## Abstract

A novel method is proposed to manipulate beam by modulating light phase through a metallic film with arrayed nano-slits, which have constant depth but variant widths. The slits transport electro-magnetic energy in the form of surface plasmon polaritons (SPPs) in nanometric waveguides and provide desired phase retardations of beam manipulating with variant phase propagation constant. Numerical simulation of an illustrative lens design example is performed through finite-difference time-domain (FDTD) method and shows agreement with theory analysis result. In addition, extraordinary optical transmission of SPPs through sub-wavelength metallic slits is observed in the simulation and helps to improve elements’ energy using factor.

©2005 Optical Society of America

## 1. Introduction

In recent years, there has been much interest in metallic subwavelength structures excited by the discovery of extraordinary optical transmission phenomena though subwavelength metallic aperture array [1–3]. Most recently, Lezec et. al. reported their experimental observation that the light emerging from a subwavelength aperture surrounded by periodic corrugation on the exit side of a metallic thin film displays highly directed beaming with a low divergence of only 2~3 degrees [4].It is generally believed that the surface plasmon polaritons (SPPs), resonantly excited in the corrugated metallic surface, are account for these phenomena [1–7]. This opens up a new avenue for new types of nano-optics device with thin metallic films. As an application example, Sun and Kim have designed a metallic nanolens with slits perforated on a thin metallic film [8], which bring different phase retardations to the light transmitted through them for variant slit depths and can be utilized to implement beam deflection and focusing if appropriately designed.

In this paper, we investigate a structure consisting of nano-slit array with variant width formed on thin metallic film and show that this structure can be employed in manipulating light, as indicated at the end of Reference 8. Each nano-slit in the metallic film is designed to transmit light with specific phase retardation controlled by the slit width instead of depth, and hence arbitrary light phase modulation becomes available. It is worth to note that the whole element is formed on a planar thin film that is convenient for miniaturization and integration. An illustrative lens design example is given and simulated by two-dimensional FDTD method. Extraordinary transmission of light through sub-wavelength metallic slits is also observed in the nano-slits lens with a transmission enhancement factor of about 1.8.

## 2. Principle

Surface plasmon polaritons are a special kind of electromagnetic field, which can propagate along metallic surfaces while keeping bounded near the surface without radiation away [9,10]. Considering two closely placed parallel metallic plates, the SPPs of each surface will be coupled and propagate in the form of a waveguide mode for TM polarized case. The complex propagation constant *β* can be calculated from the equation [11]

where *k*
_{0} is the wave vector of light in free space, *ε _{d}* and

*ε*are the relative dielectric constant for the metal and the materials between slits, and

_{m}*w*is the slit width. Fig.1 plots the complex propagation constant of SPPs in the waveguide for variant slit width. The used metal here is Ag with

*ε*= -17.36 + i0.715 at the wavelength of 650nm, and

_{m}*ε*=1 for air. It can be seen clearly that the value of

_{d}*β*, both real and imaginary parts, increases steadily with decreased slit widths but grows rapidly for slits below about 20nm, indicating the greatly increased coupling of SPPs. Because of the surface wave property, Re(

*β*/

*k*

_{0}), determining the phase velocity of SPPs in the slit, is always above the dotted line which stands for the plane EM field in air. The imaginary part of β represents the decibel loss coefficient per unit length, is usually ignorable for light propagation in short slits.

Obviously, the dispersion relation between the effective refractive index and slit width implies a potential way of phase modulation by simply tuning the slit widths. Let’s consider the process of light transmitted through the nano-scaled metallic slits. When TM polarized incident plane wave impinges the slit entrance, it will excites SPPs. Then the SPPs propagate along slit region with corresponding propagation constant until they reach the exit where the SPPs radiate into light in free space.

Considering slits as dielectric films with finite length of *d* and normal incident light, we can express the phase of light transmitted through the slit as [12]

*ϕ*
_{0} is the initial phase at the entrance. Δϕ_{1}=arg[(*n*
_{1}-*β*/*k*
_{0})/(*n*
_{1}+*β*/*k*
_{0})] and Δϕ_{2} = arg[(*β*/*k*
_{0} -*n*
_{2})/(*β*/*k*
_{0} +*n*
_{2})] are the accompanied phase changes at the entrance and exit interface, where SPPs mode are excited and reradiated into free space, *n*
_{1} and *n*
_{2} are the refractive indices of the media outside the slit and arg is the argument of complex number. In particular case where similar media are taken, Δϕ_{1} and Δϕ_{2} have same value but opposite signs and can be cancelled together. *βd* is the phase retardation of SPPs propagation in the slit. The last term *θ*, originating from the multiple reflections between the entrance and exit interfaces, can be calculated with the following equation

The complete relation of *ϕ* and slit width, length and used metal and dielectric materials can be readily obtained by combining all the above equations. Complex although it looks, both physical analysis and numerical simulation show that only the term *βd* plays a dominating role.

