grcwa 0.1.1

Rigorous coupled wave analysis supporting automatic differentation with autograd

grcwa

grcwa (autoGradable RCWA) is a python implementation of rigorous coupled wave analysis (RCWA) for arbitrarily shaped photonic crystal slabs, supporting automatic differentation with autograd

• Free software: GPL license

• Documentation: https://grcwa.readthedocs.io.

Citing

If you find grcwa useful for your research, please cite the following paper:

@article{Jin2020,
  title = {Inverse design of lightweight broadband  reflector for efficient lightsail propulsion},
  author ={Jin, Weiliang and Li, Wei and Orenstein, Meir and Fan, Shanhui},
  year = {2020},
  journal = {arXiv:2005.04840},
}

Features

RCWA solves EM-scattering problems of stacked photonic crystal slabs. As illustrated in the above figure, the photonic structure can have N layers of different thicknesses and independent spatial dielectric profiles. All layers are periodic in the two lateral directions, and invariant along the vertical direction.

• Each photonic crystal layer can have arbitrary dielectric profile on the 2D grids.

• autograd is integrated into the package, allowing for automated and fast gradient evaluations for the sake of large-scale optimizations. Autogradable parameters include dielectric constant on every grid, frequency, angles, thickness of each layer, and periodicity (however the ratio of periodicity along the two lateral directions must be fixed).

Quick Start

• Installation:

  $ pip install grcwa

  Or,

  $ git clone git://github.com/weiliangjinca/grcwa
  $ pip install .

• Example 1: transmission and reflection (sum or by order) of a square lattice of a hole: ex1.py

• Example 2: Transmission and reflection of two patterned layers: ex2.py, as illustrated in the figure below (only a unit cell is plotted)

  

  • Periodicity in the lateral direction is L_x = L_y = 0.2, and frequency is 1.0.

  • The incident light has an angel pi/10.

    import grcwa
    import numpy as np
    grcwa.set_backend('autograd') # if autograd needed
    
     # lattice constants
     L1 = [0.2,0]
     L2 = [0,0.2]
     # Truncation order (actual number might be smaller)
     nG = 101
     # frequency
     freq = 1.
     # angle
     theta = np.pi/10
     phi = 0.
    
     # setup RCWA
     obj = grcwa.obj(nG,L1,L2,freq,theta,phi,verbose=1)

  • Geometry: the thicknesses of the four layers are 0.1,0.2,0.3, and 0.4. For patterned layers, we consider total grid points N_x * N_y = 100*100 within the unit cell.

  • Dielectric constant: 2.0 for the 0-th layer; 4.0 (1.0) for the 1st layer in the orange (void) region; 6.0 (1.0) for the 2nd layer in the bule (void) region; and 3.0 for the last layer.

    Np = 2 # number of patterned layers
    Nx = 100
    Ny = 100
    
    thick0 = 0.1
    pthick = [0.2,0.3]
    thickN = 0.4
    
    ep0 = 2.
    epN = 3.
    
    obj.Add_LayerUniform(thick0,ep0)
    for i in range(Np):
        obj.Add_LayerGrid(pthick[i],Nx,Ny)
    obj.Add_LayerUniform(thickN,epN)
    
    obj.Init_Setup()

  • Patterned layer: the 1-th layer a circular hole of radius 0.5 L_x, and the 2-nd layer has a square hole of 0.5 L_x

    radius = 0.5
    a = 0.5
    
    ep1 = 4.
    ep2 = 6.
    epbkg = 1.
    
    # coordinate
    x0 = np.linspace(0,1.,Nx)
    y0 = np.linspace(0,1.,Ny)
    x, y = np.meshgrid(x0,y0,indexing='ij')
    
    # layer 1
    epgrid1 = np.ones((Nx,Ny))*ep1
    ind = (x-.5)**2+(y-.5)**2<radius**2
    epgrid1[ind]=epbkg
    
    # layer 2
    epgrid2 = np.ones((Nx,Ny))*ep2
    ind = np.logical_and(np.abs(x-.5)<a/2 and np.abs(y-.5)<a/2))
    epgrid2[ind]=epbkg
    
    # combine epsilon of all layers
    epgrid = np.concatenate((epgrid1.flatten(),epgrid2.flatten()))
    obj.GridLayer_geteps(epgrid)

  • Incident light is s-polarized

    planewave={'p_amp':0,'s_amp':1,'p_phase':0,'s_phase':0}
    obj.MakeExcitationPlanewave(planewave['p_amp'],planewave['p_phase'],planewave['s_amp'],planewave['s_phase'],order = 0)
    
    # solve for R and T
    R,T= obj.RT_Solve(normalize=1)

• Example 3: topology optimization of reflection of a single patterned layer, ex3.py

Note on conventions

• The vacuum permittivity, permeability, and speed of light are 1.

• The time harmonic convention is exp(-i omega t).

Acknowledgements

My implementation of RCWA received helpful discussions from Dr. Zin Lin. Many details of implementations were referred to a RCWA package implemented in c called S4. The idea of integrating Autograd into RCWA package rather than deriving adjoint-variable gradient by hand was inspired by a discussion with Dr. Ian Williamson and Dr. Momchil Minkov. The backend and many other styles follow their implementation in legume. Haiwen Wang and Cheng Guo provided useful feedback. Lastly, the template was credited to Cookiecutter and the audreyr/cookiecutter-pypackage.

History

0.1.1 (2020-05-18)

• Fix license

0.1 (2020-05-12)

• First release on PyPI.