Plasmonic achromatic doublet lens

Kyookeun Lee; Seung-Yeol Lee; Jaehoon Jung; Byoungho Lee

doi:10.1364/OE.23.005800

1. Introduction

Surface plasmon polaritons (SPPs) are electromagnetic waves propagating along the surface between a dielectric and a metal [1]. Due to their high carrier frequency and small modal volume, it is expected that plasmonics can encompass advantages of both electronics and photonics [2]. In order to use the advantages of SPPs and realize plasmonic integrated circuits, plasmonic elements are required that can manipulate SPPs in-plane. For example, plasmonic lenses which can focus SPPs are proposed by utilizing array of nanoparticles [3, 4], nano-slits [5], and arc-shaped slits [6–9]. Other plasmonic elements such as sources, low-loss waveguides, beam splitters, and mode converters for plasmonic applications are demonstrated [10–13].

Methodologies of designing and engineering plasmonic elements can be transplanted from general frameworks of optics, since SPPs still retain the nature of electromagnetic wave. Similarity with the scalar diffraction theory has been shown [14] and plasmonic devices are designed which have analogies to classical diffractive optical devices. For example, a Fresnel lens and Fresnel zone plates are proposed [15–17]. More elaborate design principles have been also arranged considering full-vectorial representation of surface plasmon fields [18, 19].

Moreover, principles of geometrical optics are often directly applied. For example, plasmonic Luneberg and Eaton lenses have been demonstrated. It has been shown that arbitrary refractive indices can be engineered by tuning the thickness of dielectric layer [20]. Similarly, several low-dimensional plasmonic devices, which use graphene sheets as conductive media, have been proposed. In these cases, certain substrate patterns and their corresponding electrostatic field profiles are used to tune effective indices [21, 22]. However, in most of these cases, the devices only function at a certain target wavelength, which means that the operation spectrum of the device is limited. Since SPPs are guided through metal, which is much dispersive than dielectric media, effective indices of SPP modes significantly alter with operating wavelengths. Therefore, compensating the dispersion of SPP signal is quite important. A broadband lens-like plasmonic device has been proposed [23], but its availability is insufficient because the thickness of the dielectric layer is in grey-scale and the focal point is made inside the waveguide. In the sense of plasmonic elements, more simple geometry is favored.

Here, we propose a plasmonic achromatic doublet lens (ADL). According to the principle of geometrical optics, an ADL compensates a chromatic aberration by using two lenses in contact which are made of materials with different dispersions [24, 25]. Similar to the conventional ADL in bulk optics, the proposed plasmonic lens is composed of two lens layers with different configurations. Each lens layer is modeled with plasmonic waveguides such as metal-insulator-metal (MIM) or metal-insulator-insulator (MII) waveguide, which can be made by current fabrication technology. By using the effective refractive indices of guided plasmonic modes in each layer, geometric parameters of the lens are determined according to two-dimensional (2D) full-wave simulation. During the 2D simulation, each layer is considered as a homogeneous effective medium. After that, full structural simulation is done to verify the function of the proposed device.

2. Configuration of the proposed structure and design principle

A schematic illustration of the proposed structure is depicted in Fig. 1(a). The ADL lens is composed of two lens layers. Let the front side of the lens be Layer A, and the rear side be Layer B. For reality in practical fabrication, both of layers are made of the same materials but have different geometrical configurations so that each layer shows different dispersion characteristics. In order to achieve achromatism, focal lengths $f_{A}$ and $f_{B}$ of two thin lenses in contact should follow the relation according to geometrical optics [24]:

f_{A} = \frac{Δ_{B} - Δ_{A}}{Δ_{B}} f, f_{B} = - \frac{Δ_{B} - Δ_{A}}{Δ_{A}} f,

where

Δ_{A, B}

denotes dispersive power of Layer A or B, and

f

is a focal length of the ADL. For a given focal length

f

, the dispersive powers of two lenses determine geometric parameters of each lens. That is, configuration of each layer, which determines the effective refractive index, should be carefully chosen to obtain appropriate geometric parameters. Among various types of plasmonic waveguides, planar MIM and MII waveguides are investigated as potential candidates for composing each lens considering ease of fabrication and analysis. Figure 1(b) shows effective indices of guided SPP modes along MIM and MII waveguides with different dielectric thicknesses. Field profiles of 200 nm thickness cases are illustrated in Fig. 1(c). Other modes represented in Fig. 1(b) have similar field profiles. For the MIM cases, only symmetric modes, which have symmetric magnetic field profile, are considered as fundamental modes. Anti-symmetric modes are discarded due to their cut-off characteristics and low coupling efficiency to MII waveguide modes. Silver is used for the metal layer, and silicon dioxide is used for the insulator layer. Optical constants of the materials are taken from ref [26]. Target wavelength range is chosen to cover green and red spectrum (500 nm to 700 nm range), not the whole visible range, due to very poor propagation length of SPPs at short wavelengths. All dispersion curves correspond to normal dispersion, similar to conventional lens materials like flint or crown glass.

