Transformation optics for perfect two-dimensional non-magnetic all-mode waveguide couplers

Hossein Eskandari; Oscar Quevedo-Teruel; Amir Reza Attari; Mohammad Saeed Majedi

doi:10.1364/OME.9.001320

1. Introduction

Transformation optics is a powerful tool that enables a systematic design of electromagnetic devices. The concept of transformation optics was introduced by Prof. Pendry [1] and Leonhardt [2,3]. Although the original idea was applied for cloaking [4–11], transformation optics was promptly adopted for a huge number of applications, including polarization splitters [12–17], phase transformers [18–20], flat lenses [21–31], field rotators [32,33], field concentrators [34–36] waveguide bends [37–39] and waveguide couplers [40–46]. Transformation optics is a technique that can be employed to tailor the propagation of electromagnetic fields in a given physical space. Such tailoring can also be realized using properly engineered metasurfaces [47–50] which may alleviate the need for a bulky medium [49].

One straightforward use of transformation optics is to compress/expand the space [37, 51]. This device can be considered as a waveguide coupler that transfers the energy between two waveguides with different widths. In order to compress the space, a coordinate transformation must be applied. For example, if a compression wants to be applied along the y-axis of Cartesian coordinates, the coordinate transformation must be: x′ = x, y′ = m(x)y, z′ = z with m(x) as the transition function. With this transformation, a rectangular virtual space of dimensions L and d is transformed into a semi-trapezoid shape in the physical space as illustrated in Fig. 1, where d′ indicates the length of the physical coupler, and the shape of transition between waveguides is controlled by the function m(x) that satisfies m(0) = 1 and m(d) = M. M × L is the width of the second waveguide where M indicates the amount of compression or expansion. In the particular case illustrated in Fig. 1, the transition function is a cosine with power of 2. Cosine, exponential and polynomial series can be employed as transition functions. These specific functions can be expressed as:

m_{1} (x) = (1 - M) \sum_{n = 1}^{N} α_{n} {(1 - x / d)}^{n} + M .

m_{2} (x) = (1 - M) \sum_{n = 1}^{N} α_{n} {cos}^{n} (π x / 2 d) + M .

m_{3} (x) = \sum_{n = 1}^{N} α_{n} M^{{(x / d)}^{n}} .

In the m(x) functions defined in Eqs. (1)–(3), $\sum_{n = 1}^{N} α_{n} = 1$ must be satisfied, so m(0) = 1. Additionally, in order to have a smooth variation in the transition, it is advantageous to choose a function in which m′(0) = m′(d) = 0. To have this property for all the proposed transition functions included in Eqs. (1)–(3), α₁ must be zero and $\sum_{n = 2}^{N} n α_{n} = 0$ for Eq. (1) and (3). Note that as long as the transition function satisfies the conditions at the transition with the waveguides, the device will properly operate. On the other hand, a smooth transition function will lead to smooth changes of material inside the coupler, producing a better coupling of the waves and a device that is easier to manufacture and more robust to fabrication errors.

Fig. 1 Scheme of the physical space for a waveguide coupler with a cosine transition function, m(x), with power of 2.

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When considering a TM mode, the problem can be simplified as two plane waves bouncing off the top and bottom sides with perfect electric conductor (PEC) boundaries [52]. The PEC is used here to realize the waveguide boundaries that lead to the existence of TM_n modes and to prevent wave leakage. For an optical incident beam, one may remove PEC and the device will maintain its functionality. The propagation vectors and the angle of their constitutive plane waves, namely β and θ, are not necessarily equal for both sides. This propagation is illustrated in Fig. 1 including the propagation vectors.

Since the transformation is continuous at the input boundary of the coupler, no reflection is produced at this boundary for any incident angle. It has been previously reported that this type of device suffers from reflections at the output boundary because of a metric mismatch between the squeezer and the surrounding space [37]. It has been also demonstrated that under special conditions, 3-D squeezers can be reflectionless [40]. In [40], it was demonstrated that if the compression/expansion is applied with similar rates to both transverse directions, y and z-directions in Fig. 1, the device will exhibit no reflections for normal incidence at both sides when surrounded by vacuum. In [41], the reflection and transmission coefficients of a 3-D squeezer/expander were calculated. The device proposed in [41] had no reflections for a longitudinal compression. Additionally, for the TM polarization, if the compression ratio in the z-direction is square of the compression ratio in the y-direction, the device is omnidirectional reflectionless at the output with a dielectric constant equal to the compression ratio in the z-direction. Finally, if the compression ratio is similar in two transverse directions, the device will be matched to the vacuum at both sides for normal incidence, which is the same result obtained in [40].

