1. Introduction

Polarization technology is an important branch of optical technology. Polarization dependent devices such as polarizer, beam splitter and phase retarder are widely used in the fields of optical detecting [1], beam shaping [2] and information reading-writing [3]. However, traditional polarization dependent devices have several deficiencies. First of all, the devices employing the double refraction effect of birefringent crystals are usually big and relatively heavy [4]. Secondly, in region of the infrared radiation, it is difficult to find suitable birefringent crystal. Except a kind of line grid polarizer has been used extensively for a long time [5], other polarization dependent devices are rarely reported.

In our previous work, a structure lens based on sub-wavelength square hole array on metallic film had been developed [6, 7]. This structure lens has the virtues of light weight and high compactness; it also has the potential for realizing large numerical aperture. However, due to the symmetry of the square hole, the structure lens is independent of the incident polarization. If the holes are unsymmetrical, the phase retardations of the transmitted electromagnetic waves for different polarizations will be different. In this paper, the phase retardations of a rectangular hole are analyzed and calculated. Based on the rectangular holes array, a method is proposed to realize a type of phase modulation devices which modulate the phases independently for two eigen polarizations. The devices manufactured by this method are of light weight and high compactness which are also suitable for the use in region of infrared radiation.

2. Design method

The polarization characters of a rectangular hole can be determined by solving the Helmholtz equation. Considering a rectangular hole with sides a_x and a_y perforated on the metallic film with thickness h, if h>λ_o, and λ_o/2<a_x, a_y.<λ_o, (here, λ_o is the incident wavelength) the first order TE guide mode will dominate the transmitted energy [8]. In this paper, metals are treated within the perfect conductor approximation (PCA), so our results are qualitatively accurate in the far-infrared to microwave frequency regime [9–11]. Combining the Helmholtz equation with the PCA, the electric field in the hole can be derived and expressed as:

Where k_o is the wave vector of the incident wave, x_o, y_o are the unit vectors of x, y directions, and A_x, A_y are the magnitudes of the TE₀₁ and TE₁₀ guide modes in the rectangular hole. From Eq. (1), the phase retardations for these two guide modes after they get through the hole can be obtained as:

These two equations indicate that the phase of the transmitted electric field can be modulated by varying the sides or depth of the hole. Based on the design regulation of traditional diffractive optical devices, when the phase modulated scope can cover the range of 2π, the considered structures will be suitable for realizing phase modulation devices. Figure 1 gives the phase retardation as a function of the side for different film thicknesses obtained from Eqs. (2) and (3). In order to guarantee the 2π phase modulation ability, it can be seen that the hole should be deeper than 2/√3 λ ₀. The figure also shows that the phase retardation is quite sensitive to the sides of the hole when the hole’s sides are close to 0.5λ_o, which is disadvantageous for the practical structure fabrication since it would require a rigorous fabrication tolerance. By this consideration, the minimum side is selected being larger than 0.5λ_o in our design.

 

Fig. 1. The phase retardation of the first order TE mode varies as a function of its related side a for a rectangular hole on the metallic film with various thicknesses.

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For the rectangular hole-structures, in the case of x-polarized incidence, the primary guide mode is TE₀₁ mode whose phase retardation is determined by a_y; in the case of y-polarized incidence, the primary transmitted energy is TE₁₀ mode whose phase retardation is determined by a_x. With the type of structures behaving like a birefringent crystal, the incident light phase can be modulated conveniently, which is suitable for forming various kinds of polarization dependent devices. During the investigation, a design rule can be followed for constructing a polarization dependent device. Firstly, acquiring the phase retardation functions φ_x(x,y) for x polarization incidence and φ_y(x,y) for y polarization incidence according to the phase modulation function of the device. Secondly, choosing a proper distribution pattern for the holes on the metallic film so that the coordinates (x_i,y_i) of each hole can be determined. Because the performance of this kind of device is closely related to the average number of holes per unit area [7], in principle if the hole density on the film is high enough, the holes can be arranged in periodical lattices, quasi-periodically even randomly distributed arrays. Since the surface of perfect conductor can support electromagnetic surface modes by drilling an array of holes on it [12–15], and the surface modes are considered as the origin of resonantly enhanced transmission [16–19]. Here, a periodically arranged array is employed for achieving a higher transmittance. Thirdly, the sides of all the holes in the array should be either parallel to x axis or parallel to y axis so that each hole can give the right phase retardations for x- and y- polarized incidence independently. Finally, calculating the phase retardations and getting the distributions φ_x(x_i,y_i), φ_y(x_i,y_i) for each hole, as well as their geometrical parameters based on Eqs. (2) and (3).

