## Abstract

We present a novel way to account for partially coherent interference in multilayer systems via the transfer-matrix method. The novel feature is that there is no need to use modified Fresnel coefficients or the square of their amplitudes to work in the incoherent limit. The transition from coherent to incoherent interference is achieved by introducing a random phase of increasing intensity in the propagating media. This random phase can simulate the effect of defects or impurities. This method provides a general way of dealing with optical multilayer systems, in which coherent and incoherent interference are treated on equal footing.

©2010 Optical Society of America

The study of multilayer films has gained increasing interest in recent years due to their many potential uses as optical coatings and as transparent conductive electrodes in optoelectronic devices such as flat displays, thin film transistors and solar cells [1–4]. In particular, the unique optical properties of multilayer films play a vital role in the construction of thin-film solar cells [5,6], where an important challenge is to increase the absorption of the near-bandgap light, which would allow a reduction in the thickness of the cell. Consequently, the ability to predict and tune the optical properties at the semiconductor interface can greatly contribute to reducing the cost of thin-film solar cells. In addition, multilayer systems have been used to study photon localization [7], model ion-implanted materials [8,9], and to determine the thicknesses, densities and roughness of films in combination with X-ray reflectivity measurements [10]. Therefore, developing a general method that allows the study of the optical response of multilayer systems is of fundamental importance for designing and tuning more efficient optoelectronic devices.

The reflectance and transmittance of a multilayer structure can be calculated by using the transfer matrix method [11–13]. However, the form of the conventional transfer matrix assumes coherent light propagation, which may lead to narrow oscillations in the calculated reflectance and transmittance spectra of the system. In practice, due to interference-destroying effects these oscillations may not be observable. Consequently, in order to have a realistic description of the optical properties of multilayer systems these interference-destroying effects should be introduced. There have been previous attempts to modify the transfer matrix in order to take into account incoherent interference [14,15], as well as partial coherence [16–18]. However, in the proposed methods the square of the amplitude of the electric field, or Fresnel coefficients, are used to study the incoherent case. The case of partial coherence simulates macroscopic surface or interface roughness, and is introduced by multiplying the Fresnel coefficients by correction factors.

In this article, we present a modified transfer matrix method to calculate the optical properties of multilayer systems that includes coherent, partially coherent, and incoherent interference. The modeling of the partially coherent and incoherent cases is done by adding a random phase that simulates the effects of impurities or defects in the layer. The value of the random phase can be gradually varied in order to go from coherence to incoherence. The capabilities of the method are illustrated by presenting calculations of the optical properties of Si and ZnO films. The physical meaning of the random phase as well as potential methods for modeling materials with varying impurity profiles and interface roughness are discussed. An additional advantage of this method is that the partial coherence or incoherence can be separately/individually introduced for each layer, without having to calculate the intensity matrix for all the layers as it has been previously implemented.

The transfer matrix method is used to solve Maxwell’s equations in a multilayer
system subject to a uniform incident field ** E**. The field in the medium is divided into two components, the forward (transmitted)
component

*E*^{+}and the backward (reflected) component

*E*^{-}(see Fig. 1 ).

The amplitudes of the field at the left- and right-hand side of an interface are related by:

*t*and

_{m-1,m}*r*are the transmission and reflection Fresnel coefficients respectively.

_{m-1,m}The field amplitudes at the left- and right-hand side of the *m*th layer are
related by:

*σ*is the wave number,

*n*is the complex refractive index of the

_{m}*m*th layer,

*d*is the thickness of the

_{m}*m*th layer, and

*φ*

*is the complex propagation angle following Snell’s law (*

_{m}*n*sin

_{0}*φ*

_{0}=

*n*

_{1}sin

*φ*

_{1}= … =

*n*

_{N}_{+1}sin

*φ*

_{N}_{+1}). The above matrix transformations can be applied for the

*N*layers and

*N*+ 1 interfaces resulting in:

**T**

_{0,(}

_{N}_{+1)}, defined byis the system transfer matrix.

The reflection and transmission coefficients of the multilayer system can be calculated from the elements of the transfer matrix as follows:

Since the transmission and reflection coefficients are related to the elements of the transfer matrix, the matrix can be written as follows:

In previous work [18,15], the incoherence was treated by replacing the reflection and transmission vectors by the squares of their amplitudes. In this way the conventional (coherent) transfer matrix was replaced by an intensity matrix, as:

The way in which the partial coherence and incoherence are introduced in our method is as follows. A random phase is added to the phase shift in the selected layer. Therefore, Eq. (5) is re-written as follows:

where*β*takes values between 0 and

*π*, and

*Rand*is a randomly generated number between −1 and 1. The randomly generated numbers are uniformly distributed. The final transmittance is obtained by averaging the calculated transmittances with different sequences of random numbers.

The physical meaning of this random phase is to simulate impurities or defects in the
layer, which would introduce a dephasing or loss of coherence such as the one we are
introducing by adding the term$\beta \cdot Rand$. If the concentration of defects is large then the loss of coherence
would be complete and the system would be in the completely incoherent case as it has been
dealt with before in Refs. 15 and 18. In our method, the total incoherence is represented
by introducing a random phase with *β* = *π*,
while in the partial coherence cases 0<*β*<
*π*.