Fig. 1 (a) A schematic illustration of the proposed lens. (b) Effective indices of fundamental modes in the MIM (solid lines) and MII (dashed lines) waveguides. For MIM configuration, Ag-SiO₂-Ag layers are assumed, and for MII configuration, Ag-SiO₂-Air layers are assumed. Among lines, thickness of SiO₂ layer differs as 50 nm (blue), 100 nm (green), 150 nm (red), and 200 nm (cyan), respectively. (c) Normalized $H_{y}$ field profiles of the fundamental modes of 200 nm thick MIM (above) and MII (below) at 633 nm wavelength are shown. Other modes also have similar profiles with them.

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The dispersive powers, whose reciprocals are called Abbe numbers, are set by referring to refractive indices of some customary Fraunhofer lines [24, 25]. Here, we consider effective indices at wavelengths of 500 nm, 633 nm, and 700 nm for calculating dispersive power. Another point to be noted is the effective index of single interface SPPs $n_{0}$ , which has analogy to free-space, is larger than 1. Therefore, denoting effective indices at each wavelength as $n_{500nm}$ , $n_{633nm}$ , and $n_{700nm}$ , the dispersive power can be defined by,

Δ = \frac{n_{500 nm} - n_{700 nm}}{n_{633 nm} - n_{0}} .

By the definition, the dispersive powers of lines from the blue solid one to the blue dashed one in Fig. 1(b) are 0.0866, 0.0617, 0.0552, 0.0556, 0.272, 0.390, 0.680, and 1.25 respectively. For the MIM and the MII cases in common, the dispersive powers become smaller as the dielectric layer becomes thicker.

Since all of the dispersive powers are different from each other, combination of any two configurations can compose the ADL. That is, with the cases depicted in Fig. 1(b), 28 different combinations can be made. By investigating the characteristics of each combination, we would like to determine the reasonable range of each waveguide layer to be used for ADL. For the convenience of explanation, let Layer A be a double-convex lens with a positive focal length, and Layer B be a lens with a negative focal length. Since two lenses are in contact, there are two faces whose radii should be determined. Let the radius of the front face be denoted as $R_{f}$ , and that of the rear face as $R_{r}$ . Magnitude of both radii can be obtained according to the thin lens equation:

R_{f} = 2 (n_{A} - 1) m_{A} f, R_{r} = \frac{f}{\frac{1}{2 (n_{A} - 1) m_{A}} + \frac{1}{(n_{B} - 1) m_{B}}},

where

n_{A}

and

n_{B}

are effective indices at each layer normalized by

n_{0}

, and

m_{A}

and

m_{B}

are coefficients given by

m_{A} = (Δ_{B} - Δ_{A}) / Δ_{B}

,

m_{B} = (Δ_{A} - Δ_{B}) / Δ_{A}

.

Figure 2 shows calculated radii of the 28 combinations, normalized by the focal length. The points adjacent to the zero, which are tied by the yellow line, depict homojunction cases. That is, two lenses have the same waveguide style but different dielectric thicknesses. Due to their small difference between the dispersive powers, focal lengths and their corresponding radii are relatively small. Lenses in these cases have a demerit that the aperture size has low upper limit. The other points in Fig. 2 designate heterojunction cases. The points inside the green line commonly have MII waveguide region of 50 nm dielectric thickness, which is the blue solid line in Fig. 1(b). For these cases, index mismatch between the layers is so large that power throughput can be aggravated. Besides, the cases inside the red line have relatively small index difference and show adequate radii dimension. Among them, we choose the combination of the 200 nm thick MIM and the 100 nm thick MII as the best candidate. The chosen one shows relatively small index mismatch, and the orders of the radii are the most similar with each other. In summary, Layer A is chosen to be made of MIM configuration with 200 nm thickness as a double-convex lens, and Layer B is chosen to be made of MII configuration with 100 nm thickness as a concave lens.

