Abstract

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

1. Introduction

The field of transformation optics (TO) has drawn a lot of scientific interest in the last few years [16]. 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 [812]—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 [1319]. 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 [2533] and play a key role in modern optical technology [3436]. 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 [3840].

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:

divB=0,rotE+1cBt=0,
divD=4πρ,rotH1cDt=4πcj
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 jν = (, j), the antisymmetric field strength tensor
Fμν=(0ExEyEzEx0BzByEyBz0BxEzByBx0)
and the antisymmetric displacement tensor
𝒟μν=(0DxDyDzDx0HzHyDyHz0HxDzHyHx0),
the Maxwell equations can be covariantly expressed as:
μFνσ+σFμν+νFσμ=0,
μ𝒟μν=4πcjν
where we used the partial derivative operator μ = ( 1ct, ∇) 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 𝒟μν = ημα ηνβ 𝒟αβ where the metric tensor is given by ημν = 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 Fμν and 𝒟μν is given by
𝒟μν=Fμν+4π𝒫μν
with the polarization-magnetization tensor
𝒫μν=(0PxPyPzPx0MzMyPyMz0MxPzMyMx0).
Tensor Eq. (7) is equivalent to the common constitutive material relations:
D=E+4πP,H=B4πM.
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:
𝒫μν=χμνσκFσκ+χμνσκαβFσκFαβ+χμνσκαβγδFσκFαβFγδ+=n=1χμνα1β1αnβnFα1β1Fαnβn
where the quantities χμνα1β1···αnβn 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 = dijEj + dijkEjEk + ··· [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 μν, symmetric under exchange of two pairs αiβiαjβj and antisymmetric under exchange of two indices αiβi within one pair αiβi. For the quadratic term, this means for example:
χμνα1β1α2β2=χνμα1β1α2β2=χμνα2β2α1β1=χμνβ1α1α2β2.
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:
𝒟μν=Fμν+4π𝒫μν=Fμν+4πn=1χμνα1β1αnβnFα1β1Fαnβn=4πn=1χμνα1β1αnβnFα1β1Fαnβn
where in the last line, we performed a re-definition of the linear coefficient χμνα1β1 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):

𝒟μν=4πχμνσκFσκ.
This expression can be equivalently rewritten in the more intuitive three-formalism [38]:
(DH)=(εξζμ1)(EB).
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 χμνσκ is given by:
εij=8πχ0i0j,ξij=4πgmnjχ0imn,ζij=4πgmniχmn0j,(μ1)ij=2πgmnigkljχmnkl
with the totally antisymmetric Levi-civita tensor gmni (with g231=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 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:

𝒫(2)μν=χμνσκαβFσκFαβ.
For simplicity, we restrict here to the components of the second-order electric polarization P(2)i=𝒫(2)0i (a similar derivation applies for the magnetization). To achieve a deeper physical insight, we use the symmetry properties of χμνσκαβ to split the sum into three parts that separately account for purely electric, purely magnetic and magneto-electric cross-coupling effects. This yields:
P(2)i=χ0iσκαβFσκFαβ=4χ0i0k0mF0kF0m+4χ0i0kmnF0kFmn+χ0iklmnFklFmn=aijkEjEkPockels,effect,multi-wavemixing+bijkEjBkFaradayeffect+cijkBjBk
where we have introduced the nonlinear material coefficients aijk, bijk, cijk. The substitutions in the last line follow from the fact that the tensor components F0i address the electric field while Fij addresses the magnetic field. Obviously, the expression 𝒫(2)μν=χμνσκαβ FσκFαβ 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:
P(3)i=χ0iσκαβμνFσκFαβFμν=aijklEjEkElKerreffect+bijklEjEkBl+cijklEjBkBlCotton-Moutoneffect+dijklBjBkBl,
and similarly for higher order contributions. Obviously, the introduced material coefficients χμνα1β1···αnβn 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

