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A general conformal transformation method (CTM) is proposed to construct the conformal mapping between two irregular geometries. In order to find the material parameters corresponding to the conformal transformation between two irregular geometries, two polygons are utilized to approximate the two irregular geometries, and an intermediate geometry is used to connect the mapping relations between the two polygons. Based on these manipulations, the approximate material parameters for TE and TM waves are finally obtained by calculating the Schwarz-Christoffel (SC) mappings. To demonstrate the validity of the method, a phase modulator and a plane focal surface Luneburg lens are designed and simulated by the finite element method. The results show that the conformal transformation can be expanded to the cases that the transformed objects are with irregular geometries.
©2011 Optical Society of America
1. Introduction
In recent years, there has been continuing interest in transformation optics [1–3]. By employing this design methodology, numerous novel devices can be designed [4–6]. However, most of these devices cannot be easily realized, because the resulting materials are always inhomogeneous and anisotropic. To reduce material complexity, Conformal mapping [2, 7–11] and quasi-conformal mapping [12] are proposed. By using conformal mapping, the attained material parameters are strictly isotropic for TE waves or TM waves. Consequently, these devices are more realizable. Many studies [7–11] have been carried out on conformal mapping, but most of them were restricted on waveguides or some special cases. In Ref [8], the method of solving Laplace’s equation is proposed to find the conformal mapping which maps a rectangle onto an irregular geometry. It can be used to calculate the material parameters according to the transformation from a rectangle waveguide onto a waveguide with arbitrary shapes. However, the method is basically suitable for the case that the geometry to be transformed is a rectangle. In Ref. [10, 11], SC transformation (or mapping) is used to construct the conformal mapping between a circle and a square analytically.
In conformal transformation optics, general transformation cases are required in which the two geometries before and after transformation are irregular, however in the present time, no available method can be used for the conformal mapping between them. We suppose that an existing intermediate geometry can be used to connect the two irregular geometries, then, if the conformal mappings between the intermediate geometry and the two irregular ones can be found, the conformal mapping between the two irregular geometries may be constructed as a result.
In this article, we propose a conformal transformation scheme to address the general cases. Firstly, two polygons are used to approximate the two irregular geometries, and an intermediate geometry is utilized to connect the two polygons. Then, by employing SC mapping, the conformal mappings between the intermediate geometry and the two polygons are found. Finally, on the basis of the obtained conformal relations, the approximate material parameters corresponding to the conformal mapping between the two irregular geometries are deduced for TE and TM waves.
2. The Conformal Transformation Method
The method is developed as shown in Fig. 1 . Q 1, Q 2 are the two irregular geometries in the complex plane. P 1, P 2 are the two polygons used to approximate Q 1, Q 2. In transformation optics terms, Q 1, P 1 are in the virtual space, and Q 2, P 2 are in the physical space [12]. R is the intermediate geometry (here it is a rectangle) used to connect P 1, P 2. Z, ζ, ω denote their coordinates, and A, B, C, D mark the four vertices of them. By employing SC mapping [13–15], the conformal mappings z = f 1 (ζ), ω = f 2 (ζ) which map the intermediate geometry R onto the two polygons P 1, P 2, respectively, can be calculated out. Use the same method, their inverse mapping ζ = g 1 (z), ζ = g 2 (ω) can also be achieved.
Fig. 1 The procedure of the CTM. Q 1, Q 2 are two irregular geometries, P 1, P 2 are two polygons, R is the intermediate geometry. Q 1, P 1 are in the virtual space, and Q 2, P 2 are in the physical space. In the CTM, the material parameters corresponding to the mapping from P 1 onto P 2 are used to approximate the ones corresponding to the mapping from Q 1 onto Q 2. ω = f (z) is the conformal mapping from P 1 to P 2. f 1, f 2 can be computed by SC mapping, and g 1, g 2 are the inverse mapping of f 1, f 2, respectively.
