Analysis of the Focusing Characteristics of Double Negative Binary Diffractive Lens

For a compact millimeter wave imaging system it is very important to design every component into small size, for the components in a millimeter wave system are usually much larger than those in an optical imaging system due to relatively long wave lengths. In this paper, we suggest a kind of binary diffractive lens (BDL) designed using double negative materials (DNG) as the objective lens for a millimeter wave imaging system. The DNG-BDL has not only the advantage of low profile but also small f number, which will be benefit for constructing a compact millimeter wave imaging system. Several DNG-BDL are designed and analyzed using the FDTD method. The numerical results of the focal plane field of the DNG-BDL are presented, which show that the DNG-BDL with small f number has relatively better focusing characteristic than that of a double positive BDL with same f number.


Introduction
Since Veselago's theory [1] about a homogeneous isotropic electromagnetic material in which both permittivity and permeability have negative real values (μ < 0, ε < 0) over thirty year's ago, the properties of the double negative materials (DNG) have raised a great deal of interest, many interesting phenomena and applications of the DNG have been found, such as Plano-concave DNG lens [2], negative angles of refraction [3,4], enhanced focusing [4,5], backward wave antennas [6], and electrically small antennas [7].In this paper, DNG was used to design binary diffractive lens (BDL) [8,9], whose potential application is the objective lens in a millimeter wave (MMW) imaging system.In MMW imaging, sizes of the components are usually much larger than those in an optical imaging system due to relatively long wave lengths, so reducing the component's sizes is very important for designing a compact MMW imaging system.The BDL has advantages of light weight and low profile [10,11] than ordinary curve face lens, which will accordingly benefit for building up compact imaging system.Besides the aforementioned advantages of an traditional BDL made of double positive medium (DPS), the DNG-BDL with small f number (FN) is expected to have better focusing characteristics than a DPS one, which is wellfounded for as we know that Plano-concave DNG lens show good small FN [2] characteristics, and the small FN characteristic of lens is also very important for constructing a compact imaging system especially at MMW frequencies.So in this paper, we present the design formulas of the DNG-BDL; several DNG-BDL are designed at frequency of 94GHz and analyzed using FDTD method which is powerful for analyzing DNG problems [12,13]; the focal plane fields of the DNG-BDL are presented to show the focusing characteristics of the DNG-BDL with small f number.

Design of the DNG-BDL
The basic characteristic of an imaging lens is that it can focus the plane wave to a focal point.In practice, imaging lens can not focus plane wave absolutely to a point, it can only focus the plane wave to a small region, and this will be represented by a main narrow lobe in the focal plane field (FPF) intensity distribution which is drawn versus a line perpendicular to the propagation direction, see Figure 1, there may be small side lobes which will degrade the imaging quality.The field distribution at the focal plane is actually the field sum of the electromag- netic waves passing through a lens, and the reason why energies concentrate into the main beam is that the phases of the electromagnetic waves reaching to the focal point have nearly same phases.So the basic rule for design a focusing lens is that every wave path to the focal point should be equal, for example the wave path ABCO in Figure 2 should equal to wave path DEFO.Following the design rule just mentioned, take a 1/2 2 BDL (2 2 sub-zones in each zone) as an example [14], the distance from the focal point to the lens are where r n is the distance from the focal point to the lens , n is the number index of the sub-zones, f is the focal length, λ is the wavelength.It is clear that there is λ/2 2 difference (phase difference is π/2) between two neighboring r n .
Waves propagated in DNG medium generate negative wave path differences compared to waves propagated in the air, which is opposite to that in DPS medium.Dif- ferent height of sub-zones will generate wave path difference and this will compensate for the difference among all r n , then each wave path to the focal point will be equal.Accordingly, the sub-zone step height d is determined by the following equation where k d is the wave number in the DNG medium and according to [12] For 1/2 k BDL, there are 2 k sub-zones in each zone, and the path difference between two neighboring subzones is λ/2 k , then the parameters of this BDL are:

