Coupled Mode Theory and Codirectional Coupling

Coupled-mode theory is concerned with coupling spatial modes of differing polarizations, distributions, or both. To understand codirectional coupling it is useful to have an understand of background material that builds to codirectional coupling. First, consider coupling normal modes in a single waveguide that is affected by a perturbation. Such case is single-waveguide mode coupling. The perturbation in question is spatially dependent and is represented as *Δ**P*(*r*), a perturbing polarization. Consider the following Maxwell’s equations \[ \nabla \times E=i \omega \mu_0 H\] \[\nabla \times H=-i \omega \epsilon E-i \omega \Delta P\] Consider two sets of fields (*E*_{1}, *H*_{1}) and (*E*_{2}, *H*_{2}), they satisfy the Lorentz reciprocity theorem give by ∇ ⋅ (*E*_{1}×*H*_{2}^{*}+*E*_{2}^{*}×*H*_{1}) = −*i**ω*(*E*_{1}⋅*Δ**P*_{2}^{*}−*E*_{2}^{*}⋅*Δ**P*_{1}) For *Δ**P*_{1} = *Δ**P* and *Δ**P*_{2} = 0 and integrating over the result for the cross section of the waveguide in question, we get \[ \sum_{\nu} \frac{ \partial }{\partial z} A_{\nu}(z)e^{i\left(\beta_{\nu}-\beta_{\mu}\right)} z=i \omega e^{-i \beta_{\nu} z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{\mu}^{*} \cdot \Delta P dxdy \] Evoking orthonormality, we can get the coupled-mode equation \[ \pm \frac{ \partial A_{\nu} }{\partial z} =i \omega e^{-i \beta_{\mu} z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{\nu}^{*} \cdot \Delta P dxdy \] The plus sign indicates forward propagating modes when *B*_{ν} > 0 and the minus sign indicates a backward propagating mode with *B*_{ν} < 0

Many applications are concerned with the coupling between two modes. This coupling between two modes can be within the same waveguide or can be coupled between two parallel waveguides. For a system where we are interested in coupling two modes for either the parallel waveguides case or within the same waveguide, the two modes are described by two amplitudes A and B. The coupled equations are given by \[\pm \frac{ \partial A}{\partial z}=i \kappa_{aa} A+ i\kappa_{ab} B e^{i(\beta_b-\beta_a)z} \] \[\pm \frac{ \partial B}{\partial z}=i \kappa_{bb} B + i\kappa_{ba} A e^{i(\beta_a-\beta_b)z} \] for the two modes we want to solve. The coupling coefficients are given as part of a matrix *C* = [*C*_{νμ}] and $\widetilde{\kappa}=[\widetilde{\kappa}_{\nu \mu}]$ They are given as \[C_{\nu \mu}=\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \left( E_{\nu}^{*} \times H_{\mu} + E_{\mu} \times H_\nu \right) \cdot \hat{z} dxdy=c_{\mu \nu}^{*}\] \[ \widetilde{\kappa}_{\nu \mu}=\omega \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{\nu}^{*} \cdot \Delta \epsilon_{\mu} \cdot E_{\mu} dxdy\] where *Δ**ϵ* is the perturbation applied to the system. We then simplify the math further by removing the self coupling terms in equations (5) and (6) by expressing our normal mode coefficients by \[A(z)=\widetilde{A}(z)e^{\pm i \int_{0}^{z} \kappa_{aa}(z)dz} \] \[B(z)=\widetilde{B}(z)e^{\pm i \int_{0}^{z} \kappa_{bb}(z)dz}\] For cases of interest, perturbation will either be independent of z or be a periodic funciton of z. This further reduces our coupling coefficients to \[ \pm \frac{\partial \widetilde{A}}{\partial z}=i\kappa_{ab} \widetilde{B}e^{i 2 \delta z} \] \[ \pm \frac{\partial \widetilde{B}}{\partial z}=i\kappa_{ba} \widetilde{A}e^{-i 2 \delta z}\] The parameter of 2*δ* is the phase mismatch between the two coupled modes.

