Designing a light extractor is crucial for enhancing photon collection efficiency in solid-state defect qubit systems, which are valuable for quantum information and metrology applications. In high refractive index materials like diamond, total internal reflection limits the ability to collect photons emitted from qubits. By optimizing light extraction through nanophotonic structures, such as waveguides or planar dielectric structures, one can significantly increase the emission directed toward detectors, improving photoluminescence detection.

In this notebook, we use inverse design to optimize a GaP extractor structure following a similar approach discussed in Srivatsa Chakravarthi, Pengning Chao, Christian Pederson, Sean Molesky, Andrew Ivanov, Karine Hestroffer, Fariba Hatami, Alejandro W. Rodriguez, and Kai-Mei C. Fu, "Inverse-designed photon extractors for optically addressable defect qubits," Optica 7, 1805-1811 (2020) DOI: 10.1364/OPTICA.408611. The optimization aims to maximize the minimum upward flux among three simulations with the dipole source oriented in the $x$, $y$, and $z$ directions. The final design achieves an order-of-magnitude enhancement in upward flux for an in-plane dipole source and over two orders of magnitude for an out-of-plane dipole source compared to the configuration without an extractor.

If you are unfamiliar with inverse design, we also recommend our intro to inverse design tutorials and our primer on automatic differentiation with tidy3d. For another example of light extractor optimization in Tidy3D, see this example.

Static Simulation Components¶

In this case, we are focusing on the photonic coupling to the negatively charged sharp zero-phonon line (ZPL) of nitrogen-vacancy centers at 637 nm.

For the extractor structure on top of diamond, a high-index material is preferred. GaP with a refractive index of 3.31 is used in this design.

The GaP layer thickness is 250 nm. The design region is very compact, only 1.5 μm by 1.5 μm. The design region is discretized into 10 nm grids. To ensure the FDTD grid is commensurate with the pixel, we will use the automatic nonuniform grid with min_steps_per_wvl = lda0 / (pixel_size * n_GaP).

The minimal feature size we are aiming for is 50 nm, compatible with electron beam lithography fabrication.

Now we define static components in the optimization. This includes the diamond substrate structure, the dipole sources, and the monitor that records the upward flux. The dipole source is located 100 nm below the top diamond surface.

In this case, we will place a FieldMonitor to record the upward radiation flux instead of directly using a FluxMonitor because differentiation with respect to flux monitor data is not supported at the moment.

Design Region¶

The design region is a pixelated array of permittivity values between $\varepsilon=1$ and $\varepsilon=\varepsilon_{GaP}$. To define it, we use a density array whose element values are between 0 to 1, apply a conic smoothing filter, a tanh projection function, and finally scale it linearly to $\varepsilon=1$ and $\varepsilon=\varepsilon_{GaP}$.

Next we create another function that takes the design parameters and create three simulations with the dipole polarized in the $x$, $y$, and $z$ directions.

To ensure the simulation setup is correct, we can create and visualize an initial simulation. The initial condition we are going to use is a uniform design region with permittivity $\varepsilon=\frac{1+\varepsilon_{GaP}}{2}$

Baseline Simulations¶

As a baseline, we will first simulate the scenario with no extractor. This result will be compared to that with the optimized extractor later on.

To facilitate post-processing, we will develop a function to extract the upward radiation fluxes across all simulations. When applied to the baseline simulation data, this function reveals that the flux for the dipole in the vertical direction is an order of magnitude smaller than in the horizontal directions. This occurs because most emission from a vertically oriented dipole undergoes total internal reflection at the diamond-air interface, significantly limiting the extraction efficiency.

Objective Function¶

Finally, before we start the optimization, we define the objective function as the minimum upward flux among the three simulations. Note that we don't apply a simply minimum since this discontinuous function will break the gradient tracking. Instead, we use a soft minimum function given by $$ -\tau [ln(e^{-f_x/\tau}) + ln(e^{-f_y/\tau}) + ln(e^{-f_z/\tau})], $$

where $f_i$ denotes the upward radiation flux in the $i=[x, y, z]$ direction, and $\tau$ is the parameter that controls the "softness" of the soft minimum function. The smaller $\tau$ we use, the function is closer to the true minimum.

Optimization Loop¶

Now we are ready to start the optimization loop. We will use the adam optimizer. In each iteration, the beta value for the tanh projection is increased on an exponential schedule to help gradually binarize the design, slowly initially an then more strongly as the optimization progresses. In each iteration, we output the objective function value as well as plot the design region density.

Optimization Result Analysis¶

After the optimization, we can plot the objective function as the function of iterations. A steady increase of the objective function observed.

The optimum design is not necessarily the last iteration as the objective function can fluctuate a bit. We will select the optimum design that gives the highest objective function value, re-simulate it, and calculate the flux values for different dipole orientations.

It's also helpful to calculate the relative flux enhancement compared to the baseline simulation without the extractor. The result indicates that the final design achieves an order-of-magnitude enhancement in upward flux for an in-plane dipole source and over two orders of magnitude for an out-of-plane dipole source, similar to what the authors of the referenced paper achieved.

It is important to note that the optimized design is likely not fully binarized, even with the application of a tanh projection using a large beta value. For fabrication, however, a fully binarized design is necessary. Here, we manually enforce binarization and test if that negatively affects the design performance.

In this case, the results are only marginally affected.

We would need to export a GDS file for the optimized design for fabrication. The GDS file generation will discretize the design into polygons. Depending on the algorithm, the discretization can potentially impact the design performance as well so it's good to test it.

First we export a GDS file.

Then we will import the GDS file using gdstk.

We can visualize the layout.

Create the extractor structure from the gds geometry. Here we don't apply any sidewall angles but if the fabrication is known to produce a certain sidewall angle, it can be tested here.

Simulate the imported extractor and check its performance. Compared to the design before GDS export, we do see a small decrease in the performance but it's quite minimal.

Lastly we can plot the simulation in 3D to better visualize the optimized extractor geometry.

Final Remarks¶

In this notebook, we focus exclusively on a single frequency. In the referenced paper, optimization is also conducted at a single frequency, while the extractor's performance is evaluated across a broad wavelength range. This can be achieved by specifying frequency sampling points in the FieldMonitor.

The enhancement of upward radiation flux in the optimized design can arise from two mechanisms: (1) the structure redirects the dipole radiation toward the upward direction, and (2) the total radiation from the dipole is increased due to the Purcell effect. Although we do not examine the Purcell factor in this notebook, it can be analyzed as demonstrated in the tutorial provided here. Based on the findings of the paper, the Purcell effect is not very significant in our case since the dipole is deeply buried (100 nm) under the surface.

In this example, we only used the filter to enforce the fabrication constraint. For better fabrication constraint compliance, one can also consider adding a fabrication penalty term in the objective function such as the one demonstrated in this example. Before fabrication, we should also closely examine the small features in the design to ensure fabricability.