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Abstract
Field transformation is complementary to transformation optics as it converts polarization without reflection for any incidence, despite not being impedance-matched to vacuum. However, to ease manufacture, a dielectric approximation may degrade the transmission performance due to the approximate dispersion relation for the exact reciprocal medium. In this paper, we instead propose a different dielectric equivalent medium to guarantee the exact dispersion. We also discuss the birefringence phenomenon in such dielectric media, and their unavoidable induced reflections.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
It is well known that a planar wave propagates in a homogenous medium, but it is generally analytically elusive what kind of waves can move in inhomogeneous media. Whatsoever, reflection and refraction should generally occur at the boundary of two distinct media. An exception comes around from the mathematical method of transformation optics (TO) [1–3], which reveals a general recipe to design an inhomogeneous medium aiming for predefined wave behaviors of null reflection, privileged by the state-of-the-art to manufacture artificial electromagnetic materials. The surprising point of reflectionlessness for an incidence waves from air to traversing the transformation media is attributed to the invariance of wave impedance. However, experimental constraints yet require a further dielectric approximation to remove the demand of magnetic materials [4]. Moreover, other reflectionless media are also available by making use of the spatial Kronig–Kramer method and phase-amplitude equations when the impedance match condition is relaxed [5–8]. The merit of this line of reflectionless media is not only to give fundamental gain to knowledge of wave equations, but also to pave the path towards engineering design to control waves in practical efficiency.
Nevertheless, the field transformation (FT) method also gives reflectionless media a reciprocal material design, which was proposed as complementary to TO [9]. Starting from Tretyakov’s proposal [10], it exploits the local unitary gauge symmetry for electric and magnetic fields simultaneously and provides an additional tunability for their polarization status. FT is intimately connected to the concept of near-to-far FT (Stratton–Chu formula) [11], which has yet not been exploited throughly. It formally packs the electromagnetic (EM) fields into a spinor form as a Riemann–Silberstein vector [12–14], and recasts Maxwell equations into a Dirac-like equation. Specifically, it can be adopted in two-dimensional (2D) fabric to keep energy density and the in-plane Poynting vector invariant [9,15,16]. In the 2D form, all fields are reduced to two-component spinor ${({E_z},{\textit{iH}_z})^{T}}$, from which their transverse components can be determined accordingly. This is peculiar as the ideal FT medium remains reflectionless despite not being exactly impedance-matched to vacuum. However, a dielectric approximation is still called on to ease the realization of reciprocal materials, considering the reproduction dispersion relation approximately, which will compromise performance to some extent in the case of polarization converters [17,18].
The dielectric approximation medium [17] cannot fully reproduce perfect transmission and zero reflection as the ideal (exact) FT one does. One could speculate that such imperfection comes from the unfaithful dispersion from the dielectric approximation, which is not the case as our work will show. Hereby, our paper provides a different dielectric medium by reproducing the exact dispersion, instead of dropping a small term in previous work [17]. In Section 2, our dielectric medium will be shown as a biaxial anisotropic medium, which is manufacturable with current metamaterial designs. We also discuss in Section 3 the birefringent behaviors for the light transversing the exact medium and both of its dielectric media and explain how such birefringent rays converge to single transmission after transmitting back to vacuum. The intrinsic reflective property of dielectric approximation media is also discussed to reveal the peculiar non-reflectionlessness for exact FT media designed from the FT approach. Our simulation results indicate that the effort of making the exact dispersion still gives comparable polarization conversion. It seems, in principle, that the underperformance for dielectric media is inevitable because only the exact (magnetic) FT medium is perfect in the discussed polarization transferring scenario. We note that other dielectric mediums can also be made up from tilt layered structures [19–21], which is beyond the scope of this paper.
2. EXACT DISPERSION
A. Preliminary
In this section, we start from previous derivation of FT [9,16] to set up preliminaries of our dielectric approximation method. Suppose that our system remains invariant along the $z$ direction, and we separate all EM vectors into transverse ($ T $) and longitudinal ($ z $) components so that a general anisotropic constitutive relation is written as
Maxwell equations in Heaviside–Lorentz units (H. L. U.) with time-harmonic terms ${e^{- i\omega t}}$ of free charge and current,
can be recast as the propagation of only two coupled $ z $ polarizations ${E_z}$ and ${H_z}$, which also determine the $ T $ components as below:The components ${E_z}$ and ${H_z}$ or, equivalently, three components of electric vectors are tangled in the general wave of Eq. (7),
In this paper, we take a polarization converter in Fig. 1 as an example to illustrate our dielectric approximation. Throughout such a converter, an incidence wave ${(E_z^{(0)},{\textit{iH}}_z^{(0)})^{T}}$ transverses the FT block while experiencing a unitary local phase rotation $\phi (x,y)$ and converts into a transmission wave ${({E_z},{\textit{iH}_z})^{T}}$, which is massaged into a compact matrix form,
Fig. 1. Schematic of field-transformation where the blue block serves as the FT medium to convert incidence wave (black arrow) to transmission wave (blue arrow) and reflection wave (yellow dashed arrow).