Reflection suppression in 2-D and 3-D couplers requires distinct approaches to be followed since the definition of the modes and the Brewster angle are different, being also the transformation different. Strategies for suppressing reflections in a 2-D squeezer/expander were investigated in [44]. This solution is based on an optical transformation that alters the impedance in the virtual space for a specified incidence angle. This coupler can be matched to similar dielectric constants at the input and output. In addition, the material for the normal incidence case was non-magnetic. In [45], an auxiliary function was proposed to produce a coupler that is matched to any given dielectric materials at both boundaries for normal incidence with a medium that is non-magnetic. Normal incidence means that the TEM (TM₀) mode can travel from a vacuum waveguide to a waveguide with the desired dielectric constant. Here, we extend the work in [45] for any TM_n mode by presenting a Brewster angle coupler that uses only non-magnetic materials. This means that the incidence angle of the wave is no longer normal to the boundary of the coupler. Also, differently to [45], our approach couples all modes to a dielectric constant predefined by the amount of compression.

It is worth to mention that we focus here on TM modes, however, the best strategy to couple TE modes with transformation optics reported in the literature is the use of quasi-conformal mappings that leads to a non-homogeneous solution that can be implemented with fully dielectric and isotropic materials. This implementation reduces the complexity of the manufacturing, although unwanted reflections are seen at the output of the device [30,53].

In this paper, we propose the use of transformation optics to eliminate the reflections from the output boundary of the 2-D waveguide coupler for an arbitrary incident angles, i.e. all the TM_n modes. Our generalized formulation of transformation optics provides extra degrees of freedom for a designer to achieve more suitable transition shapes. Using the Brewster angle definition, we present a coupler suitable for a single-mode transmission. This coupler is matched to an arbitrary dielectric at the output. Also, we illustrate that for a given level of compression, an all-mode coupler can be produced for a specified dielectric constant at the output. In order to impose non-magnetic materials, an auxiliary function is employed. The validity of this technique is proven with simulated results calculated with COMSOL Multiphysics.

2. Theoretical basis

2.1. Transformation optics

Transformation optics theory is based on the form-invariance of Maxwell equations under a coordinate transformation. The coordinate transformation between two spaces, named virtual and physical spaces, is defined by the Jacobian matrix J = ∂(x′, y′, z′) /∂ (x, y, z). This theory implies that the material in the virtual space (taken here as vacuum) is transformed to the following relative permittivity ε and permeability μ tensors in the physical space [1]:

ε = μ = \frac{J J^{T}}{det (J)} .

It must be remarked that Maxwell equations imply that constitutive parameters that play a role in the propagation of TE (E_z, H_x, H_y) and TM (H_z, E_x, E_y) polarizations are ε_zz, μ_xx, μ_yy, μ_xy, μ_yx and μ_zz, ε_xx, ε_yy, ε_xy, ε_yx. Furthermore, if the dispersion equation is kept intact, that is, the multiplication of ε_zz(μ_zz) by μ_xx, μ_yy, μ_xy, μ_yx(ε_xx, ε_yy, ε_xy, ε_yx) is kept unchanged, the wave propagation direction is not altered. Hence, a smooth engineered spatial variation can be applied to permittivity and permeability parameters without changing the functionality of the device. However, reflections may occur at the boundaries due to the impedance mismatch with respect to the surrounding media. This property has been employed for several purposes in the literature including simplification of the transformation material [4–6, 8, 35, 36, 42, 46] and reflection suppression [45].

The dispersion equation for TM modes and the Brewster angle condition for normal incidence TM₀ case (TEM mode) were derived in [45]. This formulation can be generalized to oblique incidence, TM_n modes, resulting in the following equation for the Brewster angle:

{cos}^{2} θ_{B} = \frac{ε_{i} ({μ_{z z} ε_{x x} |}_{i} - ε_{i})}{{ε_{u} ε_{v} |}_{i} - ε_{i}^{2}},

where θ_B is the specified Brewster angle (mode angle for our case) and subscript i is 1 or 2, corresponding to the left and right boundaries. ε_u and ε_v are the eigen values of the permittivity tensor in the x − y plane. Dielectrics in the left and right regions are denoted as ε_i.

