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In order to miniaturize and integrate conveniently in THz quasi-optical systems, binary axicons, based on binary optical ideas, are introduced in our paper and designed for generating pseudo-Bessel beams at THz frequencies. The designed binary axicons are easier to fabricate than holographic axicons, more compact and thus less lossy in the material when compared with classical cone axicons. To calculate the electromagnetic fields diffracted by binary axicons precisely, a two-dimension finite-difference time-domain (2-D FDTD) method in conjunction with Stratton-Chu formulas are employed in this paper. Applying this method, the properties of pseudo-Bessel beams produced by binary axicons are analyzed, and a brief summary is given in the end.
©2009 Optical Society of America
1. Introduction
In 1987, Durnin discovered a nondiffracting zero-order Bessel function solution to the free space scalar wave equation [1], and it can be written in the form
where J 0 is the zero-order Bessel function of the first kind, ρ 2 = x 2 + y 2 , k ⊥ 2 + k z 2 = k 2 = (2π/λ)2, k ⊥, and kz are the radial and longitudinal wave numbers, respectively, λ, is the free space wavelength and ω is the angular frequency. The ideal Bessel beams exhibit many novel properties, such as propagation invariance, long depth of field, extremely narrow intensity profile and so on. Therefore, they are useful for both the technical and physical applications [2]. In optics they are suitable for optical manipulation of particles [3, 4], collimation and measurement [5], and they may also be applicable in imaging [6, 7] and measurement in THz quasi-optical systems. Much attention has therefore been devoted to generating Bessel beams. However, the ideal Bessel beams can not be exactly generated, due to their infinite lateral extent and energy. Only their approximations known as the near or pseudo-Bessel beams can be obtained physically [2, 6-8]. Currently, numerous ways for generating pseudo-Bessel beams have been proposed, among which using axicon is the most popular method [6-12], owing to its simplicity of configuration and easy realization. However, at THz quasi-optical ranges classical cone axicons are usually bulk ones and therefore have many disadvantages, like heavy weight, large volume and thus increased absorption loss in the material. These limitations together make them extremely difficult in miniaturizing and integrating in THz quasi-optical systems. To overcome these problems, binary axicons, based on binary optical ideas, are introduced in this paper and designed for producing pseudo-Bessel beams at THz frequencies. The designed binary axicons are more convenient to fabricate than holographic axicons [11, 12] and, become thinner and less lossy in the material than classical cone axicons [6-8]. In order to analyze binary axicons accurately when illuminated by a plan wave at THz frequencies, the rigorous electromagnetic analysis method, that is, a two-dimension finite-difference time-domain (2-D FDTD) method for determining electromagnetic fields in the near region in conjunction with Stratton-Chu formulas for obtaining electromagnetic fields in the far region, is adopted in our paper. Using this combinatorial method, the properties of approximate Bessel beams generated by the designed binary axicons are analyzed, and a brief summary is presented at last.
The present paper is organized as follows. The design of a binary axicon is introduced in Section 2. Section 3 describes the rigorous electromagnetic analysis method briefly. The equivalent performance between a classical cone axicon and a binary axicon is demonstrated in Section 4. The designed binary axicons are analyzed in Section 5. In the last Section 6, our summary is given.
2. Binary axicon design
A classical cone axicon, introduced firstly by McLeod in 1954 [13], is usually a bulk one, as illustrated in Fig. 1(a), in which D is the aperture diameter and γ is the prism angle. Based on binary optical ideas, the profile of a binary axicon, whose performance required is equivalent to that of a bulk one, can be easily formed. Assuming straight-ray propagation through the bulk axicon, the relation between the phase retardation φ(ρ) and the surface height h(ρ) is given as [14]
where k is the free space wave number, n 0 and n 1 are the refractive indexes of the air and the axicon, respectively. To generate the continuous profile of the binary axicon, the equivalent transformation can be used by [15]
The continuous profile of the binary axicon produced by Eq. (3) is shown in Fig. 1(c). For the multilevel axicon, the profile is quantized into equal height step ∆. The quantized height is given by
where ∆ = h max/M, h max = λ/(n 1 - n 0) and M is the number of levels. Eq. (4) generates the multilevel profiles of the binary axicon. The schematic diagram of the 4-level binary axicon is illustrated in Fig.1 (d). It is known that the larger the number of levels is, the higher the diffraction efficiency is, however, the higher the difficulty of manufacture becomes. Therefore, the compromise between the diffraction efficiency and the difficulty of manufacture should be considered when determining the number of levels. In our work the selection of the 32-level binary axicon is made. From Figs. 1(c) and (d), we can see easily that the designed binary axicon is not only more compact than the classical cone axicon, but also simpler to fabricate than the holographic axicon.
Fig. 1. The design process of a binary axicon. (a) A bulk axicon. (b) An axicon removed the unwanted material (red part). (c) An equivalent binary axicon with continuous profile. (d) An equivalent binary axicon quantized into four levels.