xαxα(xα)
where xα = (ct,r) is the coordinate vector of the space time. By Aαα=xα/xα and |A|:=det(Aαα) we denote the Jacobian matrix and its determinant. For notational convenience, we introduce the convention that, if a product of Aαα occurs in a transformation formula, the kernel symbol A is written only once, e.g.
AααAββAγγ=Aαβγαβγ.
The electromagnetic fields Fμν and 𝒟μν in the primed and unprimed coordinate systems are related by [38, Chap. 3.2]:
Fμν=AμνμνFμν,𝒟μν=|A|1Aμνμν𝒟μν.
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:
μFνσ+σFμν+νFσμ=0,μ𝒟μν=4πcjν
with jν=|A|1Aννjν. As a consequence, the material coefficients in the constitutive relation (12), must transform as:
χμνα1β1αnβn=|A|1Aμνα1β1αnβnμνα1β1αnβnχμνα1β1αnβn.
The proof of Eq. (22) is not difficult. To see this, we multiply both sides of Eq. (12) by |A|1Aμνμν, insert in the right-hand side the identity
Aα1β1αnβnα1β1αnβnAα1β1αnβnα1β1αnβn=Aα1β1αnβnα1β1αnβn=1
(see [38], Chap. 1.4) and replace the unprimed quantities with the corresponding primed ones. This yields:
𝒟μν=4πn=1χμνα1β1αnβnFα1β1Fαnβn|A|1Aμνμν𝒟μν𝒟μν=4πn=1|A|1AμνμνAα1β1αnβnα1β1αnβnχμνα1β1αnβnχμνα1β1αnβnAa1β1αnβnα1β1αnβnFα1β1FαnβnFα1β1Fαnβn𝒟μν=4πn=1χμνα1β1αnβnFα1β1Fαnβn.
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 xα=Aααxα where the Jacobian matrix Aαα describes a Lorentz boost in the minus x-direction:

Aαα=(γγu/c00γu/cγ0000100001),|A|=1
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:
χμνσκ=Aμνσκμνσκχμνσκ.
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, εij = −8πχ0′i′0′j). In the same manner, the quadratic susceptibility transforms as
χμνσκαβ=Aμνσκαβμνσκαβχμνσκαβ
from which the nonlinear quadratic properties in the new system can be calculated. Note that the non-vanishing 0i-components of the Jacobian matrix A0i0 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 [4043]. 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′jkl ≠ 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

t=t,x=x(x,y,z),y=y(x,y,z),z=z(x,y,z)
which are of particular interest for the design of TO devices with tailored optical properties. In this case, we have
A00=tt=1andAi0=ctxi=0.
With the Einstein sum convention, this implies the useful calculation rules Aμ0Bμ=B0 and AμiBμ=AiiBi (remember: Latin indices run from 1 to 3). For example, the permittivity transforms under these conditions as
εij=8πχ0i0j=8π|A|1Aμναβ0i0jχμναβ=8π|A|1Aijijχ0i0j=|A|1Aijijεij.
Accordingly, the permeability and the magneto-electric coupling terms in Eq. (15) transform as:
μij=|A|1Aijijμij,ξij=|A|1Aijijξij,ζij=|A|1Aijijζij.
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 aijk = 4χ0i0j0k 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 χμνσκαβ is
χμνσκαβ=|A|1Aμνσκαβμνσκαβχμνσκαβ
and for the special case of a purely spatial coordinate transformation (i.e. with the conditions given in Eq. (29)), the second order susceptibility aijk transforms as
aijk=4χ0i0j0k=4|A|1Aμνσκαβ0i0j0kχμνσκαβ=4|A|1Aijkijkχ0i0j0k=|A|1Aijkijkaijk
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 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)):

r={R1R2r0rR2R3R1R3R2rR2R1R3R2R3R2rR3rotherwiseϕ={ϕ+π2cos2(πr2R3)0rR3ϕotherwise.
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:
Ag=(rrrϕϕrϕϕ)={(R1R20π24R3sin(πrR3)1)0rR2(R3R1R3R20π24R3sin(πrR3)1)R2rR3diag(1,1)otherwise.
And the determinant of Ag is:
|Ag|={R1R20rR2R3R1R3R2R2rR31otherwise.

