Simulation and Optimization Technique for a Multi-mirror MM Wave Reflector Radio Telescope with FPA V.B. Khaikin#1, M.K. Lebedev*1,2 , A.A. Nosich#3 #1
The Special Astrophysical Observatory of Russian Academy of Sciences #2 The St. Petersburg State University #3 Department of Mathematics, Kharkiv National University, Pl. Svobody 4, Kharkiv 61077, Ukraine 1
2
3
[email protected] [email protected] [email protected]
Abstract— The ray tracing and integral equation methods are used to optimize focal optics and focal plane array configuration of the single, dual and multi-mirror mm-wave telescopes. Simulation and optimization techniques are considered, results being presented for on-axis mm-wave reflector and RATAN-600 radio telescope. Near and far fields of RATAN-600 radio telescope in the multi-mirror mode are studied and optimized with the Method of Discrete Singularities (MDS), which uses the singular electric-field integral equation approach.
I. INTRODUCTION Using a Focal Plane Array (FPA) one can essentially expand the field of view (FoV) of a radio telescope and observe a rather extended sky area without any mechanical scanning. Two possible architectures for reflector antennas are phased and non-phased FPAs. In phased FPAs, many small size feeds (< λ/2) (similar to Vivaldi-antenna element) are used to form one or several antenna beams with a beamforming matrix. In the antennas field, such phased FPAs are called simply FPAs and are widely used in radar applications. The radio astronomical version of the FPA concept is more general: a detector array (coherent or incoherent) sampling the field in the telescope focal plane [1]. In the past years several attempts were made to realize phased FPAs for the needs of radio astronomy in the centimeter waveband (SKA, EVLA) as well. Techniques of phased FPA optimization for a reflector radio telescope are considered in [2]. A subject of our consideration is simulation and optimization techniques for a reflector radio telescope with incoherent (non-phased) tightly packed multibeam FPA (MFPA). A combination of ray tracing, GO and integral diffraction methods is used to optimize focal optics and MFPA configuration of a single, double, or multi-mirror MM wave reflector telescope.
problem occurs from the horn feed physical size that typically exceeds a focal spot or a diffraction disk size. A compact smooth-walled spline-profile horn with 95%-100% aperture efficiency [3] allows us to reach feed spacing less than Airy disk. III. RAY TRACING IN MULTI-MIRROR REFLECTOR RADIO TELESCOPE To reduce wave aberrations and expand the FoV of RATAN-600 radio telescope, long-focus optics has been suggested in the form of small tertiary quasi-elliptic mirror of double curvature (not a figure of rotation) which must be installed close to the secondary focus [4]. For initial simulation of a long focus multi-mirror scheme of RATAN600 we have used geometrical optics [5].
II. CHOICE OF MFPA FEED The choice of a proper feed is very important for MFPA optimization. Strip feeds (like a patch or λ/2 dipole) are small but not effective enough for such applications at a mm-wave radio telescope. This is due to losses in the dielectric substrate, the low radiation, aperture, spillover and coupling efficiency and narrow bandwidth. For the MFPA of a multi-beam mmwave radio telescope, we can only consider compact feedhorns providing high efficiency in the wide bandwidth but
495
Fig. 1. Example of ray tracing in the secondary and tertiary RATAN-600 radio telescope foci (top), optical focal spots in RATAN-600 radio telescope image plane without (bottom) and with quasi-eliptical (middle left) and quazi-hyperbolical (middle right) tertiary mirror, magnification of the optical scheme with a tertiary mirror M=2.5.
General 3D ray tracing methods can be used for modeling of multi-mirror mm-wave radio telescope with any optical scheme and shape of reflecting surface. In such a way, one can optimize focal optics, focal beams and focal spot size in the image plane. Example of ray tracing in the secondary/tertiary RATAN-600 radio telescope foci with and without an optimized tertiary mirror and focal spots in the image plane in comparison with Airy disk are given in Fig. 1. IV. MULTIBEAM SIMULATION OF A SINGLE/DUAL/MULTI MIRROR REFLECTOR To simulate Multibeam Radiation Pattern (MRP) of a single/dual/multi mirror reflector, we use Scalar Kirchhoff diffraction integral (SKDI)/Projected aperture method or direct Integration of Aperture Field (IAF), which gives us a far field of an antenna in the transmission mode [6]. Both methods provide good approximation for on/off-axis radiation patterns in the main and the nearest sidelobes that can be enough for simulation and optimization of the MFPA configuration of mm-wave radio telescope. Both methods need significantly less computational time than PO method in the mm band. Aperture illumination is calculated with the known radiation pattern of the array feed using GO method, and sub-reflector (focal reflectors), feed array and struts are included in the simulation as projected shadows on the mainreflector or sub-reflector aperture. The multibeam simulation technique can be used to calculate a multibeam pattern of a single/dual/multi mirror reflector mm-wave radio telescope in the principal polarization. Example of multibeam simulation of a dual reflector mm-wave radio telescope is given in Fig. 2.