Note that the local gauge symmetry above for Maxwell’s equations is reminiscent of the duality transformation, which leaves the in-plane Poynting vector and the spinor modulus invariant [23]. In the case of our polarization converter, the transverse electric (TE) mode converts completely into the transverse magnetic (TM) mode, i.e., ${(E_z^{(0)},{\textit{iH}}_z^{(0)})^{T}}={ (1,0)^{T}},({E_z},{\textit{iH}_z}{)^{T}} = (0, - {1)^{T}}$ if the gauge phase is defined as $\phi = - \pi /2$. This induces cross terms between transverse and longitudinal components in permittivity and permeability matrices, explicitly as
Symmetric material matrices are assumed for simplicity, although asymmetric ones are also extensible, for example, in gyromagnetic materials. In order to achieve reflectionlessness, the diagonal term is taken to be unity $n = 1$ to match with vacuum and the gauge vector ${\bf A} = ({A_x},{A_y}{)^{T}} = \nabla \phi /{k_0}$ parallel to the interface. This is the trivial case of null pseudo-magnetic force $\nabla \times {\bf A} = 0$. Resulting from FT, the anisotropic material of Eqs. (16) and (17) is remarkable because it guarantees perfect non-reflection for the incidence wave ${(E_z^{(0)},{\textit{iH}}_z^{(0)})^{T}}$, although it is not impedance-matched to vacuum [cf. Figs. 2(a) and 2(b)].
Fig. 2. (a) and (b) Dispersion contours for ideal FT media in blue solid curves, Li’s approximation medium in purple dashed curves, our dielectric media 1 and 2 in red solid curves, and vacuum in a blue dotted curve. (b) Blue arrow for incidence wave, green arrow for reflected wave, red arrows for refracted wave vectors, and purple arrows for refracted rays. Three-dimensional normal surfaces for (c) ideal FT media and (d) our dielectric media 1 along with its three principal axes marked in arrows; chained red circles for 2D dispersion contours as in (a) and (b). Parameters: $\theta ={ 45^ \circ}$, ${A_x} = 0$, ${A_y} = - 0.25$, $n = 1$.
However, such magnetic materials are always difficult to manufacture, and one often resorts to a more realizable scheme of dielectric approximation, which results from enforcing a non-magnetic material while keeping the original wave equation solution [4] or its approximate dispersion [9]. Our dielectric approximation is achieved via reproducing the exact dispersion with dielectrics, as will be detailed below. We note that when separately treating TE and TM modes to keep their dispersions we will obtain an estranged dispersion off of the original one.
B. Li’s Dielectric Approximation
It is straightforward to obtain the explicit dispersion curve for exact medium in Eq. (13), and they are quartic closed curves [24,25], shown as blue chained circles in Fig. 2(a). The convectional dielectric approximation (Li’s approximation) derives from enforcing non-magnetic materials,
The solutions for Li’s dielectric approximation are then solved by neglecting quadratic terms of wave-numbers contrasted with the quartic of them, which approximates well except for the grazing angles
Fig. 3. (a)–(f) Field plots from wave simulation for incidence angle $\theta ={ 45^ \circ}$ when the beam transverses the designed material for a thickness of $h = \lambda$ demarcated by two black lines. (a) and (b) Ideal FT medium; (c) and (d) Li’s approximation medium; (e) and (f) our dielectric medium 1. (a), (c), and (e) for ${E_z}$ and (b), (d), and (f) for ${H_z}$.
Fig. 4. Conversion ratio $\eta (\theta)$ from TE to TM for a plane wave with incidence angle $\theta$, as in Eq. (24), for four media as listed in the lengend. It samples a line-segment $l$ between the two points $(1.5,1)\lambda$ and $(2.5,1)\lambda$.
The wave amplitudes for both (a), (b) exact and (b)–(d) in Li’s approximation media are shown in Fig. 3, where TE converts to the TM mode for an incidence angle of 45° after transversing a thickness of $h = \lambda$, respectively. For Li’s approximation medium, the conversion ratio $\eta$ from TE to TM remains rather close to that of the exact medium until the incidence angle gets more than 45° [cf. red dashed curve in Fig. 4], where the conversion ratio is defined to sample only line-segment $l$ between the two points $(1.5,1)\lambda$ and $(2.5,1)\lambda$ [26]:
Nevertheless, the conversion ratio of Li’s approximation still outperforms that of the exact FT medium in black in Fig. 4 when approaching the grazing angle of 79.2°.
