ISSN 1064-2269, Journal of Communications Technology and Electronics, 2009, Vol. 54, No. 10, pp. 1097–1110. © Pleiades Publishing, Inc., 2009. Original Russian Text © S.E. Bankov, 2009, published in Radiotekhnika i Elektronika, 2009, Vol. 54, No. 10, pp. 1157–1170.
ELECTRODYNAMICS AND WAVE PROPAGATION
Two-Dimensional Beamforming Networks on the Basis of Coupled Waveguides S. E. Bankov Received February 16, 2009
Abstract—The theory and the numerical simulation of 2D microwave beamforming networks (BFNs) in the form of a system of coupled waveguides are considered. The analyzed BFNs are designed for quasi-optical multibeam antennas. The theory of coupled modes is used to analyze the BFNs with fixed phase shifters and not identical coupled waveguides whose radiators are placed in the nodes of a rectangular or a hexagonal grid. It is shown that the proposed N-channel networks form N flat-top patterns intersecting at a level close to –3 dB. The input channels of this network are matched and isolated. The theory of coupled modes is used for optimization of such devices. PACS numbers: 41.20.-q, 84.40. Az DOI: 10.1134/S1064226909100015
1. FORMULATION OF THE PROBLEM Radio-wave imaging systems use multibeam antennas containing an optical system (OS) in the form of one of several reflectors or lenses having large electric dimensions and forming the pencil-beam pattern. In addition, this multibeam antenna contains a beamforming network (BFN), which has several inputs and an array illuminating the OS (see Fig. 1). In [1, 2], 1D BFNs on the basis of systems of coupled waveguides were considered. It was shown that such BFNs much better satisfy the optimality criteria than the BFNs containing isolated channels. The requirements on the optimum BFN were formulated in [2]. The first requirement is that an N-channel BFN should form N amplitude–phase distributions (clusters) at its output (in the aperture of the array illuminating the OS) and these clusters should intersect at a level close to –3 dB. The cluster shape determines the array pattern. In the perfect case, the pattern should be flat-top in order to illuminate the OS without energy spillover beyond the boundaries and, simultaneously, ensure high aperture efficiency of the OS. A BFN pattern close to flat-top is the second requirement on the optimum BFN.
Note that, in many cases, it is necessary to have a 2D system of beams, rather that a 1D system, because precisely this system can form a valid image of the studied object. A 2D set of beams can be formed by a 2D BFN whose output is a 2D array illuminating the OS. Such BFNs are studied and optimized in this paper.
One-dimensional BFNs forming a system of flat-top patterns were considered in [3, 4]. The principle of operation of such BFNs is close to that of the BFNs studied in [1, 2]. These BFNs also use coupling of waveguides for formation of radiating clusters at the BFN outputs. However, the method of organization of this coupling proposed in [3, 4] has substantial differences, which result from the fact that the approach proposed in these papers is predominantly oriented at a particular type of waveguide, namely, a rectangular metal waveguide. The theory developed in [1, 2] is substantially independent of the type of the guiding structure. 1097
Optical system
BFN
1
2
N Fig. 1. Structure of a multibeam antenna.
1098
BANKOV Radiating array
Unit of phase shifters
Coupled waveguides Fig. 2. Structure of a BFN containing coupled waveguides and phase shifters.
