II. GEOMETRY

The two-dimensional geometry of a Cassegrain antenna is shown in Fig 2. The following parameters are shown in Fig 2:

• d _p = diameter of the main reflector,

• d _s = diameter of the subreflector,

• f _p = focal length of the paraboloid,

• 2c = distance between foci of the hyperboloid,

• ψ_o = opening half angle of the paraboloid, and

• ϕ_o = opening half angle of the hyperboloid.

Fig. 2. Geometry of the Cassegrain reflector.

The hyperboloidal subreflector has two focal points. One of the focal points coincides with the phase center of the feed horn and the other coincides with the focal point of the paraboloid [Reference Baars4]. The relation between the geometrical parameters can be found in [Reference Granet13].

To employ AF integration, it is necessary to project the main reflector surface on the aperture plane. For convenience, the aperture plane is assumed to go through the focal point of the main reflector [Reference Balanis2, Reference Baars4]. The angle ψ _o is created between the axis of the reflector and the line joining the focal point and edge of the main reflector. In general, angle ψ can be defined for any arbitrary point on the reflector surface. This angle varies from –ψ _o to +ψ _o. The normalized radius of a main reflector surface point projected on the aperture plane, a, can be related to the geometrical parameters as [Reference Baars4]:

(1)$$a=\displaystyle{{4f_{p} } \over {d_{p} }}\tan \displaystyle{\psi \over 2}.$$$a=\frac{4f_p}{d_p}\tan \frac{\psi }{2}.$

The variable a is depicted in Fig 1. It can be shown that [Reference Baars4]:

(2)$$\tan \displaystyle{{\psi _{o} } \over 2}=\displaystyle{{d_{p} } \over {4f_{p} }}.$$$\tan \frac{{\psi }_o}{2}=\frac{d_p}{4f_p}.$

From (1) and (2), it is clear that as the angle ψ varies from 0 to ψ _o, a varies from 0 to 1. Using (1), each point on the main reflector surface defined by an angle ψ can be mapped into the aperture plane defined by a normalized radius a.

III. DEFINING THE ILLUMINATION FUNCTION

The field distribution on the aperture plane is approximated by an illumination function. The phase distribution of the projected field on the aperture plane is often assumed to be constant [Reference Baars4, Reference Zaman, Choudhury, Gaffar and Matin8]. Well-designed feed horns can produce illumination with constant phase over the entire aperture, thus justifying the assumption. In such cases, the illumination function represents the variation in field amplitude over the aperture. The feed and therefore the resulting aperture distribution are usually circularly symmetric for an axially symmetric geometry. Thus, the illumination function can be expressed by the single independent variable a, and is denoted as A(a).

The illumination function can be modeled as a Gaussian function, a raised-quadratic function, or other higher order polynomial functions [Reference Zaman, Choudhury, Gaffar and Matin8, Reference Richardson9]. These functions represent a field distribution with maximum value near the bore-sight (axis of the reflector) and gradually taper at the edge. The ratio of the field value at the edge of the aperture to the maximum field value at the center of the aperture is known as the edge taper, T. Most illumination functions use the parameter T [Reference Zaman, Choudhury, Gaffar and Matin8]. However, none of the illumination functions found in the literature take into account the decrease in the AF caused by the subreflector blockage.

In this paper, a new illumination function is defined as

(3)$$A\lpar a\rpar =\left\{\matrix{g\lpar a\rpar \comma \; \quad - a_{e}< a< a_{e} \hfill \cr f\lpar a\rpar \comma \; \quad {\rm otherwise} \hfill} \right.$$$A\left(a\right)=\{\begin{array}{c}g\left(a\right),-a_e<a<a_e\\ f\left(a\right),\text{otherwise}\end{array}$

Here, a _e is defined as the normalized radius of the aperture that is shadowed by the subreflector, f(a) is the illumination function in the region where no aperture blockage has occurred, and g(a) is the illumination function in the shadowed region describing the decreased AF caused by the subreflector blockage. The shadowed region is highlighted in Fig. 1.

