I_{1} = \max [I (0, z_{0}; t_{i})] - \min [I (0, z_{0}; t_{i})] \\ I_{2} = \max {\max [I (0, z_{1}; t_{i})] \underset{}{} \max [I (0, z_{2}; t_{i})]} \\ I_{3} = \max [I (\frac{F W H M}{2}, z_{0}; t_{i})] - \min [I (\frac{F W H M}{2}, z_{0}; t_{i})] \end{cases}

S u b j e c t t o {\begin{cases} 0.9 \leq I (0, z_{0}; t_{i}) \leq 1.1,_{}^{}_{}^{} f - \frac{D_{f}}{2} \leq z_{0} \leq f + \frac{D_{f}}{2} \\ I (0, z_{1}; t_{i}) \leq 0.3_{}^{} &_{}^{} I (0, z_{2}; t_{i}) \leq 0.3,_{}^{}_{}^{} 0 < z_{1} < f - \frac{D_{f}}{2},_{}^{} f + \frac{D_{f}}{2} < z_{2} \leq z \\ 0.4 \leq I (\frac{F W H M}{2}, z_{0}; t_{i}) \leq 0.5, \overset{}{} \overset{}{} \overset{}{} f - \frac{D_{f}}{2} \leq z_{0} \leq f + \frac{D_{f}^{}}{2} \\ I (r, z_{0}; t_{i}) \leq 0.3,_{}^{}_{}^{} F W H M \leq r \leq κ F W H M,_{}^{} f - \frac{D_{f}}{2} \leq z_{0} \leq f + \frac{D_{f}}{2} \\ t_{i} \in {0, 1}_{}^{}_{}^{} i = 1, 2, \cdot \cdot \cdot, N \end{cases}

where, the electric intensity I is normalized; f represents the location of focal length; D_f is the depth of focus; t_i is the transmittance value of the ith annulus ring and N is the total number of rings contained in the mask. Ideally, we want to achieve a precisely controllable sub-diffraction-limit optical needle, so a three-objective-function and five-constraint-condition model is built to compromise the main properties of the needle. The first objective function Min.(I₁) means that the difference of the intensity distribution along the DOF changes a little, that is to ensure the excellent light uniformity of the needle; the second objective function Min.(I₂) means that the light energy contained in the optical needle is significantly higher than that contained at other locations on the optical axis, which is useful for precisely controlling the DOF; the third objective function Min.(I₃) aims to keep the FWHM along the DOF as constant as possible. The five constraint conditions are used to control the optical needle’s parameters that we prescribed before optimizing, such as the focal length, DOF, and FWHM. Because the objectives are generally conflicting, we set a specific fluctuation range of the electric intensity distribution at different locations to ensure a high-quality needle. For example, as shown in Fig. 2, we try to confine the intensity of the needle along the DOF in between 0.9 and 1.1, and the intensity of the other points on the optical axis is set to below 0.3. On the other hand, for the focusing planes perpendicular to the optical axis, the intensity at r = FWHM/2 varies between 0.4 and 0.5, and meanwhile, the radial width of the transition dark region between the central main lobe and the surrounding side lobes is set to be (κ − 1)FWHM, in which the normalized intensity of the side lobes is confined to be no larger than 30% of the peak intensity of the central lobe.

Fig. 2 The strategy to construct a controllable sub-diffraction-limit optical needle.

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2.2 Optimization

For the multi-objective and multi-constraint optimization problem (known as the Pareto-optimal problem), GA is particularly well-suited for this class of problems [18,19 ]. Compared with other optimization algorithms, such as the particle swarm optimization (PSO) and the simulated annealing (SA), GA is powerful for its parallel and global searching ability. That is to say, GA can simultaneously search a solution space in diverse regions which makes it possible to find a different solution set for the complex problems, but PSO always falls into the local optimal solution. Therefore, GA has been the most effective heuristic approach to solve the multi-objective and multi-constraint optimization problems. The classical approach of GA to solve the Pareto-optimal problem is to assign a weighted coefficient w_j to each objective function I_j, so that the problem is converted to a single objective problem [18] with the scalar objective function defined as:

\min (I) = \min {Σ_{j = 1}^{3} [ω_{j} \cdot I_{j} (r, φ; t_{i})]}

where each weighted coefficient w_j is set according to the value of I_j. Here, the weighted coefficients w_1, w₂ and w₃ are 0.5, 0.2 and 0.3, respectively. The main advantage of the approach is straightforward and it works efficiently. In order to dramatically accelerate the numerical calculation speed, a fast Hankel transform algorithm is employed taking advantages of its good accuracy, fast speed, and less storage requirement [20].

