Ultra-wideband optical leaky-wave slot antennas: errata

Yan Wang; Amr S. Helmy; George V. Eleftheriades

doi:10.1364/OE.21.013184

Abstract

The errata consist of corrections for one typo and one misinterpretation of the results regarding the antenna efficiency in the original article [Opt. Express 19, 12392–12401 (2011)].

1. Correction 1

There is a typo in Eq. (1) in [1], where a factor of 2 is missing. The correct expression should read as

H_{θ} = j \frac{2 k_{2} I_{m 0} l}{η_{2}} \frac{e^{- j k_{2} r}}{4 π r} sin θ (e^{j \frac{k_{2} l}{2} (cos θ - K)}) \frac{sin (\frac{k_{2} l}{2} (cos θ - K))}{\frac{k_{2} l}{2} (cos θ - K)}

The factor of 2 arises from the application of the field equivalence principle, such that the original structure can be interpreted as a magnetic current backed by an infinite conducting plane (for each medium). Hence the magnetic current I_m ₀ doubles due to the image theory. This typo has no influence on the radiation patterns presented in [1] since all results are normalized.

2. Correction 2

There is a mis-interpretation of the results regarding the antenna efficiency (Fig. 8) presented in the first paragraph on page 12400. The re-interpretation and derivations of the analytical expression of antenna efficiency are elucidated as the following.

The antenna efficiency, defined as η_eff = P _rad(L)/P _in × 100, is calculated based on the attenuation constant extracted (Fig. 6(b) in [1]). For nominal λ ₀ = 1550 nm, the total and radiative attenuation constants are α_t = −0.370k ₀ and α_r = −0.207k ₀, respectively, which implies that the dissipative attenuation constant is α_d = α_t − α_r = −0.163k ₀. The power attenuation along the slot due to these loss mechanisms is illustrated in Fig. 1. (Note that Fig. 1 is used to replace Fig. 8 in [1] for better clarity.)

Fig. 1 Power decay along the PLS antenna due to the total, radiative and dissipative attenuation constants, respectively.

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In order to calculate the radiated power P _rad(L) for a slot of length L, the total attenuation in Fig. 1 is discretized into L = NΔz, where Δz is assumed infinitesimally small. Therefore, the radiated power can be written as the sum of the radiation from each Δz section as shown in

\begin{array}{l} P_{rad} (L) & = lim_{Δ z \to 0} \sum_{n = 0}^{N - 1} [P_{tot} (n Δ z) (1 - e^{2 α_{r} Δ z})] \\ = lim_{Δ z \to 0} \sum_{n = 0}^{N - 1} [P_{in} e^{2 α_{t} n Δ z} (1 - e^{2 α_{r} Δ z})] \\ = lim_{Δ z \to 0} [P_{in} (1 - e^{2 α_{r} Δ z}) \sum_{n = 0}^{N - 1} {(e^{2 α_{t} Δ z})}^{n}] \end{array}

Recognizing the summation in Eq. (2) is a geometric series, the antenna efficiency can then be written as

\begin{array}{l} η_{eff} & = \frac{P_{rad} (L)}{P_{in}} \times 100 \\ = lim_{Δ z \to 0} [(1 - e^{2 α_{r} Δ z}) \frac{1 - {(e^{2 α_{t} Δ z})}^{N}}{1 - e^{2 α_{t} Δ z}}] \times 100 \end{array}

Applying L’Hôpital’s rule, the analytical expression for the antenna efficiency is obtained

η_{eff} = (1 - e^{2 α_{t} L}) \frac{1 - 2 α_{r}}{1 - 2 α_{t}} \times 100

Similarly, the dissipated power percentage can be calculated from

\frac{P_{dis} (L)}{P_{in}} \times 100 = (1 - e^{2 α_{t} L}) \frac{1 - 2 α_{d}}{1 - 2 α_{t}} \times 100

The results for the radiation and dissipation with respect to the input are shown in Fig. 2, where the sum of these two loss mechanisms is shown to yield the expected total loss, hence confirming the validity of the approach in Eqs. (4) and (5).

Fig. 2 The percentage of the total, radiated and dissipated power loss with respect to the input.

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In conclusion, the antenna efficiency can be extracted from the percentage of radiation in Fig. 2 (the blue dashed line). It is shown that it takes an antenna length of 0.5λ ₀ to lose 90% of the input power; out of this total loss, 50.5% is due to radiation and 39.8% due to dissipation. Additionally, in order to lose 99% of the input power, the PLS antenna should be at least one wavelength long. Finally, a PLS with L = 1.5λ ₀ attenuates 99.9% of the input power, which can be regarded as equivalent to an infinite slot.

References and links

1. Y. Wang, A. S. Helmy, and G. V. Eleftheriades, “Ultra-wideband optical leaky-wave slot antennas,” Opt. Express 19, 12392–12401 (2011) [CrossRef] [PubMed] .

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