The main method of investigation of the system of coupled waveguides is the phenomenological theory of coupled modes [5]. The main advantage of this theory is the simplicity and clarity of the mathematical apparatus. The theory of coupled modes allows rather easy understanding of the basic laws of operation of the BFN and can be used for optimization of this BFN. A disadvantage of this theory is that it uses several generalized parameters and the relations between these parameters and the design of a real BFN fall beyond the scope of the theory. Therefore, application of this theory can be considered as only the first stage of the network analysis. We consider here several modifications of 2D BFNs, which differ in the type of the grid (whose nodes are the positions of the BFN waveguides) and the BFN structure. We analyze two grid types: a rectangular grid and a hexagonal grid. In [1, 2], 1D networks with phase shifters and identical coupled waveguides as well as BFNs without phase shifters but with different coupled waveguides were proposed. Both design versions have advantages and disadvantages. Such versions are also analyzed in this paper, as applied to the 2D network. 2. TWO-DIMENSIONAL BFN WITH PHASE SHITERS AND A RECTANGULAR GRID The structure of a BFN with phase shifters is shown in Fig. 2. This BFN is similar to that considered in [1], except for the arrangement of the waveguides in the nodes of a rectangular grid (see Fig. 3). Let us consider operation of
the unit of coupled waveguides. Assume that indices n and m specify the position of the grid node: x n = nP x ,
(1)
y m = mP y ,
where Px, y are the array spacings along the 0ı and 0Û axis, respectively. In this case, each waveguide is characterized by two indices (n, m). Arrows in Fig. 3 show distributed couplings between the waveguides. We assume that the waveguides are coupled with each other only along the directions parallel to the coordinate axes. In the accepted theory, the waveguide coupling is described by the linear coupling coefficient (LCC), which will be denoted by ë. Wave propagation in coupled waveguides is described by differential equations. Let us write the equation for the waveguide with indices (0, 0): dU 0, 0 ------------- + iβ 0 U 0, 0 + iCU 0, 1 + iCU 0, –1 + iCU –1, 0 dz + iCU 1, 0 = 0,
(2)
where β0 is the propagation constant of an isolated waveguide and Un, m are the amplitudes of the modes in the coupled waveguides, which are the functions of coordinate z.
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x
0,1
C y
C –1,0
0
1,0
0, –1 Px Py Fig. 3. Two-dimensional system of identical coupled waveguides with a rectangular grid.
In the quasi-periodic operating mode of the system of coupled waveguides, the mode amplitudes are related as follows: U n, m = U 0, 0 exp ( – iκnP x – iβmP y ),
(4)
∫ ∫ A ( κ, β ) exp ( –iγz
π π – ----- – -----Px Py
(5)
If the operating mode of the waveguide system is aperiodic, the mode amplitudes are written as a superposition of particular quasi-periodic solutions:
(6)
– iκnP x – iβmP y )dκdβ, where A(κ, β) is an unknown function. In order to determine this function, it is necessary to specify the excitation conditions of the waveguide system. We will consider the case of excitation of one waveguide with indices (0, 0) in the section z = 0. This excitation corresponds to the following end condition: U 0, 0 ( 0 ) = 1,
Upon solving homogeneous differential equation (4), we find propagation constant γ(κ, β) of the eigenmode of the system of coupled waveguides in the quasi-periodic operating conditions: γ ( κ, β ) = β 0 + 2C ( cos ( κP x ) + cos ( βP y ) ).
U n, m =
(3)
where κ and β are the parameters specifying the phase shift between channels. Relationship (3) can be used to write Eq. (2) for amplitude U0, 0: dU 0, 0 ------------- + iβ 0 U 0, 0 + 2iCU 0, 0 dz × ( cos ( κP x ) + cos ( βP y ) ) = 0.
π π ------ -----Px Py
(7)
Un, m(0) = 0, n, m ≠ 0. Using condition (7), we find function A(κ, β) in the form ⎧ Px Py π π ⎪ -----------, κ < -----, β < -----, 2 Px Py ⎪ 4π A ( κ, β ) = ⎨ π π ⎪ 0, κ ≥ -----, β ≥ -----. ⎪ Px Py ⎩
