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Coating Materials News & More: August 2021

A Note From The Materion Team

Like many of you, I was deeply touched by the passing of mentor, instructor, innovator and leader, Angus Macleod. We have dedicated this issue of Coating Materials News & More to Angus – a tribute to his contribution to the optical coating world. In this issue, you will be treated to a personal reflection by Sam Pellicori, a discussion of key building blocks Angus used to connect to engineers by Len Traub, and a guest contribution written by Angus himself from our CMN archives. Since joining the industry, I have been most humbled and impressed by my interactions with Angus over the years, from my first job, to my BOD position and my involvement with The Society of Vacuum Coaters.

We sincerely hope you enjoy this tribute to Professor Angus Macleod. 

David A. Sanchez

A Tribute to Angus Macleod

I have been fortunate to know Angus for more than 35 years, and I will remember him as an excellent teacher, gentleman, and friend. I mentally listed the world’s top 10 thin-film physicists of all time; Angus placed high among them. 

The Essential Macleod program is one of his many legacies. I ran the early version on a Macintosh 512K computer and while waiting for the calculations to finish I had time to drive into town for errands. The current program computes complex multi-layer designs in seconds and includes many unique analysis routines related to engineering applications. The program, designed by Angus and maintained by Christopher Clark, is and will remain among the most powerful and versatile thin-film program tools available.

As a consultant in thin-film optics, I have used the program daily. If I encountered difficult design problems or analyses, Angus would always graciously and thoroughly answer my questions, no matter how elementary.

I was fortunate to experience a six-week tutorial with Prof. Macleod when he taught at the University of Arizona, during which he made time to lecture me for two hours each day. It was a lot of material to absorb, but Angus was patient with me! He included me in morning group meetings with his graduate students during which I related real-world exposure to the applications of coatings. The members of that group now occupy leading positions in industry. I was never an official “student”, but I have always considered Angus to be my mentor, from whom there was always more to learn. The study and application of his work has contributed greatly to the success of my consulting business. He had no agenda beyond the pleasure of teaching thin-film optics and that excitement is conveyed in his students.

During an evening visit to his home, Angus shared his fondness for the best Scotch whiskey with me. It was my first such taste experience. This time it was not the equations that were making my head spin! Angus was amused. Needless-to-say, we had some happy times together! 

Encountering Angus and Ann and their family at the international Optical Interference Conference (OIC) was always a pleasure. They greeted everyone with interest and importance.

His book and many publications and presentations provide only a hint of the depth of his knowledge of the different components of thin-film physics. His book is standard reference for optical coating theory and applications. These contributions have continuing impacts on all optical applications including astronomy and aerospace, medicine, and commercial.

I join the international community who has had the honor and privilege of having known this great man and experienced his brilliance. His presence will be greatly missed around the world.

Written by: Samuel Pellicori


Some Simple Tools to Aid the Design of Optical Coatings

In Memory of Prof Angus Macleod

In April 2021, the optical thin film world lost one of its finest ambassadors and teachers with the passing of Prof. Angus Macleod. Anyone who attended his lectures or courses, will have appreciated his ability to explain often complex concepts in language accessible to all. After listening to him talk on even familiar subjects, invariably one came away with a sense of not only learning something new, but also of gaining a much greater depth of understanding of the topic in question.

His approach was always to start from the basic physics of electromagnetic radiation and in a logical and consistent manner, assemble a collection of mathematical tools applicable to addressing most design challenges. While nowadays, with the power of modern software, it is tempting to place the burden of arriving at a final design on sophisticated synthesis algorithms, Angus always warned of the pitfalls of relying on such an approach, emphasising instead the importance of coming up with a good starting design prior to optimisation through computer refinement. Using simple building blocks such as periodic structures and symmetrical matching layer combinations he provided us with a comprehensive collection of design examples for almost every conceivable application.

Although Angus is sadly no longer with us, his legacy lives on in his many publications, including his seminal work “Thin Film Optical Filters,” often referred to as “the bible” by generations of thin film practitioners, and in the industry leading thin film design software package “The Essential Macleod.”

