## Abstract

An optimization method for design of SPP-based metallic nanoaperture optical elements is presented. The design process is separated into two steps: Firstly, derive the amplitude and phase modulation of isolating single slit with different width; Secondly, realize the optimal design of element by using an iteration procedure. The Yang-Gu algorithm is expanded to perform this design. Three kinds of lenses which can achieve various functions have been designed by using this method. The rigorous electromagnetical theory is employed to justify and appraise the performances of the designed elements. It has been found that the designed elements can achieve the preset functions well. This method may provide a convenient avenue to optimally design metallic diffractive optical elements with subwavelength scale.

© 2011 OSA

## 1. Introduction

Since the pioneering work of Ritchie in the 1950s [1], the surface plasmon polaritons (SPPs) have been widely recognized in the field of surface science. SPPs are charge density waves resulting from the coupling of electromagnetic waves with oscillations of electrons in a metal and propagating along the interface between the dielectric and noble metal with amplitudes decaying into both layers. One of attractive properties of SPPs is that they can be concentrated and channeled in nanostructures which makes it possible to miniaturize the photonic devices in nano-scale.

In recent years, particular propagation properties of SPPs in nanostructures and light manipulations with nano-apertured metal structures have been drawing increasing attentions for realization of miniaturized devices with subwavelength scale [2–7]. Since both phase and amplitude of light wave at the exit of each subwavelength slit in the metallic grating crucially depend on the structural and physical parameters of the grating [8–12], the phase and amplitude modulation of the light can be controlled by adjusting the slit width, slit spacing, thickness of the metallic film, as well as the pattern on the surfaces of the film. Various SPP-based diffractive optical elements have been proposed and designed [13–18]. For instance, Sun and Kim employed several nanoslits with specially designed widths or depths on a metallic film to focus, collimate, or split light effectively [13]. Xu *et al.* have demonstrated a type of planar metallic lens for subwavelength imaging in the far-field [14]. This kind of devices becomes an ideal candidate for new diffractive optical elements with subwavelength scale in many optoelectronic applications. However, the existing design procedure of this kind of elements relays on a beforehand conjecture which are mostly referred to the conventional diffraction optical device. As the basic mechanism of the SPP-based metallic diffraction elements is distinctly different from the conventional optics and it is noted that the slit spacing and width are much smaller than the operating wavelength. The design is involved with rigorous vector electromagnetic (EM) theory, for instance, the finite-difference time-domain (FDTD) method. During the design process, it requires to repeatedly adjust the related parameters and thus the simulation should be carried out again and again. Therefore, it is a tremendous and onerous work. If one can store in advance all the information of the amplitude and phase information of a individual nano-aperture obtained by using the exact EM theory and employ an approximated analytical expression under a specified structure, then he can calculate the transmitted near/far-field of the structure to avoid the complete time-consuming simulation from initial every time. It manifests that the design can be separated into two independent steps: One step is extraction of the amplitude-phase information of isolating slits in structure and the second step is to implement estimation of the transmitted near/far-field of structure. It is apparent that this separated design procedure should put the optimal design of the SPP-based metallic diffractive elements in practice.

The Yang-Gu algorithm has been widely used to optimally design diffractive optical elements [19, 20]. While the previous Yang-Gu algorithm which only adopts the amplitude or phase modulation can not be utilized here since both amplitude and phase are tuned at the same time. So it is necessary to expand the conventional Yang-Gu algorithm. Motivated by these works, we present an optimal design of SPP-based metallic nanoaperture optic elements by using expanded Yang-Gu algorithm. Multiple-slit width modulation structure has been considered here to demonstrate the complete design process. The design process is separated into two steps: First step is to calculate the amplitude and phase modulation of isolating single slits with different widths using the FDTD method and to store them into memory. The second step realizes the optimal design of element by the iteration procedure using the Yang-Gu algorithm, concerning with two planes: The exit plane of the element and the observation plane. The rigorous EM theory, *i.e.* the FDTD method, is used to justify and appraise the performances of the design element and show the deviation degree.

This article is arranged as follows: Section 2 describes the structure of the elements to be designed and design theory. Section 3 presents the design results with discussion and Section 4 draws a conclusion.