C. Our Dielectric Media 1 and 2 of Perfect Dispersion
Li’s approximation medium requires three constant diagonal terms for simplicity, but we note that a further degree of freedom can be exploited to remove the dispersion deviation in Li’s approximation medium in the hope of achieving a different approximation. Hence, our dielectric media is designed as follows. In order to achieve the dispersion for exact medium, we allow the diagonal terms of permittivity to differ to realize the perfect dispersion rather than approximately:
Our strategy taken here is to keep all coefficients for wave-numbers ${k_x}$, ${k_y}$, different from Li’s approximation in Subsection 2.B, which neglects the quadratic terms. We equate every coefficient of dispersion curves in Eq. (20) of our dielectric media and the exact FT medium, both from Eq. (13). This gives
Note that the opposite coefficients ${-}b$, ${-}l$ in the cross terms can also be effectively the same for wave propagation and omitted due to their unrealistic values. We shall name the two realistic solutions as mediums 1 and 2, respectively, in the rest of our paper. For the exemplary parameters in Fig. 2, the permittivities for media 1 and 2 are explicitly
Our dielectric media 1 and 2 shall guarantee that the dispersion (red) is perfectly aligned with the original one (blue), so that in Fig. 2(b) only the former reveals itself in the picture. However, a perfect dispersion does not assure that our dielectric media is reflectionless as the exact medium. A relook into the FT approach indicates that only the exact medium perfectly converts polarization as designed [9], and both Li’s and our dielectric media have to induce intrinsic reflection for a non-perpendicular incidence as Figs. 3(b)–3(f) demonstrate. Dielectric media always cause reflection in TM modes [panels 3(d) and 3(f)], as they encode less so as not to be completely impedance-matched to the air when enforcing a non-magnetic medium, as will be investigated further in Sec. 3.
The conversion ratio of plane waves for our dielectric medium 1 resembles Li’s approximation within the interval [0, 58.6]° and even beats Li’s around grazing angle $\theta \ge {79.2^ \circ}$, although generally is lower, as seen in Fig. 4. This is the central result of this paper. For our dielectric medium 2 (lossy medium), its conversion ratio diminishes severely for incidence angles greater than 45° [blue dotdashed] due to the complex terms in ${{\bar {\bar \varepsilon}} _2}$, which is very inefficient and thus not in our favor.
3. DISCUSSION
A. Not Perfect Conversion of Polarization
The dielectric media presented above is not an exact equivalent medium. After all, it would seem too good to be true that one could achieve the perfect polarization conversion of the ideal FT medium only via a dielectric one. All dielectric media cannot be exactly equivalent to a magnetic medium. A signature of that is the reflections, which are spotted in Figs. 3(c)–3(f), while they are absent in Figs. 3(a) and 3(b). Only the exact medium indeed perfectly guarantees no reflection, as shown in Figs. 3(a) and 3(b), which appears similar to the impedance-matched medium such as transformation media [16,27–29]. Note that although the dispersion curves align exactly, the actual normal surfaces differ when viewed from the same angle in Figs. 2(c) and 2(d), and henceforth we confirm that the dielectric medium cannot fully reproduce the exact medium for waves.
B. Indeed Birefringence
It is natural and familiar in the language of crystal optics to consider birefringence in anisotropic media [30]. It would, at first glance, seem irrelevant for birefringence in previous work on FT, as only one polarization is transferred into another. However, the three-dimensional dispersion for non-trivial electric field ${\bf E}$ in the exact reciprocal medium reveals that it is indeed the case. Considering a planar electric wave ${\bf E}$ in a general magnetic anisotropic medium,
Fig. 5. (a) Dispersion contours for the ideal FT media in blue solid curves, Li’s approximation medium in purple dashed curves, our dielectric media 1 and 2 in red solid curves, and vacuum in a blue dotted curve. Field plots from wave simulation for incidence angle $\theta ={ 80^ \circ}$ for our dielectric medium 1 (b) for ${E_z}$ and (c) for ${H_z}$. Other parameters are the same as Figs. 2(b), 3(e), and 3(f).