Fig. 2 Radii of the front and the rear faces normalized by the focal length. Signs of the radii are left out. Each point represents the radii pair of all the combinations of the candidates in Fig. 1(b), according to Eq. (1) and the thin lens equation. The points inside the yellow line represent the homojunction cases, inside the green line represent the heterojunction cases combined with 50 nm thick MIM, and inside the red line represent remaining heterojunction cases.

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Aforementioned approaches are based on the geometrical optic theories, which are valid when the feature size of optical element is far bigger than the wavelength. For the case of ADL, however, its feature size is about a few times of the wavelength, so that applicability of the geometrical optic theories should be carefully checked. In the next section, this issue will be discussed by quantifying approximation errors and introducing a proportional constant which corrects the geometric parameters. Moreover, in the section 4, we also discuss the feasibility of regarding MIM and MII plasmonic waveguides into the homogeneous media with effective refractive index, by comparing the simulation results from three-dimensional (3D) full-vectorial simulations and reduced dimensional simulations based on the effective index model.

3. Determination of the geometric parameters

In order to determine the geometric parameters, the desired focal length should be given. Noting that the propagation length of SPPs at 500 nm wavelength is about 9 μm, the focal length $f$ is set to be 4 μm. The aperture size of the ADL is another degree of freedom. The aperture size itself does not affect any other geometric parameters, but two issues are involved. The first issue is that variation of effective indices can occur. Effective indices depicted in Fig. 1(b) assume the width of waveguide, in this case the aperture size, to be infinite. When finite aperture size is given, however, effective indices can be changed. The variation of effective indices increases when the aperture size becomes smaller. The second issue is about validity of the lens equation. According to the scalar diffraction theory, the lens equation is valid under the Fresnel approximation. Under the Fresnel regime, phase due to a spherical lens approximates to parabolic function, so that wavefront converges to a point satisfying the lens equation. If the aperture size is comparable to the focal length, higher spatial frequency components start to be involved so that the lens equation does not hold. Therefore, the aperture size should be set as a moderate value not to violate these two issues.

Figure 3 quantitatively shows the effects of aperture size. Blue lines illustrate the change of the effective indices compared with the infinite aperture size case according to the aperture size. Green lines depict difference between phase due to a spherical lens and parabolic phase with a corresponding focal length according to the lens equation. Larger difference implies larger approximation error. In detail, the phase difference functions are

\begin{array}{l} Δ ϕ_{A} = n_{0} k_{0} (n_{A} - 1) [2 R_{f} (1 - \sqrt{1 - {(\frac{w}{R_{f}})}^{2}})] - n_{0} k_{0} \frac{w^{2}}{2 f_{A}}, \\ Δ ϕ_{B} = n_{0} k_{0} (n_{B} - 1) [R_{f} (1 - \sqrt{1 - {(\frac{w}{R_{f}})}^{2}}) - R_{r} (1 - \sqrt{1 - {(\frac{w}{R_{r}})}^{2}})] - n_{0} k_{0} \frac{w^{2}}{2 f_{B}}, \end{array}

where

k_{0}

is free-space wavenumber, and

w

is the aperture size. As expected, trade-off relation between the effective index variation and the approximation error appears in Fig. 3. In order to retain the both errors within tolerable scope, we set the aperture size to be 2.6 μm.

Fig. 3 Blue lines represent changes of the effective indices according to the aperture size compared with the effective indices when the aperture size is infinite. Green lines plot Eq. (4), which quantifies the approximation error of the lens equation. (a) Layer A case, and (b) Layer B case are depicted, respectively.

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In this case, however, $Δ ϕ_{A}$ and $Δ ϕ_{B}$ are still too significant to be ignored. That is, the resultant focal length is slightly shorter than the target focal length. Therefore, the focal length $f$ used for the design, should be set larger than the target focal length $f_{0}$ to match the resulted focal length at $f_{0}$ . The proportional constant, which relates $f = α f_{0}$ , is found empirically by matching resultant focal length and $f_{0}$ according to increase of $f$ . Computation is done by 2D full-wave simulation using rigorous coupled-wave analysis (RCWA) [27]. Exact field profiles of SPPs are not taken into account and each layer is dealt as an effective medium without loss. The focal length is defined as the distance between the maximum point of electric field intensity and the principal plane. Positions of the principal planes are found according to the thick lens equation and relations between the nodal points of the lens. Since the doublet lens is composed of two lenses, the principal planes of each lens are found, and then the principal plane of the compound lens is calculated [25]. $α$ is found to be 1.19. In other words, the ADL designed by the geometrical optic theories gives the focal length of $0.84 f_{0}$ . It implies that just the thin lens equation can give nice prediction for the geometric parameters of the plasmonic lens. The final geometric parameters shown in Fig. 4 are summarized in Table 1. Signs of $R_{f}$ and $R_{r}$ denote direction of the face curvature. Since Layer A is a double-convex lens, the face radius between the lenses is $- R_{f}$ .