## 2. Beaming structure and design theory

The structural diagram of a transmission grating with subwavelength slits under study is shown in Fig. 1. The coordinate system and the relevant parameters used are also indicated in the figure. The grating is a silver slab featured with a series of slits. The thickness of the grating is *h*. The air-slit width is *w* and the concrete width of each slit will be designed according to the function of the element. The interspace between neighboring slits is *d*. When a TM-polarized plane wave illuminates on the grating from the left side, the SPPs can be excited at slit entrances. Then SPPs propagate inside the slits in specific waveguide modes till they arrive at the exit plane where they can radiate into free space. The exit plane of the structure, denoted as P_{0}, is laid on the *x*
_{0} – *y*
_{0} plane with *z* = 0. Because of different slit widths, a specific phase and amplitude wavefront is generated on the exit plane. So when the diffracted light with a specific wavefront radiates into the free space, particular shape of the field distribution can be formed on the observation plane, denoted as P, on the *x* – *y* plane with *z* = *f*. Here, *f* denotes the spacing between the exit plane and the observation plane along the *z-*axis. The coordinates on the P_{0} plane (P plane) is denoted as *r⃗*
_{0} = (*x*
_{0}
*, y*
_{0}
*, z* = 0) (*r⃗* = (*x,y,z* = *f*)). The wave function of the magnetic on the P_{0} plane is denoted as *H*
_{y0} and the corresponding magnetic field on the P plane is denoted as *H*
_{y}, respectively.

For simplification, we consider that the structure along the *y* direction is infinite. The magnetic field *H⃗* is parallel to the *y-*axis and the electric field in the plane of incidence. Therefore, it is more convenient to calculate the magnetic field. The electric field is also easily deduced from Maxwell’s equations. It is free space propagation from P_{0} to P plane. So if the phase and amplitude distributions on the P_{0} plane can be derived, the field distribution on the P plane can be calculated by using the Green’s function.

#### 2.1. Extraction of complex amplitude distribution on the structure’s exit plane

In order to derive the phase and amplitude distributions on the P_{0} plane, we have employed the following approximation: (*i*) Neglecting the coupling of SPPs during the propagation in the slits provided that the metallic walls between any two adjacent slits are much thicker than the skin depth in metal. (*ii*) Omitting the coupling effect occurred on the exit surface from neighboring slits, compared with the intensity of directly radiation light through slits. So we can express the complex amplitude distributions on the P_{0} plane of the structure as the sum of phase and amplitude distributions of each slit.

Assuming the slit width is much smaller than the operating wavelength, it is reasonable to only consider the fundamental mode in the slit. Its complex propagation constant *β* in the slit is determined by the following equation:

*k*

_{0}is the wave vector of illuminating light in free space.

*ε*

_{m}and

*ε*

_{d}are the relative dielectric constants of the metal and dielectric material, respectively.

*β*is the propagation constant for SPPs in a slit. So the phase retardation of light transmitted through the slit can be approximately expressed as [21]

The amplitude of light wave at the exit of each slit crucially depends on the slit width. It has been investigated theoretically [22] and experimently [9]. When other parameters are fixed, the slit width dependent SPPs intensity shows a sinusoidal behavior and the intensity ratio between the peak and dip values can reach 10. It demonstrates that amplitude information is quit important in the design procedure. The relation between the slit width and SPPs amplitude can be approximately expressed as

_{0}plane can be derived by summing the phase and amplitude of each slit.

#### 2.2. Two dimensional Green function diffraction formula

The light transmits from P_{0} plane to P plane via free space propagation. After knowing the field distribution on the P_{0} plane, the near/far-field distribution at any point on the observation plane P is given by [23]

_{0}, and it can be expressed as

*H*

_{1}(

*k*

_{0}|

*r⃗*–

*r⃗*

_{0}|) is the first-order Hankel function of the second kind.

#### 2.3. Design procedure of the Yang-Gu algorithm

Generally, the wave functions on both P_{0} and P planes are complex and can be expressed as

*Ĝ*(

*r⃗*

_{0}

*,r⃗*) of the form To describe the closeness of the calculated wave front

*ĜH*

_{y0}to the desired wave front ${H}_{\text{y}}^{0}$, we introduce a distance measure

*D*

^{2}in an

*L*

_{2}norm,

*D*

^{2}with respect to function arguments

*ρ*

_{0},

*ϕ*

_{0}, and

*ϕ*.

