## Abstract

A design and optimization method based on vectorial angular spectrum theory is proposed in this paper for the vectorial design of a super-oscillatory lens (SOL), so that the radially polarized vector beam can be tightly focused. The structure of a SOL is optimized using genetic algorithm and the computational process is accelerated using fast Hankel transform algorithm. The optimized results agree well with what is obtained using the vectorial Rayleigh-Sommerfeld diffraction integral. For an oil immersed SOL, a subwavelength focal spot of about 0.25 illumination wavelength without any significant side lobe can be created at a distance of 184.86μm away from a large SOL with a diameter of 1mm. The proposed vectorial design method can be used to efficiently design a SOL of arbitrary size illuminated by various vector beams, with the subwavelength hotspot located in a post-evanescent observation plane.

©2013 Optical Society of America

## 1. Introduction

Since 1950s, multi-annular plates, typically Toraldo filters [1], have been extensively studied and widely used in lens-based pupil filtering engineering for tight focusing [2,3] and super-resolution imaging [4,5]. Theoretical study indicated that it would be possible to unlimitedly compress the central main lobe with a Toraldo filter [1,6]. However, there is almost no such a tiny spot that can be practically used for super-resolution imaging, because compared with the magnified side lobes the main lobe is dramatically low in intensity. In addition, it has also been found in astronomy that a bright ‘star-halo’ disc with a wide transition dark area between the central main lobe and the surrounding side lobes could be modulated [7]. However, the central main lobe is larger than the Airy disc, and so, there is no super-resolution effect. A moderate improvement in resolution can be actually achieved via pupil filtering engineering [6,8] in a conventional lens-based optical system. Though some contrast can be lost, and a significant improvement with a factor of two can be achieved in the solution of confocal scanning microscopy, there is a limit from the practical perspective [9].

The recently proposed super-oscillatory lens (SOL) directly focuses a subwavelength sharp hotspot with a binary multi-annular nanostructured plate, and it features the capability to avoid the restriction imposed by working in the lens-based optical system [10]. The highly compressed subwavelength hotspot is created by precisely modulating the interference of the diffracted beams coming out from a large number of concentric rings. Subwavelength focusing beyond the near-field region which can be achieved using a single amplitude mask, a nanohole array, or a multi-annular plate is closely related with the optical super-oscillation phenomenon [10–13]. In the process of subwavelength focusing beyond the near-field region only propagating waves contribute to the far-field super-resolution hotspot. Therefore, super-resolution focusing by optical super-oscillation is fundamentally different from those subwavelength focusing methods using near-field evanescent waves, such as superlens [14,15] and plasmonic lens [16,17]. Compared with a traditional lens-based pupil filtering strategy, SOL has the advantage that the surrounding side lobes could still be at a relatively low level when the central main lobe is highly compressed [10–12]. Even more intriguing is that SOL can be realized simply by a binary multi-annular mask, and practical super-resolution imaging with SOL has been demonstrated in confocal scanning microscopy [10].

The scalar angular spectrum theory has served so far as the theoretical basis for designing microscale SOLs in [10,11]. An oil immersion SOL of 40μm in diameter is designed to create a subwavelength focal spot at a distance of 10.3μm away from the SOL, which is sharper than 0.3 of the illumination wavelength [10]. However, an efficient vectorial design method of SOL needs to be developed because of the depolarization effect and the practical need of computation. Most importantly, the scalar theory fundamentally limits the general design and analysis of SOLs especially with high numerical aperture (NA). As for a linearly polarized beam, the longitudinally polarized component of the electric field in the observation plane becomes pronounced when the equivalent NA of SOL is high. In this case, the circular symmetry of the hotspot is destroyed, and the focal pattern may exhibit a ‘bone’ shape instead. This depolarization effect is common in focusing systems with high-NA, such as aplanatic lens [18], paraboloid mirror [19], or Fresnel zone plate (FZP) [20,21] system, and the ignorance of depolarization using a scalar theory would produce imprecise results. As for radially polarized beams (one of the cylindrical-vector beams), a tighter focal spot with predominately longitudinal polarization is readily created, which is similar to the case in the lens-based high-NA optical system [3,22,23]. With such a vector beam illuminating SOL, a full vectorial theory must be used instead of the scalar one. Moreover, in order to improve the light throughput and enlarge the focal length (also the working distance), the SOL should be large, e.g., from tens of micrometers (microscale) to several millimeters (macroscale). The optimization of a macroscale SOL is complicated, as the number of rings in the SOL might be several hundred. Additionally, angular spectrum must be numerically calculated for a macroscale SOL, so the spatial discretization of SOL requires extensive data storage, as each subwavelength ring must be sufficiently subdivided to reach an accurate light field at the observation plane. Thus, developing an efficient optimization method especially for the macroscale SOL is imperative.

