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On the perfectly matched layer and the DB boundary condition Nicola Tedeschi,1,* Fabrizio Frezza,1 and Ari Sihvola2 1

Department of Information Engineering, Electronics and Telecommunications, “La Sapienza” University of Rome, Via Eudossiana 18, 00184 Rome, Italy 2 Department of Radio Science and Engineering, Aalto University, Otakaari 5A, 00076 Espoo Aalto, Finland *Corresponding author: [email protected] Received July 10, 2013; accepted August 6, 2013; posted August 21, 2013 (Doc. ID 193218); published September 6, 2013 In this paper, we consider a particular uniaxial material able to achieve the DB boundary condition. We show how, for particular transverse electromagnetic properties, this material behaves like a perfectly matched layer (PML). Moreover, we find that, with an approximation, the material becomes passive, i.e., loses the active part of the permittivity and of the permeability typical of a PML. In this case, the uniaxial medium becomes realizable as a particular absorbing metamaterial. We present simulations with both guided and free-space waves to show the absorbing behavior of the proposed material. © 2013 Optical Society of America OCIS codes: (160.1190) Anisotropic optical materials; (160.3918) Metamaterials. http://dx.doi.org/10.1364/JOSAA.30.001941

1. INTRODUCTION In the past few years, implementations of various electromagnetic boundary conditions with anisotropic and bi-anisotropic media have been proposed in the literature [1]. In this paper, we consider the so called DB boundary condition. When a plane interface, at z 0 in a Cartesian coordinate system, is considered, the DB boundary is defined by the conditions uz · D 0;

uz · B 0;

(1)

where, uz is the unit vector perpendicular to the interface and the vectors D and B are the electric and magnetic flux densities, respectively. The realization of the boundary condition (1) has been proposed in [1] as the interface between an isotropic medium and an anisotropic one. The isotropic medium has characteristics ε1 and μ1 (medium 1, see Fig. 1). The anisotropic medium, a uniaxial medium with the optic axis perpendicular to the interface, has the following characteristics: ε¯ εt I¯ t εz uz uz ;

μ¯ μt I¯ t μz uz uz ;

(2)

where ε¯ and μ¯ are the relative permittivity and permeability tensors, respectively, and I¯ t I¯ − uz uz . Here, with the double overline, we indicate dyadics and we call I¯ t the transverse unit dyadic with respect to direction z. If εz , μz → 0, the interface with this medium becomes a DB boundary. We call this material a DB medium. The DB medium can be included in the class of the indexnear-zero metamaterials whose possible realization has been recently investigated [2,3]. The reflection and transmission at the interface between an isotropic medium and a DB medium has been studied, finding a characteristic angular selective property. In particular, it can be seen that the DB medium behaves as a perfect reflector for any incident plane wave except 1084-7529/13/101941-06$15.00/0

for the wave at normal incidence that is totally transmitted. Some applications of such metamaterial have been proposed in the literature [4]. In the present paper, we discuss a similarity between the DB medium and the perfectly matched layer (PML) proposed as an ideal electromagnetic medium and widely employed in numerical applications [5,6]. In the literature, the possibility to design a metamaterial to realize an electromagnetic absorber has been recently investigated. In [7], a perfect electromagnetic absorber in the microwave frequencies has been proposed by a metamaterial having the same impedance of a vacuum, i.e., ϵ μ, where with ϵ and μ we are representing the relative permittivity and permeability of the material, respectively. The same idea has been developed to realize electromagnetic absorbers in the terahertz and infrared regions [8,9]. The metamaterials proposed in such works are always anisotropic, being obtained by arrays of anisotropic inclusions. However, in the cited papers, the permittivity and the permeability are always considered as scalar quantities. This assumption, often made in the metamaterial community, is because the plane waves are supposed to propagate along one of the principal optic axes of the medium. In this case, the anisotropic material behaves like an isotropic medium with permittivity and permeability equal to the ones related to the considered optic axis. As a consequence, the materials presented show no reflection only at normal incidence, and consequently the absorption shows a strongly angular dependence. However, it has been shown that, by a proper design of these structures, it is possible to reach a low reflectance also in a wider angular sector [9]. In the literature, there are several contributions on the possible practical realization of a PML. A discussion on the physical realizability of the PML, in particular in the presence of curved surfaces, can be found in [10]. The realization of the PML as an artificial uniaxial medium with active inclusions © 2013 Optical Society of America