To eliminate the reflections for all modes (i.e. all incident angles), the constitutive parameters of the transformation medium must satisfy a specific condition at the boundaries with the surrounding medium. Using Eq. (5) and aiming to eliminate the dependence of the equation to the Brewster angle θ_B and hence achieving an all-mode reflectionless coupler, the conditions at the boundaries along the y-direction are:

{ε_{u} ε_{v} |}_{i} = ε_{i}^{2} .

{ε_{x x} μ_{z z} |}_{i} = ε_{i} .

From Eqs. (5)–(7), we can conclude that the reflections at a boundary along the y-direction in the transformed medium are governed by two values, ε_uε_v and ε_xx μ_zz.

2.2. Non-magnetic Brewster-angle waveguide coupler

Let’s consider the case where the goal is to match two waveguides with arbitrary dielectric constants. Similar to [45], this matching can be achieved by scaling to the constitutive parameters and multiplying (dividing) ε_xx, ε_xy, ε_yx, ε_yy(μ_zz) by the scaling function s(x). In this case, the dispersion equation (also the ε_xx μ_zz quantity) is kept unchanged while ε_uε_v is multiplied by s²(x). Using Eq. (5) and the transformation material of the coupler, the output-matched dielectric constant for the device with scaled parameters will be:

\begin{array}{l} {{cos}^{2} θ_{B} |}_{x = d} = \frac{ε_{2} (1 / M^{2} - ε_{2})}{s^{2} (d) - ε_{2}^{2}} \to \\ ε_{2} = \frac{1 \pm \sqrt{1 - M^{4} s^{2} (d) {sin}^{2} (2 θ_{B})}}{2 M^{2} {sin}^{2} θ_{B}} . \end{array}

Figure 2 illustrates the matched dielectric constants versus the corresponding Brewster angles (TM_n modes angles) for different values of M²s(d) and M = 0.5.

Fig. 2 Output matched dielectric constant versus the Brewster angle based on Eq. (8) for a device with scaled parameters and M = 0.5. (a) M²s(d) < 1 and “+” sign and (b) M²s(d) < 1 and “−” sign. (c) M²s(d) > 1 and “+” sign and (d) M²s(d) > 1 and “−” sign.

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From Eq. (8) and as illustrated in Fig. 2(a) and Fig. 2(b), we can conclude that the values of matched dielectric constant vary in the range of [∞, M⁻²] or [M²s²(d), 0] for the Brewster angle range of [0, 90]. Additionally, from Fig. 2(c) and Fig. 2(d) we observe that for the dielectric constants range of [∞, ${(M^{2} - \sqrt{M^{4} - s^{- 2} (d)})}^{- 1}$ ] or [M²s²(d), ${(M^{2} - \sqrt{M^{4} - s^{- 2} (d)})}^{- 1}$ ], the valid Brewster angle values cover the range of [0, 0.5sin⁻¹(M⁻²s⁻¹(d))].

Note that the dielectric constant of the right waveguide ε₂ in Eq. (8) and Fig. 2 is a desired design parameter and not a required value. Hence, for a given value of ε₂ and the incident angle of the TM_n mode, the value of s(d), which is used in the coupler design procedure, is derived from Eq. (8), as represented in Fig. 2. The incident angles of the TM_n modes are θ_i = sin⁻¹(f_c/f), where f_c is the cut-off frequency of the mode and f is the operating frequency [52]. After calculating the value of s(d), the full function, s(x), can be any continuous function that satisfies the values of s(0) = 1 and s(d). The proposed series functions in Eqs. (1)–(3) are good candidates for this purpose since one can also engineer the slope of the function at the beginning and the end of the coupler.

To make the medium non-magnetic, we introduce the auxiliary function f(x) that applies a transformation to x-constant lines. The transformation formula is now x′ = f(x), y′ = m(x)y, z′ = z and the transformation medium parameters can be found in Eq. (7) of [45]. This auxiliary function does not perturb the reflectionless condition as it only compresses/expands the x-constant coordinate line in the longitudinal direction. Note that the magnetic parameter of the coupler is μ_zz = 1/(s(x)m(x)f_x(x)). This indicates that if the function f(x) is selected in such a way to satisfy f_x(x) = 1/s(x)m(x), the transformation medium becomes non-magnetic. This auxiliary function can also change the coupler length if needed since d′ = f(d). Hence, in summary, s(x) produces a reflectionless waveguide coupler, while f(x) enforces only non-magnetic materials.