At the first, the material parameters corresponding to the mapping from P 1 onto P 2 expressed by f 1, f 2 is deduced based on the relations given in Ref. [8, 9]. With the scheme shown in Fig. 1, the material parameters (i.e., the permittivity ε and the permeability μ) corresponding to the conformal mapping ω = f (z) for TE and TM waves can be given by
Where u, v represent the real and imaginary part of the complex function ω = f (z), respectively, namely ω = f (z) ≡ u(x, y) + i v(x, y), and the virtual space is assumed to be vacuum. Because ω = f (z) is an analytic function, its derivative can be written as f ′(z) = ∂x u + i ∂x v [15]. Then Eq. (1) becomeswhere |∙| represents the norm of a complex function. Since ω = f (z) can be expressed in the form of composite function: ω = f 2 [g 1 (z)], where ζ = g 1 (z) is the inverse function of z = f 1 (ζ). Thus using of differential chain rule yieldsAs ζ = g 1 (z) is also an analytic function, thus g 1′(z) = 1 / f 1′(ζ). Then Eq. (3) can be rewritten as f ′(z) = f 2′(ζ) / f 1′(ζ). Consequently, the approximate material parameters for TE and TM waves expressed by Eq. (2) finally becomes
The material parameters expressed by Eq. (4) correspond to the mapping from P 1 onto P 2 rather than from Q 1 onto Q 2. However, because P 1, P 2 are approximations of Q 1, Q 2, thus the material parameters given by Eq. (4) can be used to approximate the ones corresponding to the mapping from Q 1 onto Q 2. With this idea, the procedure of the CTM can be described as following:
- Step 1: Use two polygons P 1, P 2 to approximate the two irregular geometries Q 1, Q 2.
- Step 2: Use an intermediate geometry R to connect the two polygons P 1, P 2. For instance, a rectangle or a disk can be utilized as the intermediate geometry.
- Step 3: By employing SC mapping (the SC mapping can be computed by, for example, the toolbox in Ref. [13]), calculate the conformal mapping z = f 1 (ζ) between R and P 1, and ω = f 2 (ζ) between R and P 2.
- Step 4: Compute the approximate material parameters corresponding to the transformation from Q 1 onto Q 2 by Eq. (4).
In the step 2, if a rectangle is used as the intermediate geometry, the conformal modulus [15] of the two polygons P 1, P 2 should be equal to ensure that a conformal mapping exists between them. If a disk is used as the intermediate geometry, conformal centers in P 1, P 2 should be defined. In the step 4, the enough side numbers of the two polygons P 1, P 2 should be selected to ensure an accurate calculation of the material parameters.
3. Design and Simulation for Two Types of Transformation Devices
To demonstrate the validity of the CTM, a phase modulator and a plane focal surface 2D Luneburg lens are designed and simulated. The processing shows that the transformation method proposed in the paper is effective for fulfilling the conformal transformation of irregular geometries.
3.1 Phase Modulator
The phase modulator is a typical example to illuminate how to calculate the corresponding material parameters by the CTM when the geometries before and after transformation both are irregular. As shown is Fig. 2 , in the virtual space, Q 1 is the geometry before transformation, and in the physical space, Q 2 is the phase modulator after transformation. In this case, both Q 1 and Q 2 can be considered as irregular. Because Q 1 is not a disk as it appears, but a topological quadrilateral [15]. Hence, whether its boundary ABCD is circular or irregular will not cause any difference in essence to the CTM procedure.
Fig. 2 Design for a phase modulator with the CTM. Q 1, P 1 are in the virtual space, and Q 2, P 2 are in the physical space. Q 1, Q 2 are approximated by the two polygons P 1, P 2, respectively. A rectangle R is used as the intermediate geometry to connect P 1, P 2.
In Fig. 2, suppose that in the virtual space, if a point source is located on the center O of Q 1, then the phase of the excited electromagnetic (EM) field on the boundary ABCD of Q 1 will be equal, namely ABCD is an equal phase surface. And then, if Q 1 is transformed onto Q 2, the boundary ABCD of Q 2 will also be the equal phase surface for the point source on the center of Q 2. Because the AB, CD sides of Q 2 are arcs whose centers are F 1, F 2, respectively. Therefore, as exhibited in Fig. 2, F 1, F 2 will become two focal points. To verify this idea, the proposed CTM is employed to calculate the material parameters corresponding to the conformal mapping from Q 1 onto Q 2 (Note that the material parameters cannot be calculated by the solving Laplace’s equation method proposed in Ref [8], since neither Q 1 nor Q 2 is a rectangle). In this case, the procedure of the CTM can be specified as following:
- Step 1: A regular 74-gon P 1 is selected to approximate Q 1, and a polygon P 2 is selected to approximate Q 2 with the boundaries AB, CD which are composed of 20 line segments respectively to approximate the arcs AB, CD. The conformal modulus of P 1 and P 2 both are 2.45 to ensure that a conformal mapping exists between them.
