ELG 3106 : Electromagnetic Engineering
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ELG 3106 : Electromagnetic Engineering
Design Study
Impact of Broadband Antireflection Coating Design on Solar Power Production
Fall 2022
A good conceptual understanding of the reflection and transmission of an electromagnetic uniform plane wave normally incident on the boundary between two dielectric media is required. You must also be familiar with MATLAB (see the “Introduction to MATLAB” presentation), or another computational tool or language.
Introduction
The purpose of this design study is to demonstrate the impact design constraints have on system response. The system is a solar cell covered by an antireflection coating comprising either two or three layers, as shown in Figure 1, where choice of layers, refractive indices, and thicknesses are governed by the design approach. The wavelength distribution of light normally incident on the system is given by an idealized solar spectral irradiance (the so-called “blackbody” irradiance1),
I (入) 入(10)15 ,
[1]
where I has units of power per unit area per unit wavelength (W/m2/nm, and an area of 1 m2 has been assumed), and λ is the wavelength in nm. The solar cell converts all light incident on it into electricity at 100% efficiency, regardless of the wavelength2, but the antireflection coating has a transmissivity T(λ), so the response is electrical power production given by
入
P = j T (入)I (入)d入,
入
[2]
where the limits of integration are λ1 = 200 nm and λ2 = 2200 nm3 . For T = 1, P = 1000 W/m2 .
The object of this study is to demonstrate that the “obvious” approach to the design of broadband antireflection coatings does not necessarily maximize power production, and to explain why. The approach is to numerically integrate equation [2], using the transfer matrix method (TMM) to numerically simulate the wavelength-dependent transmissivity of the planar multilayer stack that comprises the antireflection coating on the solar cell. The theoretical description of the TMM presented below is completely general. It may be used to address any number of layers of arbitrary thickness.
The Design Approach
The focus is on double-layer and triple-layer antireflection coatings, an example of which is shown in Figure 1. We will assume that the sunshine incident on these systems may be represented by uniform plane waves of a given free- space wavelength at normal incidence. The challenge is to achieve a multilayer coating with as low a reflectivity (i.e., as high a transmissivity) as possible across as broad a spectrum as possible. This depends on the number of layers used, their thicknesses and refractive indices. In general, the
Air: n0 = 1
Layer 1, n1 |
|
1 |
Layer 2, n2 |
|
2 |
Layer 3, n3 |
|
3 |
Solar cell, ncell = 3.5 |
Figure 1: A triple-layer coating
greater the number of layers used, for “optimal” choices of thickness and index, the greater the spectral range over which the reflectivity is suppressed. An example is shown in Figure 2. Note that without an antireflection coating, a reflectivity of about 30% is anticipated.
Figure 2: Impact of an antireflection coating upon the reflectivity.
In the analysis and simulations to be done, you are to presume that the thickness of each layer is equal to one-quarter the centre wavelength, 入C , adjusted appropriately for the layer’s refractive index. That is, the thickness should be one-quarter the material wavelength, not the in-air wavelength. Presume that the centre wavelength (in air) is 入C = 650 nm, that the refractive index of air is unity, and that n = 3.5 is the refractive index of the solar cell.
The Double-Layer Coating
Part 1
(a) Find the reflectivity at the center wavelength for a system with no anti-reflective layer. What is the power transmitted into the semiconductor (photovoltaic cell)?
N.B. This part must be done analytically. You do not have to use the TMM approach.
(b) Apply the TMM approach described below to a two-layer anti-reflective coating. Explain your reasoning. In this step you only have to discuss the TMM approach and how you will implement it in MATLAB or Python.
(c) Now consider a cell with a double-layer coating. Find analytically the reflectivity as a function of the refractive indices (n0, n1, n2, ncell) at the central wavelength. Analytically means that you will find a formula like: n2 = ...
(d) Find the relationship between refractive indices minimizing reflectivity at the central wavelength. If n1 = 1.4, what is the optimal choice for n2?
Part 2
(a) Make a graph of the total reflectivity spectrum as a function of wavelength in free space λ , from 400 nm to 1400 nm. For this, use the refractive index and thickness values obtained from your TMM code implemented in Python or MATLAB.
(b) In the last part you have calculated the reflectivity as a function of the wavelength. Numerically integrate this spectrum (equation [2]) to obtain the power transmitted into the solar cell. Do it twice: from λ1 = 200 nm and λ2 = 2200 nm, AND from λ1 = 400 nm and λ2 = 1400 nm.
(c) By choosing other values for refractive index and / or layer thickness, can you increase the power? If so, explain why, by comparing reflectivity spectra.
The Triple-Layer Coating
Part 3
Let’s now study a three-layer anti-reflective coating analytically. For this problem you do not need to do programming.
(a) Find analytically the reflectivity at the central wavelength. Analytically means that you will find a formula like: n2 = ...
(b) Find the relationship between refractive indices that minimizes the reflectivity at the central wavelength.
(c) If n1 = 1.4 and n3 = 3. 15, which value of n2, of two significant digits, allows you to reach the minimization result?
Part 4
Modify your program to allow you to treat the case of a triple-layer coating.