Moreover, both dielectric media are biaxial anisotropic with tilting optical axes. With off-diagonal terms in their permittivities, their principal axes are positioned tilted. For our dielectric medium 1, one can calculate the Euler angles [31] and figure out the three principal axes $({k_{{x^\prime}}},{k_{{y^\prime}}},{k_{{z^\prime}}})$ with three principal values {1.5625, 0.9375, 0.5625} marked in Fig. 2(d), so that any three vectors in $({k_x},{k_y},{k_z})$ space are rotated according to
To implement such a bi-axial dielectric medium, one can build a periodic lattice of printed board circuits for radio frequency waves, which lose their rotational symmetry in the transverse plane [32–35]. It is also of interest to see that the two intersection points of dispersion curves on the $({k_x},{k_y})$ plane happen to be two of the four dimples, a general hallmark of a biaxial medium, where the wave group velocity is undefined in the doubly degenerate wave surfaces.
One may also wonder why no bifurcating waves are seen in the transmission beams in Figs. 3(b)–3(f). To answer this question, it is helpful to pinpoint the divergent wave routes through the dispersion curve only. For a TE wave with the incidence angle of 45° [blue arrow in Fig. 2(b)], two distinct wave vectors [red arrows] and two rays [purple arrows] appear after refraction upon the bottom surface. Note that purple arrows are parallel, and hence no birefringence is observed for the transmission beams in Figs. 3(b) and 3(f). The birefringence still appears for the grazing incidence angle. For a near-grazing angle of 80°, the double-refracted wavevectors are marked in two red arrows Fig. 5(a). It is peculiar to note that a partially downward propagation, in fact, derives from refraction instead of reflection. This is confirmed in the reflection part of magnetic field (TM mode), which accounts to the severe reflection in the lower-right region in Fig. 5(c). Therefore, both the exact medium and its dielectric medium 1 hide birefringence beneath the facade of reflectionlessness.
Fig. 6. Amplitude ratio $\zeta (\phi)$ and the phase ${\tau _x}(\phi ,\theta)$ (inset) with the full range of $\phi \in [- \pi /2,0]$ and incidence angle $\theta \in [0,\pi /2]$.
For the wave fields, the two separate beams with distinct polarizations are visible from the bifurcating reflection beams in Figs. 3(d) and 3(f). They are not due to double reflection, as they might appear to at first glance, because reflection law requires a single reflection wavevector only ${k_r} = - {k_i}$, as the green arrow indicates in Fig. 2(b). Instead, another reflection beam reflects off of the top surface after transversing the dielectric medium for a short distance.
C. Remark on Polarization Status for Transmitted Waves
Our proposed method is also applicable to other FT devices in the sense to convert linear polarization to other polarization circular ones, similar to previous works by our coauthors [17,18]. Moreover, it is even possible to convert linear polarization to other elliptic polarization when one tunes the gauge phase $\phi$ in the range $[- \pi /2,0]$. We add a remark in this subsection to address the generality of polarization conversion for the ideal FT medium [9], considering the ideal FT medium in Eq. (16) and the incidence wave of TE mode ${(E_z^{(0)},{\textit{iH}}_z^{(0)})^{T}}={ (1,0)^{T}}$. From Eq. (15),
According to Eq. (5),
For Eq. (16), ${A_x} = 0$, ${A_y} = \phi /({k_0}h)$, with ${k_x} = {k_0}\sin \theta$, ${k_y} = {k_0}\cos \theta$ and planar wave approximation ${E_z} \approx {e^{i({k_x}x + {k_y}y - \omega t)}}$, it is straightforward to derive
Then, it is important to look at the amplitude ratio $\zeta$ and the phase ${\tau _x}$,
As shown in Fig. 6 and its inset, one sees that it is not completely tunable to achieve any polarization status $\zeta$, ${\tau _x}$ for transmitted light via tuning gauge phase $\phi$, because incidence angle $\theta$ however also affects the phase ${\tau _x}$ (see the inset). This may also account for the deterioration of the conversion ratio for the grazing incidence angle in Fig. 4. Therefore, we remark that with the full range of $[- \pi /2,0]$ for the gauge phase one cannot achieve arbitrary polarization generally.
4. CONCLUSION
In this article, to exploit the extra freedom of dielectric approximation media for polarization conversion purposes, we design different biaxial dielectrics to reproduce the exact dispersion for the FT medium. We also further analyze the birefringence phenomenon prevailing in our biaxial anisotropic medium and point out the inevitable reflections for all dielectric equivalent media. Hence, other types of reflectionless anisotropic media are worth future investigations.
Funding
Department of Science and Technology, Hubei Provincial People’s Government (2018CFB148); National Natural Science Foundation of China (NSFC11804087); Hubei University (030-090105).
Acknowledgment
L. Y. acknowledges useful discussion with Xu Donghui, Xu Yadong, and Xu Lin, and also the hospitality received during the single-day visit to Xi Lei’s lab.
Disclosures
The authors declare no conflicts of interest.
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