text:
Engaging students: Finding the area of regular polygons

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Conner Dunn. His topic, from Geometry: finding the area of regular polygons.

How can technology be used to effectively engage students with this topic? Note:  It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

A good way to get students into the concept and see it’s real life use is to be given a realistic problem. The natural world doesn’t typically give you perfectly regular polygons, but we certainly like making them ourselves. Better yet, we can make regular polygons of a certain area knowing the methodology behind computing their areas. Using GeoGebra, I can challenge students to construct a regular polygon of an exact area using what they know about the use of equilateral triangles to compute area.

For example, let’s say I ask for a hexagon with an area of 12√3 square units. While there’s a few strategies of constructing a regular hexagon that Geometry students may know of, the strategy to shoot for here is to recognize this means we want a hexagon with a side length of 2 then construct the triangles. GeoGebra allows for students to use line segments and give them certain lengths, as well as construct angles using a virtual compass tool. Below is the solution to this example by constructing 6 equilateral triangles (each with an area of 2√3 square units) to form the regular hexagon.

How does this topic extend what your students should have learned in previous courses?

By the end of the unit, students will have learned the formula for finding the area of a polygon (A = (1/2) * a * p, with a being the apothem, and p being the perimeter). But a big part of this unit is how we derive the formula from the process in which we solve the area using this equilateral triangle method. From many previous courses, students will have learned both the order of operations and properties of equality in equations, and we use this previous knowledge to connect a geometric understanding of area to an algebraic one. For example, when we have the idea of multiplying the area of an equilateral triangle by the number of sides, n, in the polygon, we have A = (1/2) b * h * n. It is by the use of communitive property that students can rearrange the variables like this: A = (1/2)h*b * n. And then we conclude that the b*n reveals that the perimeter of the polygon plays a role in our equation. This may seem subtle, but students being fluent in this knowledge helps them work in their geometric understandings much easier.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

A big part of the method for understanding area of polygons is seeing how we can perfectly fit equilateral triangles inside of our polygons of choice. Perfectly fitting shapes into and around other shapes is something you see in mosaic art everywhere, particularly in Islamic architecture.

While mosaic artists are not necessarily calculating the areas of their art pieces (they might but I doubt it), a big part what makes these buildings so nice to look at is how the shapes fit with one another so nicely. This is an art that’s very intentional in its aesthetically pleasing aroma. This is something I think Geometry students can take to heart when confronting Geometry problems (a just as well with future courses). It’s the overlooked skill of literally connecting pieces together in order to get something we want. In the case of the Islamic architect, what we want is a pleasing building to look at, but math, of course, brings in more possible things to shoot for and equips us with plenty of pieces to (literally) connect together.

Engaging students: Finding the area of a circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Brendan Gunnoe. His topic, from Geometry: finding the area of a circle.

History: Squaring the circle

The ancient Greeks and other groups at the time had a fascination with geometry. These cultures tended to like thinking in terms of simpler geometric shapes, such as circles, equilateral triangles and squares. One of the classic problems proposed by these ancient peoples was “Can you create a square with the same area as a circle with finitely many steps only using a compass and straightedge?”. This problem stood for thousands of years, stumping even the most brilliant of mathematicians that attempted to show it true. Eventually, in the year 1882, it was finally proven impossible because of a property of the number π. It’s not too hard to show that π isn’t an integer, nor is it rational. What was left to show is whether π was algebraic or transcendental. The proof from 1882 showed that π is in fact transcendental, proving that it cannot be made using the rules set out by the original question. If a number is algebraic, then it is a solution to a polynomial with rational coefficients.

Curriculum: Using limit of triangular approximations to get the integral

The teacher starts off class by drawing a circle with an inscribed triangle, another with a square, and so on until a hexagon is inscribed. The teacher then draws isosceles triangles that originate at the circles center and extend to the corners of the polygons. The teacher could ask questions like “What do you notice about the total area of the triangles and the area of the circle as we keep adding sides to the polygon?” and “What do you notice about the triangles we made and the little wedges of the circle, what’s the same and what’s different about them?”. Then the teacher could arrange both the triangles and wedges in an alternating up and down fashion, almost like two zippers, to line up the triangles and wedges. The teacher could ask “What’s the length of the top of the triangles? What about the tops of the wedges, what’s their length?”.