3. Design example

A bifocal lens with two focal planes for different polarizations of the incident light was designed as an example by our approach. The incident light is in the infrared range with wavelength of 10.6μm. As we know, the perfect conductor approximation will invalidate if the dimensions of the structure can compare to the skin depth [20]. In our case, the refractive index of a typical metal Au is about 12.5+55i at the wavelength of 10.6μm[21], the calculated skin depth is only 0.064μm which is very small comparing to the dimension of the hole-structure. Therefore, it is reasonable to take the metal film as a perfect conductor. The lens consists of square latticed rectangular holes with period of λ_o on the metallic film with thickness of 1.8 λ_o. Its aperture is 280μm and its focal length for the x-polarized incident plane wave is 200 μm, for the y-polarized incident plane wave is 150μm. The structure of this bifocal lens and the definition of coordinate are shown as Fig. 2(a).

 

Fig. 2. (a). Diagram of the metallic bifocal-polarization lens; (b) the corresponding phase retardation functions for y- and x- polarized incidence of the holes in the frame formed by white-dotted line in Fig. 2(a).

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For the lens with focal length f, the phase difference φ(r) between the center of the structure and the location r away from the center can be determined by the following expression:

Here an integer m is properly set to guarantee the phase modulation scope being in the range (π, 3π) (see Fig. 1). When 3π phase retardation is taken in the center of the designed bifocal lens, the obtained sides of the rectangular hole is in the range (0.52λ_o, 0.905 λ_o). Figure 2(b) shows the phase retardation for the holes at the line crossing through the lens’ center parallel to x axis. In this figure the continuous curve represents the phase distribution of an ideal lens with equivalent focal length; each red point on the curve is the phase corresponding to a rectangular hole. The abscissa represents its location and the ordinate of it represents its phase retardation. It should be noticed that the sketch map Fig. 2(a) shows only part of the holes in the designed bifocal lens.

After all the parameters were determined, we calculated the intensity distribution of the electric field behind the lens by carrying out the following two steps: Firstly, the electric field on the plane 1 μm away from the metallic film was obtained by finite difference time domain (FDTD) method. Then the electric field behind this plane was calculated by Rayleigh-Sommerfeld diffraction integrals. In the FDTD simulation, the incident is the plane wave with unit amplitude. Considering the difference in the side lengths among the holes generally being larger than 0.1 μm, the grids in xy plane are specified to be 0.1 μm; while in the z direction, since the thickness is constant, the grid length is specified to be 0.4μm. Figure 3 gives the intensity distribution of the electric field behind the bifocal lens for x- and y- polarized incident plane wave respectively. If we define the location of the brightest point on the z axis as the location of the focus, the focal length for x polarization will be 195μm, the relative error is -2.5%; for y polarization, 146μm, corresponding to relative error -2.7%. This discrepancy is consistent with the precision of our FDTD calculation and the near-far field transformation.

 

Fig. 3. The intensity (E ²) distribution of electric field behind the bifocal lens for x- and y-polarized incident plane wave respectively. (a) and (c) are the intensity for y=0 and z from 100μm to 250μm; (b) and (d) are the intensity distribution on the optical axis for z from 100μm to 250μm;

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The spot size in the focal plane (z=200μm for x polarization and z=150μm for y polarization) can be obtained from Fig. 3. The results indicate that for x-polarization, the full-width of half-maximum (FWHM) of the focus is 10.1 μm; for y-polarization, the FWHM of the focus is about 8.5μm. While the radius of the Airy disk of traditional lenses with the same aperture and focal lengths are 9.2μm and 6.9μm. We can see that the spot sizes are close to the diffractive limitation.

The calculation results show good consistence with the designed values. For the reason that this kind of devices is formed by sub-wavelength hole array on thin metal film and by using only one piece of it complex phase modulation functions can be realized. Equipments using it will naturally owe the virtues of light weight and high compactness. Moreover, its design and manufacture are simple than the traditional polarization sensitive optical devices, and it is suitable for infrared regime where the polarization sensitive devices is hard to be achieved due to the lack of usable birefringent crystals. All of these make us believe that this kind of devices will provide great potential for polarization involved applications.

4. Conclusion

In conclusion, a method has been introduced to achieve polarization dependent devices by using asymmetrical hole arrays on metallic film. In order to testify this method, we designed a bifocal lens used for 10.6μm incident wavelength and the phase modulation ability of it was studied by using numerical method. The calculation results show that this device not only possesses the designed polarization properties, it also bears the perfect focusing properties for the focus sizes on both focal planes are close to the diffraction limitation. Hence this method opens an avenue to realize various polarization dependent devices with light weight and high compactness.

This work was supported by 863 Program(2007AA03Z332),973Program (2006CB302900) of China and the Chinese Nature Science Grant (60678035, 60727006). The authors thank Haofei Shi, Lifang Shi and Qiling Deng for their kind contributions to the work.