Formulas for Beam Shift and Beam Narrowing in 1-D Leaky-Wave Antennas Due to Element Pattern

The radiation pattern of 1-D unidirectional leaky-wave antennas (LWAs) is typically derived from the analysis of the space factor (SF) only. However, when the antenna aperture is characterized by a longitudinal radiating current, a cosine-type element pattern term has to be included: this factor may considerably affect the beam properties. Specifically, the beam peak is shifted and the beamwidth is narrowed when the element pattern is included. These effects are more prominent as the antenna size decreases and the beam angle increases. Approximate but accurate analytical formulas are here provided and numerically validated to predict the beam angle and the half-power beamwidth for these kinds of 1-D unidirectional LWAs when the element pattern is included.


I. INTRODUCTION
R ECENT progress has been made to improve the estimation of the beam angle θ 0 and the half-power beamwidth Θ h in 1-D leaky-wave antennas (LWAs) for both unidirectional [1], [2], [3], [4], [5], [6] and bidirectional excitations [7], [8], [9]. Most of these works provided corrections to the asymptotic formulas derived in [10] and [11], when the antenna is truncated, but assume that the radiation pattern is based solely on the space factor (SF). This assumption holds when the antenna aperture is characterized by a transverse radiating current (see Fig. 1, top), as in LWAs with transverse slots [12], [13], [14]. When the antenna aperture is characterized by a longitudinal radiating current (see Fig. 1, bottom), as in slitted waveguide LWAs [15], the SF has to be multiplied by a cos θ factor (θ being measured from the vertical z-axis) that accounts for the dipole-like element pattern. This can be described as a polarization effect, as it depends on the polarization of the current.
The work in [5] originally investigated this problem in detail, showing that when the beam peak approaches endfire (θ 0 → 90 • ) and for electrically short antenna lengths (L 10 λ 0 ), this aspect has nonnegligible effects on the radiation pattern: i) the actual beam angle θ 0p is shifted with respect to that predicted from the SF only, viz., θ 0 ; ii) the actual beamwidth Θ hp  (measured between the −3 dB points) is narrowed with respect to that predicted from the SF only, viz., Θ h . These polarization effects are a consequence of the null at endfire enforced by the cosine term. The investigation in [5] reported numerical results for some specific cases and an experimental validation, without providing any formulas to predict the observed beam shift and the beam narrowing. Here, we propose approximate, fully analytical formulas to accurately predict both the beam angle and the beamwidth in 1-D unidirectional LWAs accounting for the polarization effect. In this regard, we should stress that the formulas proposed in [6] for 1-D unidirectional LWAs, although including the element pattern, assumed the cosine term be approximately constant near the beam peak, and therefore ignored its effect. As a result, the formula in [6, eq. (20)] is almost equivalent to the one later proposed in [3, eq. (6)], which indeed ignores polarization effects.
This letter is organized as follows. Section II describes the problem and motivates the need for the formulas. In Section III, analytical formulas are derived for both the beam angle and the beamwidth, and then validated with numerical results. Conclusions are finally drawn in Section IV.