Finally, the teacher asks “What happens when we let the number of pieces gets REALLY big? What happens to difference between the area of the triangles and wedges? What about the tops of the triangles and the tops of the wedges?”. In the limit, the upper edge converges to half of the circumference of the circle and the height of the triangles converges to the radius of the circle. Using this line of thinking, the teacher guides the students into seeing how you can derive the equation for the area of a circle by using approximating it with triangles, and then looking at what happens in the limit.

Application

A telescope’s lens is what’s used to control how much light gets into the eye piece. Suppose you’re an astronomer and want to take a photo of the full Moon on a clear night, which gives off 0.25 lumens/s-m2. Suppose your camera needs to get a total of at least 3 lumens to produce a good photo and 5 lumens to get an amazing photo. What’s the radius of a lens (in centimeters) that can take a good photo in 10 minutes? What’s the radius of a lens that can take an amazing photo in 10 minutes?

Now suppose you’re working with the Hubble space telescope in low Earth orbit trying to get photos of a nearby star system. The radius of the main telescope is 120cm and the star system you want to observe is giving off light at a rate of 10-5 lumens/s-m2. How long will it take to get a good photo with Hubble? What about a great photo?

https://en.wikipedia.org/wiki/Hubble_Space_Telescope

https://en.wikipedia.org/wiki/Squaring_the_circle

Engaging students: Computing the determinant of a matrix

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Brendan Gunnoe. His topic: computing the determinant of a matrix.

How can this topic be used in your students’ future courses in mathematics or science?

When students learn about the determinant of a matrix, they only learn about computing it and don’t learn about the applications of the determinant or what they signify. One interesting use of the determinant is finding the eigenvectors of a matrix. A visual understanding of what an eigenvector is can be done by showing what a matrix does to the any generic vector, and what it does to the eigenvectors. For a generic vector that is different from an eigenvector, the matrix knocks the vector off the span of the original vector. What makes an eigenvector special is the fact that the matrix transformation keeps the eigenvector on its span but rescales said eigenvector by its eigenvalue. For example, take the matrix

$\left[ \begin{array}{cc} 5 & 3 \\ 3 & 5 \end{array} \right]$.

This matrix’s eigenvectors are $\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$ and $\left[ \begin{array}{c} 1 \\ -1 \end{array} \right]$ with eigenvalues 8 and 2 respectively. That is,

$\left[ \begin{array}{cc} 5 & 3 \\ 3 & 5 \end{array} \right] \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] = \left[ \begin{array}{c} 8 \\ 8 \end{array} \right] = 8 \left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$

and

$\left[ \begin{array}{cc} 5 & 3 \\ 3 & 5 \end{array} \right] \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] = \left[ \begin{array}{c} 2 \\ -2 \end{array} \right] = 2 \left[ \begin{array}{c} 1 \\ -1 \end{array} \right]$.

Eigenvectors have many useful applications in future math and science classes including electronics, linear algebra, differential equations and mechanical engineering.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

The YouTube channel 3Blue1Brown has a fantastic video on determinates and linear transformations. Grant, the channel owner, uses animations to visualize what a matrix transformation does to the plane . He starts by showing what a transformation does to a single square then shows why the change of change of that one area shows what happens to the area of any region. He also gives multiple explanations for what a negative determinate means in terms of orientation of the axes. Then he explains what happens when the determinate is 0. All of this is already extremely useful for understanding what a 2×2 matrix does, but Grant continues and extends all the same things for 3×3 transformations. Lastly, Grant gives a few explanations on why the formula for the determinate is what it is and poses a small puzzle for the viewer to contemplate. This video explains what and why we use determinates and how they can be useful all while showing pleasing visual examples and other explanations.

What interesting word problems using this topic can your students do now?

Using determinates to see if a set of vectors is a basis.

The determinant can tell you when a matrix squishes space into a lower dimensional space like a line or a plane. Thus, taking the determinate of a matrix composed of a set of vectors tells you if those vectors are a basis for the matrix’s dimension.

