Covariant description of transformation optics in nonlinear media

Oliver Paul; Marco Rahm

doi:10.1364/OE.20.008982

1. Introduction

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 [7]. 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 [20], beam concentrators [15], novel types of lenses [21], omnidirectional retroreflectors [22] 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 [37]. 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 [37] where the transformation of certain classes of nonlinear materials under purely spatial transformations is studied. However, the validity of the methodology presented in [37] 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:

\begin{matrix} div B = 0, & rot E + \frac{1}{c} \frac{\partial B}{\partial t} = 0, \end{matrix}

\begin{matrix} div D = 4 π ρ, & rot H - \frac{1}{c} \frac{\partial D}{\partial t} = \frac{4 π}{c} j \end{matrix}

where E, H, D, B, ρ, j and c denote the electric and magnetic field, the electric and magnetic flux density, the charge and current density and the speed of light in vacuum, respectively. By using the four-current j^ν = (cρ, j), the antisymmetric field strength tensor

F_{μ ν} = (\begin{matrix} 0 & E_{x} & E_{y} & E_{z} \\ - E_{x} & 0 & - B_{z} & B_{y} \\ - E_{y} & B_{z} & 0 & - B_{x} \\ - E_{z} & - B_{y} & B_{x} & 0 \end{matrix})

and the antisymmetric displacement tensor

𝒟_{μ ν} = (\begin{matrix} 0 & D_{x} & D_{y} & D_{z} \\ - D_{x} & 0 & - H_{z} & H_{y} \\ - D_{y} & H_{z} & 0 & - H_{x} \\ - D_{z} & - H_{y} & H_{x} & 0 \end{matrix}),

the Maxwell equations can be covariantly expressed as:

\partial_{μ} F_{ν σ} + \partial_{σ} F_{μ ν} + \partial_{ν} F_{σ μ} = 0,

\partial_{μ} 𝒟^{μ ν} = \frac{4 π}{c} j^{ν}

where we used the partial derivative operator ∂_μ = (

\frac{1}{c} \partial_{t}

, ∇) and the Einstein summation convention of summing over repeated indices (throughout this paper, Greek indices run from 0 to 3 while Latin indices run from 1 to 3). As usual, the relation between covariant and contravariant tensors is obtained via the metric tensor according to 𝒟^μν = η^μα η^νβ 𝒟_αβ where the metric tensor is given by η^μν = diag(1, −1, −1, −1). In the covariant notation, Eq. (5) contains the two homogeneous Maxwell equations whereas Eq. (6) contains the two inhomogeneous Maxwell equations. The relation between the tensors F_μν and 𝒟_μν is given by

𝒟_{μ ν} = F_{μ ν} + 4 π 𝒫_{μ ν}

with the polarization-magnetization tensor

𝒫_{μ ν} = (\begin{matrix} 0 & P_{x} & P_{y} & P_{z} \\ - P_{x} & 0 & M_{z} & - M_{y} \\ - P_{y} & - M_{z} & 0 & M_{x} \\ - P_{z} & M_{y} & - M_{x} & 0 \end{matrix}) .

Tensor Eq. (7) is equivalent to the common constitutive material relations:

\begin{array}{l} D = E + 4 π P, \\ H = B - 4 π M . \end{array}

Since the general response of a nonlinear medium depends on the amplitude of the applied optical field [36], the induced polarization can be described as a power series in the field strength according to:

\begin{array}{l} 𝒫^{μ ν} & = χ^{μ ν σ κ} F_{σ κ} + χ^{μ ν σ κ α β} F_{σ κ} F_{α β} + χ^{μ ν σ κ α β γ δ} F_{σ κ} F_{α β} F_{γ δ} + \dots \\ = \sum_{n = 1}^{\infty} χ^{μ ν α_{1} β_{1} \dots α_{n} β_{n}} F_{α_{1} β_{1}} \dots F_{α_{n} β_{n}} \end{array}

where the quantities χ^{μνα₁β₁···α_nβ_n} are the covariant optical susceptibilities of n-th order. Note that for purely electric interactions this equation reduces to the fundamental equation of nonlinear optics Pⁱ = d^ijE_j + d^ijkE_jE_k + ··· [36]. From the antisymmetry of the tensors F_μν and 𝒫_μν and the commutativity of the products F_{α_iβ_i} F_{α_jβ_j}, it follows that the coefficients χ^{μνα₁β₁···α_nβ_n} are antisymmetric under exchange of μ ↔ ν, symmetric under exchange of two pairs α_iβ_i ↔ α_jβ_j and antisymmetric under exchange of two indices α_i ↔ β_i within one pair α_iβ_i. For the quadratic term, this means for example:

