SUSY-inspired one-dimensional transformation optics

Transformation optics aims to identify artificial materials and structures with desired electromagnetic properties by means of pertinent coordinate transformations. In general, such schemes are meant to appropriately tailor the constitutive parameters of metamaterials in order to control the trajectory of light in two and three dimensions. Here we introduce a new class of one-dimensional optical transformations that exploits the mathematical framework of supersymmetry (SUSY). This systematic approach can be utilized to synthesize photonic configurations with identical reflection and transmission characteristics, down to the phase, for all incident angles, thus rendering them perfectly indistinguishable to an external observer. Along these lines, low-contrast dielectric arrangements can be designed to fully mimic the behavior of a given high-contrast structure that would have been otherwise beyond the reach of available materials and existing fabrication techniques. Similar strategies can also be adopted to replace negative-permittivity domains, thus averting unwanted optical losses.

indistinguishable. Remarkably, the proposed formalism can be employed to synthesize photonic configurations that behave exactly the same way as high refractive index contrast devices -by only utilizing low-contrast dielectric media. Similar methodologies can be employed to substitute negative-permittivity inclusions with purely dielectric media as a means to obtain the intended functionality without introducing any additional loss. can be constructed. Whereas the broken SUSY system ( ) preserves all bound modes, unbroken SUSY ( ( ) ) removes the fundamental mode. Regardless, in both cases the intensity reflection and transmission coefficients of the superpartners are identical to those of the fundamental system. In order to maintain the full complex scattering characteristics, a family of iso-phase structures can be synthesized. Finally, a hierarchical sequence of higher-order superpartners ,2… ( ) may be utilized to obtain a scattering-equivalent structure, which requires a substantially lower refractive index contrast than that involved in the original system .