Fig. 2. Example of multibeam simulation of a dual reflector mm-wave radio telescope.
Optimization of focal optics and MFPA configuration is done using the results of ray tracing and MRP simulation with a target to reach a given FoV, admissible off-axis aberrations in peripheral beams and MFPA feed spacing less than Airy disc at the wavelength. Optimization of MFPA configuration provides the tightest array packaging and beam spacing equal to or less than 2 HPBW. MFPA final configuration is a result of a few iterations allowing us to find a trade-off between horn/step size, mutual coupling, and antenna efficiency [3]. V. NEAR AND FAR FIELD SIMULATION OF A MULTIMIRROR REFLECTOR RADIO TELESCOPE WITH MDS A. MDS introduction. Electromagnetic modelling of reflectors is usually done with methods borrowed from optics (GO, PO) combined with Gaussian beams. However, these methods, based on the ray tracing, fail to fully characterize all fine wave effects and resonances that occur in the quasioptical domain of the reflectors, especially when the reflector size exceeds hundred of wavelengths (λ) and considered is the case of a multireflector system with associated wave interactions between individual reflectors. This makes economic and accurate full-wave analysis tools for multireflector antenna systems to be highly in demand by modern R&D industry. General way to build more economic numerical algorithms is to use the singular electric-field integral equation (SIE) approach and develop efficient discrete models, i.e. fast and convergent numerical algorithms having controlled accuracy.
496
In this paper, we work with the recently developed Method of Discrete Singularities [7]-[9] which uses coupled SIEs of this sort and apply MDS to accurate modelling of an electrically-large two and three-reflector extended-focus optical schemes of the RATAN-600 telescope. Note that MDS is similar to the Nystrom method of solving the SIEs in the scattering of waves by the flat strips [10] however uses different interpolation formulas.
parameter kb, which controls its collimation: the greater kb the narrower the beam. The total filed here is a sum of the field scattered by the both reflectors and the incident field: U = Usc+U0,. The function Usc has to solve the Helmholtz equation off Lq (q=1,2) and satisfy (a) the PEC boundary condition on Lq, (b) the edge conditions at the endpoints and (c) the radiation condition. In the case of E-polarization, this problem is reduced to the set of 2 coupled SIEs for the surface currents induced on reflectors, jq(s):
i 4
2
G G G ∑ ∫ H ( k r (s) − r (s ) ) j (s)ds = −U ( r (s ) ) (1) 0
q =1 L q
p = 1, 2
q
p
0
q
0
p
0
Lp
(1)
Note that (1) have logarithmic singularities in the kernels, therefore, their discretization requires an efficient and convergent numerical algorithm.
Fig. 3a. A mock-up sketch of a considered R-600 extended-focus scheme, having a parabola front-fed by a symmetric ellipse (not to scale).
Fig. 3b. A mock-up sketch of a considered R-600 extended-focus scheme, having a parabola front-fed by a non-symmetric ellipse (not to scale).