χ^{μ ν α_{1} β_{1} α_{2} β_{2}} = - χ^{ν μ α_{1} β_{1} α_{2} β_{2}} = χ^{μ ν α_{2} β_{2} α_{1} β_{1}} = - χ^{μ ν β_{1} α_{1} α_{2} β_{2}} .

It is obvious that additional, inherent symmetries (such as the Kleinman symmetry or spatial symmetries given by the point symmetry class of the medium) further reduce the number of independent coefficients. However, in order to provide a general description, we only consider the basic symmetries given by Eq. (11). Inserting Eq. (10) into the contravariant version of material Eq. (7) yields:

\begin{array}{l} 𝒟^{μ ν} & = F^{μ ν} + 4 π 𝒫^{μ ν} \\ = F^{μ ν} + 4 π \sum_{n = 1}^{\infty} χ^{μ ν α_{1} β_{1} \dots α_{n} β_{n}} F_{α_{1} β_{1}} \dots F_{α_{n} β_{n}} \\ = 4 π \sum_{n = 1}^{\infty} χ^{μ ν α_{1} β_{1} \dots α_{n} β_{n}} F_{α_{1} β_{1}} \dots F_{α_{n} β_{n}} \end{array}

where in the last line, we performed a re-definition of the linear coefficient χ^{μνα₁β₁} to include the free-space space contribution (see 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):

𝒟^{μ ν} = 4 π χ^{μ ν σ κ} F_{σ κ} .

This expression can be equivalently rewritten in the more intuitive three-formalism [38]:

(\begin{matrix} D \\ H \end{matrix}) = (\begin{matrix} ε & ξ \\ ζ & μ^{- 1} \end{matrix}) (\begin{matrix} E \\ B \end{matrix}) .

This is the general equation of a linear bianisotropic medium where ε, μ, ξ and ζ are the permittivity, permeability and the magneto-electric coupling tensors. As shown in the appendix, the relations between the tensor components of ε, μ, ξ and ζ and the covariant four-tensor χ^μνσκ is given by:

\begin{array}{l} ε^{i j} = - 8 π χ^{0 i 0 j}, & ξ^{i j} = 4 π g_{m n}^{j} χ^{0 i m n}, \\ ζ^{ij} = - 4 π g_{m n}^{i} χ^{m n 0 j}, & {(μ^{- 1})}^{i j} = 2 π g_{m n}^{i} g_{k l}^{j} χ^{m n k l} \end{array}

with the totally antisymmetric Levi-civita tensor

g_{m n}^{i}

(with

g_{23}^{1} = 1

). These expressions will be useful, if we want to apply a general space-time coordinate transformation.

2.2. Nonlinear terms

Next, we consider quadratic contributions to the polarization-magnetization tensor 𝒫^μν, 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:

𝒫_{(2)}^{μ ν} = χ^{μ ν σ κ α β} F_{σ κ} F_{α β} .

For simplicity, we restrict here to the components of the second-order electric polarization

P_{(2)}^{i} = 𝒫_{(2)}^{0 i}

(a similar derivation applies for the magnetization). To achieve a deeper physical insight, we use the symmetry properties of χ^μνσκαβ to split the sum into three parts that separately account for purely electric, purely magnetic and magneto-electric cross-coupling effects. This yields:

\begin{array}{l} P_{(2)}^{i} & = χ^{0 i σ κ α β} F_{σ κ} F_{α β} \\ = 4 χ^{0 i 0 k 0 m} F_{0 k} F_{0 m} + 4 χ^{0 i 0 k m n} F_{0 k} F_{m n} + χ^{0 i k l m n} F_{k l} F_{m n} \\ = \underset{Pockels, effect, multi-wave mixing}{\underset{︸}{a^{i j k} E_{j} E_{k}}} + \underset{Faraday effect}{\underset{︸}{b^{i j k} E_{j} B_{k}}} + c^{i j k} B_{j} B_{k} \end{array}

where we have introduced the nonlinear material coefficients a^ijk, b^ijk, c^ijk. The substitutions in the last line follow from the fact that the tensor components F_0i address the electric field while F_ij addresses the magnetic field. Obviously, the expression