Part 1. A 3D printer’s axes are set up in such a way that the print head can only travel in the direction $\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$ and $\left[ \begin{array}{c} -1 \\ 1 \end{array} \right]$. Assume that the place where the print head is right now is the origin $\left[ \begin{array}{c} 0 \\ 0 \end{array} \right]$. Can you move the print head to the location $\left[ \begin{array}{c} x \\ y \end{array} \right]$ and $\left[ \begin{array}{c} 1 \\ -1 \end{array} \right]$ by only moving in the directions of $\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$ and $\left[ \begin{array}{c} -1 \\ 1 \end{array} \right]$?

Hint: Try to solve $\left[ \begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array} \right] \left[ \begin{array}{c} a \\ b \end{array} \right] = \left[ \begin{array}{c} x \\ y \end{array} \right]$ . Does this always have a solution $\left[ \begin{array}{c} a \\ b \end{array} \right]$?

Part 2. Next time you turn on your 3D printer, one of the motor’s broke and now the print head can only move in the direction of $\left[ \begin{array}{c} 1 \\ 0 \end{array} \right]$. Assume that the place where the print head is right now is the origin $\left[ \begin{array}{c} 0 \\ 0 \end{array} \right]$. Can you move the print head to the location  by only moving in the direction of $\left[ \begin{array}{c} 1 \\ 0 \end{array} \right]$?

Hint: Try to solve $\left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right] \left[ \begin{array}{c} a \\ b \end{array} \right] = \left[ \begin{array}{c} x \\ y \end{array} \right]$ . Does this always have a solution $\left[ \begin{array}{c} a \\ b \end{array} \right]$?

Part 3. You buy a new 3D printer that it can move in all three directions: up/down, left/right, forward/backwards. When you test out the movement of the print head, you see that each axis moves in the directions of $\left[ \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right]$, $\left[ \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right]$, and $\left[ \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right]$. Can you use your new 3D printer to go to any location $\left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$, inside the printing space?

Hint: Think about solving $\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} a \\ b \\ c \end{array} \right] = \left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$ . Does this always have a solution $\left[ \begin{array}{c} a \\ b \\ c \end{array} \right]$? How do you know?

Part 4. Your little sibling messed around with your new 3D printer and now it moves in the directions $\left[ \begin{array}{c} 1 \\ 0 \\ 1 \end{array} \right]$, $\left[ \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right]$, and $\left[ \begin{array}{c} 2 \\ 1 \\ 1 \end{array} \right]$. Is your 3D printer able to get to some point $\left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$ inside the printing space as is, or do you need to fix your printer?

Hint: Think about solving $\left[ \begin{array}{ccc} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} a \\ b \\ c \end{array} \right] = \left[ \begin{array}{c} x \\ y \\ z \end{array} \right]$. Does this always have a solution $\left[ \begin{array}{c} a \\ b \\ c \end{array} \right]$? How do you know?

Engaging students: Finding the equation of a circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Noah Mena. His topic, from Precalculus: finding the equation of a circle.

The equation of a circle relies on knowing the definition of a circle, knowing the radius and deciding where the circle is centered at. All of these come into play when a student has to find the equation of a circle. It takes basic understanding of the cartesian grid and understanding the coordinate system.  The equation of a circle also builds on students being able to manipulate the equation to get it into standard form and identifying the equation of a circle when it is expanded out. The shape of a circle should also be known, which means with the equation of a circle, students should be able to construct the perfect circle according to the given specifications in the equation.

Learning to write the equation of a circle can be difficult. For one of my teaches last semester my mentor teacher suggested the use of a desmos paired with a worksheet to allow the students to explore what changes the standard equation of a circle. The worksheet had the students enter certain coordinates into the graphing calculator and write down what they thought was the equation of a circle. The next part of the assignment was student driven by having them share their conclusions on what the equation for a circle would be when it is centered at the origin vs. centered at (h,k). The worksheet shows that the students drove their own learning and came to their own conclusions which enhanced engagement through the lesson.