Clearly, the parameter a _e is dependent on the subreflector diameter and the main reflector diameter. As a = 1 represents the radius of the main reflector (d _p/2) and a _e is related to the subreflector radius, it can be written intuitively that,

(4)$$a_e=\displaystyle{{d_{s} /2} \over {d_{p} /2}}=\displaystyle{{d_{s} } \over {d_{p} }}.$$$a_e=\frac{d_s/2}{d_p/2}=\frac{d_s}{d_p}.$

The definition of f(a) is taken from [Reference Zaman, Choudhury, Gaffar and Matin8]:

(5)$$f\lpar a\rpar =1 - qga^{2}+\lpar q - 1\rpar ga^{4}.$$$f\left(a\right)=1-{qga}^2+(q-1){ga}^4.$

where q is a parameter that is used to model a feed horn more accurately, and

(6)$$g=1 - T.$$$g=1-T.$

Here, T is the edge taper. As (5) contains even powers of a only, it is symmetric around the origin. The general shape of (5) is a function that has maximum value around the center region, and gradually decreases to T on both sides. This represents typical feed patterns. However, the overall illumination of the main reflector will have a different shape than this because of the subreflector blockage. It is expected that the subreflector will create a shadow around the axial region of the main reflector, causing a decrease in main reflector aperture illumination. This is modeled using the function g(a) in the shadowed region (−a _e < a < a _e).

To keep a consistency with (5), the illumination function in the shadowed regions, g(a), is also defined as a fourth-order polynomial:

(7)$$g\lpar a\rpar =c+ma^{2}+na^{4}.$$$g\left(a\right)=c+{ma}^2+{na}^4.$

Here, c, m, and n are parameters. The parameter c denotes the value of the illumination function at the center of the aperture where the blockage has maximum effect. Thus, c represents the maximum attenuation due to the subreflector blockage. The signs of these constants will be such that the function g(a) will have a U-shaped nature. This implied that the g(a) will have lowest value in the center of the main reflector where the blockage resulting shadowing is maximum. The value will gradually increase toward the edge of the shadowed region.

The parameters m and n can be related to other known parameters by enforcing continuity conditions. As the illumination function describes the AF distribution, it should be continuous and smooth. Hence, the two parts of A(a) defined in (3) must be continuous and smooth at the edge transition point a _e. The following two equations are used to enforce these conditions:

(8)$$f\lpar a_{e} \rpar =g\lpar a_{e} \rpar .$$$f\left(a_e\right)=g\left(a_e\right).$

(9)$$\left. {\displaystyle{{{\rm d}f\lpar a\rpar } \over {{\rm d}a}}} \right\vert _{a=a_{e} }=\left. {\displaystyle{{{\rm d}g\lpar a\rpar } \over {{\rm d}a}}} \right\vert _{a=a_{e}}.$$${\frac{\text{d}f\left(a\right)}{\text{d}a}ǀ}_{a=a_e}={\frac{\text{d}g\left(a\right)}{\text{d}a}ǀ}_{a=a_e}.$

The continuity of the derivatives ensures that the transition is smooth.

Using (5) and (7) in (8):

(10)$$ma_{e}^{2}+na_{e}^{4}=1 - qga_{e}^{2}+\lpar q - 1\rpar ga_{e}^{4} - c.$$${ma}_e^2+{na}_e^4=1-{qga}_e^2+(q-1){ga}_e^4-c.$

Calculating the derivatives from (5) and (7), and substituting them in (9):

(11)$$m+2na_{e}^{2}=- qg+2\lpar q - 1\rpar ga_{e}^{2}.$$$m+2{na}_e^2=-qg+2(q-1){ga}_e^2.$

The parameters m and n can be easily solved from (10) and (11) to yield the following results:

(12)$$\left. \matrix{m=\displaystyle{{2\lpar 1 - c\rpar } \over {a_{e}^{2} }} - qg \hfill \cr n=\lpar q - 1\rpar g - \displaystyle{{1 - c} \over {a_{e}^{4} }} \hfill} \right\}.$$$\begin{array}{c}m=\frac{2(1-c)}{a_e^2}-qg\\ n=(q-1)g-\frac{1-c}{a_e^4}\end{array}\}.$

The parameter a _e can be calculated from the geometry using (4), the parameters q and g can be calculated from the feed horn characteristics, and the value of c can be approximated from the expected field reduction caused by the blockage. Thus, using (12), m and n can be calculated.

As all the parameters are defined and related to known quantities, the definition of the illumination function is complete. Figure 3 shows the plot of the defined illumination function for some arbitrary parameters. The q parameter does not significantly affect the shape of the illumination function. The value of q = 0.4 is used for numerical analysis in this paper.

Fig. 3. Plot of the defined illumination function.