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Table 1. Distribution of the mode amplitudes at the outputs of a system of identical coupled waveguides with a rectangular grid m
n
–2
–1
–2 5.748 × –1 0.028i 0 –0.064 1 0.028i 2 5.748 × 10–3 10–3
0.028i –0.136 –0.312i –0.136 0.028i
0
1
2
–0.064 0.028i 5.748 × 10–3 –0.312i –0.136 0.028i 0.716 –0.312i –0.064 –0.312i –0.136 0.028i –0.064 0.028i 5.748 × 10–3
Table 2. Distribution of the transmission coefficients of phase shifters m
n –2 –1 0 1 2
–2
–1
0
1
2
–i 1 –i 1 –i
1 i 1 i 1
–i 1 –i 1 –i
1 i 1 i 1
–i 1 –i 1 –i
Table 3. Distribution the mode amplitudes at the output of a BFN with phase shifters and a rectangular grid n
m –2
–1
0
1
2
–2 5.748 × 10–3 –0.028 –0.064 –0.028 5.748 × 10–3 –1 –0.028 0.136 0.312 0.136 –0.028 0 –0.064 0.312 0.716 0.312 –0.064 1 –0.028 0.136 0.312 0.136 –0.028 2 5.748 × 10–3 –0.028 –0.064 –0.028 5.748 × 10–3
In view of relationship (8), the integral in (6) is calculated analytically [6]: U n, m = ( – i ) ( – i ) J n ( 2Cz )J m ( 2Cz ) exp ( – iβ 0 z ), (9) n
m
where Jn(x) is the Bessel function of order n. Relationship (9) describes the energy distribution process in the waveguide system in the case of excitation of one channel. In the following, we omit factor exp ( – iβ 0 z ) , which is common for all channels and is insignificant for the subsequent analysis. The values of amplitudes Un, m corresponding to CL = 0.4, where L is the length of the waveguide coupling section, is presented in Table 1. It can readily be seen that the phase of the waves propagating in the adjacent channels
π differ by ± --- . At the same time, it is known that, in order to 2 form a pattern close to flat-top, it is necessary to form a dissin x tribution close to ----------, (which allows only phase shifts of x ±π) in the aperture of a 1D BFN. Correction of the phase distribution is ensured by the unit of phase shifters (see Fig. 2). We assume that, in this unit, the waveguides are not coupled with each other. Some channels contain fixed phase shifters. In the analysis of the BFN structure, different design versions of the unit of phase shifters were considered. Below, the results obtained for the best version are presented. If the phase shifters are perfect (lossless and reflectionless), the effect of phase shifters on amplitudes Un, m appears as multiplication by some factor Φn, m whose absolute value is unity. This factor has the meaning of the phaseshifter transmission coefficient. The best results were obtained for the following distribution of factors Φn, m over the channels: 2 πn 2 πm Φ n, m = 1 – ( 1 + i ) cos ⎛ ------⎞ cos ⎛ -------⎞ ⎝ 2⎠ ⎝ 2 ⎠
(10)
2 πm πn – ( 1 – i ) sin ⎛ ------⎞ sin ⎛ -------⎞ . ⎝ 2⎠ ⎝ 2 ⎠ 2
The distribution of the phase-shifter transmission coefficients is presented in Table 2. The resulting distribution of the amplitudes of the modes in the BFN channels is described by the formula U n, m = ( – i ) ( – i ) J n ( 2CL )J m ( 2CL )Φ n, m . n
m
(11)
The distribution of mode amplitudes (11) is presented in Table 3. As seen from this distribution, the phases of the channel modes differ now by only ±π. An important stage in the study of the BFN characteristics is the analysis of the BFN pattern F(θ, ϕ) which can be determined from well-known relationships for antenna arrays [7]: F ( θ, ϕ ) = F 0 ( θ, ϕ )
∑∑U n
n, m exp ( ik sin ( θ )
m
(12)
× ( nP x cos ϕ + mP y sin ϕ ) ), where F0(θ, ϕ) is the pattern of the element of the BFN array and k is the free-space wave number. Relationship (12) is written in the spherical coordinate system. Angle θ is measured from the 0z axis (see Fig. 4) and angle ϕ is measured from the 0ı axis. For definiteness, we take an open-ended waveguide with dimensions ax and ay for the array element. The sides with widths ax and ay are oriented along the 0ı axis and 0Û axis, respectively. The relationship for pattern F0(θ, ϕ) is presented in [8]:
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1101 Optical system
z
θ θm
Fig. 4. Definition of the elevation angle in the spherical coordinate system.
