Multilayer Calculation

The transfer matrix method (TMM) has been applied to calculate the reflection spectra for our devices; this method inherently included a standing wave enhancement effect and a multi-reflection with the optical cavity [33]. The electric fields, together with the magnetic field inside and outside multilayer structures, have been calculated using TMM, giving us the transmittance and reflectance of this kind of structure.

For homogeneous and isotropic multilayer structures, each layer is represented by a 2 × 2 matrix *Mj* of the form:

$$M\_{\dot{j}} = \begin{bmatrix} \cos \delta\_{\dot{j}} & \frac{\left(i \sin \delta\_{\dot{j}}\right)}{\mu\_{\dot{j}}}\\ i\mu\_{\dot{j}} \sin \delta\_{\dot{j}} & \cos \delta\_{\dot{j}} \end{bmatrix} \tag{7}$$

where *μ<sup>j</sup>* is the optical admittance of *j* layer and *δ<sup>j</sup>* is the phase change of the electromagnetic radiation traversing on the *j* layer and can be written as follows:

$$\delta\_{\vec{j}} = \frac{\begin{pmatrix} 2\pi n\_{\vec{j}} \ d\_{\vec{j}} \ \cos \theta\_{\vec{j}} \end{pmatrix}}{\lambda} \tag{8}$$

where *nj* is the refractive index of *j* layer, *dj* is the physical thickness of *j* layer and *θ<sup>j</sup>* is the incidence angle at *j* layer; herein, a quarter wavelength is considered as the thickness of each layer to obtain the highest reflection [34].

$$d\_{\hat{j}} = \frac{\lambda}{4n\_{\hat{j}}} \tag{9}$$

The matrix relation defining the electric field (*B*) and magnetic field (*C*) of the multilayer structure adopted from [35] is as follows:

$$\left(\begin{array}{c} B\\ C \end{array}\right) = \prod\_{j=1}^{K} M\_{\hat{j}} \left(\begin{array}{c} 1\\ \mu\_{sub} \end{array}\right) \tag{10}$$

Using Equation (10) and considering the admittance introduced by the interfaces, which is indistinguishable from the reflectance, this idea has been applied for the reflectance calculation through the assembly of thin films. Then, the transmittance has been deduced from the reflectance *R* through *T* = 1 − *R* [36]. The reflectance R, transmittance T and the phase change on reflection Ψ are given by

$$R = \left| \frac{\mu\_0 B - C}{\mu\_0 B + C} \right|^2 \tag{11}$$

$$T = \frac{4\mu\_0 \,\mu\_{sub}}{\left|\mu\_0 B + C\right|^2} \tag{12}$$

$$\Psi = \arg\left|\frac{\mu\_0 B - C}{\mu\_0 B + C}\right|\tag{13}$$

where *μ*<sup>0</sup> and *μsub* represent the optical admittance of the emission layer and substrate layer, respectively. Mathcad software has been used to compute the data needed for the calculations.