 figure: Fig. 1

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=x2+y2 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)f(r,ϕ)g(r,ϕ)h(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
Afgh=AfAgAh.
As shown in the appendix, the Jacobian determinants of the intermediate transformations are |Af| = 1/r and |Ah| = r′, respectively. Hence, the Jacobian determinant of the composite is
|Afgh|=|Af||Ag||Ah|=rr|Ag|
with the Jacobian determinant |Ag| 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 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:

a333(r)={a00rR10otherwise
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:

Eω(x,y,t)=𝒠ωei(kωxx+kωyyωt)
where 𝒠ω denotes the amplitude of the wave and kω = (kωx,kωy) is the wave vector in the medium. For the second harmonic wave, we represent the electric field by
E2ω(x,y,t)=𝒠2ω(x,y)ei(k2ωxx+k2ωyy2ωt)
with the SHG wave vector k2ω = 2kω (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]:
(k2ωxx+k2ωyy)𝒠2ω(x,y)=κ𝒠ω2a333(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 fgh), the z-component of the electric field transforms as

Ez=F03=A03μνFμν=F03=Ez
since t′ = t and z′ = z imply A03μν=δ0μδ3ν. 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:
Eω(x,t)=𝒠ωei(kωxxωt)
with constant amplitude 𝒠ω = 𝒠ω and kωx = 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.

 figure: Fig. 2

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 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 expression

x𝒠2ω(x,y)=κc2ω𝒠ω2a333(x,y)
which can be immediately integrated to:
𝒠2ω(x,y)=κc2ω𝒠ω2xdsa333(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:
a333=|A|1Aijk333aijk=|A|1a333
since z′ = z implies that Aijk333=δi3δj3δk3. With the determinant |A| given by 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:
a333=rr|Ag|a333={(R1R2)2a00rR20otherwise.
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:

E2ω(x,y,t)=𝒠2ω(x,y)e2iω(x/ct).
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 Fμν=12(FμνFνμ)=12(ημσηνκηνσημκ)Fσκ, the linear terms in Eq. (12) can be written as

𝒟μν=Fμν+4πχμνσκFσκ=(12(ημσηνκηνσημκ)+4πχμνσκ)Fσκ.
Finally, the last line of Eq. (12) follows after the re-definition 4πχnewμνσκ:=12(ημσηνκηνσημκ)+4πχμνσκ (in Eq. (12), the label “new” has been omitted).

Proof of Eq. (15)

By using the identity

Fij=12(FijFji)=12(δijmnδjimn)Fmn=12gijkgkmnFmn=gijkBk
the components of the electric displacement field are
Di=𝒟0i=4πχ0iσκFσκ=8πχ0i0jF0j4πχ0imnFmn=8πχ0i0jEj+4πgmnjχ0imnBj
and, accordingly, the components of the magnetic fields are given by
Hi=12gmni𝒟mn=2πgmniχmnσκFσκ=4πgmniχmn0jF0j2πgmniχmnklFkl=4πgmniχmn0jEj+2πgmnigkljχmnklBj.
By comparing the expressions of Di and Hi with
Di=εijEj+ξijBjHi=ζijEj+(μ1)ijBj,
we obtain the relations given in Eq. (15).

Intermediate transformations used in Eq. (39)

The first intermediate transformation f is given by

r=x2+y2ϕ=arctan(y/x)
with the corresponding Jacobian matrix and determinant:
Af=(rxryϕxϕy)=(xryryr2xr2),|Ag|=1r.
The inverse intermediate transformation is given by:
x=rcosϕy=rsinϕ
The corresponding Jacobian matrix and determinant are:
Ah=(xrxϕyryϕ)=(cosϕrsinϕsinϕrcosϕ),|Ah|=r

References

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef]   [PubMed]  

2. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef]   [PubMed]  

3. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006). [CrossRef]  

4. S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45 (2007). [CrossRef]  

5. A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33, 43–45 (2008). [CrossRef]  

6. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008). [CrossRef]   [PubMed]  

7. N. Kundtz, D. Smith, and J. Pendry, “Electromagnetic design with transformation optics,” Proc. IEEE 99, 1622–1633 (2011). [CrossRef]  

8. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 312, 977–980 (2006). [CrossRef]  

9. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007). [CrossRef]  

10. N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105, 193902 (2010). [CrossRef]  

11. U. Leonhardt, “To invisibility and beyond,” Nature 471, 292–293 (2011). [CrossRef]   [PubMed]  

12. Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011). [CrossRef]  