𝒫_{(2)}^{μ ν} = χ^{μ ν σ κ α β}

F_σκF_αβ describes all second-order electric, magnetic and magneto-electric cross-coupling effects in a single equation. Similarly, one finds for the nonlinear polarization of third order:

\begin{array}{l} P_{(3)}^{i} & = χ^{0 i σ κ α β μ ν} F_{σ κ} F_{α β} F_{μ ν} \\ = \underset{Kerr effect}{\underset{︸}{a^{i j k l} E_{j} E_{k} E_{l}}} + b^{i j k l} E_{j} E_{k} B_{l} + \underset{Cotton-Mouton effect}{\underset{︸}{c^{i j k l} E_{j} B_{k} B_{l}}} + d^{i j k l} B_{j} B_{k} B_{l}, \end{array}

and similarly for higher order contributions. Obviously, the introduced material coefficients χ^{μνα₁β₁···α_nβ_n} are capable to describe arbitrary complex materials including bian-isotropic, magneto-electric and nonlinear media.

3. Coordinate transformation

The particular advantage of the covariant formalism is its form-invariance under coordinate transformations

x^{α} \to x^{α^{'}} (x^{α})

where x^α = (ct,r) is the coordinate vector of the space time. By

A_{α}^{α^{'}} = \partial x^{α^{'}} / \partial x^{α}

and

| A | : = det (A_{α^{'}}^{α})

we denote the Jacobian matrix and its determinant. For notational convenience, we introduce the convention that, if a product of

A_{α}^{α^{'}}

occurs in a transformation formula, the kernel symbol A is written only once, e.g.

A_{α}^{α^{'}} A_{β}^{β^{'}} A_{γ}^{γ^{'}} = A_{α β γ}^{α^{'} β^{'} γ^{'}} .

The electromagnetic fields F_μν and 𝒟^μν in the primed and unprimed coordinate systems are related by [38, Chap. 3.2]:

\begin{array}{l} F_{μ^{'} ν^{'}} = A_{μ^{'} ν^{'}}^{μ ν} F_{μ ν}, \\ 𝒟^{μ^{'} ν^{'}} = {| A |}^{- 1} A_{μ ν}^{μ^{'} ν^{'}} 𝒟^{μ ν} . \end{array}

The Maxwell 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:

\begin{array}{r} \partial_{μ^{'}} F_{ν^{'} σ^{'}} + \partial_{σ^{'}} F_{μ^{'} ν^{'}} + \partial_{ν^{'}} F_{σ^{'} μ^{'}} = 0, \\ \partial_{μ^{'}} 𝒟^{μ^{'} ν^{'}} = \frac{4 π}{c} j^{ν^{'}} \end{array}

with

j^{ν^{'}} = {| A |}^{- 1} A_{ν}^{ν^{'}} j^{ν}

. As a consequence, the material coefficients in the constitutive relation (12), must transform as:

χ^{μ^{'} ν^{'} α_{1}^{'} β_{1}^{'} \dots α_{n}^{'} β_{n}^{'}} = {| A |}^{- 1} A_{μ ν α_{1} β_{1} \dots α_{n} β_{n}}^{μ^{'} ν^{'} α_{1}^{'} β_{1}^{'} \dots α_{n}^{'} β_{n}^{'}} χ^{μ ν α_{1} β_{1} \dots α_{n} β_{n}} .

The proof of Eq. (22) is not difficult. To see this, we multiply both sides of Eq. (12) by

{| A |}^{- 1} A_{μ ν}^{μ^{'} ν^{'}}

, insert in the right-hand side the identity

A_{α_{1} β_{1} \dots α_{n} β_{n}}^{α_{1}^{'} β_{1}^{'} \dots α_{n}^{'} β_{n}^{'}} A_{α_{1}^{'} β_{1}^{'} \dots α_{n}^{'} β_{n}^{'}}^{α_{1} β_{1} \dots α_{n} β_{n}} = A_{α_{1} β_{1} \dots α_{n} β_{n}}^{α_{1} β_{1} \dots α_{n} β_{n}} = 1

