.. _theory-two-layer: Two-Layer Planar Geometry ========================== This section derives the scattering dyadic Green's function for a planar two-layer geometry — the simplest nontrivial system in which a dielectric interface modifies the electromagnetic field. We state the Fresnel reflection coefficients for *s*- and *p*-polarised waves, then give the resulting scattering Green's function as Sommerfeld integrals in cylindrical coordinates. The mathematical discussion follows [Wu2018]_; a full step-by-step derivation (boundary conditions, coordinate rotation, Bessel-function identities) can be found in [Novotny2012]_ Ch. 10 and [Sarabandi]_. A detailed derivation is also included in :download:`this PDF `. Geometry -------- .. figure:: /_static/two_layer/Two_Layer.png :align: center :width: 80% Two-layer planar geometry. The source (donor) is in medium *i* above the interface and the field point (acceptor) is at lateral separation :math:`\rho`. The source (donor) is at :math:`\mathbf{r}' = (0, 0, z')` in medium *i* (above the interface) and the field point (acceptor) is at :math:`\mathbf{r} = (\rho, \varphi, z)`, also in medium *i*. The interface sits at :math:`z = 0`. Fresnel Reflection Coefficients --------------------------------- Applying the electromagnetic boundary conditions (tangential :math:`\mathbf{E}` and :math:`\mathbf{H}` continuous across the interface) to *s*- and *p*-polarised plane waves yields the Fresnel reflection coefficients. Define the perpendicular wave vector in medium :math:`l`: .. math:: :label: eq-Kz K_{z,l}(\omega, k_\rho) = \sqrt{\varepsilon_{r,l}(\omega)\,\frac{\omega^2}{c^2} - k_\rho^2}, \qquad l = i, j, where :math:`k_\rho` is the in-plane wave vector magnitude and the branch is chosen so that :math:`\mathrm{Im}\,K_{z,l} \geq 0` (outgoing/evanescent waves). **s-polarisation** (TE — electric field parallel to the interface): .. math:: :label: eq-Rs R_s(k_\rho, \omega) = \frac{K_{z,i} - K_{z,j}}{K_{z,i} + K_{z,j}}. **p-polarisation** (TM — magnetic field parallel to the interface): .. math:: :label: eq-Rp R_p(k_\rho, \omega) = \frac{\varepsilon_{r,j}\,K_{z,i} - \varepsilon_{r,i}\,K_{z,j}} {\varepsilon_{r,j}\,K_{z,i} + \varepsilon_{r,i}\,K_{z,j}}. .. _theory-scattering-gf: Scattering Green's Function ------------------------------ The total dyadic Green's function decomposes as .. math:: \overline{\overline{\mathbf{G}}}(\mathbf{r}, \mathbf{r}', \omega) = \overline{\overline{\mathbf{G}}}_0(\mathbf{r}, \mathbf{r}', \omega) + \overline{\overline{\mathbf{G}}}_\mathrm{Sc}(\mathbf{r}, \mathbf{r}', \omega), where :math:`\overline{\overline{\mathbf{G}}}_0` is the closed-form vacuum Green's function and :math:`\overline{\overline{\mathbf{G}}}_\mathrm{Sc}` encodes the effect of the interface. Vacuum Green's Function ^^^^^^^^^^^^^^^^^^^^^^^^ The free-space dyadic Green's function has the closed-form analytical expression .. math:: \overline{\overline{\mathbf{G}}}_0(\mathbf{r}_\alpha,\mathbf{r}_\beta,\omega) = \frac{e^{ik_0 R_{\alpha\beta}}}{4\pi R_{\alpha\beta}} \Big[ \left(\overline{\overline{\mathbf{I}}}_3 - \mathbf{e}_\mathrm{R}\mathbf{e}_\mathrm{R}\right) + \left(3\mathbf{e}_\mathrm{R}\mathbf{e}_\mathrm{R} - \overline{\overline{\mathbf{I}}}_3\right) \left(\frac{1}{(k_0 R_{\alpha\beta})^{2}} - \frac{i}{k_0 R_{\alpha\beta}}\right) \Big], where :math:`R_{\alpha\beta} = |\mathbf{r}_\alpha - \mathbf{r}_\beta|` is the distance between the source and field points, :math:`\mathbf{e}_\mathrm{R} = (\mathbf{r}_\alpha - \mathbf{r}_\beta)/R_{\alpha\beta}` is the unit vector along that direction, :math:`\overline{\overline{\mathbf{I}}}_3` is the :math:`3\times 3` identity tensor, and :math:`k_0 = \omega/c` is