13. A. Novitsky, C.-W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” New J. Phys. 11, 113001 (2009). [CrossRef]  

14. D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008). [CrossRef]  

15. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008). [CrossRef]  

16. T. Han, C. Qiu, and X. Tang, “An arbitrarily shaped cloak with nonsingular and homogeneous parameters designed using a twofold transformation,” J. Opt. 12, 095103 (2010). [CrossRef]  

17. A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. 33, 1584–1586 (2008). [CrossRef]   [PubMed]  

18. X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010). [CrossRef]  

19. W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008). [CrossRef]  

20. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007). [CrossRef]  

21. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010). [CrossRef]  

22. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009). [CrossRef]   [PubMed]  

23. D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687–692 (2009). [CrossRef]  

24. E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009). [CrossRef]  

25. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965). [CrossRef]  

26. W. L. Faust and C. H. Henry, “Mixing of visible and near-resonance infrared light in gap,” Phys. Rev. Lett. 17, 1265–1268 (1966). [CrossRef]  

27. N. Bloembergen, “The stimulated raman effect,” Am. J. Phys. 35, 989–1023 (1967). [CrossRef]  

28. J. T. Manassah, P. L. Baldeck, and R. R. Alfano, “Self-focusing, self-phase modulation, and diffraction in bulk homogeneous material,” Opt. Lett. 13, 1090–1092 (1988). [CrossRef]   [PubMed]  

29. Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989). [CrossRef]  

30. G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989). [CrossRef]  

31. J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-order harmonic generation using intense femtosecond pulses,” Phys. Rev. Lett. 70, 766–769 (1993). [CrossRef]   [PubMed]  

32. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995). [CrossRef]  

33. A. Dubietis, R. Butkus, and A. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12, 163–172 (2006). [CrossRef]  

34. Y. R. Shen, “Recent advances in nonlinear optics,” Rev. Mod. Phys. 48, 1–32 (1976). [CrossRef]  

35. M. Evans and S. Kielich, Modern Nonlinear Optics (Wiley, 1997).

36. R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

37. L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the kerr effect,” Phys. Rev. B 84, 205103 (2011). [CrossRef]  

38. E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics (Dover Publications, 1997).

39. R. T. Thompson, “Transformation optics in nonvacuum initial dielectric media,” Phys. Rev. A 82, 053801 (2010). [CrossRef]  