(see [38], Chap. 1.4) and replace the unprimed quantities with the corresponding primed ones. This yields:

\begin{array}{l} 𝒟^{μ ν} & = & 4 π \sum_{n = 1}^{\infty} χ^{μ ν α_{1} β_{1} \dots α_{n} β_{n}} F_{α_{1} β_{1}} \dots F_{α_{n} β_{n}} \\ \underset{𝒟^{μ^{'} ν^{'}}}{\underset{︸}{{| A |}^{- 1} A_{μ ν}^{μ^{'} ν^{'}} 𝒟^{μ ν}}} & = & 4 π \sum_{n = 1}^{\infty} \underset{χ^{μ^{'} ν^{'} α_{1}^{'} β_{1}^{'} \dots α_{n}^{'} β_{n}^{'}}}{\underset{︸}{{| A |}^{- 1} A_{μ ν}^{μ^{'} ν^{'}} A_{α_{1} β_{1} \dots α_{n} β_{n}}^{α_{1}^{'} β_{1}^{'} \dots α_{n}^{'} β_{n}^{'}} χ^{μ ν α_{1} β_{1} \dots α_{n} β_{n}}}} \underset{F_{_{α_{1}^{'} β_{1}^{'}}} \dots F_{α_{n}^{'} β_{n}^{'}}}{\underset{︸}{A_{a_{1}^{'} β_{1}^{'} \dots α_{n}^{'} β_{n}^{'}}^{α_{1} β_{1} \dots α_{n} β_{n}} F_{α_{1} β_{1}} \dots F_{α_{n} β_{n}}}} \\ 𝒟^{μ^{'} ν^{'}} & = & 4 π \sum_{n = 1}^{\infty} χ^{μ^{'} ν^{'} α_{1}^{'} β_{1}^{'} \dots α_{n}^{'} β_{n}^{'}} F_{α_{1}^{'} β_{1}^{'}} \dots F_{α_{n}^{'} β_{n}^{'}} . \end{array}

Thus, the constitutive equation transforms covariantly as required which completes the proof. Relation (22) represents the general transformation law for linear and nonlinear materials parameters (including sophisticated magneto-electric coupling terms) and is equally valid for arbitrary space-time coordinate transformations.

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 $x^{α^{'}} = A_{α}^{α^{'}} x^{α}$ where the Jacobian matrix $A_{α}^{α^{'}}$ describes a Lorentz boost in the minus x-direction:

\begin{matrix} A_{α}^{α^{'}} = (\begin{array}{l} γ & γ u / c & 0 & 0 \\ γ u / c & γ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}), & | A | = 1 \end{matrix}

with γ = (1 − u²/c²)^−1/2. The constitutive material properties in the reference frame of the observer are obtained by applying the transformation law (22) to the material parameters in the rest frame of the medium. For the linear susceptibility, this yields:

χ^{μ^{'} ν^{'} σ^{'} κ^{'}} = A_{μ ν σ κ}^{μ^{'} ν^{'} σ^{'} κ^{'}} χ^{μ ν σ κ} .

From this expression, the usual linear material parameters, such as the permittivity ε or the permeability μ, can be calculated based on the relations given in Eq. (15) (for example, ε^i′j′ = −8πχ^{0′i′0′j′}). In the same manner, the quadratic susceptibility transforms as

χ^{μ^{'} ν^{'} σ^{'} κ^{'} α^{'} β^{'}} = A_{μ ν σ κ α β}^{μ^{'} ν^{'} σ^{'} κ^{'} α^{'} β^{'}} χ^{μ ν σ κ α β}

from which the nonlinear quadratic properties in the new system can be calculated. Note that the non-vanishing 0i-components of the Jacobian matrix

A_{0}^{i^{'}} \neq 0

due to the movement of the medium implies a mixing of the electric and magnetic material properties. For the linear susceptibility, this results in the well-known effect that a medium which is isotropic at rest becomes bianisotropic when it moves relative to the observer [40–43]. For the nonlinear coefficients, the magneto-electric coupling implies, for instance, that a Pockels medium at rest (χ^0i0j0k ≠ 0) can display a Faraday effect (χ^{0′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 with

\begin{matrix} t^{'} = t, & x^{'} = x^{'} (x, y, z), & y^{'} = y^{'} (x, y, z), & z^{'} = z^{'} (x, y, z) \end{matrix}

which are of particular interest for the design of TO devices with tailored optical properties. In this case, we have

\begin{matrix} A_{0}^{0^{'}} = \frac{\partial t^{'}}{\partial t} = 1 & and & A_{i}^{0^{'}} = \frac{c \partial t^{'}}{\partial x^{i}} = 0. \end{matrix}

