One can often use analytical methods to determine the optical response of a multilayer stack instead of direct simulations of Maxwell's equations. Lumerical's STACK optical solver provides a set of script-based functions (stackdipole, stackpurcell) for this purpose. Additionally, the STACK simulation object in FDTD provides a GUI for the script commands.

The stackrt and stackfield script commands are also useful for analyzing multilayer stacks. These commands are included with all of Lumerical's products.

## STACK for plane wave illumination

The stackrt script command returns the reflection and transmission of a plane wave through an arbitrary multilayer stack using the analytic transfer matrix method. This function returns the fraction of transmitted and reflected power (Ts, Tp, Rs, Rp), and the complex reflection and transmission coefficients (ts, tp, rs, rp), for both S and P polarizations. The stackfield script command returns the field information (Es, Ep, Hs, Hp) for the stack.

For both stackrt and stackfield, the user must specify the angle of incidence and frequency range of the illumination, as well as the thickness and refractive index of each layer. For a comparison between results obtained using the analytic transfer matrix method and 1D FDTD simulations, please see 4 layer stack application example.

### Complex Coefficients

The complex coefficients are defined as the amplitude of the output wave, after reflection or transmission, divided by the amplitude of the incidence wave. Fresnel analysis of plane waves at an interface reduces a vector problem in 3D space to 1D where all important information is contained within the complex coefficients. For a more thorough treatment of the coordinates see Source Rotations in 3D FDTD. This reduction in dimension is a very useful simplification; however, it relies on some conventions which are important to understand when interpreting the results. Two important concepts should be understood; the plane of incidence, and polarization basis states.

**Plane of Incidence**

In order to fully specify the results, we need to make a choice about the coordinate system to work in. Although this is arbitrary, it is helpful to take advantage of the fact plane waves are TEM, meaning the wave vector **k**, electrical field **E**, and magnetic field **B** field are mutually orthogonal. It should be obvious that choosing a coordinate system where these are aligned along the principal axes will simplify the analysis considerably. Historically different options were considered but most modern analysis uses the helpful concept of the plane of incidence. The plane of incidence is defined as a plane that contains the **k** vectors, and the surface normal **n**. This allows you to define two intuitive linear polarization states S-pol (TE), and P-pol(TM) which need to be treated separately in the analysis. One should remember that these basis states are defined as having either the **E** field or **B** fields out of the plane of incidence respectively. Once these are determined the third component follows from Maxwell and will be within the plane of incidence. These bases can be seen in the following figure.

**Basis Conventions**

After reflection or refraction, we see that **E, B,** and **k** go through a simple rotation. The angle of rotation is readily determined from Snell's law, and these angles are returned by the solver. Most authors agree on the definitions given in the above figure; however, there is some variation in the choice for the reflected P-polarized wave basis that should be highlighted. This essentially comes down to choosing the positive notion for **E,** and **B, **upon reflection i.e. which way the arrows point. We use the definition given in ii), but many important optics textbooks such as *Hecht*, *Born, and Wolf* use a basis where **E** points downwards and **B** points out of the page. This effect can be observed as an additional phase factor of \( \pi \), or equivalently a factor of -1 for r\(_p\) which sometimes causes confusion.

**Propagation Phasor**

The complex coefficients contain phase information from the interface as well as the additional phasor propagation between interfaces. When considering multiple reflections, such as in microcavities, it is important to consider the accumulated phase. It should be remembered that when you set the first or last layer thickness you will be observing the phase at the distance specified from the first and last interfaces. Often, as in ellipsometry measurements, one is interested in the ratio of the complex coefficients which is invariant with the distance from the interface. If you want to compare the amplitude coefficients a single interface with a references then you should make the first layer very thin to avoid this additional phase.

## STACK for dipole emission

For many thin-film applications such as OLEDs and other electroluminescent devices, both the quantum efficiency and extraction efficiency are affected by the coherent interference between layers. The stackdipole and stackpurcell script commands are ideal for studying this microcavity effect as it analytically calculates the emission characteristics of a dipole source (or dipole sources) based on its location in the stack.

The stackdipole function returns the luminance (cd/m^2) and radiance (W/steradian/m^2) as a function of emission angle, as well as the corresponding X, Y, Z tristimulus values. The CIE 1931 color functions are used for calculating X, Y, Z. To determine how the stack geometry affects the quantum efficiency and extraction efficiency separately, stackpurcell should be used.

Coherence Effects in Thick Layers

When studying quantum efficiency, and extraction efficiency in microcavities modelling the coherent interaction between multiple reflections is imperative; however, once a layer has a thickness on the order of the coherence length these coherent effects should be disabled. To do this one can use the incoherent_propagation option. In many emissive devices, the active region is covered in a thick glass protective layer. To model the spectral, and angular-dependent extraction efficiency out of the glass incoherent propagation should be enabled in this region.

### References:

- K. A. Neyts, "Simulation of light emission from thin-film microcavities", J. Opt. Soc. Am. A, Vol. 15, No. 4 (1998)
- W. Lukosz and R. E. Kunz, "Light emission by magnetic and electric dipoles close to a plane interface", J. Opt. Soc. Am. 67, 1607-1619 (1977)
- CIE Proceedings (1932), 1931. Cambridge: Cambridge University Press.

### Related topics

See Video: Optimize Your OLED/LED in One Second for examples on how to use the Stack optical solver to simulate OLEDs and LEDs.

See stackdipole for an example of a multilayer stack dipole emission calculation.

For mode analysis of an OLED layer structure, see OLED slab mode analysis.

See OLED application examples for more examples.