3. Rigorous electromagnetic analysis method
Because of rotational symmetry of the binary axicon, a 2-D FDTD method is applied to evaluate the electromagnetic fields diffracted by the binary axicon in the near region. The computational model of the 2-D FDTD method is shown schematically in Fig. 2, in which the binary axicon is utilized to convert an incident beam into a pseudo-Bessel beam. To stimulate the entire 2-D FDTD grid, a total-scattered field approach is applied to introduce a normally incident plane wave. In this approach the connecting boundary serves to connect the total and the scattered field regions, and is the location at which the incident field is introduced. Because of the limitation of computational time and memory, the computational range of the 2-D FDTD method is truncated by using perfectly matched layer (PML) absorbing boundary conditions (ABCs) in the near region. Therefore, in order to accurately determine the electromagnetic fields in the far region, Stratton-Chu integral formulas are applied and given by [16]
where r⃗ = (ρ, z) and r⃗' = (ρ′, z′) denote an arbitrary observation point in the far region and an source point on the output boundary of the 2-D FDTD model, respectively; unit vector n⃗ is the outer normal of the closed curve, L, of the output boundary; G 0 (r⃗, r⃗′) = -iH 0 (2)(k|r⃗ - r⃗′|)/4 , is the 2-D scalar Green’s function in free space and k is the wave number; ω is the angular frequency; ε and μ are the permittivity and permeability, respectively.
Fig. 2. Schematic diagram of 2-D FDTD computational model, where the 8-level binary axicon is embedded into FDTD grid.
4. Demonstration of equivalence
To demonstrate the equivalent performance between the bulk axicon and the designed binary axicon, Fig. 3 shows the on-axis intensity distributions for both the bulk axicon and the 32-level binary axicon. In this case both axicons, with the same aperture diameter D = 20λ and prism angle γ = 100, are normally illuminated by a plane wave of unit amplitude. Other parameters used in Fig. 3 are as follows: an incident wavelength is λ = 0.32mm (f = 0.94THz), the refractive indexes of the axicon and the air are n 1 = 1.4491 (Teflon) and n 0 = 1.0 , respectively. Two distributions exhibit some differences in the near region (z < 75λ). The reason is that the binary axicon suffers more from edge diffraction and truncation effects [7]. The effects can also be seen from Fig. 4(b), which has more burr than Fig. 4(a) in the near region. However, two curves show a good agreement in the region (z > 75λ), where the propagating beam can be best approximated by the Bessel beam in terms of its intensity profile. Thus, the performance of the designed binary axicon is equivalent to that of the bulk one. In order to further demonstrate the equivalent effect between two axicons, we extend our 2-D FDTD calculated region to 200λ along z-axis, and display their electric-field amplitudes in a pseudo-color representation in Fig. 4. It can also be seen that the designed binary axicon has the same performance as the bulk one.
Fig. 4. Electric-field amplitude patterns plotted in a pseudo-color representation. (a) For the bulk axicon. (b) For our designed binary axicon.
5. Properties of pseudo-Bessel beam
In order to study the properties of a pseudo-Bessel beam, the other 32-level binary axicon having aperture diameter D = 22λ and prism angle γ = 120, are examined. Other parameters used in this example are the same as in Fig .3. When this axicon is normally illuminated by a plane wave of unit amplitude, its axial and transverse intensity distributions at three representative values of z : z = 0.8Zmax , Zmax and 1.2Zmax are shown in Figs. 5 (a)-(d), respectively. It can be seen clearly from Fig. 5(a) that the on-axis intensity increases with oscillating, and reaches its maximum axial intensity then decreases quickly, as the propagation distance z increases. The maximum value of on-axis intensity in Fig. 5(a) is 10.297, located at Zmax = 125 .4λ . As shown in Fig. 5 (a), if L max is defined as the maximum propagation distance of a pseudo-Bessel beam, we can obtain L max ≈ 230λ . In addition, according to geometrical optics [7], a limited diffraction range L = 227λ is estimated by: L=D/(2 tan β) and sin(β + γ) = n 1 sin γ . We discover two results almost coincide. From Figs. 5 (b)–(d) we can observe that their transverse intensity distributions are approximations to Bessel function of the first kind. The radii of their central spot are only about 3.5λ. This indicates that the transverse intensity distribution of pseudo-Bessel beam is highly localized. It is also interesting to point out that the radius size of 3.5λ is very close to the value of 3.4λ, which is determined roughly from the first zero of the Bessel function (2.4048λ/(2π sin β)). [7].
Fig. 5. The axial and transverse intensity distributions for the designed binary axicon. (a) The on-axial intensity versus propagation distance z. (b) The transverse intensity distribution at z = 0.8Zmax plane. (c) z = 1.0Zmax . (d) z = 1.2Zmax.
6. Summary
In our paper, several binary axicons have been designed based on binary optical ideas; and they are analyzed by rigorous electromagnetic calculation method. Simulation results show that the designed binary axicons, when illuminated by a plane wave, can generate pseudo-Bessel beams at THz frequencies; and their performance are equivalent to those of the bulk axicons. The designed binary axicons have advantages over bulk axicons because of their small volume, light weight and low absorption loss in the material, and compared with holographic axicons, they are easily fabricated. With the fast development of terahertz technology and applications, we believe that pseudo-Bessel beams generated by the binary axicon have a promising application in THz quasi-optical compact systems.
Acknowledgments
This work is supported by NSFC under grant 60621002, and the Natural Science Foundation of Fujian Province of China (No.A0610027).
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