With the Einstein sum convention, this implies the useful calculation rules

A_{μ}^{0^{'}} B^{μ} = B^{0}

and

A_{μ}^{i^{'}} B^{μ} = A_{i}^{i^{'}} B^{i}

(remember: Latin indices run from 1 to 3). For example, the permittivity transforms under these conditions as

\begin{array}{l} ε^{i^{'} j^{'}} & = - 8 π χ^{0^{'} i^{'} 0^{'} j^{'}} \\ = - 8 π {| A |}^{- 1} A_{μ ν α β}^{0^{'} i^{'} 0^{'} j^{'}} χ^{μ ν α β} \\ = - 8 π {| A |}^{- 1} A_{i j}^{i^{'} j^{'}} χ^{0 i 0 j} \\ = {| A |}^{- 1} A_{i j}^{i^{'} j^{'}} ε^{i j} . \end{array}

Accordingly, the permeability and the magneto-electric coupling terms in Eq. (15) transform as:

\begin{matrix} μ^{i^{'} j^{'}} = {| A |}^{- 1} A_{i j}^{i^{'} j^{'}} μ^{i j}, & ξ^{i^{'} j^{'}} = {| A |}^{- 1} A_{i j}^{i^{'} j^{'}} ξ^{i j}, & ζ^{i^{'} j^{'}} = {| A |}^{- 1} A_{i j}^{i^{'} j^{'}} ζ^{i j} . \end{matrix}

Note that for purely spatial transformations there is no magneto-electric cross-mixing. By the general transformation law (22), we can now calculate the transformation behavior of the nonlinear material coefficients under spatial coordinate transformations. For instance, we exemplarily calculate the second order susceptibility of the electric polarization a^ijk = 4χ^0i0j0k which describes nonlinear optical effects such as second-harmonic generation or three-wave-mixing. With help of Eq. (22), the general transformation of the second order tensor χ^μνσκαβ is

χ^{μ^{'} ν^{'} σ^{'} κ^{'} α^{'} β^{'}} = {| A |}^{- 1} A_{μ ν σ κ α β}^{μ^{'} ν^{'} σ^{'} κ^{'} α^{'} β^{'}} χ^{μ ν σ κ α β}

and for the special case of a purely spatial coordinate transformation (i.e. with the conditions given in Eq. (29)), the second order susceptibility a^ijk transforms as

\begin{array}{l} a^{i^{'} j^{'} k^{'}} & = 4 χ^{0^{'} i^{'} 0^{'} j^{'} 0^{'} k^{'}} \\ = 4 {| A |}^{- 1} A_{μ ν σ κ α β}^{0^{'} i^{'} 0^{'} j^{'} 0^{'} k^{'}} χ^{μ ν σ κ α β} \\ = 4 {| A |}^{- 1} A_{i j k}^{i^{'} j^{'} k^{'}} χ^{0 i 0 j 0 k} \\ = {| A |}^{- 1} A_{i j k}^{i^{'} j^{'} k^{'}} a^{i j k} \end{array}

where we applied similar calculation steps as in Eq. (30). This shows that as soon as the relation between the material coefficient of interest (linear or nonlinear) and the general covariant tensor χ^{μνα₁β₁···α_nβ_n} 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)):

\begin{array}{l} r^{'} = {\begin{array}{l} \frac{R_{1}}{R_{2}} r & 0 \leq r \leq R_{2} \\ \frac{R_{3} - R_{1}}{R_{3} - R_{2}} r - \frac{R_{2} - R_{1}}{R_{3} - R_{2}} R_{3} & R_{2} \leq r \leq R_{3} \\ r & otherwise \end{array} \\ ϕ^{'} = {\begin{array}{l} ϕ + \frac{π}{2} {cos}^{2} (\frac{π r}{2 R_{3}}) & 0 \leq r \leq R_{3} \\ ϕ & otherwise . \end{array} \end{array}

