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In this paper the frequency dependence and focusing performance in focal plane of a diffractive lens is analyzed by FDTD method at millimeter wavelengths. Binary lens and four-level lens are considered. The field distribution on the focal plane of the diffractive lens for the incident wave at different frequency is presented, which shows the frequency dependence and focusing performance of the lens.
©2002 Optical Society of America
1. Introduction
The Fresnel diffractive lens (FDL) is a planar device that produces focusing and imaging of electromagnetic waves based on the effects of diffraction, rather than refraction. The diffractive lens transforms a normally incident plane wave into a converging wave, concentrating to a point, focal point of the lens. Therefore, at millimeter wavelengths the diffractive lens can be used as a high gain antenna instead of paraboloid antenna. The focusing performance is characterized by diffraction efficiency. FDL have been used at frequencies from the microwave range to the X-ray region for a long time. The characteristics of the FDL such as efficiency and bandwidth have been analyzed [1]. Because of the increased influence in electromagnetic coupling effects along the surface boundary of the FDL, which may affect the performance of the efficiency and bandwidth, some new techniques that account for such effects is being developed [2]. When the FDL is used in an imaging system the field-of-view of the lens is analyzed in [3]. However, these analyses only consider the diffraction efficiency of the lens at single frequency. In this letter, we will analyze the diffraction field distribution in focal plane of the lens in a frequency band to obtain the bandwidth performance of the lens.
Analysis
FDTD method is used. For simplicity only 2 dimensional problems are considered. In ref. [2,3], FDTD method has been used to get the field distribution along the surface of the lens, and then Stratton-Chu formula is used to calculate the diffraction field in the focal plane of the lens. In this paper the FDTD method is implemented directly to obtain the diffraction field in the focal plane of the lens. The configuration of the structure to be analyzed is depicted in Fig.1. Computation space is surrounded by a perfectly matched layer (PML), which is backed by conducting wall. The FDL is put in a hole in a conducting screen that is extended into PML. A planar wave is incident on the FDL. We want to see how the incident wave is transformed to a converging wave, concentrating to the focal point of the lens. FDTD formulations have been presented in [4], so it is omitted here for saving space.
A planar wave source is placed in front of the FDL along a plane as shown in Fig.1 to model the incident wave. It is given by
Where n is time step number.
Calculation results
Binary lens is considered firstly, its dimensions are given in Fig.2 from design formula. The lens radius is about 25 mm and the dielectric constant is 2.1 (Teflon). The thick of the lens d = 3.5529 mm. The f-number of the lens, the ratio of focal length of the lens to the lens diameter, is designed as 0.5.
By implementing FDTD method directly we can know how the incident wave is transformed to a converging wave, concentrating to the focal point of the lens, and much other details of the diffraction mechanism. These details may be ignored when Stratton-Chu formula is used to obtain the diffraction field of the lens. Fig.3 shows the contour pattern of the diffraction field at different time step. In Fig.3 (a) the incident wave is just arrived the FDL. The FDL is positioned between 180ds and 202ds along x-axis. In Fig.3 (b) the incident wave has passed through the FDL, and at the top and bottom of the pattern there are some reflected wave by the conducting screen. Fig.3 (c) shows the diffraction field has converged to the focal point region, however there are also obviously diffraction fields in other region in focal space of the lens. Fig.3 (d) and (e) shows the process of the diffraction wave passing through the focal point region. It can be seen that the diffraction field begin diverging after passing through the focal point. Therefore we know besides the diffraction field in the focal point region the incident wave also be diffracted to other region, which is the cause of the binary lens has relative low diffractive efficiency.
Because the field shown in Fig.3 is in time domain the bandwidth performance of the lens cannot be observed. So the field in the cross plane, which is shown in Fig.1, is recorded and transformed by Fourier transform to frequency domain. In Fig.4 the ratio of diffractive field to the incident field at different position point in the focal plane is given in the frequency band from 70GHz to 120 GHz. Ez (360,185) denote the field in the focal point that is at x=360ds, y=185ds. Ez (360, 200) and Ez (360, 250) can be explained similarly.