# Codirectional Coupling

Codirectional coupling is when the coupling of two propagating modes are in the same direction, over some length *l*, where *β*_{a} and *β*_{b} are both greater than zero. Coupling equations to be used are \[\frac{\partial \widetilde{A}}{\partial z}=i\kappa_{ab} \widetilde{B}e^{i 2 \delta z} \] \[ \frac{\partial \widetilde{B}}{\partial z}=i\kappa_{ba} \widetilde{A}e^{-i 2 \delta z}\] The general solution of this system is solved as an initial value problem in matrix form is given as $$\begin{bmatrix}
\widetilde{A}(z)\\
\widetilde{B}(z)\\
\end{bmatrix} = F(z;z_0) \begin{bmatrix}
\widetilde{A}(z_0)\\
\widetilde{B}(z_0)\\
\end{bmatrix}$$ Where *F*(*z*; *z*_{0}) is the forward coupling matrix and it relates field amplitudes at an intial value of *z*_{0} to those at z. For the most simple case when power is launched into only mode a at *z* = 0 , giving $\widetilde{B}(0)=0$. With *z* = 0 \[\widetilde{A}(z)=\widetilde{A}(0)\left( cos \beta_c z -\frac{i \delta}{\beta_c} sin \beta_c z \right) e^{i \delta z} \] \[\widetilde{B}(z)=\widetilde{B}(0)\left(\frac{i \kappa_{ba}}{\beta_c} sin \beta_c z \right) e^{-i \delta z} \] where $\beta_c=\left( \kappa_{ab}\kappa_{ba}+\delta^2\right)^\frac{1}{2}$ Then looking at the power of the two modes when they are completely phase matched, that is when *δ* = 0 \[\frac{P_a(z)}{P_a(0)}=|\frac{\widetilde{A}(z)}{\widetilde{A}(0)}|^2=cos^2 \beta_c z \] \[\frac{P_b(z)}{P_a(0)}=|\frac{\widetilde{B}(z)}{\widetilde{A}(0)}|^2=sin^2 \beta_c z\] where *κ*_{ab} = *κ*_{ba}^{*} Define the coupling efficiency for a length l as $\eta=\frac{|\kappa_{ba}|^2}{\beta_c ^2} sin^2 \beta_c z$ as well as the coupling length $l_c=\frac{\pi}{2 \beta_c}$. I have plotted the phase matching condition of *δ* = 0 in figure 1. It is seen that we can only have complete power transfer when we are phase matched between the two coupled modes.

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UVIS Calibration Notes

\begin{equation} \label{eq:general} L(\lambda_j)=\frac{\left[C(\lambda_j)\cdot N(C)-D(j)-B(j)-S'_l(\lambda_j)\right]/\Delta t}{A\cdot\Delta\lambda\cdot R_c(\lambda_j)\cdot FF_j\cdot\bar{\Omega}\cdot(1-S_l)} \end{equation}

Measuring Acoustic Normal Modes of a Rectangular and Cylindrical Geometry and Measurement of Speed of Sound

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*in vivo*half-life and reduce immunogenicity. Moreover, the incorporation of peptide crosslinkers that are only degraded by the tumor-specific proteases allows the controlled release of mAbs only at the tumor microenvironment. We also show that the administration of mAbs nanocapsules can suppress tumor growth and improve the survival rate with greater efficacy, compared to the native mAbs. Thus, the mAbs nanocapsules represent a new and more effective delivery approach for cancer immunotherapies.

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# Materials and methods

# Results

## Computational

### Electronic structures for 20 full-length A*β*40 conformer structures and inter-atomic interaction energies from these electronic structures using natural bonding orbital (NBO) formalism \cite{NBO,Louis_NBO_2013} revealed:

C-terminus:helix region interactions

N-terminus:helix region interactions

strong interactions of residues within the 15-24 region that stabilize the helix

interactions between residues 40-35, 33-27, and 3-9

## Experimental

### Structure 3 synthesis

lactam bridge between Asp23 and a Lys residue that we substituted for Val40

covalent linkage prevents interaction of the C-terminus with the core helix

predicted to preclude formation of the stable dimer state formed through concerted collapse of two monomers and produce distinct assembly behavior

\label{fig:peptide_association_scheme_v9.pdf}

### SDS-PAGE showed that structure 3 vs A*β*40

displayed no dimer band, but an intense monomer band

a band migrating below monomer

suggesting that structure

**3**is capable of forming two conformers in SDS-PAGE, one with an M_{r}consistent with wild type A*β*40 and one with a smaller M_{r}of ≈3500.

\label{fig:sdspage}

### Mass spectrometry of structure 3 vs A*β*40 revealed:

structure

**3**spectrum contained a large -7/2 dimer (formed from -4 and -3 monomers) peaksuggesting that self association of structure

**3**produces a dimer structure unique from that formed by wild type A*β*40

### Ion mobility spectroscopy (IMS) studies of structure 3 vs A*β*40 revealed:

“collision cross section” (

*σ*) value for structure**3**was larger than that of wild type A*β*40 (620 vs 607 Å^{2})supporting the explanation that structure

**3**forms a unique conformer

\label{fig:Spectral_Overlay}

### ThT fluorescence to monitor formation of extended *β*-sheet in structure 3 vs A*β*40 revealed:

an immediate monotonic increase in fluorescence that plateaued at 12 days at a level that was ≈1/2 that of A

*β*40appears to self-associate without a substantial lag phase (<<1 days)

appears to produce aggregates that display substantially less

*β*-sheet than do wild type assemblies

\label{fig:LMM-tht}

### EM to compare assembly morphology of structure 3 vs A*β*40 revealed:

far fewer fibrils

a number of structures not seen with WT A

*β*40, including short worm-like structures, longer worm-like structures, and fibrils with prominent helical twistsmultifilar rod-like structures

some of the rod-like structures had “sausage-link” morphologies with the links occurring at intervals of ≈270 nm

\label{fig:em}