As illustrated in 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 [15]. In addition, the space experiences a radius-dependent twist about the z-axis [20]. 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:

A_{g} = (\begin{matrix} \frac{\partial r^{'}}{\partial r} & \frac{\partial r^{'}}{\partial ϕ} \\ \frac{\partial ϕ^{'}}{\partial r} & \frac{\partial ϕ^{'}}{\partial ϕ} \end{matrix}) = {\begin{array}{l} (\begin{matrix} \frac{R_{1}}{R_{2}} & 0 \\ - \frac{π^{2}}{4 R_{3}} sin (\frac{π r}{R_{3}}) & 1 \end{matrix}) & 0 \leq r \leq R_{2} \\ (\begin{matrix} \frac{R_{3} - R_{1}}{R_{3} - R_{2}} & 0 \\ - \frac{π^{2}}{4 R_{3}} sin (\frac{π r}{R_{3}}) & 1 \end{matrix}) & R_{2} \leq r \leq R_{3} \\ diag (1, 1) & otherwise . \end{array}

And the determinant of A_g is:

| A_{g} | = {\begin{array}{l} \frac{R_{1}}{R_{2}} & 0 \leq r \leq R_{2} \\ \frac{R_{3} - R_{1}}{R_{3} - R_{2}} & R_{2} \leq r \leq R_{3} \\ 1 & otherwise . \end{array}

Fig. 1 Visualization of the twisted cylindrical concentrator. The mesh grid indicates lines with constant values of x and y plotted in the coordinate system of x′ and y′. The radius of the three displayed circles are R₁ = 3, R₂ = 4.5 and R₃ = 6.

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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 $r = \sqrt{x^{2} + y^{2}}$ 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:

(x, y) \overset{f}{\to} (r, ϕ) \overset{g}{\to} (r^{'}, ϕ^{'}) \overset{h}{\to} (x^{'}, y^{'}) .

According to the chain rule in higher dimensions, the Jacobian matrix of a composite function is just the product of the Jacobian matrices of the composed functions, that is

A_{f \circ g \circ h} = A_{f} A_{g} A_{h} .

As shown in the appendix, the Jacobian determinants of the intermediate transformations are |A_f| = 1/r and |A_h| = r′, respectively. Hence, the Jacobian determinant of the composite is

| A_{f \circ g \circ h} | = | A_{f} | | A_{g} | | A_{h} | = \frac{r^{'}}{r} | A_{g} |

with the Jacobian determinant |A_g| given in 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 a^3′3′3′ := 4χ^{0′3′0′3′0′3′} (see Eq. (16)). For the spatial distribution of a^3′3′3′, we suppose that the nonlinearity is only located in the inner cylindrical region of the material according to:

a^{3^{'} 3^{'} 3^{'}} (r^{'}) = {\begin{array}{l} a_{0} & 0 \leq r^{'} \leq R_{1} \\ 0 & otherwise \end{array}

with a₀ = const. This could, for example, be realized by uniformly doping the center of the concentrator with some nonlinear material.

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:

E_{ω}^{'} (x^{'}, y^{'}, t) = 𝒠_{ω}^{'} e^{i (k_{ω x^{'}}^{'} x^{'} + k_{ω y^{'}}^{'} y^{'} - ω t)}

where 𝒠′_ω denotes the amplitude of the wave and k′_ω = (k′_ωx′,k′_ωy′) is the wave vector in the medium. For the second harmonic wave, we represent the electric field by

E_{2 ω}^{'} (x^{'}, y^{'}, t) = 𝒠_{2 ω}^{'} (x^{'}, y^{'}) e^{i (k_{2 ω x^{'}}^{'} x^{'} + k_{2 ω y^{'}}^{'} y^{'} - 2 ω t)}

with the SHG wave vector k′_2ω = 2k′_ω (phase matched case). In the slowly varying amplitude approximation for the second harmonic wave and the undepleted pump approximation (i.e. 𝒠′_ω is constant), the wave equation for the SHG amplitude 𝒠′_2ω(x′,y′) is given by [36]:

(k_{2 ω x^{'}}^{'} \frac{\partial}{\partial x^{'}} + k_{2 ω y^{'}}^{'} \frac{\partial}{\partial y^{'}}) 𝒠_{2 ω}^{'} (x^{'}, y^{'}) = κ 𝒠_{ω}^{' 2} a^{3^{'} 3^{'} 3^{'}} (x^{'}, y^{'})

with κ = −16πiω²/c². This is a partial differential equation in two dimensions for the second harmonic field amplitude 𝒠′_2ω (x′,y′) which is very difficult to solve in general. However, the complexity can be significantly reduced if we apply a coordinate transformation to the uniform space (spanned by unprimed coordinates).