Therefore, a vectorial design method based on vectorial angular spectrum theory is proposed for the design of SOL with radially polarized vector beams. An efficient optimization model is established for the design of super-resolution focusing SOL using genetic algorithm and accelerated by fast Hankel transform algorithm. This optimization method has been used to design SOLs with diameters ranging from tens of micrometers (microscale) to one millimeter (macroscale). The optimized results agree well with those obtained through the vectorial Rayleigh-Sommerfeld diffraction integral. With the high-NA Fresnel zone plate used as a special structured SOL, the results obtained using the derived integral formula agree well with those obtained using the widely used vectorial Debye-Wolf diffraction integral theory. The proposed method lays a basis for the efficient design of the super-resolution focusing SOL of arbitrary size illuminated by various vector beams with the optimal subwavelength hotspot located in any post-evanescent observation plane. The vectorial diffraction integral formulae can be derived in a similar way for linearly, circularly, azimuthally polarized vector beams and SOLs which focus these vector beams can be designed using the proposed method.

## 2. Integral representations based on vectorial angular spectrum theory

Assuming a radially polarized vector beam is propagating along the positive direction of z, and illuminates the SOL mask, as shown in Fig. 1. If the electric field immediately behind the mask or aperture plane is known, the light field in any observation plane away from the surface can be accurately determined using the vectorial angular spectrum theory [21,24,25]. For a sufficiently thin mask, the electric field behind the mask plane is reasonably approximated by the multiplication of the illumination electric field in the mask plane and the transmission function of the mask [10,11,21].

In a Cartesian coordinate system, the rectangular components of the electric field for any point P(*x*,*y*,*z*) in the observation plane (*z*>0), can be described, using the vectorial angular spectrum theory [24,25], as

*m*,

*n*,

*q*are the frequency components along x, y, and z directions, and

*q*(

*m*,

*n*) = (1/

*λ*

^{2}-

*m*

^{2}-

*n*

^{2})

^{1/2};

*λ*=

*λ*

_{0}/

*η*with

*λ*

_{0}being the vacuum wavelength, and

*η*the refractive index of the immersion medium;

*q*(

*m*,

*n*) = j(

*m*

^{2}+

*n*

^{2-}1/

*λ*

^{2})

^{1/2}if

*m*

^{2}+

*n*

^{2}>1/

*λ*

^{2};

*A*

_{x}_{,}

*(*

_{y}*m*,

*n*) are the angular spectrums of the electric field components in the mask plane (

*z*= 0), determined by [24,25]

*E*

_{i}_{,}

*(*

_{x}*x*,

*y*) and

*E*

_{i}_{,}

*(*

_{y}*x*,

*y*) are the multiplications of the illumination electric components and the transmission function of SOL,

*t*(

*x*,

*y*); for a radially polarized beam, in a cylindrical coordinate system, one has [24]

*r*= (

*x*

^{2}+

*y*

^{2})

^{1/2}, and tan

*φ*=

*y*/

*x*;

*g*(

*r*) represents the amplitude distribution of the radially polarized beam, with the waist plane located in the aperture plane of SOL; for the widely used radially polarized Bessel-Gaussian beam,

*g*(

*r*) can be expressed as [3,22]

*β*

_{0}is the ratio of the mask radius,

*a*, to the beam waist,

*w*

_{0};

*J*

_{1}(·) denotes the first-order Bessel function of the first kind. The equivalent numerical aperture of SOL can also be defined as NA =

*η*sin

*α*, with

*α*being the maximum focusing semi-angle with respect to the z direction. Using Eq. (3), Eq. (2) is rewritten as

*l*= (

*m*

^{2}+

*n*

^{2})

^{1/2}and$\mathrm{tan}\varphi =m/n$; by substituting Eq. (5) into Eq. (1),

*J*(·) is the

_{n}*n*th-order Bessel function of the first kind; Eq. (5) reduces to

*E*

_{x}_{,}

*into*

_{y}*E*

_{r}_{,}

*in the transverse plane, via [22]*

_{φ}*E*and

_{r}*E*have been expressed as the first-order and zeroth-order Hankel transforms, respectively; the radially, longitudinally polarized electric energy densities are calculated using |

_{z}*E*(

_{r}*r*,

*z*)|

^{2}and |

*E*(

_{z}*r*,

*z*)|

^{2}, while the total electric energy density (or light intensity) is calculated using |

**(**

*E**r*,

*z*)|

^{2}= |

*E*(

_{r}*r*,z)|

^{2}+ |

*E*(

_{z}*r*,

*z*)|

^{2}.