Supersymmetric optical transformations.
In one-dimensional inhomogeneous settings, the propagation of TE polarized waves is known to obey the Helmholtz equation [23] � + + 0 2 ( )� ( , ) = 0 where 0 is the vacuum wavenumber and ( ) is the relative permittivity of a given (fundamental) structure to be emulated via SUSY transformations (see Fig.1). The analysis of TM waves can be carried out in a similar manner (see the supplementary information). In the TE case, the spatial dependence of the electric field can be The resulting Hamiltonian = + ( ) can be factorized as = + where the operators and are defined as = + ( ), = − ( ), and is an auxiliary constant of the problem. Here = − † , where " †" represents the Hermitian conjugate. The superpotential can then be obtained as a solution of the Riccati equation [7]: in terms of the fundamental permittivity profile ( ). Once has been determined, one can establish a partner Hamiltonian = + , which corresponds to a new distribution in the electric permittivity As a direct consequence of this construction, the modes of the partner potential are related [24] to the ones of the fundamental through the following expressions ∝ ( + ) and ∝ ( − ) . These latter relations hold for guided waves as well as for radiation modes, and each such pair of states is characterized by a common eigenvalue. We note that two options for choosing exist: (a) Assuming that the structure supports at least one bound state, one may opt to set α equal to the fundamental mode's eigenvalue, i.e., = 0 .
(b) The other possibility is to choose > 0 , irrespective of whether the system supports bound states or not. The first case corresponds to an unbroken SUSY: The two potentials share the guided wave eigenvalue spectra, except for that of the fundamental mode, which does not have a corresponding state in the partner. In the second case, however, SUSY is broken, and the two arrangements share an identical eigenvalue spectrum, including that of the fundamental mode. As an example, Fig. 2(a) depicts the relative permittivity distribution corresponding to a step-index-like waveguide; its unbroken and broken SUSY partners are shown in Figs.
2(b,c) respectively. can also be found analytically [7] via = − ln( 0 ) in the unbroken SUSY case, i.e. for = 0 , when supports at least one bound state 0 . In either regime, Eq. (2) can always be solved numerically to obtain the superpotential . An alternative approach is to start with an arbitrary superpotential and construct the two superpartner structures and according to Eqs. (2,3). In this scenario, it still remains to be determined whether SUSY is unbroken or broken. This question can be resolved by the so-called Witten index [6]. In general if, ( ) approaches ± at → ±∞, unbroken SUSY requires + = − − , while a broken SUSY demands that + = − . , dashed) coefficients of the structures in (b-d) compared to the fundamental system (a) as a function of the incident angle . The scattering characteristics were evaluated by means of the differential transfer matrix method [23].
It is important to note that more than one superpotential can exist for any given distribution ( ). As shown in the supplementary information, one can actually find a parametric family of viable superpotentials that satisfy Eq. (3). Whereas all members of this family lead to the same superpartner , each of them describes a different permittivity distribution according to Eq. (2). The resulting parametric family [10,25] of structures ( ; ) is associated with the fundamental distribution and its ground state 0 as follows: where represents a free parameter. Figure 2(d) depicts such family members for the fundamental structure ϵ shown in Fig. 2(a) when = 0.1, 0.5 or 2.0, respectively. Note that the original permittivity distribution ϵ is in itself a member of this family, since → for → +∞. All the modes of any other member are related to its states according to Scattering characteristics. Let us now turn our attention to the scattering characteristics of structures connected by SUSY transformations.
Consider a plane wave exp ( 0 cos + 0 sin ) incident from the left, i.e. → −∞, as shown schematically in Fig. 2(e). For reasons of simplicity we assume a uniform background medium, + = − = 0 at → ±∞; the general case of + ≠ − is discussed in the supplementary information. Here, the reflected and transmitted waves in the far field are given as exp(− 0 + 0 ) and exp ( 0 cos + 0 sin ) in terms of the complex reflection and transmission coefficients and [24]. By adopting similar solutions for the partner scatterer , its respective reflection and transmission coefficients and can readily be found (see Table 1 and supplementary information). Interestingly, the SUSY transformation yields a partner structure with exactly the same absolute values in reflection and transmission, as illustrated in Fig. 2(j). Evidently, all the permittivity distributions from Figs. 2(a-d) display identical reflectivities = | | 2 = � � 2 and transmittivities = 1 − for all angles of incidence. In contrast, the scattering phases depend on whether supersymmetry is broken or not (see Table 1). In the case of unbroken SUSY, both reflection and transmission coefficients acquire additional phases with respect to the fundamental scattering potential. If on the other hand SUSY is broken, the transmission coefficient is the same in both amplitude and phase. Finally, one can show that each member of the parametric family directly inherits all scattering properties of the original structure, i.e. they are phase-equivalent to . Figures 2(k-m) illustrate these relations.
Wavelength dependence. So far the performance of these systems has been examined at a given operating wavelength 0 . Of importance would be to investigate to what extend their supersymmetric properties persist when the wavelength varies around 0 . As one would expect, even if two dissimilar profiles exhibit the same phases at a given wavelength, their internal light dynamics may gradually undergo different changes with . To elucidate this structural dispersion, we provide the spectral dependence of the difference in transmittivities (or reflectivities ) between the fundamental structure ( Fig. 2(a)) and its superpartners ( Fig. 2(b-d)) as a function of the incidence angle , as shown in Figs. 3(a-c). As these figures indicate, this difference only becomes notable in the unbroken SUSY regime ( Fig. 3(a)), while it is almost absent under broken SUSY and iso-phase conditions (Figs. 3(b,c)). The difference in the corresponding reflection phases is similarly presented in Figs. 3(d-f). The dashed lines trace the abrupt phase jumps of , which mark the resonances in the two partners and intersect at the design wavelength 0 . Evidently, the iso-phase design displays the greatest resilience with respect to spectral deviations. Note that resonances play no role in the transmission phases, as can be seen in Figs. 3(g-j). In this latter case, the iso-phase system again proves to be the least susceptible to spectral deviations. These results demonstrate that SUSY transformations can be robust over a broad spectral range around the design wavelength. Index-contrast reduction. One of the main challenges in designing optical systems is the limited dynamic range of refractive indices associated with available materials. This issue becomes particularly acute when high contrast arrangements are desirable. For example, the number of grating unit cells required to achieve a certain diffraction efficiency grows with the inverse logarithm of the index contrast 2 / 1 between the individual layers [23]. As it turns out, SUSY optical transformations can be utilized to reduce the index contrast needed for a given structure. This can be done through a hierarchical ladder of superpartners, i.e. sequentially removing the bound states of the original high-contrast arrangement (Fig. 4a). As a general trend, each successive step demands less contrast in the corresponding index landscape than the previous one (Fig. 4b). The ultimate result is a low-contrast equivalent structure that fully inherits the reflectivity and transmittivity of the original configuration (Figs. 4c,d). Replacing negative-permittivity features. Finally, SUSY transformations can provide a possible avenue in replacing negative-permittivity inclusions (typically accompanied by losses) by purely dielectric materials. In this respect, inverse SUSY transformations, which now add modes with certain propagation constants to a given structure, can instead be used to locally elevate the permittivity (see supplementary information). Along similar lines, it is possible to find superpotentials that relate a structure with metallic or negative permittivity regions to an equivalent arrangement with entirely positive ϵ, as depicted in Fig. 5. Here we make use of the fact that in a broken-SUSY transformation, the spatial average of ϵ happens to be a conserved quantity. Therefore, changes in the broader vicinity of the original metal-dielectric structure can be used to achieve this goal.