2.3. Non-magnetic all-mode waveguide coupler

In this section, we enforce the use of only non-magnetic materials while maintaining the reflectionless property at the output boundary for all modes. An all-mode reflectionless device implies that different waveguide modes, or waves with different incident angles, passing through the coupler do not have reflections. As in the previous section, we apply here again the scaling function s(x), being Eq. (6) and Eq. (7) changed to:

{\begin{cases} {ε_{u} ε_{v} |}_{x = 0} = s^{2} (0) \\ {ε_{u} ε_{v} |}_{x = d} = s^{2} (d) \end{cases}

{\begin{cases} {ε_{x x} μ_{z z} |}_{x = 0} = 1 \\ {ε_{x x} μ_{z z} |}_{x = d} = 1 / M^{2} \end{cases}

Hence, for an all-mode coupler, s(x) should satisfy s(0) = 1 and s(d) = 1/M² conditions. Such a device is impedance-matched to ε₁ = 1 at the input port and to ε₂ = 1/M² at the output port. As previously stated, the scaling function s(x) can be any continuous function that satisfies the above conditions. One choice is s(x) = 1/m²(x) that is both smooth and differentiable and also meets the above requirements.

To make the medium non-magnetic, the magnetic parameter is μ_zz = 1/(s(x)m(x)f_x(x)) after the scaling. Considering s(x) = 1/m²(x), then μ_zz = m(x)/f_x(x). Hence, the auxiliary function can lead to the non-magnetic property if f_x(x) = m(x) condition holds. In summary, it is possible to have an all-mode reflectionless non-magnetic coupler by choosing s(x) = 1/m²(x) and f(x) = ∫ m(x)dx. Note that if another scaling function s(x) is used, the auxiliary function f(x) changes too.

3. Simulation results

To confirm the functionality and the reflectionless property of our proposed designs, we carried out numerical simulations. In all simulations, the width ratio of the waveguides is 2:1 meaning that M = 0.5. TM modes are excited and the normalized real part and magnitude of H_z component are represented. The simulation frequency for all cases except the last one is $f = 8 f_{c}^{{TM}_{1}}$ where $f_{c}^{{TM}_{1}} = c_{0} / 2 L$ is the cut-off frequency of the vacuum left waveguide for the TM₁ mode and c₀ is the speed of light in the vacuum. The incident wave is excited with a port on the left side boundary.

In the first case, we aim to compare the coupling performance between two vacuum waveguides with and without the non-magnetic coupler. The design parameters are L = 0.7m and d′ = 0.4m. Based on Eq. (1), the following transition function is used.

m (x) = (1 - M) (3 {(1 - x / d)}^{2} - 2 {(1 - x / d)}^{3}) + M,

which is a polynomial series and satisfies m(0) = 1, m(d) = M conditions as well as m′(0) = m′(d) = 0. Note that we assign later a value to the parameter d in order to achieve the desired coupler length d′. The left port is excited with the TM₀ (TEM) mode which corresponds to the incidence angle of zero. Eq. (8) shows that s(d) = 2 ensures the reflectionless coupling for this mode. Since the scaling function must satisfy two conditions, s(0) = 1 and s(d) = 2, one smooth choice can be the following equation:

s (x) = (1 - s (d)) (3 {(1 - x / d)}^{2} - 2 {(1 - x / d)}^{3}) + s (d) .

The auxiliary function should now fulfill f(x) = ∫ 1/s(x)m(x) to make the material non-magnetic. The calculation is done numerically. Under these conditions, d = 0.42m is needed in order to have d′ = f(d) = 0.4m. Figure 3 represents the effect of non-magnetic coupler between the vacuum waveguides.

Fig. 3 Simulated magnetic field for the TM₀ mode. The medium in both left and right waveguides is a vacuum. (a) Real part and (b) magnitude of H_z without the non-magnetic coupler. (c) Real part and (d) magnitude of H_z for the non-magnetic coupler with the Brewster angle of zero.

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The reflection coefficient in the absence of the coupler medium is Γ ≃ 0.21 whereas Γ ≃ 0.004 is achieved in the presence of the non-magnetic coupler. Figure 3(a) and Fig. 3(b) illustrated that the wave does not only have reflections, but the phase is also distorted.