the free-space wave number. Scattering Part ^^^^^^^^^^^^^^^^ In cylindrical coordinates the scattering part is given by the Sommerfeld integral: .. math:: \overline{\overline{\mathbf{G}}}_\mathrm{Sc}(\rho, \varphi, z, z', \omega) = \int_{0}^{+\infty} \frac{i\,dk_{\rho}}{4\pi} \Big[ R_s\,\overline{\overline{\mathbf{M}}}^{(s)} + R_p\,\overline{\overline{\mathbf{M}}}^{(p)} \Big]\, e^{iK_{z,i}(k_{\rho},\,\omega)(z+z')}. :math:`\mathbf{M}^{(s)}` — s-wave contribution ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: :label: eq-Ms \overline{\overline{\mathbf{M}}}^{(s)}(k_{\rho}, \omega) = \frac{k_{\rho}}{2\,K_{z,i}} \begin{pmatrix} J_0 + \cos 2\varphi\;J_2 & \sin 2\varphi\;J_2 & 0 \\ \sin 2\varphi\;J_2 & J_0 - \cos 2\varphi\;J_2 & 0 \\ 0 & 0 & 0 \end{pmatrix}, where :math:`J_n = J_n(k_\rho \rho)` are Bessel functions of the first kind. The *s*-wave has no :math:`z`-component because the electric field lies entirely in the interface plane. :math:`\mathbf{M}^{(p)}` — p-wave contribution ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: :label: eq-Mp \overline{\overline{\mathbf{M}}}^{(p)}(k_{\rho}, \omega) = \frac{-k_{\rho}\,K_{z,i}}{2\,k_i^2} \begin{pmatrix} J_0 - \cos 2\varphi\;J_2 & -\sin 2\varphi\;J_2 & \dfrac{2i\,k_{\rho}}{K_{z,i}}\cos\varphi\;J_1 \\[6pt] -\sin 2\varphi\;J_2 & J_0 + \cos 2\varphi\;J_2 & \dfrac{2i\,k_{\rho}}{K_{z,i}}\sin\varphi\;J_1 \\[6pt] -\dfrac{2i\,k_{\rho}}{K_{z,i}}\cos\varphi\;J_1 & -\dfrac{2i\,k_{\rho}}{K_{z,i}}\sin\varphi\;J_1 & -\dfrac{2\,k_{\rho}^2}{K_{z,i}^2}\;J_0 \end{pmatrix}. The *p*-wave carries all the :math:`z`-components of the scattered field. Sommerfeld Integrals (Implementation) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For numerical evaluation, the Fresnel coefficients and common prefactors are absorbed into six Sommerfeld integrals over the in-plane wave vector :math:`k_\rho`, each involving the phase factor :math:`e^{i K_{z,i}(z + z')}`: .. math:: :label: eq-sommerfeld \begin{aligned} I_1 &= \int_0^\infty \frac{R_s\,k_\rho}{2\,K_{z,i}}\; J_0(k_\rho\rho)\;e^{iK_{z,i}(z+z')}\,dk_\rho, \\[4pt] I_2 &= \int_0^\infty \frac{R_s\,k_\rho}{2\,K_{z,i}}\; J_2(k_\rho\rho)\;e^{iK_{z,i}(z+z')}\,dk_\rho, \\[4pt] I_3 &= \int_0^\infty \frac{R_p\,k_\rho\,K_{z,i}}{2\,k_0^2}\; J_0(k_\rho\rho)\;e^{iK_{z,i}(z+z')}\,dk_\rho, \\[4pt] I_4 &= \int_0^\infty \frac{R_p\,k_\rho\,K_{z,i}}{2\,k_0^2}\; J_2(k_\rho\rho)\;e^{iK_{z,i}(z+z')}\,dk_\rho, \\[4pt] I_5 &= \int_0^\infty \frac{i\,R_p\,k_\rho^2}{k_0^2}\; J_1(k_\rho\rho)\;e^{iK_{z,i}(z+z')}\,dk_\rho, \\[4pt] I_6 &= \int_0^\infty \frac{R_p\,k_\rho^3}{K_{z,i}\,k_0^2}\; J_0(k_\rho\rho)\;e^{iK_{z,i}(z+z')}\,dk_\rho, \end{aligned} where :math:`k_0 = \omega/c`. In the implementation (:mod:`mqed.Dyadic_GF.GF_Sommerfeld`) the integration is split at :math:`k_\rho = k_0` (the propagating/evanescent boundary) for numerical accuracy and evaluated with :func:`scipy.integrate.quad_vec`. Implementation -------------- The class :class:`mqed.Dyadic_GF.GF_Sommerfeld.Greens_function_analytical` evaluates these integrals for arrays of energies and lateral separations :math:`\rho`. See the :doc:`/tutorials/GF_Sommerfeld` tutorial for usage and the :ref:`config-dyadic-gf` section for all available configuration parameters. References ---------- .. [Wu2018] J. S. Wu, Y. C. Lin, Y. L. Sheu, and L. Y. Hsu, "Characteristic distance of resonance energy transfer coupled with surface plasmon polaritons," *J. Phys. Chem. Lett.* **9**, 7032–7039 (2018). .. [Novotny2012] L. Novotny and B. Hecht, *Principles of Nano-Optics*, 2nd ed. (Cambridge University Press, 2012), Ch. 10. .. [Sarabandi] K. Sarabandi, "Dyadic Green's function," Lecture Notes, University of Michigan.