F 0 ( θ, ϕ ) sin ϕ ( 1 + V cos θ )⎞ cos ϕ ( V + cos θ ) ⎛ ----------------------------------------- + ⎛ ----------------------------------------⎞ ⎝ ⎠ ⎝ ⎠ 1+V 1+V 2
=
2
(13) ka ka sin ⎛ --------y sin θ sin ϕ⎞ cos ⎛ --------x sin θ cos ϕ⎞ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛ π⎞2 ----- , × --------------------------------------------- ---------------------------------------------------ka y π⎞2 2 ⎝ a x⎠ ⎛ ----- – ( k sin θ cos ϕ ) -------- sin θ sin ϕ ⎝ a x⎠ 2 γ π 2 2 where V = -----w and γ w = k – ⎛ -----⎞ is the propaga⎝ a x⎠ k tion constant of the waveguide mode. Relationship (13) is the normalized pattern of an open-ended waveguide excited by the fundamental mode polarized along the 0Û axis. In calculation of this pattern, it is assumed that the waveguide supports only the fundamental mode with one polarization. Coupling with orthogonally polarized modes is absent. The square of this pattern is proportional to the absolute value of the Poynting vector of the field radiated in the corresponding direction. Normalized BFN patterns are shown in Fig. 5. Figures 5a–5c correspond to CL = 0.35, 0.50, and 0.65, respectively. Curves 1 and 2 correspond to different azimuth angles, ϕ = 0° and 45°, respectively. The array spacings Px and Py are 20 mm and frequency f is 8 GHz. In calculations, it was assumed that the waveguide walls are infinitely thin, ax, y = Px, y = 20 mm. In the principal planes, which correspond to ϕ = 0° and 90°, the BFN patterns are almost identical, irrespective of the fact that the patterns of array elements corresponding to
these planes are substantially different. In the diagonal plane (ϕ = 45°), the BFN pattern is wider than the patterns in the principal planes. It is seen from Fig. 5 that large values of parameter CL increase the approach of the BFN pattern to the flat-top shape; however, this approach is accompanied by an increase in the sidelobe level. The OS illumination efficiency is conveniently described by the OS aperture efficiency (EFF), which indicates the degree of closeness of the BFN pattern to the perfect flat-top pattern: EFF θ m 2π
2
∫ ∫ F ( θ, ϕ )F ( θ, ϕ ) sin θ dθ dϕ p
0 0
-, = ------------------------------------------------------------------------------------------------------------θ m 2π π 2π
∫ ∫ F ( θ, ϕ )
2
sin θ dθ dϕ
0 0
∫ ∫ F (θ) p
2
(14)
sin θ dθ dϕ
0 0
Here, Fp(θ, ϕ) is the perfect flat-top pattern: ⎧ 1, F p ( θ, ϕ ) = ⎨ ⎩ 0,
0 ≤ θ ≤ θm , θ > θm ,
0 ≤ ϕ < 2π,
0 ≤ ϕ < 2π,
(15)
where θm is the maximum value of angle θ at which the OS edge is seen from the origin of coordinates. If the OS shape is not circular, parameter θm is a function of angle ϕ. Note that, if the BFN pattern coincides with the perfect pattern, EFF = 1.
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BANKOV F(θ), dB 0
F(θ), dB 5
(a)
(b)
0
–5
–5 –10
1
2
2
–10
–15
–20
1
–15 –20 0
20
40
60 θ, deg F(θ), dB 5
80
0
100
20
40
60 θ, deg
80
100
80
100
(c) 0 –5 2 –10
1
15 –20
0
20
40
θ, deg
60
Fig. 5. Patterns of a BFN with coupled waveguides and phase shifters arranged in a rectangular grid for CL = (a) 0.35, (b) 0.50, and (c) 0.65.
EFF, dB 3
(a)
EFF, dB 3
3 35
39
2
2
1
20
30
40
50
1
60 θm, deg
0
2
3
2
1
0
(b)
1
20
30
40
50
60 θm, deg
Fig. 6. Aperture efficiency of a BFN containing coupled waveguides and phase shifters as a function of angle θm for (a) a square OS and (b) a circular OS. JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS
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1103
x
4
3 β2
1 β0
β0 β1 2 y
Fig. 7. Two-dimensional system of not identical coupled waveguides with a rectangular grid.
The values of parameter EFF as a function of angle θm are shown in Figs. 6a and 6b for a square OS and a circular OS, respectively. In the case of a square OS, angle θm is determined in the principal planes. Curves 1–3 correspond to ëL = 0.35, 0.50, and 0.65, respectively. The results were obtained for ‡x, y = Px, y = 20 mm. It is seen from Fig. 6 that parameter EFF has extrema in two variables, θm and CL. As a function of argument CL, parameter EFF reaches its minimum (in decibels) at CL ≈ 0.5, irrespective of OS shape. The absolute values of parameter EFF at the minima corresponding to the circular OS and the square OS are close to each other: 0.777 dB and 0.652 dB, respectively. The optimum value of angle θm corresponding to the circular OS is slightly greater than the optimum value for the square OS: 39° and 35°, respectively. Thus, we come to the conclusion that the analyzed BFN can be sufficiently efficient for illumination of both circular and square OSs. It can be assumed that application of BFNs with different values of spacings Px and Py allows efficient illumination of a rectangular OS.
