# Quantum Eigenstates of Curved and Varying Cross-Sectional Waveguides
## 3 Transmission Studies through a Straight Waveguide with a Varying Thickness  
In this section, we use the developed 1D model to compute transmission and reflection coefficients of a particle confined to a varying thickness quantum waveguide with a straight axis defined as the \(x\) axis. The waveguide cross-sectional dimension in one direction (\(z\)) is assumed to be much larger than the dimensions along the \(x,y\) directions. In this case, the 3D problem reduces to a 2D problem in the \(x-y\) plane. In the region confined by the waveguide, the Schrodinger equation reads  
\[-\frac{\hbar^{2}}{2m}\
abla^{2}\psi=E\psi\,, \tag{26}\]  
and the wavefunction satisfies \(\psi=0\) on the surface of the waveguide corresponding to infinite barriers. We will attempt to compute the transmission and reflection coefficients for the case where the particle approaches the central region from the far left, and the waveguide thickness is assumed to vary in a continuous (and piecewise two times differentiable) manner. The waveguide thickness is considered constant to the left and the right of the central region (for simplicity, the same thickness \(h_{0}\) and a zero curvature are assumed in the left and right regions). In this case the wavefunction part \(\phi\) and its derivative \(\frac{\mathrm{d}\phi}{\mathrm{d}s}\) must be continuous everywhere, and we may write  
\[ \begin{split}\phi(s)&=\exp\left(i\alpha s\right)+B \exp\left(-i\alpha s\right)\,,&\text{to the left of the central region,}\\ \phi(s)&=C\exp\left(i\alpha s\right)\,,& \text{to the right of the central region,}\end{split}  \tag{27}\]  
where  
\[\alpha=\sqrt{\frac{2 mE}{\hbar^{2}}-\frac{\pi^{2}}{4\epsilon^{2}h_{0}^{2}}}>0\,. \tag{28}\]  
The latter inequality guarantees that wave solutions are propagating in the left and right regions. We can now obtain the transmission \(T\) and reflection \(R\) coefficients as  
\[T=C\,,\qquad R=B\,. \tag{29}\]  
Evidently, since the wave problem is lossless, we must require  
\[|T|^{2}+|R|^{2}=|C|^{2}+|B|^{2}=1\,. \tag{30}\]  
In order to determine the coefficients \(B\) and \(C\), Equation (15) must be solved in the central region by matching to the wave solutions in the left and right regions. This has been done numerically. We shall consider the central region to have a trigonometrically varying thickness, i.e.,  
\[ \begin{split} h(s)&=h_{0}+b\sin\left(\frac{n\pi}{L_ {m}}s\right)\,,\\ \frac{\mathrm{d}h}{\mathrm{d}s}&=b\frac{n\pi}{L_{m}} \cos\left(\frac{n\pi}{L_{m}}s\right)\,,\\ \frac{\mathrm{d}^{2}h}{\mathrm{d}s^{2}}&=-b\left( \frac{n\pi}{L_{m}}\right)^{2}\sin\left(\frac{n\pi}{L_{m}}u\right)\,,\end{split}  \tag{31}\]  
where \(L_{m}\) defines the size of the central region and \(n\) is an integer. In Figure 2 (upper plot), we show the waveguide thickness and its first- and second-order derivatives as a function of the centerline coordinate \(u\) for the case where \(h_{0}=1\,\mathrm{nm}\), \(b=2\,\mathrm{\AA}\), \(L_{m}=10\,\mathrm{nm}\), and \(n=1\), \(\epsilon=1\). In Figure 2 (lower plot), the transmission and reflection curves as a function of the energy of the quantum-mechanical particle are shown. In Figure 3 (upper plot), we show the waveguide thickness and its first- and second-order derivatives as a function of the centerline coordinate \(u\) for the case where \(h_{0}=1\,\)nm, \(b=-2\,\)A, \(L_{m}=10\,\)nm, and \(n=1\), \(\epsilon=1\). In Figure 3 (lower plot), the transmission and reflection curves as a function of the energy of the quantum-mechanical particle are shown. The mass of the particle is the free-electron mass. Note that a threshold energy is required for the particle to be transmitted to the right. This comes about as the thickness in the central region is lower than the thickness to the far left and right. Hence, the particle will essentially only propagate to the right if its energy is higher than the energy associated with the smallest thickness \(h_{\text{min}}=h_{0}-b\) in the central region.