*E*(

_{r}*r*= 0,

*z*) = 0, and the on-axis electric field becomes purely longitudinally polarized.

The focus distribution for a conventional linear polarized beam of SOL can be derived similarly. For a conventional linearly polarized Gaussian beam (polarized along the x direction) perpendicularly illuminating the mask plane of SOL, one has

*g*(

*r*) = exp(-

*r*

^{2}/

*w*

_{0}

^{2}), with

*w*

_{0}being the waist radius of the Gaussian beam in the aperture plane of SOL. It is obvious that when

*φ*=

*π*/2 or 3

*π*/2,

*E*(

_{z}*r*,

*φ*,

*z*) = 0, i.e., the longitudinally polarized component vanishes along the y direction. When Eq. (13) is compared with the result obtained using the scalar angular spectrum theory [10], the longitudinally polarized component,

*E*, disappears in the regime of a scalar theory, and thus the circular symmetry of the transverse light field is maintained; however, when

_{z}*E*in Eq. (13) becomes pronounced, the vectorial nature of the incident beam must be considered. The derived integral formula, Eq. (13), is consistent with that in [21], where the depolarization effect has been reexamined and experimentally validated by using a high-NA phase Fresnel zone plate. For circularly or azimuthally polarized vector beams, the integral formulae could be similarly derived. As for the circularly polarized beam, the light field in the observation plane is still circularly symmetric, and for the azimuthally polarized beam, the light field only contains the azimuthally polarized electric component. These general properties are similar to those results under lens-based optical systems [22], which are analyzed based on the vectorial Debye-Wolf diffraction integral [18].

_{z}## 3. Optimization of super-oscillatory lens for super-resolution focusing

For a radially polarized beam, the subwavelength light pattern is circularly symmetric and can be much sharper than that for a linearly polarized beam, because the radially polarized electric component, *E _{r}*, is remarkably suppressed for the former with a high-NA SOL. The tighter focusing effect is similar to that in the lens-based optical system with a radially polarized beam [3,23]. In order to characterize the required subwavelength hotspot (or a ‘star-halo’ disc) [10,11], a constrained, single-objective optimization model is established as

*d*

_{0}restrains the full-width-at-half-maximum (FWHM) of the subwavelength hotspot (central main lobe); the radial width of the transition dark region between the central main lobe and surrounding large side lobes is set to be (

*κ*-1)

*d*

_{0}, among which the normalized maximum intensity is constrained to be no larger than 20% of the peak intensity of the central lobe;

*N*is the total number of rings contained in the SOL; the optimal distance is constrained to make the SOL have a high NA, e.g., between 0.60 and 0.95 in air, which is useful for rapidly reaching a subwavelength SOL. Following the basic configuration in [10], concentric rings contained in one SOL are supposed to be equidistant, either transparent or opaque.

Genetic algorithm (GA), is a widely used stochastic evolution algorithm with the powerful parallel and global searching capability [26], and so it is used to solve the optimization problem described by Eq. (15). The binary amplitude transmission is coded straightforward using binary digits {0,1}; while for the distance *z*, it is coded using ten binary digits; the total length of the binary digits contained in one individual is hereby (*N* + 10). For a stable and fast convergence, GA is configured using two-point crossover, two-point mutation, and the elite selection strategy. Further, GA is set to hold a population of 40~60, with a crossover probability of 0.8, and a mutation probability of 0.12. It is found through numerical calculations that, using the above configurations, various microscale or macroscale SOLs can be steadily reached within several hundred iterations.

From Eqs. (9) and (12), the intensity distribution in the given observation plane for one specific SOL is determined by two first-order and one zeroth-order Hankel transforms; thus during each iteration, a large number of Hankel transforms should be implemented in order to evaluate the fitness of each individual. To accelerate the optimization, a fast Hankel transform algorithm is programmed to operate with efficient computation and good accuracy especially useful for designing a macroscale SOL with a narrow ring width and many rings [27]. The key point of this algorithm is converting an M-point discrete approximation of Hankel transform into a procedure of computing two 2M-term fast Fourier transforms and their multiplication [27]. Advantages of the algorithm include fast speed, less storage requirement, and good accuracy.

