PyATMM : Dyakonov - Nikitin 2009

This example repeats the results reported in “Polarization conversion spectroscopy of hybrid modes” by A. Yu. Nikitin et. al., 2009 doi:10.1364/OL.34.003911.

Probably because the dielectric permittivities presented in the paper were rounded, an exact repeat of the results turned out to be difficult to achieve. This examples seeks to repeat the graph shown in Figure 2c. of the paper.

The modeled system consists of 3 layers 1. a ZnSe prism 2. a Ta2O5 isotropic spacer 3. and a semi-infinite YV04 anisotropic layer

Imports

    import numpy as np
    import matplotlib.pyplot as plt

    from PyATMM.uniaxialTransferMatrix import *
    from PyATMM.isotropicTransferMatrix import *
    from PyATMM.transferMatrix import solve_transfer_matrix

Setup

    c = 299792458.  # m/c
    w = c

    # Angle of incidence sweep
    theta = np.linspace(54.20, 54.22, 50, endpoint=True)

    # Angle between optic axis and projection of k-vector
    phi = np.deg2rad(46.25)

    # Start layer (prism), ZnSe
    eps_ZnSe = 6.6995
    ran_kx = np.sqrt(eps_ZnSe) * (w/c) * np.sin(np.deg2rad(theta))
    ky = 0

    # Second layer, Ta_2O_5
    eps_Ta2O5 = 4.41
    ran_d = np.linspace(5, 0, 50, endpoint=True)

    # Anisotropic layer, YV04
    eps_YVO4_per = 3.9733
    eps_YVO4_par = 4.898

Since everything is defined in terms of wavelength, we set the frequency of the incident radiation, w equal to c
theta defines the range of angles of incidence for which we do the simulation
ran_d defines the range of Ta2O5 thicknesses for which we do the simulation

Calculations

    a_refl_pp = []
    a_refl_ss = []

    a_refl_ps = []
    a_refl_sp = []
    for d in ran_d:
        a_refl_pp_ = []
        a_refl_ss_ = []

        a_refl_ps_ = []
        a_refl_sp_ = []

        d *= 2.*np.pi

        for kx in ran_kx:
            # Layer 0
            D_ZnSe = build_isotropic_dynamic_matrix(eps_ZnSe, w, kx, ky)

            # Layer 1
            D_Ta2O5 = build_isotropic_dynamic_matrix(eps_Ta2O5, w, kx, ky)
            D_Ta2O5_p = build_isotropic_propagation_matrix(eps_Ta2O5, w, kx, ky, d)

            # Layer 2
            D_YVo4 = build_uniaxial_dynamic_matrix(eps_YVO4_per, eps_YVO4_par, w, kx, ky, opticAxis=[np.cos(phi), np.sin(phi), 0])

            # Multiply Matricies
            D = np.dot(np.dot(np.dot(np.linalg.inv(D_ZnSe), D_Ta2O5), D_Ta2O5_p), np.dot(np.linalg.inv(D_Ta2O5), D_YVo4))

            # Solve Matrix
            r_ss, r_sp, r_ps, r_pp, t_ss, t_sp, t_ps, t_pp = solve_transfer_matrix(D)

            a_refl_pp_.append(np.abs(r_pp**2))
            a_refl_ss_.append(np.abs(r_ss**2))

            a_refl_sp_.append(np.abs(r_sp**2))
            a_refl_ps_.append(np.abs(r_ps**2))

        a_refl_pp.append(a_refl_pp_)
        a_refl_ss.append(a_refl_ss_)

        a_refl_sp.append(a_refl_sp_)
        a_refl_ps.append(a_refl_ps_)

The calculation is done with two loops: the outer loop is for the spacer thickness sweep, and the inner loop is for the angle of incidence sweep (recalculated into wavevector projection). For each spacer thickness & angle of incidence, the transfer matrix is built as follows:

The prism is modeled by one isotropic dynamic matrix, D_ZnSe
The spacer is modeled by two matrices - one isotropic dynamic matrix, D_Ta2O5 and one isotropic propagation matrix, D_Ta2O5_p.
The uniaxial crystal is modeled by one uniaxial dynamic matrix, D_YVo4.

Since the incident light comes from inside the prism, we only need to model the interface between the prism and the spacer. The spacer is a finite layer, so it needs to have both interfaces and internal propagation defined. The uniaxial layer is considered to be semi-infinite, meaning that only one interface needs to be defined.

Having defined all the constituent matrices, the whole transfer matrix for the system is built by multiplying (from left to right)

the inverse of D_ZnSe
D_Ta2O5
D_Ta2O5_p
the inverse of D_Ta2O5
D_YVo4

Where 1. and 2. create the first interface (prism-spacer), 3. is propagation inside the spacer, 4. and 5. create the second interface (spacer-uniaxial crystal).

Plotting

    plt.figure(figsize=(9, 6))
    plt.imshow(a_refl_ps, interpolation='none', extent=[np.min(theta), np.max(theta), np.min(ran_d), np.max(ran_d)],
               aspect="auto",  cmap=plt.get_cmap("jet"))

    plt.colorbar()
    plt.xlabel("$\Theta$, degrees")
    plt.xticks([54.20, 54.205, 54.21, 54.215, 54.22])
    plt.ticklabel_format(style = 'sci', useOffset=False)
    plt.ylabel("$\Delta / \lambda$")
    plt.title("R_ps")

    plt.show(block=True)

PyATMM Generalized transfer matrix method implementation

Dyakonov - Nikitin 2009

Imports

Setup

Calculations

Plotting