II. DESCRIPTION OF THE PROBLEM
The reference structure is the one reported in Fig. 1 and consisting of a 1-D unidirectional LWA of length L, whose radiating aperture is dominated by a single leaky mode characterized by a complex leaky wavenumber k x = β − jα. In this case, the SF is obtained by Fourier-transforming the radiating current distribution, which reads exp(−jk x x)Π(x/L − 0.5), where Π(·) is the centered unit rectangular pulse. When a transverse radiating current is assumed, the radiation power pattern is obtained by just taking the modulus squared of the SF and thus reads [1] where we have conveniently defined the normalized variables t = b − l sin θ and a =αl, with b =βl and l = πL/λ 0 , and the normalized phaseβ := β/k 0 and attenuationα := α/k 0 constants, k 0 = 2π/λ 0 being the free-space wavenumber.
When a longitudinal radiating current is assumed, the SF has to be multiplied by a cosine term, and thus the radiation power pattern reads From these two expressions, it is possible to numerically evaluate the beam angles and the half-power beamwidths. The results are reported in Fig. 2(a) and (b), where the deviations of the beam angles (namely, the beam shift) and the beamwidths, defined as Δθ = θ 0 − θ 0p , and ΔΘ = Θ h − Θ hp are both normalized to the half-power beamwidth Θ hp (and then multiplied by 100 to get percent) and shown as functions of L and θ 0 to highlight how important the beam shift and the change in the beamwidth really are. This normalization is relevant to highlight the practical importance of the changes in the beam angle and the beamwidth.
In both cases, the radiation efficiency e r = 1 − exp(−4a) is fixed to 20% (this choice will be further motivated next), and hence the leakage rate is in the range 10 −4 ≤α ≤ 10 −2 for 1 ≤ L/λ 0 ≤ 100, which is rather practical. Results do not substantially change as long as e r < 0.99 (for e r > 0.99 and L < 5 λ 0 , thenα > 0.1 and the leaky mode might no longer be dominant [10], [11], thus controverting the initial assumption).
The results in Fig. 2(a) and (b) clearly highlight that polarization effects are negligible for θ 0 < 45 • regardless of the antenna size, whereas they are increasingly more appreciable for θ 0 > 45 • and more so as θ 0 increases and L/λ 0 decreases. The boundary between these two regions is reported as a black dashed line in Fig. 2(a) and (b). Remarkably, this boundary can be predicted by the following formula for the "discontinuity angle" θ 0d = arcsin(1 − t h /l) discussed in [3], where t h = 1.39156[1 − tanh(0.021a)] + a tanh(0.21a). This formula gives the beam angle θ 0 for which the right-sided −3 dB point would be at endfire, ignoring polarization effects. As can be inferred from Fig. 2(a) and (b), polarization effects become important for θ 0 > θ 0d .
We note that the results reported so far are consistent with those obtained in [5] for a limited number of cases. Moreover, they clearly show that polarization effects cannot be neglected for practical LWAs radiating close to endfire, thus motivating the interest in having analytical formulas for a correct prediction of both the beam angle and the beamwidth. The following Section III is thus devoted to this task.

A. Beam Shift
We start by deriving a formula for correctly predicting the beam angle from the expression for P (θ) in (2). A straightforward application of the first derivative test to find the local maximum does not allow for a closed-form solution, thus we first use the following "key" approximation for P 0 (θ): which is obtained from (2) by means of a second-order Taylor expansion with respect to t and near θ 0 (for which t = 0), which is the exact beam peak for P 0 . Note that t = ±t h corresponds to the left/right −3 dB points of P 0 . By exploiting the definition of t, and using (3), (2) can be recast as a function of t only The application of the first derivative test to this expression with respect to the variable t leads to a cubic polynomial equation a 3 t 3 + a 2 t 2 + a 1 t + a 0 = 0 whose coefficients have the following expressions: Although there exist three roots, Descartes' signs rule tells us that there exist only two positive roots, but does not provide any guess about which one should be the correct one. With the help of numerical calculations and the use of Cardano's formula, it is found that the correct root in t is given by where with Δ 0 = a 2 2 − 3a 3 a 1 and Δ 1 = 2a 3 2 − 9a 3 a 2 a 1 + 27a 2 3 a 0 and where the cubic and square roots are interpreted as the principal branches. The beam angle is finally found by using the definition of t, which yields θ 0p = arcsin(β − t p /l).
The accuracy of (6) and (8) can be inferred from Fig. 3(a), where numerical results for the SF only (solid line) and for the polarization effect (dashed lines) are compared with those given by the analytical formula (circles) for different antenna lengths when θ 0 is allowed to vary from broadside to endfire. The formula is remarkably accurate as confirmed also by Fig. 3(b) and (c), where its error relative to the beamwidth is seen to be under a few percent. For a few cases CST full-wave validations have been performed. Namely, we considered four different LWAs based on rectangular waveguides operating at f 0 = 10 GHz in their fundamental TE mode and with one of the side walls replaced by a PRS to allow for radiation. The simulation setup is based on the ideal model of 1-D PRS-based LWAs described in [16, Sec. V.A], which is not elaborated on here for the sake of brevity. As opposed to the examples discussed in [16] that are optimized for maximum gain, here the structures are designed with a rather low radiation efficiency, viz., 20%, to accentuate the effects of the element pattern and, at the same time, have a practical value ofα with a short antenna.
For a fixed radiation efficiency of 20%, we thus designed four different antennas radiating at θ 0 = 65 • , 70 • , 75 • , 80 • when L/λ 0 = 2, 5, 10, 20, respectively, using the equations provided in [16,Sec. V(a)]. The angles of maximum radiation are then extracted from the simulated, full-wave 3-D radiation patterns and reported as colored asterisks in Fig. 3(a). In all cases, an overall good agreement is obtained.

