The design of a polarised optical filter is more complicated than that of a filter where the polarisation effect does not exist (at a normal angle
of incidence). An error in the optical parameters, such as the physical thickness or refractive index of a layer, results in a change in the
spectral performance of the multilayer structure. The correlation between error sensitivity and the polarisation effect of light in structures
designed at an oblique angle was investigated. To illustrate the correlation, a perpendicular (S) and parallel (P) polarised beam splitter, at
0.9818 µm central wavelength, designed by genetic algorithm, was used. The beam splitter changes its state of polarisation according to
the error in thickness simultaneously induced in each of the layers. The error was calculated by optimising the original design. The observation of
the change of the state of polarisation as a result of error sensitivity leads to a different method of designing pure Spolarised or Ppolarised
optical filters.
In the design and manufacture of a multilayer structure, the sensitivity of layers to error needs to be given special consideration. Errors in the
optical parameters, such as thickness and refractive index, can have various causes. The sensitivity of the layers to the induced error results in a
change in the spectral performance of the structure as a result of a change in the interference pattern at the layer interfaces where the error is
induced. The error effect on a multilayer structure designed at an oblique angle can be intense as a result of polarisation. A light incident at an
oblique angle has two sets of beams: Spolarised and Ppolarised beams. Because of a difference in the phase shift in the Spolarised and
Ppolarised beams, the effect of the error seems noticeable in structures designed at an oblique angle of incidence.
^{1} We analysed the error sensitivity of a multilayer structure designed at an oblique angle of incidence. The error sensitivity of each of the layers
of a multilayer structure gives information on the effect it has on the change in the optical behaviour of S and P components of the beam. The
errorinduced behavioural change of Spolarised and Ppolarised components leads to a change in the state of polarisation of the structure as a
system. This observation gives an important clue to the method of designing a pure Spolarised multilayer structure.
Sensitivity analysis
A thin film structure with n number of layers can be represented by a 2×2 matrix, which contains the optical parameters (refractive
index, physical thickness, etc.) of a thin film.
^{2,3,4,5}Reflectance and transmittance of an optical filter can be calculated by using the 2x2 matrix of the multilayer structure surrounded by two
substrates.^{3} For this study, a Matlab code computer program,^{6} which calculates
reflectance and transmittance, was prepared using the matrix representation method of a multilayer structure. Sensitivity is often related to errors that are induced during the manufacturing process, as a result of human or instrumental error, as one loses
control of precision.^{4,7} Hence, it is useful to study the sensitivity factor, as it has a
significant effect on the final spectral
performance of the manufactured thin film. The optical thin film parameters that are prone to error are the physical thickness and the refractive
index. A sensitivity study helps to reveal the tolerance level of the thickness or refractive index of each layer.^{8}
By investigating the
sensitivity, it is possible to see whether the design developed is manufacturable in the existing depositing conditions. Based on the concept of
sensitivity, it is possible to induce errors in one of the parameters deliberately and adjust the design to obtain a desired spectral performance.
In this study, we exploited the concept of sensitivity in converting an Spolarised design into a Ppolarised one and vice versa. This conversion
can be done by introducing a random change to all the layers simultaneously. Using Beumeister’s sensitivity analysis method, it is possible to analyse the sensitivity effect of only one layer at a
time.^{4,8}
Onelayerat–atime variation involves a partial derivative of reflectance. The partial derivative of the matrix with respect to quarter wave
thickness can be evaluated as in Thelen^{8}. The sensitivity of the whole system, as a result of parameter variation in a single layer, can be calculated by the derivative of the reflectance
over a desired spectral range.^{5,8,9} The result gives a good indication
of how the spectral performance is affected by a parameter change.
If the parameter involved is simultaneously changed in all the layers, the effect will be high and the result can be a complete change of the
spectral performance. It is important to have a different form of analysis if we are interested in simultaneously changing the parameter in all the layers. For this
study the method used was slightly different. Firstly, the variation in thickness needed was determined. Secondly, by simultaneously adjusting the
thickness of each of the layers with the determined variation, the polarisation state of the design was changed. The determination of the exact
thickness variation values appropriate for each layer can be rather difficult, but with the help of the optimisation computer program developed in
Matlab code using the Matlab optimisation tool box, the difficulty can be minimised.^{10} For this study, three computer programs were developed. The first program calculated the transmittance, reflectance and merit value of the
multilayer structure.^{11,12} The second program is a classical optimisation program that
calls programs from the Matlab optimisation
toolbox and the developed program that calculates merit values. The third program is a genetic algorithm program that calls programs from the
genetic algorithm tool
box^{6,13,14,15,16} and the program
that calculates merit values. An original design is prepared using the genetic
algorithm based program. The result of the original design is optimised using the classical optimisation based program. Simultaneous variation of thickness or refractive index needs a different kind of treatment. We investigated a different way of varying all the
layers simultaneously, resulting in the change of the state of polarisation. The calculation of thickness variation involves the optimisation of the
original design. For instance, the original Spolarised design undergoes Ppolarised optimisation to obtain a second set of parameters. The
variation between the original and the second set is then determined and used as the variation in thickness of each layer. When all the layers of
the original design are simultaneously subjected to this variation, the design changes its state of polarisation.
To illustrate the method of designing polarised filters using the concept of sensitivity, a genetic algorithm designed beam splitter at an oblique
angle of incidence was considered. Two 50/50 (percentage transmittance/percentage reflectance) beam splitters were designed with the following
characteristics: a central wavelength of 0.9818 µm, a bandwidth of 0.2 µm, an angle of incidence of 45°, and surrounding substrates
with a refractive index value of 1.46. The first original design was the Swave beam splitter with a thickness of 24.5072 µm (Figure 1) and
the second original design was a Pwave beam splitter with a thickness of 20.3629 µm (Figure 2). In the genetic algorithm system, the physical
thickness is used as a variable and allowed to vary within the boundaries of 0.9818 µm and 0 µm. The refractive index is not
allowed to vary because the optimisation is conducted on only one of the parameters. The design consists of two materials – one with a high
refractive index (H) and one with a low refractive index (L) – in alternating layers with a structure of HLHL. The two materials used in the
designs were SO_{x}N_{y} materials with refractive index values of 3.2 (H) and 1.7 (L). The optimisation was run for 15 000
generations.

