Let us examine a smooth, convex surface lacking sharp curves, which, in general case, represents an IS with an array of CWSR above it. We consider that all of the electrodynamics heterogeneities (radiators, parallel-plate layers of magneto-dielectrics etc.) are located between the surfaces and , where the surface is placed above at some distance , and all coordinate systems introduced below have a common origin located on the surface . Suppose that a primary monochromatic plane electromagnetic wave is incident to the array along the negative direction of the –axis of the Cartesian coordinate system at angles , . As a result of the diffraction of this wave, the electric and magnetic currents, as sources of secondary electromagnetic wave (scattered field , ), are induced on the surface of the CA. In the paper [2], using the method of Fok – Filippov [3], the asymptotic solution of the problem for bi-dimensional periodic CA of CWSR above a slowly-changing cylindrical ISC of arbitrary form and large electrical size have been constructed. In this paper, the results obtained in [3], have been generalized for convex surfaces of double curvature. We’ll attribute to them both closed surfaces (real ellipsoid) and unclosed indefinite surfaces (elliptical paraboloid).

On the surface the general electromagnetic field can be written as a continuous expansion of terms of plane waves [4]:

(1) |

where , , is the covariant basis of some orthogonal curvilinear coordinate system , , , the metrical tensor of which is:

In the coordinate system on the surface , we will refer as “partial” vector harmonics to the vectors

(2) |

In order to determine their specific kind, let us introduce a semigeodesic (polar [5]) coordinate system , on the surface in such a way that the following differential correlations are valid:

(3) |

Here is some arbitrary function which never vanishes and which makes sure the integration requirement is fulfilled, . The covariant vectors , tangent to the –axis, satisfy the relation [10]:

(4) |

where is a surface gradient [6], is a contra-variant basis of the coordinate system , , , and is an eikonal of "partial" harmonic of incident wave on the surface . In the expression (4) , for *k*=1,2 are the components of the co-variant metric tensor (=1, =0, ) [7]:

Let’s denote by , the curvature radii of the surface along the coordinate lines and (=0). We will consider that the physical components of the electromagnetic fields (incident and scattered) satisfy the condition that cross diffusion must be small and longitudinal diffusion must be negligible [8]. Then, the Maxwell’s equations can be asymptotically (when , for *m*=1,2) reduced to the system of coupled parabolic Leontovich-Fok’s equations relative to the components of the electrical field and in the , , coordinates [3]. The solution of the latter in homogeneous space region near the array (when ) permits to find all the remaining components of the electric and magnetic fields using the formulas

with an accuracy up to [3], where , and is the curvature radius of the surface along the coordinate line . For the periodic array on the generalized cylindrical surface, this solution is presented in [2]. But, in the case of an arbitrary surface of large size and arbitrary spacing of radiators, the solution obtained becomes unwieldy and difficult in computation, since it requires performing of numerical differentiation and numerical double-integration by contours in the complex region. It can be obtained from the expressions presented in the paper [3] (formulas (2.56)…(2.58) and (2.71)…(2.74)). We will confine ourselves to the case of a bi-dimensional periodic array, with periods satisfying the condition:

(5) |

where is the period of the array along the –axis (for *i* = 1,2). The condition (5) is typical for a CA of CWSR. In formula (1), let’s proceed to discrete Fourier transform and consider that, under the conditions , and (5) within the given period:

— the curvature radius does not depend on and

— the metrical tensor components do not depend on and

— the metrical tensor components do not depend on and .

Under the restrictions (3), assuming , we find out that within any single period:

If, in addition to the above conditions, another one is satisfied

then we can assume in all the expressions. Moreover, we can consider the electromagnetic field near the array upon “partial” excitation (2) to be local-periodical [3], and the scattered field in the coordinate system , , can be represented for the zero cell in the region in the form [3]:

(6) |

where , , and the coefficients are slowly-changing functions of coordinates , . The covariant components of eigenvectors can be obtained using the formulas (2.70), (2.65), (2.56) and (2.58) from the paper [3] and have the following form:

— for the electric field

(7) |

(8) |

In the expressions given:

And:

, are the Airy functions, in the definition and notation of V. A. Fok; ; the stroke at the Airy functions denotes an argument derivative; , and are the elements of the second square form of the surface . The lower index at and corresponds to the –coordinate. The harmonics (7) and (8), are analogous to the Floquet harmonics for the plane case [9] and uniformly reduced into them when .

In chapter 4.4 of the paper [3], the asymptotic formulas for the “partial” harmonics have been obtained in regard to cylindrical periodic structures. The expressions (7) and (8) of this paper are more general since they are oriented towards periodical structures connected with bi-dimensional-convex surfaces.