Fig. 4 Schematic illustrations of the final ADL in (a) top-view (xy-plane) and in (b) side-view (xz-plane) along the dashed line in part (a). Exact parameters in detail are provided in Table 1.

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Table 1. Determined geometric parameters of the plasmonic achromatic lens

View Table

4. Simulation results

For the comparison of achromatic characteristics, a double-convex lens made of 200 nm thick MII configuration is introduced as a reference. A focal length and an aperture size are set to be the same with the ADL, and radii are determined according to the effective index and the thin lens equation. Figure 5 shows focal length change according to the wavelength. In both cases, the focal length is measured from the principal plane to the maximum point of electric field intensity. For the case of ADL, the principal plane is located at 892 nm left from the right most vertex. For the case of double-convex lens, it is located at 532 nm left. Blue lines depict the results calculated by 2D RCWA simulations and green lines by finite element method (FEM) based 3D full-vectorial simulations. It is shown that results from the real device are well-matched with the results based on reduced dimension. It means that the assumption addressing the proposed MIM and MII configurations to the effective media is reasonable one. Comparing Figs. 5(a) and 5(b), it is shown that the variation of the focal length according to the wavelength is much smaller for the achromatic lens than the double-concave lens. Standard deviations are 168 nm and 668 nm, respectively. That is, chromatic aberration is clearly reduced.

Fig. 5 Focal length changes of (a) the achromatic lens and (b) the double-convex lens according to the variance of the wavelength are shown. In both cases, target focal length is set to be 4 μm at 633 nm wavelength. Blue lines represent reduced dimensional computation results based on the effective indices, and green lines represent 3D results applying exact SPP field profiles.

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Detailed field profiles for some cases are shown in Fig. 6. Each column depicts electric field intensity at 500 nm, 633 nm, and 700 nm wavelength, respectively. All of the profiles are normalized by the same scale. Besides decrease of the chromatic aberration, it is shown that a blurring of focal spot is also reduced especially for the longer wavelength cases when the ADL is used. Full-width at half maxima (FWHMs) of the first row, which are the cases of the achromatic lens, are given as 540 nm, 765 nm, and 720 nm each, while FWHMs of the second row are 545 nm, 935 nm, and 985 nm. Among the whole spectral range in average, the FWHM is reduced from 937 nm to 747 nm.

Fig. 6 Electric field intensity profiles of the ADL (the first row) and the double-convex lens (the second row) at 50 nm above the substrate are shown. The fields shown in first column are those at 500 nm, those for the second column are at 633 nm, and those for the third column are at 700 nm wavelength, respectively. All profiles are normalized with the same unit. White scale bars denote 2 μm length.

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5. Conclusion

Starting from the principles of the geometrical optics, a plasmonic ADL is proposed. According to 3D FEM based simulation results, reduction of the achromatic aberration is verified compared with a double-convex plasmonic lens. Among 200 nm of spectral range, the focal length shifts by only 0.47 μm, while it moves 1.67 μm for the double-convex lens. Furthermore, blurring of the focal spot is reduced. When the proposed device is integrated with broadband nanostructures and optical nanoantennas [28, 29], it is expected that higher field enhancement can be achieved. Some plasmonic topics based on ultra-short SPP pulse [30, 31], which requires broadband plasmonic elements, can be another promising area for the application. Additionally, we hope that our methodology and design procedure can provide a neat example that bridges between theories of classical optics and plasmonics.

Acknowledgment

This work was supported by the National Research Foundation of Korea funded by Korean government (MSIP) through the Creative Research Initiatives Program (Active Plasmonics Application Systems).

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$f$	4.76 μm	$Δ_{A}$	0.0556	$Δ_{B}$	0.680	$R_{f}$	5.46 μm
$w$	2.60 μm	$t_{A}$	200 nm	$t_{B}$	100 nm	$R_{r}$	−7.44 μm
		$d_{A}$	1.32 μm	$d_{B}$	0.66 μm

Plasmonic achromatic doublet lens

Abstract

1. Introduction

2. Configuration of the proposed structure and design principle

3. Determination of the geometric parameters

4. Simulation results

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (6)

Tables (1)

Equations (4)

Optics Express