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Fig. 1. Geometry of the problem.

has been treated in [11]. To avoid the relevant realizability problems, other media with similar characteristics at the interface have been proposed, e.g., a uniaxial omega medium with zero reflection only at normal incidence [12] and a realizable broadband approximate PML [13]. The principal aim of the present work is to discuss the strange analogy between a DB medium and a PML, considering how, for fixed longitudinal properties, the behavior can be driven by the transverse characteristics. Moreover, we propose an approximation on the electromagnetic characteristics to make the DB medium an approximate perfectly matched layer (a-PML) by eliminating the active part of the tensors that is a well known characteristics of the PML. We show how the a-PML, being a passive anisotropic medium, can behave as a perfect electromagnetic absorber for an extremely wide angular sector of the incoming wave. In Section 2, an overview on the DB medium reflectance properties and the analogy with the PML is discussed. In Section 3, an approximation on the electromagnetic properties of the medium is performed to make it a passive medium and simulations on the behavior as an electromagnetic absorber are presented. Finally, in Section 4, the conclusions are drawn.

2. DB MEDIUM AS A PML The wave propagation in the DB medium and its properties have been analyzed in [1], where the reflection coefficient is obtained for both the TE and the TM polarizations. Here, we call TE or TM polarization the case in which the electric or magnetic field of the incident plane wave is perpendicular to the plane of incidence, respectively. The angular selective properties of these coefficients, when εz , μz → 0, are pointed out in [1]. We write below the expressions of the reflection coefficient for both the polarizations: ΓTE

ζ t cos θi − ζ 1 1 − sin2 θi ϵ1 μ1 ∕μz ϵt 1∕2 ; ζ t cos θi ζ 1 1 − sin2 θi ϵ1 μ1 ∕μz ϵt 1∕2

(3)

ΓTM

ζ 1 cos θi − ζ t 1 − sin2 θi ϵ1 μ1 ∕ϵz μt 1∕2 ; ζ 1 cos θi ζ t 1 − sin2 θi ϵ1 μ1 ∕ϵz μt 1∕2

(4)

where ζ1 μ1 ∕ϵ1 1∕2 and ζ t μt ∕ϵt 1∕2 , and θi is the angle that the incident plane wave forms with the normal to the interface, as shown in Fig. 1. In Fig. 2, the behavior of the amplitude of the reflection coefficient at the interface between a vacuum and a matched medium is shown. Here, with matched medium, we mean a medium with the same impedance of a vacuum. Three different cases are considered: the case of an isotropic medium with ϵ μ, as the medium modeled in [7], and two different DB media, having the transverse parameters

Fig. 2. Amplitude of the reflection coefficient at the interface between a vacuum and (solid line) a matched isotropic medium with ϵ μ 2, (dashed line) a DB medium with ϵt μt 2 and ϵz μz 0.1, or (dotted line) ϵz μz 0.01.