According to Eq. (20) and the Jacobian matrix of the transformation given by Eq. (38) (with omitted subscript f ○ g ○ h), the z-component of the electric field transforms as

E_{z^{'}}^{'} = F_{0^{'} 3^{'}} = A_{0^{'} 3^{'}}^{μ ν} F_{μ ν} = F_{03} = E_{z}

since t′ = t and z′ = z imply

A_{0^{'} 3^{'}}^{μ ν} = δ_{0}^{μ} δ_{3}^{ν}

. Consequently, the electric fields of both the fundamental and SHG wave remain unchanged under coordinate transformation (but the functional dependence on the underlying coordinate system may be different). Since the unprimed system is homogeneous and isotropic with a refractive index equal to one, the electric field of the fundamental wave in the unprimed system has the form of a uniform plane wave propagating in the x-direction:

E_{ω} (x, t) = 𝒠_{ω} e^{i (k_{ω x} x - ω t)}

with constant amplitude 𝒠_ω = 𝒠′_ω and k_ωx = ω/c. From this expression, the electric field in the primed coordinate system is readily obtained by substituting x by x = x(x′,y′). The real part of the fundamental wave in the primed and unprimed system is plotted in Figs. 2(a) and 2(b), respectively.

Fig. 2 Second harmonic wave generation in a nonlinear cylindrical concentrator. (a) Electric field of the fundamental wave in the uniform space (unprimed system) and (b) in the inhomogeneous, physical space (primed system). (c) Electric field of the second harmonic wave in the unprimed system and (d) in the primed system.

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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 k_2ωx = 2k_ωx = 2ω/c and k_2ωy = 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 expression

\frac{\partial}{\partial x} 𝒠_{2 ω} (x, y) = \frac{κ c}{2 ω} 𝒠_{ω}^{2} a^{333} (x, y)

which can be immediately integrated to:

𝒠_{2 ω} (x, y) = \frac{κ c}{2 ω} 𝒠_{ω}^{2} \int_{- \infty}^{x} d s a^{333} (s, y) .

The remaining unknown is the nonlinearity a³³³ = 4χ⁰³⁰³⁰³, i.e. the relevant tensor component for the SHG process in the unprimed system. According to the transformation law derived in Eq. (33), the nonlinearity transforms as:

a^{3^{'} 3^{'} 3^{'}} = {| A |}^{- 1} A_{i j k}^{3^{'} 3^{'} 3^{'}} a^{i j k} = {| A |}^{- 1} a^{333}

since z′ = z implies that

A_{i j k}^{3^{'} 3^{'} 3^{'}} = δ_{i}^{3} δ_{j}^{3} δ_{k}^{3}

. With the determinant |A| given by Eqs. (39) and (36), the nonlinearity a^3′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:

a^{333} = \frac{r^{'}}{r} | A_{g} | a^{3^{'} 3^{'} 3^{'}} = {\begin{array}{l} {(\frac{R_{1}}{R_{2}})}^{2} a_{0} & 0 \leq r \leq R_{2} \\ 0 & otherwise . \end{array}

As expected, the nonlinearity in the unprimed system is reduced by a factor of (R₁/R₂)² compared to the nonlinearity in the primed system (see 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:

E_{2 ω} (x, y, t) = 𝒠_{2 ω} (x, y) e^{2 i ω (x / c - t)} .

By applying the inverse transformation (i.e. expressing x and y by x = x(x′,y′) and y = y(x′,y′)), we finally obtain the SHG wave in the physical space of the primed coordinates. The real part of the resulting SHG field distribution in the two coordinate systems is plotted in 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.

5. Conclusion

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.