3. TWO-DIMENSIONAL BFN WITH UNEQUAL WAVEGUIDES AND A RECTANGULAR GRID Let us consider a BFN composed of not identical waveguides without phase shifters. The principle of operation of this BFN is similar to that of a BFN with phase shifters, except for the fact that the required phase shifts in the BFN channels are ensured by the differences in the propagation constants of the waveguide modes. Here, distribution of the energy between channels and phase changes occurs simultaneously in the section of distributed coupling. The structure of this BFN is shown in Fig. 7. The rectangle in Fig. 7 selects one period of this BFN, which now contains four waveguides with numbers 1 through 4. These waveguides have different propagation constants: the first and the third waveguides have propagation constant β0 while the second and fourth waveguides have propagation constants β1 and β2, respectively. We will consider the case of symmetric changes in the propagation constants, which is described by the following equality:
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BANKOV
Let, as before, the BFN waveguides be coupled only along the coordinate axes. Indices n and m describe now the position of the BFN period rather than the position of a waveguide. Index n corresponds to the 0ı axis and index m corresponds to the 0Û axis. Thus, the amplitudes of ( j) waveguide modes are described by three indices: U n, m , where j = 1, …, 4. The subscripts determine the period in which the waveguide is situated and the superscript indicates the number of the waveguide within the period. The spacing between the periods along the 0ı axis is 2Px and the spacing along the 0Û axis is 2Py. In the quasi-periodic operating mode, the amplitudes of waveguide modes are related as follows: ( j)
( j)
U n, m = U 0, 0 exp ( – 2inκP x – 2imβP y ).
Omitting intermediate calculations, we present the system of four differential equations for the analysis of the eigenmodes of a system of coupled waveguides: (1)
dU 0, 0 (1) (2) ------------- + iβ 0 U 0, 0 + 2iC ( cos ( βP y ) exp ( iβP y )U 0, 0 dz (4)
+ cos ( κP x ) exp ( iκP x )U 0, 0 ) = 0, (2)
dU 0, 0 (2) (1) ------------- + iβ 1 U 0, 0 + 2iC ( cos ( βP y ) exp ( – iβP y )U 0, 0 dz (3)
+ cos ( κP x ) exp ( iκP x )U 0, 0 ) = 0, (3) dU 0, 0
π β < --------- . 2P y
(4)
------------- + iβ 0 U 0, 0 + 2iC ( cos ( βP y ) exp ( – iβP y )U 0, 0 dz
(17)
(18)
3
(2)
+ cos ( κP x ) exp ( – iκP x )U 0, 0 ) = 0, (4)
Parameters κ and β have the same meaning as before; however, their ranges of variation differ from the preceding values: π κ < --------- , 2P x
(19)
dU 0, 0 (4) (3) ------------- + iβ 2 U 0, 0 + 2iC ( cos ( βP y ) exp ( iβP y )U 0, 0 dz (1)
+ cos ( κP x ) exp ( – iκP x )U 0, 0 ) = 0. Using system (19), we obtain propagation constants γi, i = 1, …, 4 of waveguide eigenmodes:
∆β 2 2 2 --------- + 4C ( cos βP y + cos κP x ) 2 2
γ i = β0 ±
∆β 2 2 2 2 2 2 4 ± --------- + 4∆β C ( cos βP y + cos κP x ) + 64C cos βP y cos κP x 4 4
where i is the eigenmode index. Eigenmode νi is treated as the four-component vector
( j)
νi =
∑∫ ∫
i=1
(1) νi
(3)
νi
,
(21)
(4)
νi
whose elements are the amplitudes of waveguide modes. In the general case, when the quasi-periodic operating ( j) mode is not kept, amplitudes U n, m of waveguide modes are written as a sum of eigenmodes:
( j)
A i ν i exp ( – iγ i z – 2iκnP x
π π – --------- – --------2P x 2P y
(22)
– 2iβmP y )dκdβ,
(2)
νi
(20)
π π --------- --------2P x 2P y
4
U n, m =
.
where Ai are the eigenmode amplitudes whose values depend on κ and β. For determination of the eigenmode amplitudes, we introduce the end conditions, which follow from BFN excitation conditions: (2)
U 0, 0 ( 0 ) = 1, ( j)
U n, m ( 0 ) = 0,
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n, m ≠ 0,
j ≠ 2.