IV. AF INTEGRATION AND FAR-FIELD FORMULATION

The far-field radiation pattern can be calculated by performing the AF integration [Reference Munro, Whitaker and McInnes14]. An observation point P(r, θ, φ) in the far-field region is defined by its spherical coordinates (r, θ, φ). The point is shown in Fig. 1. For a circularly symmetric aperture distribution with constant phase, the far-field radiation pattern can be approximated by the following integral [Reference Baars4]:

(13)$$f\lpar \theta\comma \; \phi \rpar =\displaystyle{{\pi d_{p}^{2} } \over 2}\int\limits_{a=0}^{1} {A\lpar a\rpar J_{o} \left({\displaystyle{{kd} \over 2}a\sin \theta } \right)a\, {\rm d}a}.$$$f(\theta ,\varphi )=\frac{\pi d_p^2}{2}\underset{a=0}{\overset{1}{\int }}A\left(a\right)J_o\left(\frac{kd}{2}a\sin \theta \right)a\text{d}a.$

Here, k =2π/λ=wave number, λ = wavelength and, J_p(·) = Bessel function of the first kind and order p. The angular coordinate variable, u is defined as:

(14)$$u=\displaystyle{{kd} \over 2}\sin \theta.$$$u=\frac{kd}{2}\sin \theta .$

Using (14) and substituting the value of the illumination function from (3) in (13), the integral becomes:

(15)$$f\lpar \theta\comma \; \phi \rpar =\displaystyle{{\pi d_{p}^{2} } \over 2}\left[{\int\limits_{a=0}^{a_{e} } {g\left(a \right)J_{o} \lpar ua\rpar a\, {\rm d}a} } \;{+\int\limits_{a=a_{e} }^{1} {f\left(a \right)J_{o} \lpar ua\rpar a\, {\rm d}a} } \right].$$$f(\theta ,\varphi )=\frac{\pi d_p^2}{2}\left[\underset{a=0}{\overset{a_e}{\int }}g\left(a\right)J_o\right(ua)a\text{d}a+\underset{a=a_e}{\overset{1}{\int }}f\left(a\right)J_o(ua\left)a\text{d}a\right].$

The value of f(a) and g(a) can be substituted from (5) and (7). The resulting formulation involves integrating product of polynomials and Bessel function. The necessary integration results are [Reference Polyanin and Manzhirov15]:

(16)$$\left. \matrix{\int {a\, J_{o} \lpar ua\rpar \, {\rm d}a=} \displaystyle{{a\, J_{1} \lpar ua\rpar } \over u} \hfill \cr \int {a^{3} J_{o} \lpar ua\rpar \, {\rm d}a=} \displaystyle{{a^{3} J_{1} \lpar ua\rpar } \over u} - \displaystyle{{2a^{2} J_{2} \lpar ua\rpar } \over {u^{2} }} \hfill \cr \int {a^{5} J_{o} \lpar ua\rpar \, {\rm d}a=} \displaystyle{{a^{5} J_{1} \lpar ua\rpar } \over u} - \displaystyle{{4a^{4} J_{2} \lpar ua\rpar } \over {u^{2} }}+\displaystyle{{8a^{3} J_{3} \lpar ua\rpar } \over {u^{3} }} \hfill} \right\}.$$$\begin{array}{c}\int a{J}_o\left(ua\right)\text{d}a=\frac{a{J}_1\left(ua\right)}{u}\\ \int a^3{J}_o\left(ua\right)\text{d}a=\frac{a^3{J}_1\left(ua\right)}{u}-\frac{2a^2{J}_2\left(ua\right)}{u^2}\\ \int a^5{J}_o\left(ua\right)\text{d}a=\frac{a^5{J}_1\left(ua\right)}{u}-\frac{4a^4{J}_2\left(ua\right)}{u^2}+\frac{8a^3{J}_3\left(ua\right)}{u^3}\end{array}\}.$

To simplify the expressions, Bessel functions are replaced by Lambda function Λ_p(.) defined as [Reference Polyanin and Manzhirov15]:

(17)$$\Lambda _{p} \lpar u\rpar =p!\displaystyle{{J_{p} \lpar u\rpar } \over {\lpar u/{2}\rpar ^{p} }}.$$${\Lambda }_p\left(u\right)=p!\frac{J_p\left(u\right)}{(u/2{)}^p}.$

After evaluating the integrals in (15) and normalizing the results, the obtained expression of normalized far-field radiation pattern, f_N(θ, ϕ) is:

