Toward nanophotonic optical isolation via inverse design of energy transfer in nonreciprocal media

In this paper we generalize the adjoint method of inverse design to nonreciprocal media. As a test case, we use three-dimensional topology optimization via the level-set method to optimize one-way energy transfer for pointlike source and observation points. To achieve this we introduce a suite of tools, chiefly what we term the “Faraday-adjoint” method which allows for efficient shape optimization in the presence of magneto-optical media. We carry out an optimization based on a very general equation that we derive for energy transfer in a nonreciprocal medium, and link finite-different time-domain numerics to analytics via a modified Born series generalized to a tensor permittivity. This paper represents a stepping stone towards practical nanophotonic optical isolation, often regarded as the “holy grail” of integrated photonics.

Figure 1

The general idea of optical isolation, illustrated in terms of the system we will consider. Both panels show a schematic of two atoms or molecules (modeled here as pointlike dipoles) in the presence of an arbitrarily shaped nonreciprocal environment of asymmetric tensor permittivity $\overline{\varepsilon }\ne {\overline{\varepsilon }}^{\mathrm{T}}$. When the donor and acceptor positions are reversed as shown in (b), the rate of energy transfer will not in general be the same due to the nonreciprocal nature of the medium. Isolation (simultaneous increase of the “forward” rate and decrease of the “backward” rate) via topological optimization of the intervening structure is the overall goal pursued in this paper.

Figure 2

The two distinct time orderings appearing in the matrix element (13). Time runs from bottom to top, and thick lines represent excited states of the donor D and acceptor A. The resonant contribution comes from the diagram on the left. While both diagrams have off-resonant contributions, these cancel as discussed in detail in the main text.

Figure 3

Illustration of the geometry used for test and verification.

Figure 4

Comparison of RET rate using FDTD (points) and using the Born series approach (lines). The parameters for the nonreciprocal medium are ${\varepsilon }_{\infty }=1.444, {\omega }_n=2.32\omega$ (with $\omega$ being the angular frequency corresponding to the donor transition wavelength ${\lambda }_0=1.55\mu \mathrm{m}$), $b=1.855, {\sigma }_m=0.1$, and ${\gamma }_n={10}^{-6}$ in the system of natural units defined by meep [68] with a length scale of $1\mu \mathrm{m}$ (although we emphasize that the scale invariance of Maxwell's equations makes this choice somewhat arbitrary). This results in off-diagonal elements of $\overline{\overline{\varepsilon }}$ being given by ${\varepsilon }_{xy}\approx 0.2$ at the donor frequency. For the reciprocal medium, $\varepsilon =1.3$. The donor and acceptor are placed on the $z$ axis at $z=\pm z_0=\pm 1.5\mu \mathrm{m}$, either side of a sphere of radius $S=1\mu \mathrm{m}$ (see Fig. 3). The wavelength of the donor transition is ${\lambda }_0=1.55\mu \mathrm{m}$. All three results are normalized to the case for parallel donor and acceptor dipole moments in vacuum.

Figure 5

Computational setup. The computational domain is bounded by perfectly matched layers, whose nonreflecting property ensures no spurious reflections from the boundaries. Within that we define a region between donor and acceptor into which the algorithm is allowed to deform the nonreciprocal medium. We choose the initial geometry to be the same as used in the test case discussed in Sec. 3 and shown in Fig. 3. The distance unit is $\mu \text{m}$.

Figure 6

(a) Iteration 3 and (b) iteration 4 of the advected geometry. The boundary (zero-level set) of the Faraday medium is represented by a solid black line. The red dots indicate the position of the acceptor and donor dipoles. The extended velocity field $v_{\mathrm{ext}}$ is shown as a colored background. The distance unit is $\mu \text{m}$.

Figure 7

Slices of the advected geometry at $x=0$ showing the evolution of the boundary (zero contour) as the iteration number increases.

Figure 8

The (normalized) isolation strength $\mathrm{\Gamma }$ of the gyromagnetic optical isolator increases during the iterative optimization process. ${\mathrm{\Gamma }}_0$, the initial isolation strength (iteration zero), is used as the normalization value.

Figure 9

(a) Initial geometry and (b)–(d) final geometry of the inverse-designed nonreciprocal RET isolator. The donor (D) and acceptor (A) dipoles are represented as gold spheres.