This topic can come up again in trigonometry, upper level calculus and in math modeling. In my TNTX math modeling course, we took a closer look at the derivation of this equation and the subtleties of the standard form. This topic may also be used in physics calculations or in general, science labs.  For a physics word problem, it may ask you to calculate the net force and acceleration of a moving object around a circle. In this instance, it would suffice to just know the definition and general shape of a circle to complete these calculations. The definition of a circle is also needed to calculate centripetal force.

Engaging students: Solving exponential equations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Austin Stone. His topic, from Precalculus: solving exponential equations.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Exponential equations can be used in lots of different kinds of word problems. One that is pretty common but is very useful for students involves interest rate. “Megan has $20,000 to invest for 5 years and she found an interest rate of 5%. How much money will she have at the end of 5 years if the interest rate compounds monthly?” I would give them the formula A=P(1+r/n)rt. It is pretty easy to convince students that this is a real-world problem and would get the students engaged about exponential equations. You can also reword the problem to ask for how much Megan started with, what the rate is, or how much time the money was in there. That way students get used to solving equations when the variable is in the exponent and when it is not. This also can lead into or us prior knowledge of natural log to solve for the variable in the exponent. How could you as a teacher create an activity or project that involves your topic? Using the basis of the problem I mentioned above, a teacher could create a Project Based Instruction lesson using this idea. The teacher can set up a scenario where, over the course of a week or two, the students would have to decide which bank to make an investment in by calculating how much money they would profit at each bank. The students would have to research different banks and their interest rate. The teacher could also give each group different scenarios where some groups have more money to invest. Students would have to figure out how long they would like to invest. The teacher would give Do It Yourselves and Workshops that deal with solving exponential equations and also getting used to natural log. They would then make a presentation explaining what bank they have chosen and why. They would also have to explain the math that they would have used. How has this topic appeared in the news? To say that exponential equations have been in the news lately would be an understatement. It has virtually been the news this year. COVID-19 is a virus and viruses spread exponentially. This would get students engaged immediately because the topic would be relatable to their own lives. Doctors and scientists try to figure out different ways to “flatten the curve”, which essentially means to make the spread of the virus not exponential anymore. We have all heard people on the news telling the public how to stop the virus from spreading and how not make people around you at risk of contracting it (contributing the exponential spread). We all have most likely seen a doctor or scientist show a graph of the virus’s spread and their predictions on how it will look in the upcoming weeks. This would give students a chance to see that what they are learning can be applied to very crucial things going on in the world around them. References Exponential Functions Engaging students: Compound interest In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission comes from my former student Mason Maynard. His topic, from Precalculus: compound interest. What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.) • A deposit of$3000 earns 2% interest compounded semiannually. How much money is in the bank after for 4 years?
• A deposit of $2150 earns 6% interest compounded quarterly. How much money is in the bank after for 6 years? • A deposit of$495 earns 3% interest compounded annually. How much money is in the bank after for 3 years?

These word problems are some of the basic compound interest problems that your students learn how to do where you just plug in the correct values for their corresponding variables.

• If you invested $1,000 in an account paying an annual percentage rate (quoted rate) compounded daily (based on a bank year of 360 days) and you wanted to have$2,500 in your account at the end of your investment time, what interest rate would you need if the investment time were 1 year, 10 years, 20 years, 100 years?
• If you invested $500 in an account paying an annual percentage rate compounded quarterly , and you wanted to have$2,500 in your account at the end of your investment time, what interest rate would you need if the investment time were 1 year, 10 years, 20 years, 100 years?

These are the types of problems that get more difficult for the students. You want them to use compound interest to solve but then they must incorporate logs into their solutions because they are looking for time instead of interest.

How does this topic extend what your students should have learned in previous courses?

With compound interest, students first learn about the simple interest formula. The only main difference is that you start to include exponents with compound interests. Then when you introduce your students to compound interest, you start to get into some more complicated problems. After they learn about compound interest and its basic problems, then you transition into logs with your students. This is used in compound interest and instead of just looking for the interest that will be accumulated after a specific amount of time, you then shift the variable around that you are looking for. The most coming type of problem that refers to this is they give you all of the information except for the amount time it takes to get a certain amount of interest. The last thing that leads up to compound interest in Calculus is when you transition from calculating the amount of interested over specific time intervals and a specific amount of times you compound it to calculating it with compounding it continuously over a specific time interval.