(18)$$\eqalign{f_{N} \lpar \theta\comma \; \phi \rpar & =\displaystyle{1 \over N}\left[{\Lambda _{1} \lpar u\rpar \left({\displaystyle{{1 - g} \over 2}} \right)+\Lambda _{2} \lpar u\rpar g\left({\displaystyle{{2 - q} \over 4}} \right)} \right. \cr & \quad +\Lambda _{3} \lpar u\rpar g\left({\displaystyle{{q - 1} \over 6}} \right)+\Lambda _{2} \left({ua_e } \right)\left\{{\displaystyle{{qga_{e}^{4} \left({1 - a_{e}^{3} } \right)} \over 4}} \right\}\, \, \cr & \left. \quad {+\Lambda _{3} \left({ua_{e} } \right)\left\{{\displaystyle{{a_{e}^{2} \lpar 1 - c\rpar } \over 6}} \right\}} \right].}$$$\begin{array}{cc}{f}_N(\theta ,\varphi )& =\frac{1}{N}\left[{\Lambda }_1\right(u)\left(\frac{1-g}{2}\right)+{\Lambda }_2(u)g\left(\frac{2-q}{4}\right)\\ & +{\Lambda }_3\left(u\right)g\left(\frac{q-1}{6}\right)+{\Lambda }_2\left({ua}_e\right)\left\{\frac{{qga}_e^4(1-a_e^3)}{4}\right\}\\ & +{\Lambda }_3\left({ua}_e\right)\left\{\frac{a_e^2(1-c)}{6}\right\}].\end{array}$

Here, the normalizing factor, N is given by:

(19)$$N=\displaystyle{1 \over {12}}\left\{{6 - 2g - 2a_{e}^{2} \lpar 1 - b\rpar } \right.\left. {+\, qg\left({3a_{e}^{4} - 3a_{e}^{7} - 1} \right)} \right\}.$$$N=\frac{1}{12}\{6-2g-2a_e^2(1-b)+qg(3a_e^4-3a_e^7-1)\}.$

Equation (18) gives the approximate closed-form expression of the far-field radiation pattern of the Cassegrain antenna, which is expected to be sufficiently accurate near the main-lobe region of the antenna.

V. NUMERICAL RESULTS

For numerical analysis, a standard Cassegrain antenna with the main reflector diameter d _p = 2 m, primary focal length f _p = 0.8 m, subreflector diameter d _s = 0.5029 m, and distance between subreflector foci 2c = 0.7 m is assumed. The relation between these parameters and other geometrical parameters are calculated using the method described in [Reference Granet13]. A typical edge taper of 10 dB is assumed, implying T = 10^−10/20 = 0.3162. The parameter a _e is found from (4) as 0.25145. The operating frequency is assumed to be 10 GHz. It is found that for the defined geometry and edge taper, q = 0.4 accurately describes a conical corrugated horn feed. The q parameter value is calculated using the method described in [Reference Zaman, Choudhury, Gaffar and Matin8].

The far-field radiation pattern calculated using (18) and using the standard PO method [Reference McNamara, Pistorius and Malherbe5] is shown in Fig. 4. It can be seen from Fig 4 that the main-lobe patterns obtained from the two methods are exactly the same. The first few sidelobe levels obtained from the derived expression differ from the results of the PO method only by a few dB. The results are summarized in Table 1. It can be observed from the data that the gain and half-power beamwidth (HPBW) computed from the derived method are almost identical to the results obtained from the PO and PO + PTD methods, showing a maximum error of only 0.2% (accuracy of 99.8%). The proposed method has an error around 2–3% in identifying the angular position of the first sidelobe. The error in computing the first sidelobe level is around 8–9%. This implies that the prediction of the first sidelobe is up to 91% accurate. The proposed method also satisfactorily predicts the position and the amplitude level of the second significant level with errors as little as 0.3% and 3.35%, respectively, when compared with the PO + PTD method. This error is higher when compared with the PO method (without PTD correction). However, PO + PTD methods are expected to have more accuracy in sidelobe regions compared with the PO method. Thus, the derived equations also have excellent agreement in the second sidelobe region as well.

Fig. 4. Far-field radiation pattern.

Table 1. Comparison of numerical results obtained from the derived method, PO method, and PO + PTD method.

For a simplified closed-form expression, the accuracy of the derived equations can be considered satisfactory. The closed-form expression can be used for approximating the main-lobe and the first few sidelobes for cases where quick calculations are required. The PO method and PO + PTD method are computationally demanding and require significant simulation time for large reflectors [Reference McNamara, Pistorius and Malherbe5]. As the derived expression is closed form, the simulation time is drastically reduced.