In memory of Angus, this article describes some of the design tools he frequently referenced, and which are contained within the software, specifically transformed admittance, reflectance circle diagrams and Herpin Equivalent Indices and explores their applicability to the design of antireflection (AR) coatings.

Transformed Admittance

The optical admittance of a surface Y, which is numerically equal to its refractive index ns, tells us how easy or hard it is for light to pass through it. If we know the optical admittance of the medium (y0) and surface (Y), it is possible to calculate the intensity reflectance using the following simple expression.

Now let us extend this concept to a smooth surface coated with a single layer of a transparent dielectric material. An incident beam of light striking the coating will be divided into two reflected beams, one reflected from the outer surface and the other from the substrate/coating interface, as well as a transmitted beam. If the thickness of the coating is correct, the two reflected beams will be 180 degrees out of phase with one another and will partially cancel each other out, a phenomenon known as destructive interference. (Fig 1)

Fig 1 The principle of destructive interference


This condition is satisfied when the optical thickness of the coating (refractive index × physical thickness) is a quarter of the wavelength of the incident light. Furthermore, if the refractive index of the layer (nl) is the square root of the product of the indices of the incident medium (nm) and the substrate (ns), then the amplitude of the two reflected waves will be the same and complete cancellation will occur. A perfect antireflection coating will result, but only at the wavelength at which the phase condition is satisfied.

Looking at this another way, if no reflection occurs it must be the case that the admittance of the substrate/layer combination is identical to that of the medium.   

Mathematically, if 
In other words, the combination of the quarter-wave layer and substrate behaves just like a single surface of transformed admittance 


What relevance does this have to the design of antireflection coatings?

The problem is that an ideal material which would act as an antireflection coating for a 1.52 index glass needs to have an index of 1.23, which is lower than that which either nature or chemistry can provide. In the real world, magnesium fluoride (MgF2), with a refractive index of 1.38, is the most popular candidate, but this would require the substrate to have an index of 1.9 for it to act as a perfect antireflection coating.

Conveniently, we can create a transformed admittance of 1.9 by depositing a quarter-wave thick layer of index 1.7 onto the glass substrate. Now our outer layer of MgF2 will be ideal.

The reflectance of this two-layer structure is shown in comparison to that of a single layer of MgF2 in Fig 2. In each case the optical thickness of the layers is one quarter-wave at 510nm.

Fig 2 Comparison of single layer and two-layer AR designs


The Reflection Coefficient Circle Diagram

The transformed admittance is a convenient concept for calculating and optimising the reflectance of designs at the wavelength at which their layers are quarter-wave thick. However, for other wavelengths the calculations get a little more complicated. We need to consider not just the amplitude of the waves reflected from each interface but also their relative phases. Let us again consider a thin transparent layer of thickness d and refractive index n1 deposited on a substrate of index n2. (Fig3)

Fig 3 Multiple reflections from a thin film coated substrate


The general expression for the amplitude reflectance, or reflection coefficient, (r) at wavelength λ is:


The intensity reflectance (R) is given by 

This expression may seem a little complicated, however, for small values of r, it can be depicted graphically in a simple circle diagram with real and imaginary axes, an example of which is shown in Fig 4.

When the film thickness is zero, the reflection coefficient r, measured from the origin to the tip of the reflectance coefficient vector, is simply that of the uncoated substrate. As the film thickness increases, the tip follows a clockwise circular path, reaching a minimum for low index films (or a maximum for high index films) at an optical thickness corresponding to a quarter-wave and returning to that of the substrate at the half-wave point. 

Fig 4 Approximate reflectance circle diagram for small values of r1and r2


This simple diagram not only allows the designer to visualise the change in reflectance of the growing film at a single wavelength, as might be seen for instance on an optical monitor, but also importantly to predict what happens when further layers are added.

This is illustrated in Fig 5 for our 2-layer design derived earlier. In this case the final reflectance for layer 1 becomes the starting reflectance for layer 2.