40. R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011). [CrossRef]  

41. D. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968). [CrossRef]  

42. L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984).

43. G. Rousseaux, “On the electrodynamics of minkowski at low velocities,” Europhys. Lett. 84, 20002 (2008). [CrossRef]  

References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [Crossref] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [Crossref] [PubMed]
  3. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
    [Crossref]
  4. S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45 (2007).
    [Crossref]
  5. A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33, 43–45 (2008).
    [Crossref]
  6. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008).
    [Crossref] [PubMed]
  7. N. Kundtz, D. Smith, and J. Pendry, “Electromagnetic design with transformation optics,” Proc. IEEE 99, 1622–1633 (2011).
    [Crossref]
  8. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 312, 977–980 (2006).
    [Crossref]
  9. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
    [Crossref]
  10. N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105, 193902 (2010).
    [Crossref]
  11. U. Leonhardt, “To invisibility and beyond,” Nature 471, 292–293 (2011).
    [Crossref] [PubMed]
  12. Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
    [Crossref]
  13. A. Novitsky, C.-W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” New J. Phys. 11, 113001 (2009).
    [Crossref]
  14. D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
    [Crossref]
  15. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
    [Crossref]
  16. T. Han, C. Qiu, and X. Tang, “An arbitrarily shaped cloak with nonsingular and homogeneous parameters designed using a twofold transformation,” J. Opt. 12, 095103 (2010).
    [Crossref]
  17. A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. 33, 1584–1586 (2008).
    [Crossref] [PubMed]
  18. X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
    [Crossref]
  19. W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
    [Crossref]
  20. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
    [Crossref]
  21. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
    [Crossref]
  22. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
    [Crossref] [PubMed]
  23. D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687–692 (2009).
    [Crossref]
  24. E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
    [Crossref]
  25. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
    [Crossref]
  26. W. L. Faust and C. H. Henry, “Mixing of visible and near-resonance infrared light in gap,” Phys. Rev. Lett. 17, 1265–1268 (1966).
    [Crossref]
  27. N. Bloembergen, “The stimulated raman effect,” Am. J. Phys. 35, 989–1023 (1967).
    [Crossref]
  28. J. T. Manassah, P. L. Baldeck, and R. R. Alfano, “Self-focusing, self-phase modulation, and diffraction in bulk homogeneous material,” Opt. Lett. 13, 1090–1092 (1988).
    [Crossref] [PubMed]
  29. Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
    [Crossref]
  30. G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
    [Crossref]
  31. J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-order harmonic generation using intense femtosecond pulses,” Phys. Rev. Lett. 70, 766–769 (1993).
    [Crossref] [PubMed]
  32. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
    [Crossref]
  33. A. Dubietis, R. Butkus, and A. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12, 163–172 (2006).
    [Crossref]
  34. Y. R. Shen, “Recent advances in nonlinear optics,” Rev. Mod. Phys. 48, 1–32 (1976).
    [Crossref]
  35. M. Evans and S. Kielich, Modern Nonlinear Optics (Wiley, 1997).
  36. R. W. Boyd, Nonlinear Optics (Academic Press, 2008).
  37. L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the kerr effect,” Phys. Rev. B 84, 205103 (2011).
    [Crossref]
  38. E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics (Dover Publications, 1997).
  39. R. T. Thompson, “Transformation optics in nonvacuum initial dielectric media,” Phys. Rev. A 82, 053801 (2010).
    [Crossref]
  40. R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
    [Crossref]
  41. D. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
    [Crossref]
  42. L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984).
  43. G. Rousseaux, “On the electrodynamics of minkowski at low velocities,” Europhys. Lett. 84, 20002 (2008).
    [Crossref]

2011 (5)

N. Kundtz, D. Smith, and J. Pendry, “Electromagnetic design with transformation optics,” Proc. IEEE 99, 1622–1633 (2011).
[Crossref]

U. Leonhardt, “To invisibility and beyond,” Nature 471, 292–293 (2011).
[Crossref] [PubMed]

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[Crossref]

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the kerr effect,” Phys. Rev. B 84, 205103 (2011).
[Crossref]

R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
[Crossref]

2010 (5)

R. T. Thompson, “Transformation optics in nonvacuum initial dielectric media,” Phys. Rev. A 82, 053801 (2010).
[Crossref]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[Crossref]

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105, 193902 (2010).
[Crossref]

T. Han, C. Qiu, and X. Tang, “An arbitrarily shaped cloak with nonsingular and homogeneous parameters designed using a twofold transformation,” J. Opt. 12, 095103 (2010).
[Crossref]

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[Crossref]

2009 (4)

A. Novitsky, C.-W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” New J. Phys. 11, 113001 (2009).
[Crossref]

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[Crossref] [PubMed]

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687–692 (2009).
[Crossref]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[Crossref]

2008 (7)

D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
[Crossref]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[Crossref]

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
[Crossref]

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. 33, 1584–1586 (2008).
[Crossref] [PubMed]

A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33, 43–45 (2008).
[Crossref]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008).
[Crossref] [PubMed]

G. Rousseaux, “On the electrodynamics of minkowski at low velocities,” Europhys. Lett. 84, 20002 (2008).
[Crossref]

2007 (3)

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45 (2007).
[Crossref]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
[Crossref]

2006 (5)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 312, 977–980 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
[Crossref]

A. Dubietis, R. Butkus, and A. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12, 163–172 (2006).
[Crossref]

1995 (1)

1993 (1)

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-order harmonic generation using intense femtosecond pulses,” Phys. Rev. Lett. 70, 766–769 (1993).
[Crossref] [PubMed]

1989 (2)

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[Crossref]

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[Crossref]

1988 (1)

1976 (1)

Y. R. Shen, “Recent advances in nonlinear optics,” Rev. Mod. Phys. 48, 1–32 (1976).
[Crossref]

1968 (1)

D. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[Crossref]

1967 (1)

N. Bloembergen, “The stimulated raman effect,” Am. J. Phys. 35, 989–1023 (1967).
[Crossref]

1966 (1)

W. L. Faust and C. H. Henry, “Mixing of visible and near-resonance infrared light in gap,” Phys. Rev. Lett. 17, 1265–1268 (1966).
[Crossref]

1965 (1)

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[Crossref]

Agrawal, G.