B. Problem formulation. The geometry of a generic 2-D two-element structure viewed as a RATAN-600 extendedfocus optical scheme can be seen in Figs. 3a,b. We consider two slightly different geometries with varied orientations of the elliptic subreflector and feed in the case of E-polarisation. Reflectors are assumed to be perfectly electrically conducting (PEC) and have zero thickness. The feed is a line current placed at the complex-valued source point (CSP) [7] and has time dependence e − iω t omitted in the analysis. The field generated by such a feed can be characterized by the zcomponent of the electric field, which is given by a Hankel function of a complex argument. This function simulates a beam looking in the direction φ = β and has two branch points, which lead to a cut in the real space that can be considered as a model of the real horn aperture with a length 2b. CSP has a
C. MDS Analysis. Applying a contour parameterization to Lq, one can transform SIEs (1) to another set of Cauchy-singular SIEs with smooth supplementary conditions and with a new unknown current function. This new set of SIEs is further discretized by using the quadrature formulas of interpolation type with the nodes in the nulls of the Chebyshev polynomials of the first and second kind. As a result, in the multi-reflector case we obtain coupled sets of algebraic equations, the solution of which gives us the sought surface current functions. The mathematical details of this method of numerical solution of SIEs can be found in [8]-[10]. It enables one to study the effects of the wave radiation, guidance and scattering in reflector antennas and waveguides with high accuracy using desktop computer resources. Surface currents, near and far field patterns, radiated and scattered power and gain behaviour can be readily computed for various reflector shapes, feed locations, etc. D. Numerical simulation results. In this section, the considered geometry concept is based on two elements of the RATAN-600 extended-focus optical scheme. It consists of an electrically-large offset parabolic mirror L2 with the aperture size of d1 = 816λ (f/d1 = 0.4), and an elliptic-arc fragment L1. We consider two geometries of the scheme with different orientations of the ellipse and feed. The elliptic arc is symmetric in the first case (Fig. 3a) and non-symmetric in the second (Fig. 3b), having the apertures of d2 = 57λ and d2 = 48λ, respectively. In the both cases, the ellipse is front-fed by the CSP from its first focus F1, and the second focus of the ellipse coincides with the focus F2 of the parabola. Edge illumination of the both ellipse edges is -10dB, which is provided by the CSP having kb=22 and kb=12, respectively, for symmetric and non-symmetric ellipse cases. Fig. 4 illustrates radiation patterns for the both of considered geometry cases. A narrow main pencil-like beam is observed. The spillover sidelobes of the both reflectors are all below the level of -30dB.
497
Fig. 4. Radiation pattern comparison for RATAN-600 extended-focus geometries illustrated in Figs. 1a and 1b.
Figs. 5a and 5b illustrate near field amplitude patterns for RATAN-600 with symmetric and non-symmetric ellipses, respectively. A well collimated main beam is observed in both cases, as well as small spillover lobes on the ellipses edges that correspond to -10dB edge illumination. Fig. 5c illustrates the near field of a single parabolic dish which serves in the current short-focus vertical optical scheme of RATAN-600.
Fig. 5c. Near field amplitude pattern for a single offset parabolic dish (d1 = 816λ, f/d1 = 0.4) of the RATAN-600 current short-focus optical scheme, kb = 3, β = 580, edge illumination = -10dB on both edges.
Figs. 6a, 6b and 6c demonstrate the near field phase patterns for the geometries in Figs. 5a, 5b and 5c respectively. In all cases one can see the domain of the quasi-plane-wave wavefront of the main beam reflected from the offset parabola (the Secondary mirror of RATAN-600).
Fig. 5a. Near field amplitude pattern for the considered 2-element R-600 extended-focus optical scheme. Reflectors are: offset parabolic (d1 = 816λ, f/d1 = 0.4) and symmetric ellipse (d2 = 57λ); kb = 22, θ1 = 530.
Fig. 6a. Near-field phase pattern for the geometry considered in Fig. 3a.
Fig. 5b. Near field amplitude pattern for the considered 2-element R-600 extended-focus optical scheme. Reflectors are: offset parabolic (d1 = 816λ, f/d1 = 0.4) and non-symmetric ellipse (d2 = 48λ); kb = 12, θ2 = 150.
Fig. 6b. Near-field phase pattern for the geometry considered in Fig. 3b.
498
periscope case. The main flat mirror is rotated by 450, and symmetric elliptic reflector (as in Fig. 5a) is used as a tertiary mirror. The resulting near field can be seen in Fig. 8, which, despite good visual beam guidance, calls for the basic geometrical parameter optimisation. This can be considered as a future step in the modelling of RATAN-600 radio telescope with MDS.
Fig. 6c. Near-field phase pattern for the geometry considered in Fig. 3c.