Fig 5 Reflectance circle diagram for two-layer AR


The first quarter-wave layer causes the reflection coefficient to increase from 0.21 to 0.31 whilst the second (outer) layer reduces it to close to zero.

Unfortunately, an index of 1.7 is difficult to obtain from standard coating materials, although aluminum oxide (Al2O3) comes close with an index of approximately 1.65. The semi-circular locus of the reflectance coefficient corresponding to a quarter-wave of this index can however be closely simulated by two sub-quarter-wave layers of tantalum pentoxide (Ta2O5) and MgF2 as shown in Fig 6:

Fig 6 Reflectance circle diagram for three-layer AR


The Herpin Equivalent Index

An alternative means of synthesising an intermediate refractive index is to use a symmetrical combination of higher and lower index materials with a total phase thickness of one quarter-wave. By adjusting the relative thicknesses of each material within this symmetrical period whilst keeping the total thickness constant it is possible to create a Herpin Equivalent Index with any value between those of the two materials. This task can be greatly simplified by using the symmetrical period design tool contained within the Essential Macleod. This yields two solutions of the form HLH or LHL. Either can be selected, with the choice being mainly dictated by which has the more favourable index dispersion characteristics.

The circle diagram corresponding to the HLH solution for the index of 1.7 using Ta2O5 for the high index and MgF2 for the low index is shown in Fig 7. Again, the addition of an outer quarter-wave of MgF2 results in close to zero reflectance at the design wavelength.

Fig 7 Reflectance circle diagram for four-layer AR


A comparison of the reflectance spectra of these two designs is shown in Fig 8. It is evident that while both perform well at the design wavelength of 510nm, the reflection increases quite sharply at both longer and shorter wavelengths.

Fig 8 Comparison of three-layer and four-layer AR designs


Use of Half-wave Layers

The bandwidth of the low reflectance region can however be dramatically increased by a simple trick – the insertion of a half-wave of high index material as the penultimate layer. This results in the reflectance circle diagram shown in Fig 9:

Fig 9 Reflection circle diagram for four-layer AR with additional half-wave


The reflectance spectrum of this design, before and after Simplex refinement is shown in Fig 10:

Fig 10 Comparison of unrefined and refined designs with additional half-waves


It is immediately evident that the addition of the half-wave of high index material has extended the region of low reflectance to cover most of the visible spectrum. But why is this so? Again, circle diagrams can give us some insight.

The following diagrams (Fig 11) are for the refined design at wavelengths 450nm (blue) and 600nm (red). In both cases, the final reflectance remains close to zero, effectively pinned to the origin by the regulating presence of the half-wave layer.

Fig 11 Reflectance circle diagrams for refined design at shorter and longer wavelengths


We are indeed fortunate that software such as the Essential Macleod allows almost instantaneous performance computation and optimisation of designs such as these. Nevertheless, the simple tools described in this article still play an important role in aiding our understanding as to why specific layer sequences behave the way they do as well as helping us come up with viable starting designs and design improvements.


About The Author

Len Traub is Technical Director of P&T Consulting Ltd, Materion’s representative for optical PVD materials in the UK, Ireland and the Nordic Countries. P&T Consulting is also the European Agent for Thin Film Center Inc, the provider of the Essential Macleod software.

Over the past 20 years, Len has provided consultancy and training services to manufacturers and users of optical thin film products throughout the world. Prior to that he worked in several senior technical and business development roles at OCLI Ltd in Hillend, Scotland. Len is a Physics graduate from the University of St Andrews. He is married with one son and two granddaughters and in his spare time enjoys hill-walking and furniture restoration.


Coating Materials News & More Volume 31, Issue 3 - August 2021

  • From the Archives: Pitfalls in Thin-Film Optical Property Measurement
    Reaching back into the archives, this Macleod article explores common pitfalls in thin film optical property measurement. These pitfalls include poorly matched film and model behavior, inadequate, and even incorrect, measurements.


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