Since there exists a serious convergence problem, it is not easy to solve the complex mathematical model through an ordinary optimization algorithm. To this end, we designed an efficient optimization procedure based on the Matlab programming language for the GA and fast Hankel transform algorithm. For the binary amplitude annular mask, the contained concentric rings are initially set to be equidistant and each ring can have either unit or zero transmittance, so the binary amplitude transmittance is encoded straightforward using the two digits {0, 1}. Genetic operator is configured using the elite selection strategy, multi-point crossover, and two-point mutation. In order to guarantee the global and stable convergence, the initial number of mask is set to 1000, with a selection probability of 0.8, a crossover probability of 0.7, and a mutation probability of 0.5. It has been demonstrated by the numerical calculations that the desired results can be achieved with the above configurations. The optimization procedure is illustrated in Fig. 3 . The major steps are as follows: first, to create 1000 random masks as an initial population and calculate objective functions of each mask served as the fitness value; second, to iteratively perform the genetic operation among the population (selection, crossover and mutation) and screen the masks satisfying the multi-constraint conditions and reinsert the offspring population to the original parental population according to the fitness value of each mask until the termination criterion is satisfied; after that, the best mask in the population is achieved.

Fig. 3 The optimization procedure of designing super-oscillatory lenses for sub-diffraction-limit optical needles.

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3. Results and discussions

3.1 Numerical computation

In the following examples, a linearly polarized plane wave normally illuminates the mask with a wavelength of 532 nm and oil is used as the output medium (n = 1.515, the objective lens is working in an immersion mode for a higher-resolution imaging capability). The diameter of the mask is set to 40 μm with a total ring number of 200, so each annular width is 100 nm (also as the minimum feature size). The maximum number of iterations is set to be 400. Considering the computation speed and accuracy, totally 16384 points are sampled discretely with an exponentially increased value along the radius of SOLs in the fast Hankel transform. For multiple-objective problems, the objectives are generally conflicting, which means that it is difficult to keep all the FWHMs at different locations along the DOF far below the diffraction limit and maintain the excellent light uniformity at the same time. According to the optimized model for the optical needle, the targeted maximum FWHM is set to λ/3.03, which is the theoretical diffraction limit in a medium of refractive index n = 1.515. Since the requirement of the mask is adequately thin but opaque to the working wavelength, e.g. 100 nm thick for the aluminum film, it can be easily fabricated by the state-of-the-art focused ion beam (FIB) milling or electron beam lithography (EBL). In order to describe the SOL (might contain several hundred rings) more compactly, the transmittance value t_i is encoded from the first ring (innermost) to the N^th ring (outermost) by continuously transforming every four successive binary digits into one hexadecimal digit [16]. For the SOL₁ in Table 1 , the first hexadecimal digit ‘9’ denotes the real transmittance values of ‘1001’ and the letter ‘A’ represents ‘1010’.

Table 1. Targeted parameters and transmittance functions of the optimized binary amplitude SOLs.

View Table

According to the optimization procedure, the transmittance functions of SOLs are achieved according to the different targeted parameters of the optical needle, as shown in Table 1. The actual numbers of the transparent rings contained in the optimized SOLs are 55, 49, 54 and 48 for SOL₁, SOL₂, SOL₃ and SOL₄, respectively. For all the four SOLs, the two main changing parameters are the focal length and DOF. From the optimally achieved results of different SOLs, as shown in Figs. 4 and 5 , we can clearly see that the optical needles’ properties of all the SOLs have reached the prescribed characteristics accurately. Comparing the optical needle of SOL₁ with the result in [10], the intensity distributes more uniformly, and the needle contains a significant light energy compared with other points (all within ~0.25) on the optical axis, as illustrated in Fig. 5(a). As shown in Figs. 5(e)-(h), the FWHM of all the optical needles along the DOF is relatively constant and below the calculated diffraction limit of λ/2NA at the focal plane. For example, for the SOL₃ with the designed focal length of 6μm, the spot sizes at z = 4, 5, 6, 7 and 8 μm are λ/3.325, λ/3.331, λ/3.799, λ/3.791 and λ/2.910, respectively, all below the calculated diffraction limit of λ/2.9. Additionally, there are no significant side lobes, which is useful for the practical applications. In order to show the flexible control over the optical needle with the optimization model, we set the different focal length and DOF. Comparing the characteristics of SOL₂ with those of SOL₁, the focal length shifts from 4 μm (for SOL₁) to 6 μm (for SOL₂). The similar controllable feature can also be found by comparing SOL₄ with SOL₃. For SOL₃, the DOF extends to 5 μm (~9.4λ), compared to 3 μm (~5.6λ) of SOL₂.

Fig. 4 Optimization results of sub-diffraction-limit optical needles with different focal lengths and DOFs. (a), (b), (c) and (d) are the simulated intensity of the diffraction patterns along the propagation direction (left) with the corresponding intensity distributions at the focal plane (center) and line-scan profiles across the focal spot (right), for SOL₁, SOL₂, SOL₃ and SOL₄, respectively.