(a) With the same refractive indices, calculate the power for values of n2 ranging from 1.4 to 3.0. (b) Which value of n2 allows you to reach a maximum of power?
(c) Is this value consistent with that found by the analytic approach? Explain.
In your conclusions, please discuss the change in power transmitted for each concept and the reasons for these changes.
The Report
Your design study is to be structured as a formal technical report. Consult the document “Writing a Technical Report” for a review. You must show the application of the theory for both the double-layer and the triple-layer calculations, as described below.
You must completely specify each system with a diagram, fully labelling all parameters (cf. Figure 1). Report the full transfer matrix for each system, as a product of all the individual matrices (see TMM theory below), for arbitrary incident wavelength. Evaluate analytically each system at the central wavelength, present the formula for the reflectance at this wavelength, and determine the relationship between the index values that minimizes this reflectance. Provide the requested refractive index values.
You must construct an algorithm using the frequency-dependent TMM theory (use wavelength as a proxy for frequency). An algorithm is suggested at the end of this document. Code up the algorithm in MATLAB or Python and numerically evaluate the power production as described above. Provide plots of the nominally optimal reflectivity (i.e., the reflectivity for the quarter- wave stack whose indices were analytically determined). Tabulate (or plot) the dependence of the power production on layer indices and thicknesses to ascertain if the design approach followed does indeed maximize power production. Discuss and suggest other optimization approaches.
Submission of the report may be done online. Softcopy submissions of the code are required and should include a plot command to produce an example plot of reflectivity (or transmissivity) vs wavelength for the optimal indices indicated by the TMM.
The Transfer Matrix Method (TMM)
An arbitrary multilayer is shown in Figure 3 below, annotated by the electric field components at each interface. Note that, for convenience, this figure is oriented such that the incident wave travels to the right (in Figure 1, the incident wave travels downward). The subscripts indicate the layer, the + and – signs distinguish between forward and backward waves, respectively, and the prime is used to distinguish waves on the right hand side of an interface from those on the left.
|
|
Figure 3: An arbitrary multilayer, used to illustrate the TMM theory.
The multilayer structure is composed of N layers, not including the unbounded media to the left (usually air, n0 = 1) and to the right (denoted the substrate – silicon in our case), and N+1
interfaces. The layers have the refractive indices, n, as noted. The boundary conditions on the electric field vectors E on either side of an interface permit simple description by means of a 2 人 2 matrix for the mth interface. The relationship between the field components is
(E+ ) (E '+ )
| m--1 | = Qm-1,m | m- | ,
\Em-1 ) \E 'm )
where
1 「 1 Tm-1,m ]
Qm-1,m = tm-1,m L(|)Tm-1,m 1
is the dynamical matrix, which is defined in terms of the usual reflection and transmission coefficients
Tm-1,m = nm-1 - nm and tm-1,m = 2nm-1 ,
nm-1 + nm nm-1 + nm
respectively. The field components on the left and right hand sides of the mth layer are related by the propagation matrix
(E'+ ) (E+ ) E'm-m = Pm Em-m ,
where
Pm = exp (jbm ) 0
and 6m =入(2冗) nm dm is the phase thickness of the mth layer, whose physical thickness is dm ; here
入 is the wavelength in free-space (air). The repeated application of the above transformations for the N layers and the N+1 interfaces leads to a product of (N+1) 2 〉2 matrices that then relates the total field in the left hand unbounded medium to that of the total field in the right hand unbounded medium:
(E+ ) (E'+ )
| 0- | = T | N-+1 | \E0 ) \E'N +1 )
where T is the system transfer matrix
「T1,1 T = |
LT2,1
T2,2 = Q0,1 PmQm,m+1 .
We set the incident, reflected and transmitted field components as Ei = E0(+) , Er = E0(-) and Et = EN(+)+1 , respectively. In terms of the transfer matrix components, r and t may be written as
Note that these quantities determined below.
E T E 1
r = r = 2,1 and t = t =
i 1,1 i 1,1
are in general complex. The reflectance and transmittance are
Assuming E'N(-)+1 = 0 , conservation of energy yields
2
E+
S+ = S- + S+
2n
where n= 仍 Since
c .
r = =
then
2 2
0
=
2n0(*)
t = =
不 r
E'+ 1
N+1
=
E+ T ,
0 1,1
2 ( n0(*) )
\nN(*)+1 ) .
For a lossless, nonmagnetic, medium, we then have
2 2 ( n )
and hence may define
R = 装 and T = T n(N)1 .
Suggested Functional Algorithm
(1) Set material parameters
Refractive indices
Reflection and transmission coefficients for each interface
Elements of the dynamical matrix – define as matrix
(2) Set design parameters
Centre wavelength
Layer thickness
(3) Calculate transfer matrix
In a loop over the wavelength (ranging from 400 to 1400 nm), do...
Define the propagator matrices
Do matrix multiplication for that wavelength
Extract reflectivity and transmissivity for that wavelength
(4) Plot reflectivity vs wavelength, calculate power production.
Note that this algorithm requires you to manually update the refractive indices. Alternatively, you could place steps (1) – (4) within loops over the relevant refractive index and/or layer thicknesses, modifying step (4) to test for maximum power production.
2022-11-07