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In relationships (23), we assume that the waveguide with n = m= 0 and j = 2 is excited. Using conditions (23), we obtain the following system for amplitudes Ai: 4
∑A ν
( j) i i
i=1
Px Py = δ j, 2 -----------, 2 π π
(24)
1105
where δi, j is the Kronecker delta. In calculation of the BFN pattern, it is convenient to introduce a new enumeration of the BFN channels (see Fig. 3). In view of this enumeration, expression (22) can be rewritten in the new form:
π
- --------⎧ 4 -------2P x 2P y ⎪ (2) A i ν i exp ( – iγ i z – iκnP x – iβmP y ) dκ dβ, ⎪ ⎪i = 1 π π --------- – --------⎪ – 2P x 2P y ⎪ π π ⎪ --------- --------⎪ 4 2P x 2P y (1) ⎪ A i ν i exp ( – iγ i z – iκnP x – iβ ( m + 1 )P y ) dκ dβ, ⎪ ⎪ i = 1 – -------π π - – --------⎪ 2P x 2P y = ⎨ π π - --------⎪ 4 -------2P x 2P y ⎪ (3) ⎪ A i ν i exp ( – iγ i z – iκ ( n – 1 )P x – iβmP y ) dκ dβ, ⎪i = 1 π π ⎪ – -------- – --------2P x 2P y ⎪ π π ⎪ --------- --------⎪ 4 2P x 2P y ⎪ (4) A i ν i exp ( – iγ i z – iκ ( n – 1 )P x – iβ ( m + 1 )P y ) dκ dβ. ⎪ ⎪i = 1 π π --------- – --------⎩ – 2P x 2P y
∑∫ ∫ ∑∫ ∫
U n, m
(25)
∑∫ ∫ ∑∫ ∫
The first line in (25) corresponds to the new enumeration of the channels with even n and even m, the second line corresponds to even n and odd m, the third line corresponds to odd n and even m, and the fourth line corresponds to odd n and odd m. Using formula (25), we can, as before, describe the BFN pattern with relationship (12). The patterns of the BFNs with not identical waveguides are shown in Fig. 8. The results were obtained for ax, y = Px, y = 20 mm, f = 8 GHz, and ∆βL = 1.9. Figures 8a–8c correspond to CL = 0.55, 0.70, and 0.85, respectively. Curves 1 and 2 correspond to ϕ = 0 and 45°, respectively. Note that, qualitatively, the pattern of the BFN with unequal waveguides is very similar to the pattern of the BFN with phase shifters. This similarity occurs despite the fact that the amplitude–phase distribution over the BFN channels differs from that obtained in Section 2. This fact is confirmed by the data in Tables 4 and 5, which present the corresponding amplitude and phase distributions. The data in these tables were obtained for the same parameters as the curves in Fig. 8c. Thus, we can conclude that the BFN with unequal waveguides also solves the OS illumination problem and the illumination characteristics are close to optimal or, at least, to the characteristics that are not worse than the illumination characteristics of the BFN with phase shifters.
4. TWO-DIMENSIONAL BFN WITH PHASE SHIFTERS AND A GEXAGONAL GRID The structure of a BFN with phase shifters and a hexagonal grid is shown in Fig. 9. The spacings between waveguides are identical and are denoted by P. Arrows show the mutual coupling between channels. The system of coupled waveguides is considered similarly to the case of the BFN with rectangular grid; however, in this case, indices n and m specify coordinates x˜ , andy˜ , which are related to coordinates x and y as y = y˜ cos θ 0 ,
(26)
x = x˜ + y˜ sin θ 0 . Angle θ0 is shown in Fig. 9. For a hexagonal grid, this angle is π/6. In this approach, the mode amplitudes corresponding to the quasi-periodic operating conditions are again related by formula (3). Variables κ and β vary from –π/P to π/P. Condition (3) reduces the system of differential equations to one equation dU 0, 0 ------------- + iβ 0 U 0, 0 + 2iCU 0, 0 ( cos ( κP ) + cos ( βP ) dz (27) + cos ( ( κ – β )P ) ) = 0.