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Fig. 5 (a)-(d) are the axial intensity distributions for SOL₁, SOL₂, SOL₃ and SOL₄, respectively. (e)-(h) are the FWHMs of the focal spot along the propagation direction for SOL₁, SOL₂, SOL₃ and SOL₄, respectively, and the upper red lines denote the calculated diffraction limit of λ/2NA at the focal plane along the DOF of the optical needles, and the lower blue lines represent the value of λ/4.

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In summary, a controllable optical needle can be achieved with the prescribed characteristics, but the quality of the needle still needs to be improved, such as the uniformity of the needle’s size and intensity. For the multi-objective problem, the weighted coefficient approach is used to convert it to a single objective problem. Though this approach is straightforward and computationally efficient, not all the Pareto-optimal solutions can be investigated when the true Pareto front is non-convex. Therefore, other approaches of GA for the Pareto-optimal problem could be considered, such as the altering objective functions approach and the Pareto-ranking approach [18]. Overall, the three-objective-function and five-constraint-condition model has been realized to control the optical needles’ properties accurately. A more reasonable multi-objective and multi-constraint model might be established to get an arbitrarily long DOF, an adjustable focal length and a higher-resolution optical needle far beyond the diffraction limit since the SOL has no physical limits on resolution in principle [2]. To the best of our knowledge, the hotspot of the SOL can be as small as we wish, providing we do not care how much energy it contains, but the side lobes would contain a significant fraction of total energy which might be not useful for practical applications. Therefore, there exists a tradeoff of the light energy between the designed optical needle and the side-lobes.

3.2 Theoretical comparisons

In order to validate the above optimized results, the electric field at the focal plane is also calculated by another approach of the vectorial Rayleigh-Sommerfeld integral [21]. For a linearly polarized plane wave normally illuminating the binary mask and propagating along the + z direction, as shown in Fig. 1, the cylindrical components of the electric field can be described as

{\begin{cases} E_{x} (r, φ, z) = - \frac{1}{2 π} \iint_{R^{2}} t (ρ) g (ρ) [\frac{z \exp (j k n R)}{R^{2}} (j k n - \frac{1}{R})] ρ d ρ d θ \\ E_{y} (r, φ, z) = 0 \\ E_{z} (r, φ, z) = \frac{1}{2 π} \iint_{R^{2}} t (ρ) g (ρ) (r \cos φ - ρ \cos θ) [\frac{\exp (j k n R)}{R^{2}} (j k n - \frac{1}{R})] ρ d ρ d θ \end{cases}

where,

R^{2} = z^{2} + ρ^{2} + r^{2} - 2 ρ r cos (φ - θ)

, and the wave number k = 2π/λ ₀. g(ρ) denotes the amplitude distribution of the linearly polarized beam, with (ρ, θ, 0) and (r, φ, z) representing the points from incident and observation planes, respectively. t(ρ) is the transmittance function of the binary mask. Here, we take SOL₂ as an example for comparison, and the axial intensity in the propagating direction and the transverse intensity distribution at the focal plane are calculated using Eq. (1) and Eq. (7), as given in Fig. 6 . The results agree well with each other, and the slight divergence might be caused by the insufficient discretization of the mask structure and the problem is much alleviated by the fast Hankel transform algorithm.

Fig. 6 Comparison of the (a) axial and (b) transverse intensity distributions of the optical needle for SOL₂ calculated using the angular spectrum theory (black solid line) and Rayleigh-Sommerfeld diffraction integral (red dot), respectively.

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Additionally, the accuracy of the angular spectrum theory has been testified by the rigorous electromagnetic simulation result calculated by the finite-difference time-domain (FDTD) method [22], which is a powerful numerical solution of Maxwell’s equations. For a circularly polarized beam, the results calculated by the vectorial Rayleigh-Sommerfeld integral also have been checked by the experiment [11]. Both the vectorial angular spectrum theory and the vectorial Rayleigh-Sommerfeld integral are proved to be adequately precise to describe the electric field.

4. Conclusion

In conclusion, we have presented a multi-objective and multi-constraint optimization model to control the optical needles’ properties based on the angular spectrum theory under the normal illumination of a linearly polarized plane wave. GA is particularly well-suited for the multi-objective and multi-constraint optimization problem for its powerful parallel and global searching ability; therefore, the optimization procedure is designed according to the GA and the computational processes are accelerated by the fast Hankel transform algorithm. Numerical examples show that the light uniformity of the optical needles is excellent, and meanwhile, the different focal lengths and DOFs of the needles are precisely controlled. In the end, the optimized results are theoretically validated by the vectorial Rayleigh-Sommerfeld integral. The proposed multi-objective and multi-constraint model can also be applied to other polarized beams in a similar way. A further improvement of the optimization model and its solving algorithm are needed to achieve a higher-quality needle. The sub-diffraction-limited optical needles can be widely used for the far-field super-resolution microscopy and nanofabrication.

