The technique of transformation optics (TO) is an elegant method for the design of electromagnetic media with tailored optical properties. In this paper, we focus on the formal structure of TO theory. By using a complete covariant formalism, we present a general transformation law that holds for arbitrary materials including bianisotropic, magneto-optical, nonlinear and moving media. Due to the principle of general covariance, the formalism is applicable to arbitrary space-time coordinate transformations and automatically accounts for magneto-electric coupling terms. The formalism is demonstrated for the calculation of the second harmonic wave generation in a twisted TO concentrator.
© 2012 Optical Society of America
The field of transformation optics (TO) has drawn a lot of scientific interest in the last few years [1–6]. By this design methodology, the form-invariance of Maxwell’s equations under coordinate transformation is used to tailor the optical properties of an electrodynamic medium. The majorities of TO applications focus on the design of the linear material parameters. In that case, a coordinate transformation is used to engineer the linear constitutive parameters of a medium such that the wave trajectory follows a desired path . An interesting application of this concept is the realization of electromagnetic invisibility cloaks [8–12]—devices in which light is guided around a certain region of space rendering the interior of the region invisible for an external observer. Cloaking devices belong to the most prominent applications of TO and have been extensively reported in the literature [13–19]. Further examples in which the concept of TO is successfully applied for designing the linear material properties of a medium include optical rotators , beam concentrators , novel types of lenses , omnidirectional retroreflectors  and the mimicking of celestial mechanics [23, 24].
In contrast, only little work in the research of TO media addresses the transformation of nonlinear material properties. Media with nonlinear response, however, provide a number of interesting effects including sum- and difference-frequency generation, parametric amplification and oscillation, stimulated scattering, self-phase modulation and self-focusing [25–33] and play a key role in modern optical technology [34–36]. Consequently, the extension of the TO concept to nonlinear media is expected to offer a variety of new opportunities for engineering optical media and for the construction of novel electromagnetic devices . Apart from that, the TO concept also provides a promising calculation method in modeling complex nonlinear systems whenever it is possible to find a coordinate transformation such that the geometry of the system takes a much simpler form in the transformed space.
A basic prerequisite for a successful integration of nonlinear effects into the TO concept is a general transformation law for linear and nonlinear material parameters under space-time coordinate transformation. A first step in this direction is suggested in  where the transformation of certain classes of nonlinear materials under purely spatial transformations is studied. However, the validity of the methodology presented in  is very limited; magneto-electric coupling effects, for example, cannot be described. The proposed formalism is further expressed in a non-covariant form and thus, is inherently restricted to purely spatial coordinate transformations that are only valid for media at rest. The aim of the following paper is to generalize this approach and to provide a rigorous theoretical framework for arbitrary nonlinear materials and for both temporal and spatial coordinate transformations. The formalism is presented in a manifestly covariant form that allows the simultaneous treatment of electric, magnetic and magneto-electric cross-coupling effects and encompasses all types of space-time transformations with a particular reference to moving media [38–40].
The paper is organized as follows: In the first part, we introduce the used tensor notation and explain how the defined tensors are related to the common linear and nonlinear material parameters such as the permittivity, permeability or the quadratic electro-optic coefficients. Subsequently, we exploit the fundamental principle of relativity to derive a general transformation law for nonlinear media under space-time coordinate transformations. In this context, a special emphasis is given to the transformation of the nonlinear constitutive parameters in moving media and the class of time-independent, spatial transformations which play a central role for the design of TO devices. In the final part of the paper, we illustrate and explain the derived expressions by means of an explicit calculation example. By determining the second harmonic wave generation in a nonlinear, twisted field concentrator, we show that an appropriate coordinate transformation can provide a significant alleviation in the formal treatment of nonlinear phenomena in a medium with highly complex nonlinear material properties and, thus, allows a convenient calculation of an otherwise sophisticated physical problem.
2. The covariant material equation
In order to establish a common basis for the following discussion and to introduce the used notation, we start with a short review of the covariant description of the electrodynamic theory. In this paper, all quantities are expressed in Gaussian units. In this case, the Maxwell equations in matter take the form:Eq. (5) contains the two homogeneous Maxwell equations whereas Eq. (6) contains the two inhomogeneous Maxwell equations. The relation between the tensors Fμν and 𝒟μν is given by Eq. (7) is equivalent to the common constitutive material relations: 36], the induced polarization can be described as a power series in the field strength according to: 36]. From the antisymmetry of the tensors Fμν and 𝒫μν and the commutativity of the products Fαiβi Fαjβj, it follows that the coefficients χμνα1β1···αnβn are antisymmetric under exchange of μ ↔ ν, symmetric under exchange of two pairs αiβi ↔ αjβj and antisymmetric under exchange of two indices αi ↔ βi within one pair αiβi. For the quadratic term, this means for example: Eq. (11). Inserting Eq. (10) into the contravariant version of material Eq. (7) yields: Appendix).