In this work, we consider a metal film perforated with a series of slits with different widths, which corresponds to amplitude-phase modulation on the exit surface. We denote the amplitude-phase modulation for a given slit-width configuration *w*(*x*
_{1}) is

*δw*as

*δϕ*we have

*Â*

_{D}(

*Â*

_{ND}) is the diagonal (off-diagonal) element of

*Â*.

The optimal design of element using the Yang-Gu algorithm is realized by the iteration process. The iteration processes are: (i) Giving any initial *w* distribution, thus known *H*
_{y0} (i.e., *p*(*w*)*,q*(*w*)), *p*
^{′}(*w*)*,q*
^{′}(*w*); (ii) Using Eq. (13), calculate *H*
_{y}; (iii) Using Eq. (22) and desired distribution
${H}_{\text{y}}^{0}$, estimate the next generation *ϕ*
_{0} in the inner loop, thus from *w* = *q*
^{−1}(*ϕ*
_{0}), calculating *ρ*
_{0} = *p*(*w*), obtaining new *H*
_{y0} for the next generation *H*
_{y0}; (vi) Repeat step (iii) and fixed *H*
_{y} until two adjacent generation *H*
_{y0} results are close or the derivation is less than a given value; then (v) go to the outer loop, using Eq. (13), estimate the new *H*
_{y}; (vi) Repeat the step (ii)–(v), until the *D*
^{2} approaches zero. The calculation is terminated.

## 3. Design and numerical simulations

To illustrate the idea of the expanded Yang-Gu algorithm, three kinds of SPPs lenses to realize different functions have been designed. The parameters of the lenses are as follows: The used metal is silver whose dielectric constant is *ε*
_{m} = −17.36 + 0.715*i* at the wavelength of *λ* = 650nm, and the surrounding medium is air with *ε*
_{d} = 1.0. The thickness of the structure is *h* = 450nm and the slit interspacing is *d* = 200nm (center to center) while each slit width is uncertain and will be determined by the Yang-Gu algorithm. The number of nano-slits is chosen as 32 to ensure that the nanolens is in the scale needed.

The design process is as follows: At first, define the objective intensity on the observation plane according to the function of the element; Then, utilize the expanded Yang-Gu algorithm to optimize the width of each slit and derive the optimal structure; At last, simulate the performance of the achieved sturcutre using the FDTD method to verify and appraise the properties of the designed nanolens.

#### 3.1. SPPs lens with one focal spot

As the first demonstration, the SPPs lens with one focal spot is designed to compare with the former method: Equal optical path principle (EOPP) [16]. The focal length of the desired lens is *f*
_{0} = 600nm. The objective intensity distribution on the observation plane P is given in Fig. 2(a) with the black line. The Yang-Gu algorithm is utilized to decide the width of each slit. The iteration will not be terminated until the given objective is achieved. The normalized intensity distribution on the observation plane which is obtained using Eq. (9) is shown in Fig. 2(a) with the blue line. The design result corresponds to the required one well except some small oscillations. The distribution of the slit widths is shown in Fig. 2(b) by the star symbols. For comparison, the distribution of slit widths designed by the EOPP method is also presented in Fig. 2(b) with circle symbols. Because the Yang-Gu algorithm considers not only the phase modulation but also the amplitude modulation of the silt during the design procedure, the result obtained by this method is different with that obtained by the EOPP method. Furthermore, the propagation from the structure’s exit plane to the observation plane is calculated by the 2D Green function method instead of the Fresnel transformation in the EOPP.

In order to show the performance of the designed structure, the FDTD method is utilized to character the structure. The normalized intensity distribution of |*H*
_{y}| is presented in Fig. 2(c), where the exit plane of the structure is laid on *z* = 0nm. The cross section of the focal spot along the *z* direction is given in Fig. 2(d). The full width at half maximum (FWHM) along the *z* direction is 544nm and the focal spot is at *z* = 608nm which has been denoted by the dashed line in Fig. 2(c). The real focal length is *f* = 608nm, which deviates from our objective *f*
_{0} = 600nm very little. The cross section of focal spot along the *x* direction is given in Fig. 2(a) with the red line. The FWHM along the *x* direction is 230nm. It is less than the diffraction limit. These results demonstrate that the performance of the optimally designed nanolens agrees well with the preset objective.

The chromatic dispersion of the designed nanolens has also been investigated. The intensity distributions around focal spot of the designed lens operating at different wavelengths are shown in Fig. 3. The up-side is close to the lens exit plane and the dashed line indicates the plane of *z* = 608nm which is real focal plane for working wavelength *λ* = 650nm. It shows that the designed nanolens can achieve good focusing at wavelength with a range of ±100nm with the reference wavelength *λ* = 650nm, the focal length becomes short with the wavelength increasing.