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BANKOV F(θ), dB
F(θ), dB
(a)
0
(b)
0
2
–10
2
–10
1
–20
0
20
1
40
60 80 θ, deg F(θ), dB
100
–20
0
20
40
60 θ, deg
80
100
(c) 0
2
–10 1 –20
0
20
40
60 θ, deg
80
100
Fig. 8. Patterns of a BFN containing coupled waveguides arranged in a rectangular grid for CL = (a) 0.55, (b) 0.70, and (c) 0.85.
The eigenmode propagation constant determined from Eq. (27) is γ ( κ, β ) = β 0 + 2C ( cos ( κP ) + cos ( βP ) + cos ( ( κ – β )P ) ).
π π --- --P P
U n, m =
(28)
∫ ∫ A ( κ, β ) exp ( – iγz – iκnP
π π – --- – --P P
– iβmP )dκdβ.
The amplitudes of waveguide modes are determined as in Section 1:
Table 4. Amplitude distribution at the output of a system of unequal coupled waveguides with a rectangular grid
(29)
Table 5. Phase distribution at the output of a system of unequal coupled waveguides with a rectangular grid m
m
n
n
–2
–1
0
1
2
–2
2.185
4.119
4.234
4.119
2.185
0.306
–1
4.119
6.052
–0.116
6.052
4.119
0.761
0.402
0
4.234
–0.116
–0.116
4.234
0.761
0.58
0.306
1
4.119
6.052
–0.116
6.052
4.119
0.402
0.306
0.161
2
2.185
4.119
4.234
4.119
2.185
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–2
–1
0
1
2
–2
0.161
0.306
0.402
0.306
0.161
–1
0.306
0.58
0.761
0.58
0
0.402
0.761
1
1
0.306
0.58
2
0.161
0.306
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x, x~
y~
1,1 1,0 1, –1 0,1
C
θ0 0
y
C –1,1
0, –1 –1,0 –1, –1
Fig. 9. Two-dimensional system of identical coupled waveguides with a rectangular grid.
1.002
2.352 2.695
2.352
2.695 –2.008
–2.008
–2.008 2.695
1.002
2.352
0
2.695 –2.008
–2.008
–2.008
2.352
1.002
2.695
2.352 2.695
2.352
1.002
Fig.10. Phase distribution at the output of a system of identical coupled waveguides with a hexagonal grid.
Function A(κ, β) is found with the use of the relationship similar to relationship (8) and is written as follows: ⎧ P2 ⎪ --------, κ < --π-, P ⎪ 2 A ( κ, β ) = ⎨ 4π ⎪ π ⎪ 0, κ ≥ --P-, ⎩
π β < ---, P
tion of the structure of the unit of phase shifters. The distribution obtained for CL = 0.35 is shown in Fig. 10. The channel phases are normalized so that the phase in the channel with n = m = 0 is zero.
(30)
The amplitude distribution at the outputs of coupled waveguides is shown in Fig. 11.
Integral (29) cannot be calculated analytically and should be determined numerically. Let us find the phase distribution at the outputs of coupled waveguides. These data are necessary for determina-
As seen from Fig. 10, the channel phases are not the multiples of π/2, as in the case of a rectangular grid. Therefore, it is impossible to obtain an inphase–antiphase distribution. If the BFN has one active channel, this transformation is possible. An active channel is treated as a waveguide that can be either the BFN output of the BFN input. However, the BFN contains a system of periodically placed
π β ≥ ---. P
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0.08 0.148
0.148
0.08
0.395
0.08
0.395
0.395 0.148
1
0.08
0.148 0.395
0.395
0.027
0.027
0.08
0.395 0.148
0.148 0.08
0.027
Fig. 11. Amplitude distribution at the output of a system of identical coupled waveguides with a hexagonal grid.
Fig. 12. Structure of active channels in a BFN with a hexagonal grid.