In the following examples, an illumination wavelength of 532nm is used either in air (*η* = 1) or in oil immersion medium (*η* = 1.515). The maximum iteration number is set to be 300. In Table 1, four SOLs from microscale to macroscale are optimized to create subwavelength hotspots far beyond the near-field region and all surpassing the Abbe’s diffraction resolution limit of 0.5*λ*_{0}/NA. SOL_{1} is designed in air and the other three are designed in oil immersion medium. In order to describe the SOL (might contain several hundred rings) more concisely, the transmission *t _{i}* is coded from the first ring (innermost) to the

*N*th ring (outermost) by continuously converting every four successive binary digits into one hexadecimal digit. For SOL

_{1}in Table 1, the transmissions of the first four rings, ‘1110’, has been coded to be ‘E’; while for SOL

_{2}, the first hexadecimal digit ‘9’ denotes the real transmissions of ‘1001’.

*D*and Δ

*r*represent the diameter of the SOL and the annular width (as also the minimum feature size), respectively.

*z*denotes the distance at which the optimal focal pattern is found. The diffraction patterns corresponding to SOL

_{o}_{1}and SOL

_{4}have been plotted in Figs. 2(a) and 2(b), respectively. The electric energy density distributions along the radial direction are shown in Figs. 2(c) and 2(d), corresponding to SOL

_{1}and SOL

_{4}, respectively. The real transmissions for SOL

_{1}and SOL

_{4}are plotted in Figs. 2(e) and 2(f), respectively; there are 22 and 55 transparent rings contained in SOL

_{1}and SOL

_{4}, respectively. When compared with the example in [10], FWHM is between 0.24~0.26 of the illumination wavelength for SOL

_{2~4}, which is sharper than the result in [10] (0.29 of the illumination wavelength); there are almost no significant side lobes with SOL

_{2~4}(less than 20% of the central peak intensity), in contrast to a pronounced side lode in [10] (~40% of the central peak intensity). Moreover, the minimum feature sizes for SOL

_{2~4}are much larger (at least 1μm) than 200nm in [10]; thus, the designed SOLs are more easy-to-fabricate with current FIB (focused ion beam) or EBL (electron beam lithography). As indicated in [10], sharper subwavelength hotspots are also possible to be created with magnified side lobes beyond the transition dark region. It should be noted that the optical unit and the real unit are two commonly used units, and we choose the latter in the presented figure.

For a linearly polarized beam (polarized along the x direction), as described by Eq. (13), the light intensity along the y direction is purely contributed from *E _{x}*. While in the x direction, both

*E*and

_{x}*E*contribute to the light field; thus, when |

_{z}*E*|/|

_{z}*E*| becomes pronounced, the circular symmetry of the hotspot is broken. The transverse focal pattern is generally in a ‘bone’ shape [21]. In order to reach a super-resolution focusing SOL with a linearly polarized beam, the light intensity along the x direction should be used as the merit function, rather than the y direction. For a circularly polarized beam, the light intensity is circularly symmetric; however, the obtainable minimum beam size is much broader than that for a radially polarized beam.

_{x}## 4. Theoretical validations and discussions

In order to validate the optimized results in Section 3, the electric field in the optimal focal plane is further calculated using the vectorial Rayleigh-Sommerfeld diffraction integral [28]. For a radially polarized beam, as shown in Fig. 1, after a series of trivial derivations, the cylindrical components of the electric field can be expressed as

*x*=

*r*cos

*φ*-

*ρ*cos

*ψ*, Δ

*y*=

*r*sin

*φ*-

*ρ*sin

*ψ*,

*u*= (Δ

*x*

^{2}+ Δ

*y*

^{2}+

*z*

^{2})

^{1/2}, and the wave number

*k*= 2

*π*/

*λ*.

*g*(

*ρ*) denotes the amplitude distribution of the radially polarized beam, with

*ρ*being the normalized radial coordinate; from Eq. (4), let

*β*

_{0}= 1,

*g*(

*ρ*) = exp(-

*ρ*

^{2})

*J*

_{1}(2

*ρ*). Numerical calculation using Eq. (16) is rather inefficient, entailing extensive data storage as well as long computing time, and so a microscale SOL, i.e., SOL

_{2}, is chosen for comparison. The subwavelength focal spots calculated using Eq. (12) and Eq. (16) are shown in Fig. 3. The results agree well with each other, and the slight divergence among the higher-order side lobes might be caused by the insufficient discretization of the microstructure plate; however, the problem is much alleviated by using the fast Hankel transform algorithm in Section 3.