equal to ϵt μt 2 and the axial parameters equal to ϵz μz 0.1 or ϵz μz 0.01, like the media proposed in [1]. The uniaxial media considered are affinely isotropic, i.e., the permittivity and the permeability tensors are both propor¯ The impedance tional to the same tensor: ϵ¯ ϵA¯ and μ¯ μA. of the medium can be considered as the square root pof the ratio between the proportionality constants ζ μ∕ϵ. For the considered media, the reflection coefficients are polarization insensitive. In Fig. 2, we see that in both cases there is no reflection at normal incidence. However, while for the isotropic matched medium the reflection coefficient slowly grows with the incident angle, for the DB media the reflection coefficient suddenly grows to unity. In the limits of zero axial parameters, it can be seen that the reflection coefficient is zero only at normal incidence, while the interface behaves like a perfect reflector for any incident angle. This is the strong angular filtering property of the DB medium presented in the literature. From this result, we see that the isotropic medium seems to be more suitable for absorbing applications. However, as we discuss in the following, it is possible to make the reflection coefficient of the DB medium equal to zero for any incident angle by varying the transverse parameters. Now, let us consider the PML proposed in the literature for numerical applications [5]. The interface with this medium is characterized by having the reflection coefficient identically equal to zero for any angle and polarization of the incident plane wave. The PML is characterized by the presence of an active part in its components that makes it an ideal medium. The PML, matched with the medium 1, can be thought as a uniaxial medium with characteristics equal to (2), and with the properties [6] μ¯ ζ 21 ε¯ ;

(5)

εz εt ε21 :

(6)

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When these conditions are met, the reflection coefficients become identically zero for both polarizations, with an exception in the case of θi π∕2, when the reflection coefficient is always equal to −1, as pointed out in [5]. We can consider separately the effects of the two conditions (5) and (6): when only the property (5) is met, the expressions of the reflection coefficient for both polarizations coincide. In fact, this is the case of matched media, i.e., ζ ζ 1 . Therefore, the reflection coefficient for both the polarizations can be written as Γ

cos θi − 1 − sin2 θi ε21 ∕εz εt 1∕2 : cos θi 1 − sin2 θi ε21 ∕εz εt 1∕2

(7)

Now, when the condition (6) is met, the coefficient in Eq. (7) also becomes zero for any angle of incidence and the uniaxial medium becomes a PML. At this point, we consider the uniaxial medium in Eq. (2), matching the properties (5) and (6), i.e., μt ζ 21 εt and μc ζ 21 εc , and with the following components of the permittivity: εt Mε1

and εz

ε1 ; M

(8)

where M is a dimensionless parameter related to the coordinate-stretching parameter s used in the PML context [14]. When the parameter M becomes large (M → ∞), this medium becomes concurrently a PML and a DB medium. Moreover, this is a particular DB medium, because the quantity εt goes to infinity while εz goes to zero. Possible realizations of media with this extreme anisotropic behavior have been proposed in [15,16], as dielectric substrates with cylindrical inclusions. Therefore, a PML can be obtained with a particular DB medium. This is an unexpected result; in fact, as emphasized before, the DB medium realize a perfect reflection condition and this is far away from the perfectly absorbing condition that we are now discussing. To understand the behavior of the DB medium, we can consider the trend of the reflection coefficient, away from the normal incidence, as a function of the transverse parameters. We suppose the condition (5) be matched, so we can consider the reflection coefficient in Eq. (7). In Fig. 3, the amplitude of the reflection coefficient at the interface between an isotropic medium and a matched uniaxial medium as a function of the transverse permittivity of the uniaxial medium is shown for an incident angle θi π∕3. Three different values for the axial permittivity are considered. We can see that the reflection coefficient is equal to unity when ϵt < ϵ−1 z , and it suddenly decreases near this value. We can see that in a neighborhood of the value ϵt ϵ−1 z , the reflection coefficient is almost zero. This behavior is the same for any incident angle. We note that the medium behaves as a DB medium only for small values of the transverse permittivity with respect to the axial one. This result was not previously emphasized in [1]. Until now, we did not consider the losses in the medium. This point is extremely important; in fact, if we consider a medium with no reflection but without losses, it would be totally useless as an absorber. In the literature, the absorbance of a layer is defined as A 1 − jΓj2 − jTj2 , where T is the transmission coefficient of the considered layer, strongly connected to the losses of the material. We can take into account the losses by considering a complex expression of the

Fig. 3. Amplitude of the reflection coefficient at the interface between an isotropic medium and a matched uniaxial medium as a function of the transverse permittivity of the uniaxial medium. The incidence is at an angle θi 60°. The uniaxial medium is considered with an axial permittivity equal to (solid line) ϵz 0.1, (dashed line) ϵz 0.06, and (dotted line) ϵz 0.03.