From Eq. (2), variant phase retardation is obtainable by simply tuning the slit depth *d*, as in Reference [8] and conventional micro optics dielectric lens design process. Similar phase modulation occurs for variant slit widths. Let *d*=300nm, for example, slits of widths 40nm and 20nm yield the phase retardations of 0.49π and 0.87π, respectively. If a collection of slits of variant widths is arranged closely in one direction and illuminated from left side, the emitted light can be modulated with desired quasi-continuous phase distribution. The nano-slit’s independent control of phase at each slit offers great flexibility in designing optical structures with arbitrary phase distribution at the exit surface.

## 3. Illustrative lens design and simulation

To illustrate the above idea of modulating phase, a metallic nano-slits lens is designed. The parameters of the lens are as follows: *D* = 4 μm, *λ* = 0.6 μm, *λ*= 0.65 μm, *d*=0.5 μm, where *D* is the diameter of the lens aperture, *f* the focus length, *λ* the wavelength and *d* the thickness of the film. The two sides of the lens is air. The schematic of lens is given in Fig.2, where a metallic film is perforated with a great number of nano-slits with specifically designed widths and transmitted light from slits is modulated and converges in free space.

The required phase distribution of emitted light at position *x* can be obtained readily according to the equal optical length principle

where *n* is an arbitrary integer number. A set of slits is then designed with matched widths determined by combining Eq. (1) and Eq. (4), as plotted in Fig. 3. The space between every two adjacent slits is assumed to be larger than the metal’s skin depth, about 24nm for Ag at 650nm, to prevent the interaction of light. To realize the phase modulation crossing the region of 2π, a wide range of slit width is needed. As seen from Fig.1, narrower slits provide larger propagation constant, hence greater phase retardation and shorter slit length, but leads to great fabrication difficulty and low filling factor as well. On the other hand, wider slits may be easily manufactured, but usually require deep slit and result in increased phase discontinuity. In our practical design, the used slit width ranges from 10nm to about 70nm with slit depth of 500nm, as a compromise of acceptable manufacture difficulty and optical quality.

FDTD simulation is performed to demonstrate the metallic lens’s validity. The simulation region size is 4 μm× 3 μm with uniform cell of Δ*x* = Δ*z* =2nm. Around the simulation region is the perfect matched layer absorber. The wavelength of incident TM polarized light is 650nm and the dielectric constant used for silver at this wavelength is *ε* = -17.36 + i0.715 . After 6000 steps of calculation, the resulting Poyinting vector is obtained and showed in Fig. 4 (a). A clear-cut focus appears about 0.6 micron away from the exit surface, which agrees with our design. The cross section of focus spot in x direction is given in Fig. 4 (b), indicating a full-width at half-maximum (FWHM) of 270nm. Another point worth to note is that the extraordinary transmission phenomena reported in Reference [1–3] also occur in our simulation. The calculated enhancement factor of transmission efficiency through the metallic film is about 1.8, with a measured total light transmission of 60.28% and the fraction of slits area of 33.35%. Clearly, this extraordinary optical transmission is mainly due to SPPs, and this property is very useful in its potential practical application.

## 4. Conclusion

In this article, the structure based on nano-slits of different width formed on thin metallic film is proposed as a new method of modulating light phase. A focusing lens with numerical aperture of 0.95 is designed as an illustrative example. Numerical simulation through FDTD shows a clear focus spot with FWHM width of 270nm. The extraordinary light transmission effects of SPPs through sub-wavelength slits is also observed in the simulation with a transmission enhance factor of about 1.8 times. These advantages promise this method to find various potential applications in nano-scale beam shaping, integrate optics, date storage, near-field imaging, ect.

## Acknowledgments

This research was supported by the Foundation of the State Key Laboratory of Optical Technologies for Microfabrication. The authors thank their colleagues for their discussions and suggestions to this research.

## References and links

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