- Step 2: A rectangle R with conformal modulus equals to 2.45 is used to connect P 1 and P 2. (In mathematics, one method to calculate conformal modulus of a topological quadrilateral is to map it onto a rectangle by SC mapping, and the ratio of height to width of the rectangle equals to the conformal modulus of the topological quadrilateral, and of course, it also equals to the conformal modulus of the rectangle.)
- Step 3: The conformal mapping f 1, f 2 are calculated numerically by employing the SC mapping program.
- Step 4: The approximate material parameters corresponding to the mapping from Q 1 onto Q 2 are calculated by Eq. (4), and the resulting refractive index distribution in the phase modulator is shown in Fig. 3(a) .
In order to demonstrate the functionality of the phase modulator designed by the CTM, simulation is performed by the finite element method in the mode of TE waves. Figure 3(b) is the z component of electric filed in the wavelength of 0.12cm, it shows that the EM wave excited by a point source on the center of the phase modulator focuses on the two points F 1, F 2. This result agrees well with our design expectations.
3.2 Plane Focal Surface 2D Luneburg Lens
The Luneburg lens is a widely used antenna, and it finds broad applications in modern mobile communications, radio astronomy, etc. Recently, the operating wavelength of the Luneburg lens is even extended to the near-infrared region [16, 17]. The 2D Luneburg lens is a cylindrical lens whose permittivity can be described by
where R is the radius of the lens and r is the distance from a point on the lens to the lens center. The focusing property of the Luneburg lens is that every point on the cylindrical surface is the focal point for the incident plane waves on the opposite side of the lens.We will transform one half of the cylindrical focal surface of the 2D Luneburg lens into plane. The lens after transformation is called a plane focal surface 2D (PFS) Luneburg lens, and because of its plane focal surface, the emitters or acceptors on the focal surface used to transmit or receive signals are arranged in a plane, rather than arranged in a cylindrical surface. Due to this property, the advantage of PSF Luneburg lens becomes quite evident when the emitters or acceptors are more suitable to arrange in a plane, such as when charge-coupled device (CCD) or complementary metal–oxide–semiconductor (CMOS) arrays are used as the signal acceptors. Additionally, PFS Luneburg lens are more convenient to fix on its carriers in practical applications.
As shown in Fig. 4 , in the virtual space, Q 1 is the 2D Luneburg lens before transformation. If Q 1 is mapped onto Q 2, then accordingly, the cylindrical focal surface AB of Q 1 (marked in green) will be mapped to the plane focal surface AB of Q 2. Consequently, in the physical space, Q 2 becomes a PFS Luneburg lens. To this end, the CTM is also utilized to find the approximate material parameters (The solving Laplace’s equation method is invalid in this case either, since neither of Q 1, Q 2 is a rectangle). In this case, the CTM procedure can be specified as following:
Fig. 4 Design for a PSF Luneburg lens with the CTM. Q 1 is in the virtual space. Q 2, P 2 are in the physical space. P 2 is a polygon and its MN boundary (marked in blue) is composed of 60 line segments used to approximate the MN arc of Q 2.
- Step 1: A polygon P 2 is used to approximate Q 2. The MN boundary of P 2 which is composed of 60 line segments is used to approximate the MN arc of Q 2. And the conformal center of P 2 is (u, v) = (0, 0).
- Step 2: The conformal mapping f 2 are calculated numerically by employing SC mapping program.
- Step 3: The approximate material parameters of the PFS Luneburg lens are calculated by ε = εL / | f ′|2, μ = 1 for TE waves, where εL is the permittivity of the 2D Luneburg lens defined by Eq. (5). And the resulting refractive index distribution is shown in Fig. 5(a) .
Fig. 5 Simulation results of the PFS Luneburg lens. (a) is the refractive index distribution in the PFS Luneburg lens. (b)(c)(d) are the Z component of the electric field when the point source is located at (u, v) = (−0.5, 0), (−0.5, 0.1), (−0.5, 0.2), respectively. (e)(f) are the Z component of the electric field when plane waves incident on the PSF Luneburg lens.