FIGURE 1:
Spectral performance of the Swave beam splitter original design. The
Swave curve is indicated by the continuous line, the Pwave curve in dashdash
and the desired curve in dashdot.



FIGURE 2:
Spectral performance of the Pwave beam splitter original design. The
Pwave curve is indicated by the continuous line, the Swave curve in dashdash
and the desired curve in dashdot.



FIGURE 3:
Spectral performance of the Swave beam splitter design after
thickness alteration. The Pwave curve is indicated by the continuous line, the
Swave curve in dashdash and the desired curve in dashdot.


Figure 1 shows that the Swave was split in a 50/50 proportion while the Pwave was split in an 80/20 proportion. In a design using a genetic
algorithm, the program affects only the Swave and does not control the Pwave. To study the effect of sensitivity, the thickness of all layers of
the original design need to be altered simultaneously. The effect of the alteration results in the spectral behaviour change of the original design.
This result is shown in Figure 3, in which the Pwave was split in a 50/50 proportion, whereas 80% of the Swave was reflected. It can be seen here
that the Swave beam splitter design was changed into a Pwave beam splitter as a result of the sensitivity of the original design to the induced
thickness errors.

FIGURE 4:
Spectral performance of the Pwave beam splitter design after
thickness alteration. The Swave curve is indicated by the continuous line, the
Pwave curve in dashdash and the desired curve in dashdot.