2.1. The linear term
To become familiar with the tensor notation, it is instructive to first consider only the linear term on the right-hand side of Eq. (12):38]: appendix, the relations between the tensor components of ε, μ, ξ and ζ and the covariant four-tensor χμνσκ is given by:
2.2. Nonlinear terms
Next, we consider quadratic contributions to the polarization-magnetization tensor 𝒫μν, i.e. contributions that depend on the product of two components of the field strength tensor. These are given by the second summand in Eq. (10) and have the form:
3. Coordinate transformation
The particular advantage of the covariant formalism is its form-invariance under coordinate transformations38, Chap. 3.2]: Eqs. (5) and (6) have the same form in the primed coordinate system as in the unprimed system due to their natural form-invariance [38, Chap. 3.2], that is: Eq. (22) is not difficult. To see this, we multiply both sides of Eq. (12) by , insert in the right-hand side the identity 38], Chap. 1.4) and replace the unprimed quantities with the corresponding primed ones. This yields:
The broad generality of the transformation formula enables us to design novel nonlinear media with tailored optical properties. An interesting application of this design tool is is the guidance of light during a frequency conversion process for walk-off compensation, phase-matching or for the enhancement of local field amplitudes. Further applications include the engineering of magneto-electric nonlinearities such as the Kerr- or Faraday-effect, the calculation of relativistic corrections in moving media and the control of nonlinear self-interaction effects such as self-phase modulation or self-focusing. The latter, for instance, allows a controlled steering of soliton pulses in a TO-medium which offers new alternatives for optical waveguides and communication systems.
3.1. Moving media
A typical physical situation where coordinate transformations play a role occurs when a medium moves relative to an observer. In such cases, the transformation law allows the calculation of the material parameters in the reference frame of the observer if the material parameters are known in the rest frame of the medium. If the medium moves with constant velocity u, say along the x-direction, the corresponding coordinate transformation is given by where the Jacobian matrix describes a Lorentz boost in the minus x-direction:Eq. (15) (for example, εi′j′ = −8πχ0′i′0′j′). In the same manner, the quadratic susceptibility transforms as 40–43]. For the nonlinear coefficients, the magneto-electric coupling implies, for instance, that a Pockels medium at rest (χ0i0j0k ≠ 0) can display a Faraday effect (χ0′i′0′j′k′l′ ≠ 0) if it is moved relative to the observer and vice versa. In other words, time transformations result in new nonlinear optical properties that are not present in the same medium at rest. The corresponding nonlinear optical coefficients can be calculated with the transformation rule of Eq. (22).
3.2. Time-independent transformations
We now focus on the special case of time-independent, spatial transformations, i.e. transformations withEq. (15) transform as: Eq. (22), the general transformation of the second order tensor χμνσκαβ is Eq. (29)), the second order susceptibility aijk transforms as Eq. (30). This shows that as soon as the relation between the material coefficient of interest (linear or nonlinear) and the general covariant tensor χμνα1β1···αnβn has been determined, the transformation law immediately follows from Eq. (22).
4. Twisted, nonlinear field concentrator
In the following, we demonstrate that nonlinear problems in complex media with sophisticated linear and nonlinear optical properties can take a much simpler form in an appropriately transformed space.
To provide an illustrative example of this calculation method, we consider a nonlinear, inhomogeneous material that is illuminated by a strong laser field. We assume that the polarization of the fundamental wave and the nonlinearity of the material match the condition for second harmonic generation (SHG) and, as a goal, we want to calculate the spatial field distribution of the SHG wave inside the medium. If the material is highly inhomogeneous, both the fundamental and the SHG wave follow a complicated, distorted trajectory through the medium which generally hampers a numerical calculation and reliable prediction of the SHG progress. However, if it is possible to find a coordinate transformation such that the wave propagates uniformly along straight lines in the new coordinate system, the wave equation can be readily solved by an ordinary integration along the field lines. A subsequent back-transformation then yields the SHG field in the physical space. In the following, we demonstrate and explain this calculation method for a specific example.
As a hypothetic, inhomogeneous material, we consider the special case of a TO medium, i.e. an artificial material whose permeability and permittivity tensors were obtained by applying a coordinate transformation to a homogeneous, isotropic space. We suppose that the optical properties of the homogeneous space are similar to that of vacuum with a permittivity and permeability equal (or close) to unity. In the following, the coordinates of the uniform, isotropic space are indicated by unprimed indices while the coordinates of the physical space of the inhomogeneous material are indicated by primed indices.
For the transformation between the primed and unprimed system, we consider the following transformation (expressed in cylinder coordinates using r2 = x2 + y2 and ϕ = arctan(y/x)):Fig. 1, the transformation compresses space of a cylindrical region with radius R2 into a region with radius R1 at the cost of an expansion of space between R1 and R3 . In addition, the space experiences a radius-dependent twist about the z-axis . Note that the transformation addresses only the x- and y-coordinates while the z-coordinate remains unchanged. For this reason, we can restrict the following discussion to the xy-submanifold of the space-time manifold. The corresponding Jacobian matrix is:
The transformation (34) is expressed in cylindrical coordinates. For the calculation of the SHG, however, it is more advantageous to formulate the transformation in Cartesian coordinates in the form x′ = x′(x,y) and y′ = y′(x,y). This is achieved by applying an intermediate transformation and ϕ = arctan(y/x) in the unprimed system and an inverse intermediate transformation x′ = r′ cosϕ′ and y′ = r′ sinϕ′ in the primed system. With these expressions, the transformation (34) can be re-expressed in Cartesian coordinates by applying the following three subsequent transformations:appendix, the Jacobian determinants of the intermediate transformations are |Af| = 1/r and |Ah| = r′, respectively. Hence, the Jacobian determinant of the composite is Eq. (36). Once the Jacobian matrix and its determinant are known, we can easily transform the fields and material parameters from one coordinate system to the other.