active channels. In this structure, the problems of phase transformation with the use of fixed phase shifters contradict each other in the case of excitation of different active channels, because solutions of such problems require different phase shifts in the same waveguides. Numerical investigations have shown that good results are obtained with the distribution of active channels schematically shown in Fig. 12. Active channels are colored in gray. It can easily be seen that, for this arrangement of active channels, only the phases in six channels surrounding an active channel can be corrected. Hence, the transmission coefficients of phase shifters are written as follows: 2 2 πm πn Φ n, m = cos ⎛ ------⎞ cos ⎛ -------⎞ + T ψ ⎝ 2 ⎠ ⎝ 2⎠
× ⎛ 1 – cos ⎝
2
⎛ πn ------⎞ cos ⎝ 2⎠
2
⎛ πm -------⎞ ⎞ , ⎝ 2 ⎠⎠
(31)
T ψ = exp ( iψ ), where ψ is the phase shift of the phase shifters placed in passive (inactive) channels. It is seen from formula (31) that there are no phase shifters in active channels and corresponding transmission coefficients take the unit value. The expression for the BFN pattern is
F ( θ, ϕ ) = F 0 ( θ, ϕ )
∑∑U n
n, m exp ( ik sin ( θ )
m
(32)
× ( x n, m cos ϕ + y m sin ϕ ) ), where x n, m = nP + mP sin θ 0 ,
(33)
y m = mP cos θ 0 . Differences between formulas (32) and (12) are related to the differences between the hexagonal and rectangular grids. The BFN patterns obtained for P = 20 mm and f = 8 GHz are shown in Fig. 13. The plots in Figs. 10a–10c correspond to CL = 0.30, 0.35, and 0.40, respectively. Curves 1 and 2 were obtained for ϕ = 0 and π/6, respectively. The function describing the BFN pattern is almost periodic with a period of π/3. The deviation from a periodic function is caused by the properties of the pattern of the array element. It is seen from Fig. 13 that, even in the middle of the period (at ϕ = π/6), the pattern only slightly differs from the pattern corresponding to ϕ = 0. This fact points to a high degree of azimuthal symmetry of this pattern. The dependence of the BFN pattern on the LCC is qualitatively the same as in the considered cases. The optimum LCC corresponds to CL = 0.35. The optimum value of
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F(θ), dB 5 (b)
(a) 0
–5
–5 –10 –10 –15
–15 1
–20
0
20
1
2 40 60 θ, deg
2
–20 80
0
100
F(θ), dB 5
20
40
60 θ, deg
80
100
(c)
0 –5 –10 2
15 1 –20
0
20
40
60 θ, deg
80
100
Fig. 13. Patterns of a BFN containing coupled waveguides with phase shifters arranged in a hexagonal grid for CL = (a) 0.30, (b) 0.35, and (c) 0.40.
phase shift ψ can be approximately obtained from the following condition: ψ = –arg ( U 0, 1 )..
(34)
In the case of illumination of a circular OS, the optimum value of parameter EFF is close to 0.8 dB. CONCLUSIONS The results presented above indicate that 2D versions of the BFNs on coupled waveguides proposed in [1] can be designed. Such BFNs ensure high level of intersection of adjacent OS beams and high aperture efficiency.
Note that, in the course of investigations, the design version of a BFN with unequal waveguides and a hexagonal grid was analyzed and the results similar to those presented in Section 4 were obtained. Since these results do not contain any fundamentally new information, they were omitted. We should only note that this BFN does not contain phase shifters. It is formed from waveguides with different propagation constants. The structure of active and passive channels coincides with that shown in Fig. 12. In this structure, the propagation constant in active channels (β2) is greater than the propagation constant in passive channels (β1). The analysis of such a BFN is identical with the analysis described in Section 3.
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ACKNOWLEDGMENTS This study was supported by the Russian Foundation for Basic Research, project no. 08-08-12200-ofi. REFERENCES 1. S. E. Bankov and T. I. Bugrova, Microwave Opt. Technol. Lett. 6, 782 (1993). 2. S. E. Bankov, Radiotekh. Elektron. (Moscow) 54 (7) (2009) [J. Commun. Technol. Electron. 54, 000 (2009)].
3. S. P. Skobelev, Radiotekhnika, No. 10, 44 (1990). 4. S. P. Skobelev, Radiotekhnika, No. 7, 15 (1996). 5. H.-G. Unger, Planar Optical Waveguides and Fibres (Mir, Moscow, 1980; Clarendon Press, Oxford, 1977). 6. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Fizmatgiz, Moscow, 1962; Academic, New York, 1980). 7. D. M. Sazonov, Microwave Circuits and Antennas (Vysshaya Shkola, Moscow, 1988; Mir, Moscow, 1990). 8. A. Z. Fradin, Microwave Antennas (Sovetskoe Radio, Moscow, 1957; Pergamon, New York, 1961).
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