The spectral performance of the Pwave beam splitter original design is shown in Figure 2. The Pwave was split in a 50/50 proportion and almost
85% of the Swave was reflected. This design is intended to control only the Pwave, which is why we did not see any control over the reflectance
of the Swave. Each layer in this design is subjected to a simultaneous alteration. The alteration results in the complete change of the optical
behaviour of the original design in the same fashion as the first design does. The changed spectral performance is shown in Figure 4. The Swave
was split in a 50/50 proportion and the Pwave in an 80/20 proportion. When studying the original design of the Pwave and the resulting design
after thickness alteration, it could be seen that the Pwave beam splitter had been transformed into an Swave beam splitter. It is important to know that neither of the two original designs purely polarise the Swave or Pwave, but are intended to split the Swave or
Pwave into a 50/50 proportion, depending on the design. For instance, the Swave beam splitter design is only intended to control the Swave in a
50/50 proportion while the Pwave is uncontrolled. The designs, after undergoing thickness adjustment, change their spectral behaviour depending on
their sensitivity to error. Any design will react to an error in any of the parameters by changing the spectral behaviour. The two examples
illustrate the concept of sensitivity in the design of Spolarised or Ppolarised beam splitters. This concept is one we used in the process of
finally obtaining a pure polariser. The spectral performance of a beam splitter that splits the Pwave into a 50/50 proportion is illustrated in Figure 3. It results from the
thickness adjustment of the Swave original design. The spectral performance of a beam splitter that splits the Swave into a 50/50 proportion
results from the thickness adjustment of the Pwave original design and is shown in Figure 4. We have also shown that a relatively pure Swave beam splitter can be obtained when the beam splitter of the original design, either the Swave or
the Pwave, and the designs obtained after thickness alteration are placed in series.^{1} The reflectance of a
thin film that is constructed
with a seriesorder arrangement of the first original design with its adjusted design is illustrated in Figure 5. It is possible to see that the
Pwave was almost 95% transmitted, whereas the Swave was split in a 55/45 proportion. Although the Swave was not split in an exact 50/50
proportion, the 55/45 result is promising.

FIGURE 5:
Spectral performance of the Swave beam splitter design in series
arrangement with the same design after thickness alteration. The Swave curve
is indicated by the continuous line, the Pwave curve in dashdash and the
desired curve in dashdot.


The overall reflectance of a thin film constructed with a seriesorder arrangement of the Pwave original design with its adjusted design is shown
in Figure 6. The Swave was split close in an almost 50/50 proportion and the Pwave was 95% transmitted. It is very interesting to see that this
result is far better than the previous one (Figure 5) when the rejection region is analysed. Hence, a better, pure Swave polarised beam splitter
can be constructed using the Pwave original design and its adjusted design.

FIGURE 6:
Spectral performance of the Pwave beam splitter design in series
arrangement with the same design after thickness alteration. The Swave curve
is indicated by the continuous line, the Pwave curve in dashdash and the
desired curve in dashdot.


It also is interesting to see from the results that the concept of sensitivity is a powerful tool in understanding the factor responsible for the
polarisation state change of a thin film design. Sensitivity is an important factor in the design of a pure polariser. This work can be extended to
investigate the effect of refractive index parameter variation, but we do not expect the results would be as promising as those obtained from
physical thickness parameter alteration.
We have successfully related the concept of sensitivity to the design of a polarised multilayer structure. It is important for a designer to
control each layer’s sensitivity to parameter error. The importance of considering the sensitivity effect, when preparing a design, is
demonstrated in the design of the Swave and Pwave beam splitter. The change in the polarisation state of a design, from an error deliberately
induced, illustrates the use of the concept of sensitivity in designing polarisers. The concept of sensitivity is also demonstrated when a pure
Spolarised beam splitter is designed. The findings of this study can be used in the design of pure polarised optical filters for optical
communication applications.
We would like to thank the Department of Electrical and Electronics Engineering Science in the Faculty of Engineering and the Built Environment at
the University of Johannesburg, the National Research Foundation (South Africa), and the Technology and Human Resources for Industry
Programme for funding the research.
Competing interests
We declare that we have no financial or personal relationships which may have inappropriately influenced us in writing this paper.
Authors’ contributions
E.K.E. performed the work and B.M.L. supervised the work.
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