It is worth noting that, different from the general case as illustrated in Fig. 1, Q 1 itself is a disk which can be used as the intermediate geometry, thus no other intermediate geometry is used. For the same reason, no polygon is used to approximate Q 1. Consequently, Eq. (2) is used instead of Eq. (4) to compute the approximate material parameters in the step 3. Additionally, due to the property of SC mapping, if a disk is used as the intermediate geometry, a conformal center should be defined as in the step 1.
The PFS Luneburg lens is also simulated in the mode of TE waves by the finite element method. Figure 5(b), 5(c), 5(d) are the z component of the electric field in the wavelength of 0.2 cm when the point sources are located at (−0.5, 0), (−0.5, 0.1), (−0.5, 0.2), respectively. It is evident that when the point source is moved along the plane focal surface, the plane waves generated on the other side will change its direction correspondingly, thus the PFS Luneburg lens can be used as a transmitting antenna. Figure 5(e) shows that a plane wave whose propagation direction parallels the u axis incidents on the PSF Luneburg lens, and then focuses on the point (−0.5, 0). Similarly, Fig. 5(f) shows that when the angle between the propagation direction of the plane wave and u axis is 30 degrees, the wave focuses on another point whose location is (−0.5, 0.14). Therefore, the PSF Luneburg lens can also be used as a receiving antenna. Note that in Fig. 5(e), 5(f), because the impedance on the MN boundary of the PSF Luneburg lens is not perfectly matched with the impedance of vacuum, thus a small part of energy of the incident wave is reflected.
4. Discussion of the Numerical Errors
In the CTM procedure described in section 2, the numerical errors may be introduced in the step 1 when irregular geometries are approximated by polygons which will distort the material parameters distribution in the region near the boundary of the polygon P 2 (in Fig. 1). In order to achieve accurate material parameters, the polygons should be set with enough sides so that the width of the distorted region is much smaller than the operating wavelength of the EM wave.
Take the PFS Luneburg lens in Fig. 4 for example, when the MN arc of Q 2 is approximated by 15 line segments, 60 line segments, and 90 line segments, respectively. Then, the widths of the distorted regions of the material parameters as shown in Fig. 6 are about 0.2cm, 0.02cm, and 0.01cm, respectively. Since the operating wavelength of the PFS Luneburg lens in our simulation is 0.2cm which is already equivalent to the width of the distorted region in the 15 line segments approximation, thus the 15 line segments approximation is unacceptable. In the 60 line segments and 90 line segments approximations, the widths of the distorted regions are ten percent and five percent of the operating wavelength, respectively. In these cases, the macroscopic EM fields will not sensitive to the detail material parameters distributions in the distorted regions, but mainly depend on the average material parameters in the regions. As a result, these approximations are acceptable. The simulation results in Fig. 5 based on the 60 line segments approximation suggest that the approximation is already accurate enough for the 0.2 cm operating wavelength. Based on the analysis, we believe that the enough polygons sides should be taken to ensure the width of the distorted region being less than ten percent of the operating wavelength for achieving accurate transformation.
Fig. 6 The dependence of the width of the distorted region on the approximation in the PFS Luneburg lens. (a)(b)(c) are contours of constant value of the resulting permittivity near the MN arc when the arc is approximated by 15 line segments, 60 line segments, and 90 line segments, respectively. And the widths (marked with red line segments) of the distorted regions of the permittivity distributions are about 0.2cm, 0.02cm, and 0.01cm, respectively.
5. Conclusion
In conclusion, we have presented a conformal transformation method (CTM) to determine the material parameters corresponding to the mapping between two irregular geometries which provides an effective approach for realizing the required conformal transformation of metamaterial devices. Compared with other conformal transformation method, the CTM is more general because it can be employed to find the conformal mapping between any two irregular geometries if there exists a conformal mapping between them. By using the CTM, a phase modulator and a plane focal surface 2D Luneburg lens are designed and validated by numerical simulations. It hopes that such method can serve as a convenient tool to design more useful and realizable devices.
Acknowledgments
This work was supported by the Chinese Nature Science Grant (61007024, 11074251, 60878031, 61078047). Authors would like to thank Dr. Haofei Shi, Dr. Zheng Chang and Dr. Yungui Ma for their valuable suggestions.
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