The calculation of the optical response in the incoherent limit is illustrated in Fig. 2
. The figure shows the calculated transmittance vs. wavelength for a crystalline Si
film of 150 nm thickness in air. The curves corresponding to the coherent case,
*β* = 0, and the incoherent limit, *β* =
*π*, are displayed. Also, as a reference, the curve for the
incoherent limit calculated from the intensity matrix method is shown. It can be observed
that the curve corresponding to the coherent limit presents oscillations while these
oscillations have vanished in the incoherent limit. In addition, the agreement between this
method and the intensity matrix method can be clearly observed from the superposition of
the two curves representing the incoherent limit.

In our calculations of the incoherent limit using the random phase method, a random phase
with *β* = *π* was used. The calculations of
the transmittance vs. the wavelength were repeated 30 times, with 30 sequences of randomly
generated numbers, and the results of the 30 runs were averaged to obtain the incoherent
curve. The calculated curve (β = π) displayed in Fig. 2 was further smoothed by using a filter of 10 moving averages.
The application of this filter did not distort the original results and it was only applied
to reduce the noise. This noise is due to the limited number of runs we averaged to obtain
the incoherent curve. It can be clearly observed from Fig.
2 that with only 30 runs the transmittance curve calculated using the random
phase method has converged to the curve calculated using the intensity matrix. This figure
clearly displays that by introducing a random phase one can successfully simulate the
coherence-destroying effects that lead to complete incoherence.

The partial coherence cases can be simulated by introducing a random phase of smaller magnitude. In these cases a certain loss of coherence is introduced; the partial coherence cases represent an intermediate situation between the coherent limit and the complete incoherence. Figure 3 shows the transmittance for a 150 nm film of crystalline Si as a function of the wavelength. The coherent limit and the complete incoherent limit are shown as a reference. It can be observed from this plot two intermediate cases of partial coherence, one with β = π/3 and one with β = π/4. It can be seen that as the magnitude of β decreases the curves approach that of the complete coherent limit. Therefore, by changing β from π to 0 one can calculate the transmittance from the complete incoherent case to the coherent limit.

Another interesting feature of this method is the capability of gradually varying the
incoherence degree independently on individual layers. The loss of coherence is introduced
in each layer by introducing a random phase to each phase shift as shown in Eq. (12). In this case δ_{1}
and δ_{2}, the phase shifts of layer 1 and layer 2 respectively, are
written as:

The capability of introducing different incoherence degrees in each layer is displayed in Fig. 4 , where the transmittance of a two layer system is presented. This figure shows the transmittance of a 150 nm ZnO film on a 150 nm Si film vs. wavelength. It shows the complete incoherent limit, where ${\beta}_{1}$ = ${\beta}_{2}$ = π, the coherent limit where ${\beta}_{1}$ = ${\beta}_{2}$ = 0, and two intermediate cases. For the first case, ${\beta}_{1}$ = 0 and ${\beta}_{2}$ = π, which means that the ZnO layer is treated as coherent, and in the second case ${\beta}_{1}$ = 0 and ${\beta}_{2}$ = π/2. It can be seen from this last case that the transmittance is rapidly approaching the coherent limit.

This method presents a unique way to study the optical response of multilayer systems since the transition from coherence to incoherence can be easily achieved. For instance, this method provides the potential to simulate layered films with impurities or scattering center defects. Moreover, as the degree of incoherence can be modified in each layer independently, it can be very useful to study systems with varying profiles of impurity concentrations by varying the value of β in each film. Also, this method could present an alternative route to study multilayered nanoporous films with different degrees of nanoporosity [19].

In addition, the introduction of partial coherence is expected to be very useful for simulating the effects of interface roughness on the scale of the wavelength. For instance, the interface morphology can be a key variable for optimizing the performance of thin film solar cells since the reflectivity of the semiconductor interface can critically affect the performance of the cell [20]. The surface roughness in a multilayer system can, if large enough, cause phase differences between the reflected and transmitted beams. This effect can also be simulated by the inclusion of a random phase of varying intensity.

In summary, we have developed a method for calculating the optical response of multilayer systems, which can deal with coherent, partially coherent, and incoherent interference on equal footing. This method is based on the transfer matrix method employed in its usual way via Fresnel coefficients in a 2x2 matrix configuration. The novel feature is that there is no need to use modified Fresnel coefficients or the square of their amplitudes to work in the incoherent limit. The transition from coherent, to partially coherent, to incoherent interference is achieved by introducing a random phase of increasing intensity in the propagating media. This random phase can account for the effect of defects or impurities in the layer. The capabilities of the method were presented by calculating the optical properties of Si and ZnO films from the coherent to the incoherent limit.

## Acknowledgments

This work was supported in part by NSF (Grant No. DMR-0906025), DOE (the Division of Material Sciences and Engineering, Office of Basic Sciences, and BES-CMSN), and DOE (Office of Energy Efficiency and Renewable Energy, Industrial Technologies Program) under contract DE-AC05-00OR22725.

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