Engineering Hybrid Guided Modes in Subwavelength Uniaxial Metamaterial Waveguides

Abstract

We report anomalous dispersion properties of hybrid guided modes (HGMs) and their group velocity in a subwavelength uniaxial metamaterial waveguide with metal cladding. We derive exact dispersion relations and modal fields of HGMs by solving eigenvalue equation based on basic electromagnetic field theory in detail. Numerical results show that two fundamental HGMs and two types of high-order HGMs can be excited, and their exciting conditions are clarified. In addition, such HGMs can be engineered to belong to normal dispersion or anomalous dispersion. Importantly, the HGMs may be controlled to be forward or backward, and their group velocities may be very small in a certain frequency band. These properties make such metamaterial waveguides have many potential applications in integrated optics, information storage and biosensing.

Appendix A: Proof of Δ>0 and ω 2/c 2<β 2 cos2𝜃/ε 1 μ 1

According to the necessary and sufficient conditions (2) of optically uniaxial metamaterials, we can get μ ₁ ε ₂−μ ₂ ε ₁≠0, where ε ₁ μ ₁>0.

The sign of the expression, ω ²/c ²−β ² cos2𝜃/ε ₁ μ ₁, would affect the roots of Eq. 12. In fact, the expression is only less than zero at the discussed two kinds of uniaxial metamaterial structures at present.

We can use reduction to absurdity to prove the expression is less than zero. Supposing ω ²/c ²−β ² cos2𝜃/ε ₁ μ ₁>0, we can obtain cos2𝜃<ω ² ε ₁ μ ₁/(c ² β ²), where ε ₁ μ ₁>0.

If μ ₁/μ ₂>ε ₁/ε ₂, two roots of Eq. 12 can be written as

$$ r_{1}=\beta^{2}\sin^{2}\theta +\beta^{2}\cos^{2}\theta \mu_{2}/\mu_{1}-\mu_{2}\varepsilon_{1}\omega^{2}/c^{2}, $$

(A1)

$$ r_{2}=\beta^{2}\sin^{2}\theta +\beta^{2}\cos^{2}\theta \varepsilon_{2}/\varepsilon_{1}-\mu_{1}\varepsilon_{2}\omega^{2}/c^{2}. $$

(A2)

According to the analyses about r ₁ and r ₂, one can know that only when r ₁<0 or r ₂<0, electromagnetic modes can propagate in the uniaxial metamaterial waveguide structures. If r ₁<0, from Eq. A1, we can obtain cos2𝜃+ sin2𝜃 μ ₁/μ ₂>ω ² μ ₁ ε ₁/(c ² β ²). If r ₂<0, from Eq. A2 ),we can get cos2𝜃+ sin2𝜃 ε ₁/ε ₂>ω ² μ ₁ ε ₁/(c ² β ²). On account of ε ₁/ε ₂<0, μ ₁/μ ₂<0, so we can get ω ²/c ²−β ² cos2𝜃/ε ₁ μ ₁ <0, which is mutually contradictory with the previous assumption.

If μ ₁/μ ₂<ε ₁/ε ₂, similarly, we can get ω ²/c ²−β ² cos2𝜃/ε ₁ μ ₁ <0, which is also mutually contradictory with the previous assumption.

In addition, supposing ω ²/c ²−β ² cos2𝜃/ε ₁ μ ₁=0, that is to say, β ² = ω ² ε ₁ μ ₁/(c ² cos2𝜃). On account of the constraint of frequency we discussed, we can get r _1,2= tan2𝜃 ε ₁ μ ₁ ω ²/c ²>0. Only r _1,2<0, however, the waveguide modes can exist, so we can get ω ²/c ²−β ² cos2𝜃/ε ₁ μ ₁≠0.

Then due to ω ²/c ²−β ² cos2𝜃/ε ₁ μ ₁<0 and μ ₁ ε ₂−μ ₂ ε ₁≠0 , from Eq. 13, we can get the discriminant Δ>0, that is to say, Eq. 12 has two different roots.