Discussion.
In conclusion, we have introduced a new type of supersymmetric optical transformations for arbitrary one-dimensional refractive index landscapes. Compared to conventional transformation optics, our approach poses significantly less stringent requirements on the constituent parameters, and does not involve any modifications to the magnetic response of the materials involved. This method can be utilized to construct photonic arrangements that faithfully mimic the behavior of high-index-contrast or even metal-dielectric structures.
SUSY transformation optics may have potential applications in a wide range of scenarios that rely on engineered scattering and transmission properties, such as for example optical metasurfaces, anti-reflection coatings and diffraction gratings. Of interest will be to explore how the aforementioned strategies could be paired up with recently developed transformation schemes for guided-wave photonics based on dielectric materials [26,27].

S1. Iso-phase potential families
For any given distribution of the relative electrical permittivity ( ), an infinite number of viable superpotentials exist.
To show this, let us start from Eq. (3), which relates the superpartner to the superpotential . Starting from a particular , this solution can be generalized by adopting the form = + 1/ , in which case the unknown function satisfies ( − 2 ) = 1. Direct integration readily leads to = +2 ∫ � + ∫ −2 ∫ �, where is an arbitrary constant, giving rise to a parametric family of superpotentials Whereas all members of this family correspond to the same superpartner , each of them describes a different distribution according to Eq. (2). The resulting family of structures is related to the fundamental one according to If the superpotential has been specifically obtained from the bound state 0 , one obtains Eq. (4). Note that since → for → +∞, the fundamental structure is itself a member of this family. The modes of any other can be found by transforming the states of according to Based on this relation, which holds for all bound modes as well as scattered states, one can show that the family of iso-phase potentials exhibit identical reflection-and transmission coefficients, down to the phase, as the fundamental structure .

S2. Reflection/transmission coefficients of superpartner structures
In order to relate the scattering characteristics of a superpartner to that of its associated fundamental structure, let us first consider an incident plane wave described by exp( 0 cos + 0 sin ) impinging on both structures from the left side. The reflected and transmitted waves in the fundamental system are then described by exp(− 0 cos + 0 sin ) and exp( 0 cos + 0 sin ), respectively. Accordingly, the corresponding waves in the superpartner geometry are given by exp(− 0 cos + 0 sin ) and exp( 0 cos + 0 sin ). By applying the relation ∝ (∂ + ) (that holds between the wave functions of any state pairs in the two structures) to these radiation states, one finds � + 0 cos + − 0 cos � ∝ �(+ 0 cos + − ) + 0 cos + (− 0 cos + − ) − 0 cos �, for → −∞, and + 0 cos ∝ (+ 0 cos + + ) + 0 cos , for → +∞. In these two equations ± denotes the limit of at → ±∞ respectively. Taking into account that both equations should have the same proportionality constant, one can show that the reflection and transmission coefficients are related via: Obviously, the reflectivity = | | 2 = | | 2 as well as the transmittivity = 1 − = | | 2 = | | 2 of the superpartner structures is identical. The specific relations for the complex coefficients given in Table 1 in turn follow from + = − − in the case of unbroken supersymmetry and + = − in the broken supersymmetry regime.
To derive similar expressions for the family of iso-phase structures ( ; ), we label reflected and transmitted waves as exp(− 0 cos + 0 sin ) and exp( 0 cos + 0 sin ) respectively. One can then relate and to and via the transformation ∝ �∂ − �(∂ + ) . On the other hand, one can write the family of associated superpotentials as by applying Eq. (5) of the main text and using the fact that can be written as = − ∂ ln( 0 ) in terms of the ground state of the fundamental structure. Consequently, and have the same asymptotic behavior, i.e., ,± = ± . In the far field → ±∞, the transformation simplifies to ∝ (∂ − )(∂ + ) . Note that (∂ − )(∂ + ) = (∂ + ′ − 2 ) = − , and therefore ∝ ( − ) = ( − ) . Since the scattered waves depend on according to exp( 0 sin ), the corresponding eigenvalue in the Helmholtz equation is given by = 0 2 sin 2 . Hence, ∝ ( 0 2 sin 2 − ) , and therefore: for → −∞, and for → +∞. Given that both equations should be normalized with respect to the same constant, it directly follows that in the iso-phase scenario = , (S10.a) = . (S10.b)