How have different cultures throughout time used this topic in their society?

Interest is something you have to pay on a load. Depending on what side you are and how thinks go, you are either getting some more money back that what you invested or you are paying off a massive debt. Some think that the idea behind charging loans on interest came from the early days of neighbors loaning there cattle to one another. What is really unique about this is that the words in the Egyptian, ancient Greek and Sumerian languages is connected to cattle and their offspring. This leads some to believe that interest came about due to the natural increase of the herd that occurred when you loaned out your cattle.

The first evidence that comes of a compound interest problem dates back to 2000-1700 B.C. in Babylon. A clay tablet was found and the unique thing is that the interest rate use to solve it was not written. Some researchers assume that the rate was 20% due to that mainly all the other compound interest problems dating back closer to this used it. What is really crazy is that 20% worked to solve the problem. The only thing that was wrong was that the time was corresponding to the Babylon calendar of 360 days instead of our 365 days.

In 50 B.C. Cicero writes to a friend in Rome. The letter tells that he would not normally recognize more than 12 percent interest on a loan, even though a decree was passed which required money lenders to charge no more than 12 percent. Cicero would then write a few days later that they will pay back the loan in 6 years will 12 percent interest and more money will be added each year.

Resources:

Compound Interest History:

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/799CB1D40CDD46F3010767BFC60F24DB/S1357321719000254a.pdf/emergence_of_compound_interest.pdf

Word Problems:

https://www.basic-mathematics.com/compound-interest-word-problems.html

http://www.sosmath.com/algebra/logs/log5/log51/log51.html

Engaging students: Infinite geometric series

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Brendan Gunnoe. His topic, from Precalculus: infinite geometric series.

Curriculum:

Students can use the formula for an infinite geometric series to discover the formula for a finite geometric series. The teacher would start by posing the question “Can we use the infinite geometric series to come up with a formula for the finite version?” and writing out a series like so

$\displaystyle \sum_{i=0}^\infty ar^i = ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n + ar^{n+1} + \dots$

Next, the instructor could ask questions like “If we’re looking for the sum up to the nth term, where do we need to chop off the terms to get what we want?,” “Does the ending part look familiar?”, and “How can we rewrite the chopped off part so that it looks like what we already know?”. The teacher guides the students into manipulating the formula to get this result

$\displaystyle \sum_{i=0}^\infty ar^i = ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n + ar^{n+1} + \dots$

$\displaystyle \sum_{i=0}^\infty ar^i = ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n + \sum_{j=n+1}^\infty ar^j$

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle \sum_{i=0}^\infty ar^i - \sum_{j=n+1}^\infty ar^j$

The teacher notes that the last sum can be simplified to make it easier to see by doing a substitution of $k = j -n-1$. Adjusting the bounds and substituting in the new index, we get

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle \sum_{i=0}^\infty ar^i - \sum_{k=0}^\infty ar^{n+1+k}$

$= \displaystyle \sum_{i=0}^\infty ar^i - \sum_{k=0}^\infty ar^{n+1}r^k$

$= \displaystyle \sum_{i=0}^\infty ar^i - r^{n+1} \sum_{k=0}^\infty ar^k$

Note that the two sums are identical, besides the index name, so we can factor and get

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = a(1-r^{n+1}) \displaystyle \sum_{i=0}^\infty r^i$

Lastly, we utilize our formula for an infinite geometric series and get

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = a(1-r^{n+1}) \displaystyle \frac{1}{1-r}$

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle a\frac{1-r^{n+1}}{1-r}$

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle a\frac{r^{n+1}-1}{r-1}$

Although the infinite series requires $|r|<1$ , the finite version works for all real $r$. Although the formal proof that this is the correct formula might be beyond the scope of the intended class, it can easily be done with induction.