Visualizing the patterns in Figs. 4, 5 and 6 enables us to study the main beam behaviour for further applications in the full RATAN-600 model. By including the ellipse size, orientation angle and depth into the set of tuneable parameters and keeping the directivity of the main beam as the target function, one can significantly improve the performance of this system and keep the spillover losses at minimum level.
VI. CONCLUSIONS A combination of ray tracing, GO and integral diffraction methods has been used to optimize focal optics of a single, double, or multi-mirror MM wave reflector telescope with tightly packed multibeam FPA. Simulation and optimization techniques is demonstrated for a MM wave on-axis reflector and RATAN-600 radio telescope. We have presented a 2-D albeit accurate and efficient numerical analysis of the RATAN-600 telescope extendedfocus scheme made of two and three electrically large reflectors: offset parabolic, elliptic-arc and flat mirrors with sizes over 800λ each, and compared two designs of the tworeflector scheme, which differ in the ellipse and the feed orientations. The analysis has been done with the SIE-MDS approach that has earlier been applied to the multireflector antennas and open beam waveguides. We have studied far, near and phase patterns of the considered geometries. An average computational time for the demonstrated 2-reflector near-field amplitude pattern of 600x600 points in Figs. 3 is 7 hours on the desktop PC (2.6GHz Duo, 4Gb RAM) platform. REFERENCES [1]
Fig. 7. 3D focusing scheme of antenna system “South sector +Periscope” of the RATAN-600 radio telescope.
Fig. 8. Near-field amplitude pattern for the 3-element RATAN-600 extendedfocus optical scheme and . Reflectors are: offset parabolic (d1 = 816λ, f/d1 = 0.4), symmetric ellipse (d2 = 57λ) and flat mirror (d3 = 800λ).
Further, we have simulated a sample near-field amplitude pattern for a 3-element RATAN-600 long-focus scheme. As a basis we have used an antenna system “South sector +Periscope” prototype scheme (Fig. 7) but without the
R. Padman, "Optical fundaments for array feeds. Multi-feed systems for radio telescopes", ASP Conference Series, Vol. 75, 1995. [2] M. V. Ivashina, M. N. Kehn, P.-S. Kildal, "Optimal number of elements and element spacing of wide-band focal plane arrays for a new generation radio telescope", in Proc. EuCAP 2007, Edinburgh, Nov. 2007. [3] Ch. Granet, V. Khaikin, T. S. Bird, "A high-efficiency spline-profile smooth-walled 34-38 GHz horn as an array feed for the long-focus optics of the RATAN-600 radio telescope", in Proc. EuCAP 2009, Berlin. [4] V. Khaikin. "About possibility to expand FoV of RATAN-600 radio telescope in survey tasks with the use of a tertiary mirror," in Proc. Int. Con. "Sakharov oscillations and CMB problems". Nizhniy Arkhyz, Sep.2007. [5] M. Lebedev,V. Khaikin,V. Bogod. "Simulation of the working mode of antenna system "South sector + Periscope" of RATAN-600 with a correcting mirror," in Proc. Conf. "Upgrading of efficiency and modernization of radio telescopes, Sept.2008, Nizhniy Arkhyz, Russia [6] V. Khaikin, M. Lebedev, "Simulation of radiation pattern of a large MM wave radio telescope with offset feed", in Proc. EuCAP-2006, Nice, 2006. [7] A. A. Nosich, Y.V. Gandel, “Numerical analysis of quasioptical multireflector antennas in 2-D with the method of discrete singularities: Ewave case,” IEEE Trans. Ant. Prop., 2007, vol. 55, no 2, pp. 399-406. [8] A. A. Nosich, Y.V. Gandel, T. Magath, A. Altintas, “Numerical analysis and synthesis of 2-D quasioptical reflectors and beam waveguides based on an integral-equation approach with Nystrom's discretization,” J. Optical Soc. America A, 2007, Vol. 24, Issue 9, pp. 2831-2836. [9] Y.V. Gandel, Introduction to the Methods of Computation of Singular and Hyper-Singular Integrals, KhNU Press, 2001 (in Russian). [10] J. Tsalamengas, “Exponentially converging Nystrom’s methods for systems of SIEs with applications to open/closed strip or slot-loaded 2D structures,” IEEE Trans. Ant. Prop. 54, pp. 1549–1558, 2006.
499