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[Crossref]

Alfano, R. R.

Alitalo, P.

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the kerr effect,” Phys. Rev. B 84, 205103 (2011).
[Crossref]

Baldeck, P. L.

Bergamin, L.

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the kerr effect,” Phys. Rev. B 84, 205103 (2011).
[Crossref]

Bloembergen, N.

N. Bloembergen, “The stimulated raman effect,” Am. J. Phys. 35, 989–1023 (1967).
[Crossref]

Bosenberg, W. R.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

Briane, M.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
[Crossref]

Butkus, R.

A. Dubietis, R. Butkus, and A. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12, 163–172 (2006).
[Crossref]

Byer, R. L.

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

Chan, C. T.

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
[Crossref]

Chen, H.

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
[Crossref]

Cheng, D.

D. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[Crossref]

Cheng, Q.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
[Crossref]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

Chin, J. Y.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
[Crossref]

Cui, T. J.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
[Crossref]

Cummer, S. A.

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Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
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Photon. Nanostruct. Fundam. Appl. (2)

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M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008).
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[Crossref] [PubMed]

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N. Kundtz, D. Smith, and J. Pendry, “Electromagnetic design with transformation optics,” Proc. IEEE 99, 1622–1633 (2011).
[Crossref]

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[Crossref]

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Figures (2)

Fig. 1
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.
Fig. 2
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.

Equations (60)