Acknowledgment

We acknowledge the financial support by the National Natural Science Foundation of China (NSFC) (51375400), the Program for the New Star of Science and Technology of Shaanxi Province (2014KJXX-38), the Fundamental Research Funds for the Central Universities (3102014JC02020504), the 111 Project (B13044), the Program for the New Century Excellent Talents in University, and the Specific Project for the National Excellent Doctorial Dissertations (201430).

References and links

1. Z. Gan, Y. Cao, R. A. Evans, and M. Gu, “Three-dimensional deep sub-diffraction optical beam lithography with 9 nm feature size,” Nat. Commun. 4, 2061 (2013). [CrossRef] [PubMed]

2. E. T. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11(5), 432–435 (2012). [CrossRef] [PubMed]

3. S. Payeur, S. Fourmaux, B. E. Schmidt, J. P. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. C. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012). [CrossRef]

4. H. Wang, G. Yuan, W. Tan, L. Shi, and T. Chong, “Spot size and depth of focus in optical data storage system,” Opt. Eng. 46(6), 065201 (2007). [CrossRef]

5. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]

6. K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010). [CrossRef] [PubMed]

7. D. Panneton, G. St-Onge, M. Piché, and S. Thibault, “Needles of light produced with a spherical mirror,” Opt. Lett. 40(3), 419–422 (2015). [CrossRef] [PubMed]

8. H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express 20(14), 14891–14905 (2012). [CrossRef] [PubMed]

9. Y. Yu and Q. Zhan, “Optimization-free optical focal field engineering through reversing the radiation pattern from a uniform line source,” Opt. Express 23(6), 7527–7534 (2015). [CrossRef] [PubMed]

10. E. T. Rogers, S. Savo, J. Lindberg, T. Roy, M. R. Dennis, and N. I. Zheludev, “Super-oscillatory optical needle,” Appl. Phys. Lett. 102(3), 031108 (2013). [CrossRef]

11. G. Yuan, E. T. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2014). [CrossRef] [PubMed]

12. F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5, 9977 (2015). [CrossRef] [PubMed]

13. M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. Math. Gen. 39(22), 6965–6977 (2006). [CrossRef]

14. F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009). [CrossRef] [PubMed]

15. V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 52(3), 330–339 (2013). [CrossRef] [PubMed]

16. T. Liu, J. Tan, J. Liu, and H. Wang, “Vectorial design of super-oscillatory lens,” Opt. Express 21(13), 15090–15101 (2013). [CrossRef] [PubMed]

17. T. Grosjean and D. Courjon, “Polarization filtering induced by imaging systems: effect on image structure,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046611 (2003). [CrossRef] [PubMed]

18. A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: A tutorial,” Reliab. Eng. Syst. Saf. 91(9), 992–1007 (2006). [CrossRef]

19. C. M. Fonseca and P. J. Fleming, “Multiobjective optimization and multiple constraint handling with evolutionary algorithms. II. Application example,” IEEE Trans. Syst. Man Cy. A 28(1), 38–47 (1998). [CrossRef]

20. A. E. Siegman, “Quasi fast Hankel transform,” Opt. Lett. 1(1), 13–15 (1977). [CrossRef] [PubMed]

21. H. Ye, C. W. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. P. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an ultra-thin planar lens: vectorial Rayleigh–Sommerfeld method,” Laser Phys. Lett. 10(6), 065004 (2013). [CrossRef]

22. J. Li, S. Zhu, and B. Lu, “The rigorous electromagnetic theory of the diffraction of vector beams by a circular aperture,” Opt. Commun. 282(23), 4475–4480 (2009). [CrossRef]

Controllable design of super-oscillatory planar lenses for sub-diffraction-limit optical needles

Abstract

1. Introduction

2. Design and optimization

2.1 Multi-objective and multi-constraint modeling

2.2 Optimization

3. Results and discussions

3.1 Numerical computation

3.2 Theoretical comparisons

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (6)

Tables (1)

Equations (7)

Optics Express

	f	D_f	FWHM	Optimization result of the transmittance value t_i
SOL₁	4μm	3μm	λ/3.03	94254A6D95D44CACEECC677DE4F008C2669A427D676A5F38CD
SOL₂	6μm	3μm	λ/3.03	C2C674D28DB6DD8C4633301C68A7027104F3EBA1769A1C2785
SOL₃	6μm	5μm	λ/3.03	0631B6D6649DD4C646473167C733334503A6345A68A2DD4A3A
SOL₄	10μm	5μm	λ/3.03	4C583177F646D640BCA4BEE74C0E67339FF7F3ABB094B250EF