Further, expressing with the help, for example, of the Lorentz lemma, the unknown coefficients in the form of quadratures of the current in the radiator in the unit cell and using the boundary conditions of electrodynamics on the IS and on the surface of the radiator, we can obtain the system of operator equations, solution to which allows us determine the coefficients and, therefore, all the characteristics of the convex array. This method was explored in detail in the paper [2].

On the basis of the expressions obtained in this paper, the algorithm and programs have been developed, which allows calculation of the characteristics of CWSR in CA.

The influence of the shape of the CA on the characteristics of an azimuth- and axial- oriented CWSR in CA is illustrated by the curves shown in Fig.1…6. We will give the form of cross–section by the means of the canonic ellipse equation:

and examine the characteristics for the radiator located in the period with the coordinates *x*=*y*=0, *z*=b. The CWSR are excited in the middle by the –generators, have length *l*=0.05 ( is the wavelength at lower frequency ) and located on the layer of magneto-dielectric having thickness of 0.016 and permeability =2, =10. The width of the radiator is 0.015, =0.224, the geometry of the array is a square grid with period 0.05.

The pattern of the radiator of the elliptical array having *a*=5, dependent on *b*, is shown in Fig.1, and with *b*=5, dependent *a*, is shown in Fig.2. The results of calculations correspond to the physical meaning: when the radius of the equivalent circular tangent to the point of position of the radiator in question increase, the pattern oscillations decrease until the scan angle . The same is valid for the radiation level in the “shadow” region at ().

The absolute values of reflection coefficients (RC) of the azimuth strip radiators of elliptic array are shown in Fig.3 and in Fig.4 within the frequency band. The parameters of the array are: *a*=5, *b*=20 (Fig.3) and *a*=20, *b*=5 (Fig.4). The rest of the sizes are unchanged.

The behavior of absolute values of RC of axial SR located on a magneto-dielectric layer of the elliptic array, with the same parameters as in the previous case, in the frequency bound, are shown in Fig.5 and Fig.6.

Fig. 1 Directivity pattern of an azimuth wide-band radiator on the elliptic cylinder, which has *a*=5 (1 − *b*=10; 2 − *b*=20; 3 is a plane array)

Fig. 2 Directivity pattern of an azimuth broadband radiator on the elliptical cylinder, which has *b*=5 (1 − *a*=10; 2 − *a*=20; 3 is a plane array)

Fig. 3 The behavior of the reflection coefficient modulo of an azimuth broadband radiator on the elliptical cylinder (*a*=5, *b*=20) in the frequency band (1 − *f*=; 2 − *f*=1.5; 3 − *f*=2; 4 is the radiator in a plane array with *f*=)

Fig. 4 The behavior of the amplitude of the reflection coefficient of an azimuth broadband radiator on the elliptical cylinder (*a*=20,
*b*=5) in the frequency band (1 − *f*=; 2 − *f*=1.5; 3 − *f*=2; 4 is the radiator in a plane array where *f*=)

Fig. 5 The behavior of the of the reflection coefficient modulo of an axial broadband radiator on the elliptical cylinder (*a*=5,
*b*=20) in the frequency band (1 − *f*=; 2 − *f*=1.5; 3 − *f*=2; 4 is the radiator in a plane array with *f*=)

Fig. 6 The behavior of the reflection coefficient modulo of an axial broadband radiator on the elliptical cylinder (*a*=20, *b*=5) in the frequency band (1 − *f*=; 2 − *f*=1.5; 3 − *f*=2; 4 is the radiator in a plane array with *f*=)

Based on the results obtained in this paper and the numerical experiment we can conclude:

1. The mathematical models of CWSR taking into account the interaction between adjacent radiators in a CA on a convex surface of double curvature have been constructed, on the condition that the array is bi-dimensionally-periodic, infinite along the generating line and has a large slowly changing curvature radius. The CA can have multi-layer coating and the screen can have losses.

2. The principal difference between conformal and plane arrays is the behavior of the reflection coefficient in the scan angle region near ±90°. While the plane array has |Г|=1, the conformal array has |Г|<0.75 at scan angle 90° within the three-repeated frequency band. This fact should be accounted for when examining the problems related to e.g. decoupling of CA.

3. While designing the wide-band CA of CWSR we should keep in mind that curving of the antenna array aperture until the curvature radii does not lead to significant changes in the internal and external characteristics of radiators of the system, as compared to the plane aperture, to the angles . In this sector, good matching (voltage standing-wave ratio ) in the twofold frequency band for the two main planes of scanning can be provided.