Next, we excite the left waveguide boundary with a TM₃ mode. This means that the coupler should be designed for the incidence angle of θ_B = sin⁻¹(f_c/f) = sin⁻¹(3/4) ≃ 49°, and based on Eq. (8), s(d) = 2.82. The transition function m(x) and scaling function s(x) follow the same formula as Eq. (11) and Eq. (12), respectively. The auxiliary function need to be recalculated for the new values. In this case, d = 0.52m leads to d′ = f(d) = 0.4m.

Figure 4 illustrates the comparison between a coupler designed for θ_B ≃ 49° and a coupler designed for the normal incidence case.

Fig. 4 Simulated magnetic field for the non-magnetic Brewster-angle coupler for the TM₃ mode. The medium in both left and right waveguides is a vacuum. (a) Real part and (b) magnitude of H_z for the design with Brewster angle of 49°. (c) Real part and (d) magnitude of H_z for the design with Brewster angle of 0 (normal incidence).

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The reflection coefficient of the Brewster-angle design is Γ ≃ 0 whereas the normal incidence design has a Γ ≃ 0.17 reflection coefficient. The reflections are actually easy to notice since the magnitude plot is highly modulated as a sign of the reflected wave presence in Fig. 4(d). Clearly, the performance of the normal incident design degrades when the incident angle deviates highly from zero.

Next, the non-magnetic all-mode design is simulated. As previously derived, for M = 0.5, the device is impedance-matched to ε₂ = M⁻² = 4. Here, the following transition function is used, corresponding to Eq. (2):

m (x) = (1 - M) {cos}^{2} (π x / 2 d) + M .

This function has a zero slope when connecting with the waveguides and also satisfies m(0) = 1, m(d) = M conditions. As a result, the scaling function equals s(x) = 1/m²(x) and the auxiliary function is derived as follows:

f (x) = \int m (x) d x = x (M + 1) / 2 - d (M - 1) sin (π x / d) / (2 π) .

To have the same coupler length as the previous simulation case, d = 0.533m is chosen leading to d′ = f(d) = d(M + 1)/2 = 0.4m. Figure 5 and Fig. 6 illustrate the simulated results for this case. The left waveguide is excited with TM₀ and TM₁ in Fig. 5 and with TM₂ and TM₃ modes in Fig. 6. These TM modes correspond to the incident angles of 0, 7, 14 and 22 degrees. The reflection coefficient of these cases is Γ ≃ 0.004. This indicates that the input power is delivered almost perfectly to the output.

Fig. 5 Simulated magnetic field for the non-magnetic all-mode coupler impedance-matched to ε₂ = M⁻² = 4. (a) Real part and (b) magnitude of H_z for TM₀ mode. (c) Real part and (d) magnitude of H_z for TM₁ mode.

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Fig. 6 Simulated magnetic field for the non-magnetic all-mode coupler impedance-matched to ε₂ = M⁻² = 4. (a) Real part and (b) magnitude of H_z for TM₂ mode. (c) Real part and (d) magnitude of H_z for TM₃ mode.

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The coupling efficiency of this coupler is tested for different incidence angles for the TM₃ mode. To reach high incidence angles, the frequency should get close to the cut-off frequency of the mode. Figure 7 represents the coupling efficiency versus the incidence angle. Note that in case of the incidence angle of 70 degrees, the simulation frequency is $f = f_{c}^{{TM}_{3}} / sin (70) = 1.06 f_{c}^{{TM}_{3}}$ , which is near cut-off and is not commonly used in practical applications.

4. Conclusion

Unwanted reflections from the output boundary of a waveguide coupler are investigated and mitigated in this work and a non-magnetic medium is obtained for TM polarization. Using a suitable parameter scaling function, a Brewster-angle design is proposed that suppresses the reflection for a specified incident angle (mode) and has the refractive index of the output waveguide as a free parameter. Also, an all-mode coupler is introduced that is reflectionless for all modes but its impedance is matched to a predefined refractive index at the output. For both devices, the auxiliary function makes the material non-magnetic for TM polarization and eases the fabrication for optical applications.

Fig. 7 Coupling efficiency of the non-magnetic coupler versus the incidence angle for TM₃ mode.

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Transformation optics for perfect two-dimensional non-magnetic all-mode waveguide couplers

Abstract

1. Introduction

2. Theoretical basis

2.1. Transformation optics

2.2. Non-magnetic Brewster-angle waveguide coupler

2.3. Non-magnetic all-mode waveguide coupler

3. Simulation results

4. Conclusion

References

Cited By

Figures (7)

Equations (14)

Optical Materials Express