#### 3.2. SPPs lens with two focal spots

Then, the SPPs lens with two focal spots is designed to show the validity of new proposed method. Two spots are set on the same plane with distance *z* = 600nm from the structure. The objective intensity distribution on the observation plane P is given in Fig. 4(a) with the black line. The intensity ratio for two spots is 1 : 2 and the positions of them are at *x* = ±885nm. After optimization, the structure of nanolens is derived. The normalized intensity distribution on the observation plane which is obtained using Eq. (9) is shown in Fig. 4(a) with the blue line. The intensity ratio of the two peaks is 1:1.92. The corresponding distribution of slit widths is shown in Fig. 4(b) by the star line.

The FDTD method is also used to check the performance of the designed lens. The normalized intensity distribution of calculated *|H*
_{y}
*|* is presented in Fig. 4(c). The exit plane of the structure is laid on *z* = 0nm and the real focal plane is at *z* = 584nm. The cross section of focal spots along the *x* direction is given in Fig. 4(a) with the red line. The positions deviate from the preset value *x* = ±885nm to *x* = ±910nm. The intensity ratio of two peaks is 1:1.96. The FWHM along the *x* direction is 320nm. These values demonstrates that the designed lens can achieve the preset objective well. The cross sections of focal spots along the *z* direction are shown in Fig. 4(d). The red line corresponds to the first spot along *x* = −910nm and the blue line to the second spot along *x* = 910nm. In order to achieve the preset intensity ratio, the maximum of two spots are not on a same *x – y* plane.

The chromatic dispersion of the designed two spots nanolens is shown in Fig. 5. Only in the range of 650 ± 20nm, the designed nanolens can keep its performance. Since the structure of this lens is more complex than that of one focal spot lens, it is more sensitive to the working wavelength.

#### 3.3. SPPs lens with three focal spots

At last, we design the SPPs lens with three focal spots. All of spots are set on the plane of *z* = 600nm. The objective intensity distribution on the observation plane P is given in Fig. 6(a) with black line. The coordinates are (−1680nm, 600nm), (0nm, 600nm), and (1680nm, 600nm) for three focal spots, respectively. The intensity ratio of three focal spots is 2:3:4. The normalized intensity distribution of the designed structure on the observation plane obtained using Eq. (9) is shown in Fig. 6(a) with the blue line. The intensity ratio of three focal spots is 2:3.08:4.17. The optimal distribution of the slit width is shown in Fig. 6(b).

The designed nanolens has been checked by the FDTD method and the normalized intensity distribution of |*H*
_{y}| is demonstrated in Fig. 6(c). The real focal length is *f* = 610nm, which deviates from the objective with 1.7%. The cross section of focal spots along the *x* direction are given in Fig. 6(a) with the red line. It demonstrates that the real coordinates of the three focal spots are (−1700nm, 610nm), (66nm, 610nm), and (1590nm, 610nm), respectively. The good agreement between the calculation result and the required one can be found. The cross sections of focal spots along the *z* direction is given in Fig. 6(d). The red line corresponds to the first spot along *x* = −1700nm, the blue line to the second spot along *x* = 66nm, and the black line to the third spot along *x* = 1590nm. In order to achieve the preset intensity ratio, the maximum of three spots are not on a same *x – y* plane.

The chromatic dispersion of the designed three spots nanolens is shown in Fig. 7. Only in the range of 650 ± 10nm, the designed nanolens can keep its performance well. These results demonstrate that the device is sensitive to the operating wavelength and it should be specially designed for a fixed operating wavelength.

## 4. Summary

In summary, a method for optimal design of SPP-based metallic nano-optic elements is proposed. The Yang-Gu algorithm is expanded to carry out the iterative procedure. Three kinds of lens which can preform different functions have been designed and the designed structures have been validated by the FDTD method. It is found that the designed lenses can achieve the preset objectives well. This method may constitute a new basis for potential applications in photonic and plasmonic devices and for achieving optimal design of metallic diffractive optical elements with subwavelength scale.

## Acknowledgments

This work was supported by the 973 Program of China (No. 2011CB301801) and the National Natural Science Foundation of China (No. 10904099).

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