permittivity components. Therefore, being the medium matched with the isotropic one, from Eq. (5), it means that also the permeability would present losses. Here, we assume a time dependence like e−iωt . We can write the expressions εt ε1 Meiα

and εz

ε1 −iα e ; M

(9)

where α ∈ 0; π∕2. These parameters could represent the permittivity components of a PML, because they match the condition (6), where we suppose that the transverse component of the permittivity presents losses. However, the longitudinal component of the permittivity has a negative imaginary part, which can be interpreted as an active behavior of the medium. The presence of this active part decreases the appeal of the PML in practical applications, and limits its use only for computational implementations. It is important to note that the parameters in Eq. (9), when M → ∞, still represent a DB medium. In the following section, we will discuss how an approximate PML can be realized by means of a DB medium.

3. APPROXIMATE PML We consider the expressions in Eq. (9), with the hypothesis α ≪ 1. Therefore, the permittivity components can be approximated as εt ≈ ε1 M1 iα

and εz ≈

ε1 1 − iα: M

(10)

At this point we note that the active part in the longitudinal permittivity is equal to α∕M. If we consider a DB medium, this component becomes equal to zero; in this case we can write the permittivity components as

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εt ≈ ε1 M1 iα and εz ≈

ε1 : M

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(11)

A medium with these characteristics is not a PML, because the condition (6) is not met anymore. However, the reflection coefficient at the interface with this medium, for sufficiently small values of α, is not so different from zero. Moreover, it could be noted that when α ≪ 1, also the losses of the medium become negligible. However, we must remember that in a DB medium, the parameter M goes to infinity, so the product Mα can be considered as a value different from zero. As a consequence, a uniaxial medium with permittivities [Eq. (11)] is a passive medium with non-negligible losses. Now we show how the reflection behavior of this medium is not so far from a PML. The behavior of the reflection coefficient for this interface can be obtained from the expression (7), making use of the components of the permittivity in Eq. (11). We obtain the expression cos θi − 1 − 1 iα−1 sin2 θi 1∕2 Γ : cos θi 1 − 1 iα−1 sin2 θi 1∕2

(12)

The reflection coefficient does not depend on the characteristics of the medium 1. Moreover, it does not depend on the magnitude M of the components of the permittivity, but only on the angle of incidence and on the parameter α. In Fig. 4, the amplitude of the reflection coefficient (12) is shown when the values of the parameter are α f0.1; 0.01; 0.001g. The behavior of the reflection coefficient allows us to call this medium an a-PML. The expression (12) and Fig. 4 lead us to draw some considerations on the parameters M and α. In fact, the parameter M does not affect the reflection coefficient, i.e., the interaction of the medium with the outside. On the other hand, the parameter α is connected with the reflection coefficient and when α → 0, the medium approaches a PML. Furthermore, the smaller α is, the smaller the losses of the medium are. However, we can control the losses by varying M without affecting the reflection coefficient. The phase of the reflection coefficient (12) is shown in Fig. 5, for the same α values of Fig. 4. We can see that the phase tends to −π as fast as the amplitude tends to zero.

Fig. 4. Amplitude of the reflection coefficient in Eq. (12), when (solid line) α 0.1, (dashed line) α 0.01, and (dotted line) α 0.001.

Fig. 5. Phase of the reflection coefficient in Eq. (12) when (solid line) α 0.1, (dashed line) α 0.01, and (dotted line) α 0.001.