Appendix

Redefinition of the linear susceptibility in Eq. (12)

With $F^{μ ν} = \frac{1}{2} (F^{μ ν} - F^{ν μ}) = \frac{1}{2} (η^{μ σ} η^{ν κ} - η^{ν σ} η^{μ κ}) F_{σ κ}$ , the linear terms in Eq. (12) can be written as

\begin{array}{l} 𝒟^{μ ν} & = F^{μ ν} + 4 π χ^{μ ν σ κ} F_{σ κ} \\ = (\frac{1}{2} (η^{μ σ} η^{ν κ} - η^{ν σ} η^{μ κ}) + 4 π χ^{μ ν σ κ}) F_{σ κ} . \end{array}

Finally, the last line of Eq. (12) follows after the re-definition

4 π χ_{new}^{μ ν σ κ} : = \frac{1}{2} (η^{μ σ} η^{ν κ} - η^{ν σ} η^{μ κ}) + 4 π χ^{μ ν σ κ}

(in Eq. (12), the label “new” has been omitted).

Proof of Eq. (15)

By using the identity

\begin{array}{l} F_{i j} & = \frac{1}{2} (F_{i j} - F_{j i}) = \frac{1}{2} (δ_{i j}^{m n} - δ_{j i}^{m n}) F_{m n} \\ = \frac{1}{2} g_{i j}^{k} g_{k}^{m n} F_{m n} = - g_{i j}^{k} B_{k} \end{array}

the components of the electric displacement field are

\begin{array}{l} D^{i} & = - 𝒟^{0 i} = - 4 π χ^{0 i σ κ} F_{σ κ} \\ = - 8 π χ^{0 i 0 j} F_{0 j} - 4 π χ^{0 i m n} F_{m n} \\ = - 8 π χ^{0 i 0 j} E_{j} + 4 π g_{m n}^{j} χ^{0 i m n} B_{j} \end{array}

and, accordingly, the components of the magnetic fields are given by

\begin{array}{l} H^{i} & = - \frac{1}{2} g_{m n}^{i} 𝒟^{m n} = - 2 π g_{m n}^{i} χ^{m n σ κ} F_{σ κ} \\ = - 4 π g_{m n}^{i} χ^{m n 0 j} F_{0 j} - 2 π g_{m n}^{i} χ^{m n k l} F_{k l} \\ = - 4 π g_{m n}^{i} χ^{m n 0 j} E_{j} + 2 π g_{m n}^{i} g_{k l}^{j} χ^{m n k l} B_{j} . \end{array}

By comparing the expressions of Dⁱ and Hⁱ with

\begin{array}{l} D^{i} = ε^{i j} E_{j} + ξ^{i j} B_{j} \\ H^{i} = ζ^{i j} E_{j} + {(μ^{- 1})}^{i j} B_{j}, \end{array}

we obtain the relations given in Eq. (15).

Intermediate transformations used in Eq. (39)

The first intermediate transformation f is given by

\begin{array}{l} r = \sqrt{x^{2} + y^{2}} \\ ϕ = arctan (y / x) \end{array}

with the corresponding Jacobian matrix and determinant:

\begin{matrix} A_{f} = (\begin{array}{l} \frac{\partial r}{\partial x} & \frac{\partial r}{\partial y} \\ \frac{\partial ϕ}{\partial x} & \frac{\partial ϕ}{\partial y} \end{array}) = (\begin{array}{l} \frac{x}{r} & \frac{y}{r} \\ - \frac{y}{r^{2}} & \frac{x}{r^{2}} \end{array}), & | A_{g} | = \frac{1}{r} . \end{matrix}

The inverse intermediate transformation is given by:

\begin{array}{l} x^{'} = r^{'} cos ϕ^{'} \\ y^{'} = r^{'} sin ϕ^{'} \end{array}

The corresponding Jacobian matrix and determinant are:

\begin{matrix} A_{h} = (\begin{array}{l} \frac{\partial x^{'}}{\partial r^{'}} & \frac{\partial x^{'}}{\partial ϕ^{'}} \\ \frac{\partial y^{'}}{\partial r^{'}} & \frac{\partial y^{'}}{\partial ϕ^{'}} \end{array}) = (\begin{array}{l} cos ϕ^{'} & - r^{'} sin ϕ^{'} \\ sin ϕ^{'} & r^{'} cos ϕ^{'} \end{array}), & | A_{h} | = r^{'} \end{matrix}

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