Fig.5 shows the ratio of the power density of diffraction field to that of the incident wave along the focal plane for different frequencies, where the y coordinate of the focal point has been shifted from y=185ds to y=0ds. It can be seen that besides the focal point there are also other stronger diffraction field point.
If we consider the bandwidth as 3dB band, that is, the power density of diffraction field decrease from its maximum to the about half of the peak value, this lens has bandwidth from 87GHz to 115GHz approximately.
Now four-level lens, which has the same radius and permittivity as above binary lens, is considered. The outer radius of the nth zone rn is given in Table 1. The depth of zone d=1.7764 mm.
Table 1. outer radius of the nth zone
The planar wave source is also given by Equation (1). Fig.6 shows the contour pattern of the diffraction field at different time step. In Fig.6 (a) the incident wave has passed through the FDL, and at the top and bottom of the pattern there are some reflected wave by the conducting screen. Fig.6 (b) shows the diffraction field has converged to the focal point region. And different from the binary lens there are only a little diffraction fields in other region in focal space of the lens, which is the cause of the four-level lens has higher diffractive efficiency than the binary lens. Fig.6 (c) shows that the diffraction field begins diverging after passing through the focal point (x=360ds,y=200ds).
In order to get the bandwidth performance of the lens in frequency domain, Fourier transform is used to obtain the field distribution along the focal plane in the band from 70GHz to 120 GHz, which is depicted in Fig.7 in the form of contour pattern. There the y coordinate of the focal point has been shifted from y=200ds to y=0ds. In the band between 80GHz and 90GHz there are another diffraction field region besides the focal point region as shown in Fig.7. Fig.8 shows the power density of the diffraction field at the focal point. The peak of the diffraction field is at 99GHz. In the band between 80GHz and 105 GHz, the strength of the diffraction field is nearly same except the peak at 99GHz. Comparing Fig.8 with Fig.4 we can see that the four-level lens can provide a maximum diffraction field at single frequency. The binary lens cannot give this performance. Fig.9 shows the power density of the diffraction field in the focal plane for the different frequencies interested. For this lens 3dB bandwidth is about from 82GHz to 102 GHz. However if higher diffractive efficiency is required the bandwidth is only from 97 GHz to 101 GHz.
2. Conclusion
In this paper the FDL is analyzed with FDTD method. The diffraction process of the incident wave by the FDL is depicted by the contour pattern of diffraction field at different time. The diffraction fields along the focal plane of the lens for the different frequencies are also presented.
From above analysis following conclusions can be obtained.
- For the binary lens it has been seen that besides the diffraction field in the focal point region the incident wave also be diffracted to other region, which is the cause of the binary lens has relative low diffractive efficiency. This lens cannot provide a peak of diffraction field at single frequency. Though the binary lens works based on the effects of diffraction, rather than refraction, it has relative broad bandwidth at millimeter wavelength that can meet the requirement of most millimeter wave system at the present time.
- Four-level lens can focus the incident wave to the focal point region more efficiently than the binary lens, it can provide maximum of diffraction field at single frequency. For this lens the diffraction field in the region out of the focal point region is reduced. If we want the lens works with high efficiency, it will has a narrower bandwidth than the binary lens.
Acknowledgement
The author gratefully acknowledges the comments and good suggestion of reviewers. It is supported by the Foundation of Nature Science of China (69971009) and State Education Ministry.
References
1. J.C. Wiltse and J.E. Garrett, “The Fresnel Zone Plate Antenna,” Microwave J. 34,101–114 (1991).
2. D.W. Prather and S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A , 16, 1131–1142 (1999). [CrossRef]
3. W.B. Dou and C. Wan, “An analysis of diffractive lenses at millimeter wavelengths,” Microwave Opt. Technol. Lett. 31, 396–401 (2001). [CrossRef]
4. W.B. Dou and E.K.N. Yung, “Diffraction of an electromagnetic beam by an aperture in a conducting screen,” J. Opt. Soc. Am. A , 18, 801–806 (2001). [CrossRef]