Numerical Results of the DNG-BDL
Among numerous methods for solving DNG problems, FDTD method [15] is a relatively convenient and powerful one, so FDTD method is used to analyze the DNG-BDL in this paper.There existing three types of frequency dispersive FDTD methods: the ecursive convolution (RC) method [16], the auxiliary differential equation (ADE) method [17] and the Z-transform method [18].
The ADE FDTD method is selected here and the Drude model [12,15] is used to model the DNG medium where permeability μ (ω) and permittivity ε (ω) take the following forms: where ω p is the material's resonant frequency.The perfect lens (μ r = -1, ε r = -1) proposed by Pendry in [12] is simulated at first to test our code.The thickness of the DNG slab (denoted by the dashed line) was d = 100 cells in the direction of propagation and the source-toslab distance was 50 cells in front of the slab, according to Pendry's theory the focus in the slab would occur at the center of the slab and the focus outside the slab would occur at 50 cells beyond the back of the slab.The electric field intensity obtained by our program at frequency 94 GHz after 2000 time steps is shown in Figure 3, where S denotes the point source; the focus occurring in and outside the slab agrees well with Pendry's theory.
The BDL designed and analyzed here are axisymmetric, so a symmetry boundary (see Figure 4) is set in the computation model to save computation space; the total FDTD region is enclosed by the PML and divided into total-field and scattered-field, the plane wave source is generated by 1D-FDTD method [19] and introduced through the connection boundary of the total-field and scattered-field by a combined total-scattered field formulation [20]; the focal plane field (FPF) is then computed through Stratton-Chu integral of the field at the output-boundary.
The FPF (f = 30e-3) of a 1/2 DPS-BDL (ε r = 2.2, contains five zones) was computed using the FDTD code and compared with that obtained by boundary element method (BEM) as shown in Figure 5, the two results agree well with each other.
Then several DNG-BDL are designed and analyzed using the FDTD method.Shown in Figure 6(a) is the     structure of a 1/2 2 DNG-BDL (μ r = -1, ε r = -1) with four zones, focal length f = 105 mm and diameter D = 105 mm (FN = 1).The FPF of the DNG-BDL was computed using the FDTD method and compared with that of a DPS-BDL (ε r = 2.2), see Figure 6(b).The structure of the DPS-BDL with same number of zones, focal length and diameter as that of the DNG-BDL is given in Figure 6(c).It is clear from Figure 6 that the FPF side lobes of the DNG-BDL are lower than that of the DPS-BDL, as has been explained in section 2 that lower side lobes of the DNG-BDL will be benefit to the imaging qualities.
For BDL of small FN, the FPF side lobes of the DNG-BDL are relatively even much lower than that of the DPS-BDL, an example is shown in Figure 7 where the focal length of the 1/2 2 DNG and DPS BDL (five zones) are f = 10 mm, diameter D = 47 mm (FN = 0.21).The FPF of two DNG-BDL with μ r = ε r = -2 and two DPS-BDL with ε r = 3.78 were also computed.Shown in Figure 8 are the results of the BDL with focal length f = 105 mm and diameter D = 105 mm (FN = 1); shown in Figure 9 are results of the small FN BDL with focal length f = 10 mm, diameter D = 47 mm (FN = 0.21).The results in Figure 8 and Figure 9 also support the conclusion that smaller FN DNG-BDL has relatively lower side lobes than that of DPS-BDL with same FN, and will get better imaging quality as a result.

Conclusions
The DNG-BDL which has good characteristics of low profile and small f number is studied in this paper.The design formulas of the DNG-BDL are presented and several DNG-BDL are designed and analyzed using the FDTD method.The focal plane fields of several DNG-BDL are given and compared with that of DPS-BDL; the compared results show that the DNG-BDL with small f number has better focusing characteristics than that of DPS-BDL, and this property makes it very useful to be taken as the objective lens to construct a compact millimeter wave imaging system.

Figure 1 .
Figure 1.Focal field distribution for a typical imaging lens incident by plane wave.
μ r and ε r are both negative, so from equation (2) we get 2

Figure 3 .
Figure 3.The electric field intensity over the DNG slab simulation space.