To understand the behavior of the reflection coefficient in Eq. (12), a simple approximate expression for Γ can be calculated. We consider the case jθi j < π∕2 and α ≪ 1; with some algebra we obtain

Γ≈

−iα tan2 θi : 4 iα tan2 θi

(13)

Looking at expression (13), we understand why the phase of the reflection coefficient, in Fig. 5, tends to be near −π∕2 when jθi j < π∕2. For small values of θi , in fact, the expression (13) tends to be a real and positive quantity, proportional to α and multiplied by −i. To test the performance of the approximation, we consider the radiation of a short dipole placed in the center of a square box with an edge of 6λ. The simulations are performed on Comsol Multiphysics. In Fig. 6, the real part of the component of the electric field parallel to the dipole (y component), on a line parallel to the y axis at a distance λ from the edge of the box, is shown. We consider different situations: when the box

Fig. 6. Normalized electric field at a distance λ from the edge of a box layered with (solid line) a vacuum, (dashed line) a PML, and (circles) an a-PML.

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Fig. 7. Difference between the fields in Fig. 6. The difference is between the field obtained with a vacuum and the field obtained with the a-PML.

is absent and the dipole radiates in a vacuum; when the box is layered with a PML, described by the parameters in Eq. (9); and when the box is layered with an a-PML with α 0.02 and M 20. The fields are normalized with the maximum value of the field jE M j in the first case. The layer’s thickness around the box is λ and the computational domain is covered with a perfectly electric conductor, in order to maximize the reflection. Let us note that, if M were smaller, then the a-PML would not attenuate the field. However, we can increase M as much as we want without affecting the reflection coefficient of the interface and obtain the attenuation that we need. From Fig. 6, it is clear that the a-PML slab is not reflective and the electromagnetic radiation is absorbed by it. Hence, we implemented an electromagnetic absorber for a wide angular sector of incident waves through a passive uniaxial layer with a thickness of only one wavelength.

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Fig. 9. Difference between the fields in Fig. 8. The difference is between the field obtained with a vacuum and the fields obtained with (solid line) the PML or (dashed line) the a-PML.

To emphasize our point, we consider the difference ϵ between the field in a vacuum and the fields with the a-PML, see Fig. 7. The fields are in a very good accordance, with errors below 5%. On the other hand, when jθi j grows, the error grows. This behavior is typical of the PML, as it is well established in the literature [5,6]. Finally, we consider the case of a guided wave, to test the approximation with a larger angular spectrum. In Fig. 8, the real part of the electric field propagating in a symmetric slab waveguide in air is shown. The slab has refractive index n 1.7 and width 0.731 μm, and operates in the TE1 mode at a wavelength λ 1.55 μm. The field is taken when the slab is not ended, when it is ended on a PML, and when it is ended on the a-PML with M 20 and α 0.01. The fields are normalized with the maximum value of the field in the not-ended slab. In the last two cases, the field is taken at λ∕2 from the interface. Both the PML and the a-PML have thickness 2λ and are closed by a ground plane. As done before, we found optimum accordances, with errors below 3% as can be seen in Fig. 9. The error in this case is less than the one in Fig. 7, although the angular spectrum of the electromagnetic radiation is wider. This is because, in the case of a dipole in a layered box, the presence of the box’s corners increases the reflections, reducing the absorbing efficiency.

4. CONCLUSIONS

Fig. 8. Normalized electric field in a symmetric slab (solid line) without ending and at a distance λ∕2 from (crosses) a PML and (circles) an a-PML.

In this paper, the analogy between the DB medium and the PML has been investigated. We show how, by varying the transverse parameters, a matched uniaxial medium can reach a zero reflection for any angle of incidence of the impinging wave. Moreover, we propose an approximation on the material’s parameters to neglect the active part typical of the PML, obtaining a passive uniaxial material that works as an absorber in a wide angular sector. Some simulations of the proposed medium have been shown, emphasizing the good absorbing properties both in the presence of a corner, where a grazing incidence is present, and in the presence of guided waves, with a wide angular spectrum. In both cases, the layer

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absorbs the incident radiation, showing small errors with respect to the unbounded radiation. The results found suggest the possibility to realize a uniaxial metamaterial where both the axial and the transverse permittivity and permeability could be designed to match the perfectly absorbing conditions. This can be a way to reduce the angular dependence of the absorbing metamaterials now available.

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