B. Beamwidth
The derivation of the beamwidth formula is based on the same approximation used for the beam shift in the previous Section III-A. In addition, the first term in square brackets in (4) is further expanded with a Taylor series at t = 0. Depending on whether a first-or a second-order expansion is used, a third-order or a fourth-order polynomial equation is obtained as we will discuss later.
In order to obtain a beamwidth equation, the power pattern is usually normalized to its maximum and then equated to a half. Here, the pattern is first normalized with respect to its value at t = 0 (which is equal to cos 2 θ 0 ), which is not exactly where the maximum occurs due to the element pattern. The normalized pattern is then equated not to a half, but to half the value of the normalized pattern at t = t p [which is now known from (6)].
The expressions of the coefficients of the third-order polynomial equation b 3 t 3 + b 2 t 2 + b 1 t + b 0 = 0 obtained from a first-order Taylor expansion are as follows: whereas those of the fourth-order polynomial equation c 4 t 4 + c 3 t 3 + c 2 t 2 + c 1 t + c 0 = 0 obtained from a secondorder Taylor expansion are as follows: with where P (1) m and P (2) m are the normalized power patterns evaluated at t = t p and obtained after a first-order and a second-order Taylor expansion, respectively. (The reason why we keep both approximations will be soon clarified.) Following the discussion in [3], the expression of the beamwidth of a 1-D unidirectional LWA when polarization effects are accounted for can be expressed as where t r hp < 0 and t l hp > 0 notably refer to the right and left −3 dB points, respectively (note that in [3, eq. (6)] only the SF is taken into account and thus t r hp = −t l hp = −t h ). However, from Descartes' signs rule we can expect at least two positive roots in both cases from (9) and (10). A criterion is thus needed for choosing the correct determination. For the third-order polynomial equation, we found that one can use (6) as is (obviously replacing the coefficients a i with b i , ∀i). However, one should be careful with the root definition for C in (7): for t r hp the expression for C is the same as in (7), whereas for t l hp the expression for C in (7) should be multiplied by e −j2π/3 to get the correct root.
For the fourth-order polynomial equation, one has to resort to Ferrari-Cardano's formulas. Among the four possible roots, the two correct choices are (15) with p = (8c 4 c 2 − 3c 2 3 )/(8c 2 4 ), q = (c 3 3 − 4c 4 c 3 c 2 + 8c 2 4 c 1 )/ (8c 3 4 ), and the known expressions for S and Q read with Δ 0 = c 2 2 − 3c 3 c 1 + 12c 0 c 4 , and Δ 1 = 2c 3 2 − 9c 3 c 2 c 1 + 27c 2 3 c 0 + 27c 4 c 2 1 − 72c 4 c 2 c 0 . The accuracy of the beamwidth formula with the third-order and the fourth-order polynomial expressions can be inferred from Fig. 4(a) where they are reported as triangles and squares, respectively. Notably, the third-order polynomial provides a slight overestimation, whereas the fourth-order polynomial provides a slight underestimation. The average between the values provided by these two expressions (in circles) turns out to be an excellent estimation of the beamwidth. For comparison purposes, the beamwidth as predicted by the SF only is also reported (dashed lines), showing that polarization effects considerably affect the beamwidth as the beam peak increases beyond 45 • , especially for short antenna sizes. The accuracy of the formula obtained as the average value of the third-order and fourth-order polynomial expressions is further highlighted by the error plots in Fig. 4(b) and (c), where the error is shown to be at most a few percent. As per the beam shift, full-wave results from the same examples of Section III-A are reported with colored asterisks, showing very good agreement. For the sake of completeness, Table I also reports some numerical values for a few practical cases of a LWA of 20 λ 0 with a radiation efficiency of about 92%, which is approximately the value required to have optimum gain [3]. Table I shows that for a moderately large LWA length (L = 20 λ 0 ), the effects of the element pattern on the beam shift and the beamwidth are only significant when the beam is scanned past about 75 • . At 75 • , the beam shift due to the element pattern is about 23% of the beamwidth, and the element pattern has also caused the beam to have narrowed to about 76% of that predicted by the SF alone. We should note that neither the formula in [3] nor that in [6] are capable of predicting the beam narrowing effect, although the radiation power pattern in [6] includes the element pattern (whereas [3] does not).

IV. CONCLUSION
The effects of the element pattern in 1-D unidirectional LWAs is demonstrated to be important when the beam angle approaches endfire and the antenna size is short. Approximate, accurate, fully analytical formulas are derived here for correctly estimating both the beam angle and the beamwidth when the element pattern is included. Although derived for an untapered aperture, the formulas are also accurate for apertures having a fairly gradual taper (results not shown for brevity). Previous formulas, based on the SF only, are instead shown to be no longer accurate when the beam angle approaches endfire; a criterion to determine the region where the formula loses accuracy is also provided. Results are extensively validated numerically to verify the accuracy of the new formulas.