We now intend to calculate the SHG wave generated in the concentrator. For this purpose, we suppose that the material used for the construction of the concentrator exhibits a non-vanishing nonlinear susceptibility. To simplify matters, we further assume that this nonlinearity obeys the phase matching condition for frequency doubling if the fundamental wave and the second harmonic wave are both polarized in the z-direction. In our notation, the corresponding tensor component of the nonlinear susceptibility is a3′3′3′ := 4χ0′3′0′3′0′3′ (see Eq. (16)). For the spatial distribution of a3′3′3′, we suppose that the nonlinearity is only located in the inner cylindrical region of the material according to:
As fundamental wave, we assume an incident monochromatic plane wave that initially propagates along the x-axis with the electric field vector polarized in the z-direction. In the physical space of the inhomogeneous material (spanned by primed coordinates), the fundamental wave takes the form:36]: Figs. 2(a) and 2(b), respectively.
Since the fundamental wave propagates straightly along the x-direction in the unprimed system and since we assumed phase matching for the SHG process, it follows that the wave vector of the SHG wave is given by k2ωx = 2kωx = 2ω/c and k2ωy = 0. On the considered length scale, beam divergence due to diffraction can be neglected. Consequently, the wave Eq. (43) for the SHG process reduces in the unprimed system to the simple expressionEq. (33), the nonlinearity transforms as: Eqs. (39) and (36), the nonlinearity a3′3′3′ defined in Eq. (40) and the relation between r and r′ given by Eq. (34), we can calculate the nonlinearity in the unprimed system to be: Eq. (40)) because the cylindrical region in which the nonlinear substance is located experiences a space expansion from radius R1 to R2 if we perform a coordinate transformation from the primed to the unprimed system.
Now we can (even analytically) evaluate the integral (47) for the amplitude 2ω (x,y) of the SHG wave. A subsequent multiplication with the propagation phasor yields the SHG wave in the unprimed system in the form:Figs. 2(c) and 2(d), respectively.
The proposed twisted nonlinear concentrator is certainly a somewhat constructed example since the exploited uniform space was already presumed in the design of the concentrator. However, the decisive step in the calculation—the straightening of the wave trajectories inside the medium by applying an appropriate coordinate transformation—is in principle always possible in any inhomogeneous media with continuously varying material properties. The proposed technique of simplifying nonlinear processes in inhomogeneous media is therefore not restricted to the presented example, but covers a wide application range.
We proposed a theoretical framework for the incorporation of nonlinear effects within the concept of transformation optics (TO). In this context, we derived a general expression for the calculation of linear and nonlinear electromagnetic material parameters under arbitrary space-time coordinate transformations. The transformation law is formulated in a manifestly covariant form that allows the simultaneous treatment of electric, magnetic and magneto-electric cross-coupling terms and is applicable to both temporal and spatial transformations. Based on the generality of our approach, a much larger range of applications can be addressed than by any other technique presented in the literature so far. Since our approach accounts for all kinds of (nonlinear) magneto-electric coupling effects, the transformational approach provides the tools to actively design effects like the Faraday effect, Voigt effect etc. in a transformation-optical medium. The inclusion of time transformations, which translate into magneto-electric coupling effects not only for the linear but also for the nonlinear properties of the transformation medium, imply that moving media at modest velocities well below the speed of light can manifest new nonlinear optical properties that are not present in the same medium at rest. This means, for example, that a Pockels medium at rest can display a Faraday effect if the material is moved relative to the observer, and vice versa. This carries the potential to be demonstrated in subsequent experiments.
In the final part of the paper we focused on time-independent, spatial coordinate transformations which are of particular interest for the design of nonlinear TO devices. As an illustrative example of such a device, we presented a twisted nonlinear field concentrator and calculated the second harmonic wave that is generated when the concentrator is illuminated by a strong laser field. In this respect, we demonstrated that sophisticated nonlinear phenomena in complex media can take a much simpler form if an appropriate coordinate transformation is applied.
The considerations have shown that the incorporation of nonlinear susceptibilities in the TO approach offers new opportunities for the design of novel optical devices with tailored nonlinear properties and provides a promising computation method for calculating nonlinear effects in moving or inhomogeneous media.
Redefinition of the linear susceptibility in Eq. (12)
With , the linear terms in Eq. (12) can be written asEq. (12) follows after the re-definition (in Eq. (12), the label “new” has been omitted).
Proof of Eq. (15)
By using the identityEq. (15).
Intermediate transformations used in Eq. (39)
The first intermediate transformation f is given by
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