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div B = 0 , rot E + 1 c B t = 0 ,
div D = 4 π ρ , rot H 1 c D t = 4 π c j
F μ ν = ( 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 )
𝒟 μ ν = ( 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 ) ,
μ F ν σ + σ F μ ν + ν F σ μ = 0 ,
μ 𝒟 μ ν = 4 π c j ν
𝒟 μ ν = F μ ν + 4 π 𝒫 μ ν
𝒫 μ ν = ( 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 ) .
D = E + 4 π P , H = B 4 π M .
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 + = n = 1 χ μ ν α 1 β 1 α n β n F α 1 β 1 F α n β n
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 .
𝒟 μ ν = F μ ν + 4 π 𝒫 μ ν = F μ ν + 4 π n = 1 χ μ ν α 1 β 1 α n β n F α 1 β 1 F α n β n = 4 π n = 1 χ μ ν α 1 β 1 α n β n F α 1 β 1 F α n β n
𝒟 μ ν = 4 π χ μ ν σ κ F σ κ .
( D H ) = ( ε ξ ζ μ 1 ) ( E B ) .
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
𝒫 ( 2 ) μ ν = χ μ ν σ κ α β F σ κ F α β .
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 = a i j k E j E k Pockels , effect , multi-wave mixing + b i j k E j B k Faraday effect + c i j k B j B k
P ( 3 ) i = χ 0 i σ κ α β μ ν F σ κ F α β F μ ν = a i j k l E j E k E l Kerr effect + b i j k l E j E k B l + c i j k l E j B k B l Cotton-Mouton effect + d i j k l B j B k B l ,
x α x α ( x α )
A α α A β β A γ γ = A α β γ α β γ .
F μ ν = A μ ν μ ν F μ ν , 𝒟 μ ν = | A | 1 A μ ν μ ν 𝒟 μ ν .
μ F ν σ + σ F μ ν + ν F σ μ = 0 , μ 𝒟 μ ν = 4 π c j ν
χ μ ν α 1 β 1 α n β n = | A | 1 A μ ν α 1 β 1 α n β n μ ν α 1 β 1 α n β n χ μ ν α 1 β 1 α n β n .
A α 1 β 1 α n β n α 1 β 1 α n β n A α 1 β 1 α n β n α 1 β 1 α n β n = A α 1 β 1 α n β n α 1 β 1 α n β n = 1
𝒟 μ ν = 4 π n = 1 χ μ ν α 1 β 1 α n β n F α 1 β 1 F α n β n | A | 1 A μ ν μ ν 𝒟 μ ν 𝒟 μ ν = 4 π n = 1 | A | 1 A μ ν μ ν A α 1 β 1 α n β n α 1 β 1 α n β n χ μ ν α 1 β 1 α n β n χ μ ν α 1 β 1 α n β n A a 1 β 1 α n β n α 1 β 1 α n β n F α 1 β 1 F α n β n F α 1 β 1 F α n β n 𝒟 μ ν = 4 π n = 1 χ μ ν α 1 β 1 α n β n F α 1 β 1 F α n β n .
A α α = ( γ γ u / c 0 0 γ u / c γ 0 0 0 0 1 0 0 0 0 1 ) , | A | = 1
χ μ ν σ κ = A μ ν σ κ μ ν σ κ χ μ ν σ κ .
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
t = t , x = x ( x , y , z ) , y = y ( x , y , z ) , z = z ( x , y , z )
A 0 0 = t t = 1 and A i 0 = c t x i = 0.
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 .
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 .
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
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
r = { R 1 R 2 r 0 r R 2 R 3 R 1 R 3 R 2 r R 2 R 1 R 3 R 2 R 3 R 2 r R 3 r otherwise ϕ = { ϕ + π 2 cos 2 ( π r 2 R 3 ) 0 r R 3 ϕ otherwise .
A g = ( r r r ϕ ϕ r ϕ ϕ ) = { ( R 1 R 2 0 π 2 4 R 3 sin ( π r R 3 ) 1 ) 0 r R 2 ( R 3 R 1 R 3 R 2 0 π 2 4 R 3 sin ( π r R 3 ) 1 ) R 2 r R 3 diag ( 1 , 1 ) otherwise .
| A g | = { R 1 R 2 0 r R 2 R 3 R 1 R 3 R 2 R 2 r R 3 1 otherwise .
( x , y ) f ( r , ϕ ) g ( r , ϕ ) h ( x , y ) .
A f g h = A f A g A h .
| A f g h | = | A f | | A g | | A h | = r r | A g |
a 3 3 3 ( r ) = { a 0 0 r R 1 0 otherwise
E ω ( x , y , t ) = 𝒠 ω e i ( k ω x x + k ω y y ω t )
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 )
( k 2 ω x x + k 2 ω y y ) 𝒠 2 ω ( x , y ) = κ 𝒠 ω 2 a 3 3 3 ( x , y )
E z = F 0 3 = A 0 3 μ ν F μ ν = F 03 = E z
E ω ( x , t ) = 𝒠 ω e i ( k ω x x ω t )
x 𝒠 2 ω ( x , y ) = κ c 2 ω 𝒠 ω 2 a 333 ( x , y )
𝒠 2 ω ( x , y ) = κ c 2 ω 𝒠 ω 2 x d s a 333 ( s , y ) .
a 3 3 3 = | A | 1 A i j k 3 3 3 a i j k = | A | 1 a 333
a 333 = r r | A g | a 3 3 3 = { ( R 1 R 2 ) 2 a 0 0 r R 2 0 otherwise .
E 2 ω ( x , y , t ) = 𝒠 2 ω ( x , y ) e 2 i ω ( x / c t ) .
𝒟 μ ν = F μ ν + 4 π χ μ ν σ κ F σ κ = ( 1 2 ( η μ σ η ν κ η ν σ η μ κ ) + 4 π χ μ ν σ κ ) F σ κ .
F i j = 1 2 ( F i j F j i ) = 1 2 ( δ i j m n δ j i m n ) F m n = 1 2 g i j k g k m n F m n = g i j k B k
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
H i = 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 .
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 ,
r = x 2 + y 2 ϕ = arctan ( y / x )
A f = ( r x r y ϕ x ϕ y ) = ( x r y r y r 2 x r 2 ) , | A g | = 1 r .
x = r cos ϕ y = r sin ϕ
A h = ( x r x ϕ y r y ϕ ) = ( cos ϕ r sin ϕ sin ϕ r cos ϕ ) , | A h | = r

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