S3. Configurations with dissimilar backgrounds
Even though the formalism of the main text was developed by assuming that the background medium is the same ( − = + = 0 ), superpartners obeying Eqs. (2,3) can be generated even if − ≠ + (here ± = � (±∞) and ). Figure  S1(a) shows an example of a step-like distribution in the relative permittivity; the corresponding superpartner is depicted in Fig. S1(b). The superpotential mediating between them no longer has the same absolute value at → ±∞, but rather is shifted by a certain offset related to the height of the potential step. Neither of the two partner structures does support any guided modes, therefore supersymmetry is necessarily broken. As depicted in Fig. S1(c), numerical results obtained by differential transfer matrix method [S2] show that the angle-dependent power reflectivities = | | 2 = | | 2 and transmittivities = 1 − = | | 2 ⋅ ( + / − )(cos + / cos − ) = | | 2 ⋅ ( + / − )(cos + / cos − ) remain identical for both systems (Fig. S1(c)), while the phases do not (Fig. S1(d)). Analytical relations of the scattering coefficients can be found by following analogous steps to those in the previous section. The results are where − and + represent the incident and transmitted wave angles related through Snell's law − sin − = + sin + . It follows that | | 2 = | | 2 and | | 2 = | | 2 .

S4. Inverse SUSY transformation
In the unbroken symmetry regime, the conventional SUSY transformation may remove a mode from a given fundamental structure . In doing so, the total area of the relative permittivity is reduced (see Fig. 2(a,b)). This can be shown easily by noting that in the unbroken supersymmetry regime the two superpartners are related via After integrating both sides of this equation we get Since in the unbroken supersymmetry regime − ≠ + , the SUSY transformation cannot preserve the total area of the relative permittivity distribution.
On the other hand, one can utilize an inverse SUSY transformation and add a bound state to a given structure , and in doing so elevate the total area of a given permittivity distribution. We factorize the fundamental Hamiltonian as = + and define the partner Hamiltonian as elev = + . Consequently, the two superpartner permittivity distributions can be written as: Equation (S14.a) can be solved numerically to obtain the superpotential elev , and from that the partner structure elev can be constructed through Eq. (S14.b). As expected in this case the partner structure elev exhibits all the guided mode eigenvalue spectrum of the fundamental structure , as well as an additional guided mode, which takes the place of its previous ground state. As it turns out, the eigenvalue of this additional state is given by the factorization parameter . Note that any value > 0 can be chosen, where 0 represents the ground state eigenvalue of the fundamental structure. Figure S2 illustrates the inverse SUSY transformation for the specific example of the fundamental structure discussed in Fig. 2 of the main text.

S5. SUSY formalism for the TM polarization
Under TM polarization conditions the magnetic field component satisfies the equation: In this case again by considering stationary solutions of the form = ( ) and after using normalized coordinates = 0 , = 0 and assuming = 2 / 0 2 we reach at: ( − (ln ) + ) = . (S16) As is, this is not a Schrödinger-like equation, and hence the factorization technique cannot be directly applied. On the other hand, by using the transformation ( ) = √ ( ), this equation can be converted to the desired form: where eff is an effective potential that can be expressed in terms of the relative permittivity as eff = − (S18) Following the SUSY formalism, the two superpartner effective potentials can now be written in terms of the superpotential via eff ( ) = + ′ − 2 + , (S19.a) eff, ( ) = − ′ − 2 + . (S19.b) One can then reconstruct the relative permittivity of the partner structure from its corresponding effective potential eff, by numerically solving the nonlinear equation (S20) Of importance will be to show that these transformations preserve the reflection/transmission properties of SUSY partners in the TM case. For this reason note that for effective potentials described in Eqs. (S19), we have the intervening relations ∝ ( + ) and ∝ ( − ) therefore: Obviously presence of the relative permittivities in the denominator makes these last equations different than those of the TE case. However for calculating the reflection/transmission coefficients (see Supplementary section S2) only the asymptotic behavior of these equations at → ±∞ are of our interest. On the other hand, according to Eqs. (S18-S20), the SUSY partners have the same asymptotic behaviors. Therefore one can use the exact same analysis of the Supplementary section S2 based on Eqs. (S21) to show that the reflection/transmission coefficients of SUSY partners constructed for the TM polarization are related via equations given in Table. 1 of the main text.