Besides, the accuracy of the vectorial angular spectrum theory has been checked [29] with the rigorous electromagnetic simulation result calculated by FDTD (finite difference time domain), which is a powerful numerical solution of Maxwell’s equations. For a linearly polarized beam, the diffraction light field through a circular aperture is also calculated. Both the vectorial angular spectrum theory and vectorial Rayleigh-Sommerfeld diffraction integral are found to be sufficiently accurate to describe the electric field when the observation plane is slightly away from the diffraction screen (several wavelengths) [29].

Fresnel zone plate (FZP) might be considered as a specially structured super-oscillatory lens, which has many important applications, especially in X-ray microscopy; in [20], the electric field focused by a high-NA phase FZP illuminated by a radially polarized beam, is described, according to the widely used vectorial Debye-Wolf diffraction integral [18], as [20,30]

*α*is the maximum focusing semi-angle of FZP, with tan

*α*=

*a*/

*f*, and NA =

*η*sin

*α*;

*f*is the focal length of FZP. The transmission function of a binary phase FZP is described as

*m*= 0, 1, …,

*N*/2-1; tan

*θ*= (

_{n}*nλf*+

*n*

^{2}

*λ*

^{2}/4)

^{1/2}/

*f*,

*n*= 0, 1, …,

*N*; the phase function takes the form

*l*

_{0}(

*θ*) denotes the amplitude distribution of a radially polarized beam; as for FZP,

*r*=

*f*tan

*θ*, Eq. (4) thus becomes

*λ*

_{0}= 633nm,

*N*= 16,

*f*= 0.5μm, and

*β*

_{0}= 1. Both are consistent with the rigorous electromagnetic simulation using FDTD [20].

Actually, Debye-Wolf integral is a coherent superposition of plane waves, which is the same idea behind the angular spectrum theory, therefore it is logical for the results obtained using them to be close to each other. The result of Debye-Wolf diffraction integral for SOL verifies the validity of the proposed method.

It is also interesting to compare the spot size focused by SOL with that by an ultra-high NA aplanatic lens system. The subwavelength hotspots created in Section 3 all surpass the diffraction resolution limit of *λ*_{0}/(2NA). Under a conventional high-NA aplanatic lens system with a radially polarized vector beam, the electric field in the focal region can be expressed as [3,22]

*l*

_{0}(

*θ*) should be written as

*r*=

*f*sin

*θ*in the pupil plane, to obey the sine condition [3]. For the objective lens of NA 0.95 in air and NA 1.4 in oil (

*η*= 1.515), FWHM of the radial focal spot is 0.68

*λ*

_{0}and 0.50

*λ*

_{0}, respectively. The subwavelength spots in Table 1 are about half of the above results. It should also be indicated that, SOL provides a practical way to beat the theoretical limit of 0.36

*λ*

_{0}/NA, which can be obtained under the known far-field lens- or mirror-based focusing systems [20,30,31].

## 5. Conclusion

In conclusion, based on the vectorial angular spectrum theory, an efficient vectorial design method of SOL is proposed for subwavelength focusing particularly with the radially polarized vector beam. The structures of SOLs (might contain several hundred concentric rings) are optimized by genetic algorithm and the computational processes are accelerated with fast Hankel transform algorithm. For example, an oil immersion SOL of 1mm large (diameter) could create a subwavelength focal spot of about one quarter of the illumination wavelength at a distance of 184.86μm away from the mask surface of SOL, which is of dominantly longitudinal polarization. The derived integral formulae are theoretically validated by the vectorial Rayleigh-Sommerfeld diffraction integral. Further, Fresnel zone plate is taken as a specially structured SOL, and the spot simulated using the derived integral formula is very close to that simulated using the widely-used vectorial Debye-Wolf diffraction integral theory. The proposed vectorial design method can be used to efficiently design a super-resolution focusing SOL of arbitrary size illuminated by various vector beams, and the optimal subwavelength hotspots can be located in any post-evanescent observation plane flexibly. The vectorial design method proposed can be equally applied to the design of a binary phase SOL, where the coding strategy for the transmission function of SOL should be slightly modified. The structured SOL can be used to achieve subwavelength focus in confocal scanning microscopy. The design of other pattern with SOL, e.g. light tunnel with cylindrical-vector beams [3], can also be done in the similar way.

## Acknowledgments

The authors thank the funding from National Natural Science Foundation of China (NSFC) under grants no. 51275121 and 51205089. Valuable discussions with Edward T. F. Rogers, University of Southampton, are gratefully acknowledged. It is a pleasure to thank Dr. Jie Lin and Min Ai for helpful discussions.

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