Technology:

Sal Khan, one of recent history’s most well-known STEM educators, has a fantastic video that shows the relationship between a fractal known as the Koch snowflake and the geometric series. Khan works through the derivation of the formulas for the perimeter and area of an the nth iteration of the Koch snowflake. It turns out that both the area and perimeters for each iteration can be expressed using a geometric series, but the perimeter diverges to infinity while the area converges. Such a result makes sense intuitively since you can fit every iteration inside of a finite box that is slightly larger than the snowflake, and thus bounding the area, yet it would require an infinitely long wire to go around the perimeter of the limiting shape. Since fractals are not normally included in the math curriculum, showing how math can be used in interesting and different ways to solve problem can be very engaging for students.

Culture:

There is a strong connection between geometric series, fractals, and self-similarity, all with a relatively simple nature. Fractals have been used in architecture and art for a very long time. Examples of self-similarity seen in ancient cultures include Hindu temples, with their structure being composed of self-similar units, and Islamic geometric art found in the domes of mosques.

Since the invention of the computer in the mid-20th century, more detailed and intricate digital art has been made popular. Although not exactly a geometric series, the Mandelbrot set acts very much like a fractal and was among the first of the uses of a computer to investigate the properties of fractals. It has been used in many ways to make animations, photos and other digital arts.

Another link between fractals and art can be found in the Legend of Zelda games. One of the iconic symbols of the game is called the triforce, which is an equilateral triangle that’s been cut into 4 smaller triangles with the middle piece removed. Such a shape is the first iteration of a fractal known as the Sierpinski triangle. As you can see, fractals can be found in all kinds of art, coming in many different forms.

https://en.wikipedia.org/wiki/Fractal_art

Engaging students: Computing trigonometric functions using a unit circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Alizee Garcia. Her topic, from Precalculus: computing trigonometric functions using a unit circle.

How can this topic be used in your students’ future courses in mathematics or science?

Being able to compute trig functions using a unit circle will be the base of knowledge for all further calculus classes, as well as others. Being able to understand and use a unit circle will also allow students to start to memorize the trigonometric functions. One of the most important things from pre-calculus to all other calculus classes was being able to solve trig functions and having the unit circle memorized was very useful. Although there are trig functions and values outside of the unit circle, the unit circle almost is like the foundation for trigonometry. Most, if not all, calculus classes after pre-calculus will expect students to have the unit circle memorized. Although it can be solved using a calculator, this will allow equations and problems to be solves easier with less thought when a student knows the unit circle. Even outside of calculus classes, the unit circle is one of many important aspects in math classes.

How does this topic extend what your students should have learned in previous courses?

Before students learn how to compute trigonometric functions using a unit circle, they learn about the trig functions by themselves. This usually starts in high school geometry where students learn sine, cosine, and tangent, yet they do not use them in the way a unit circle does. Most schools only teach the students how to use the calculator to compute the functions to solve sides or angles for triangles. As students enter pre-calculus, they use what they have learned about the trig functions in order to apply them to the unit circle. This will allow students to see that using trig functions can still be used to solve triangles, but it can also be used to solve many other things. Once they learn the unit circle, they will see more examples in which they will apply the functions and make connections to real-world scenarios that they can also be applied to.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

There are probably many simulations and websites that can help students compute trig functions using the unit circle, but I think something that will engage the students is a Kahoot or Quizziz that will help the students memorize the unit circle. Giving students an opportunity to apply what they learned into a friendly competition not only gives them practice but will also let them be engaged. Other technology resources such as videos or a website that is teaching the lesson does not really allow the students to apply what they know rather than just being lectured. Although some websites and technology can be useful, I personally, enjoy giving students the opportunity to work out problems as well as being engaged. Also, using calculators could be helpful to check answers but if they have a unit circle it might not be necessary unless they do not have the unit circle in front of them.

Engaging students: Using a recursively defined sequence

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Enrique Alegria. His topic, from Precalculus: using a recursively defined sequence.

How can this topic be used in your students’ future courses in mathematics or science?

Recursion is heavily emphasized within the branches of computer science. The technique can be used more than just in arithmetic and geometric sequences for finding the next term. Within computer science, recursion techniques can be utilized for sorting algorithms. The content will be able to transfer easily. Instead of finding the previous term to use to find the current term, within sorting algorithms, a set of numbers is chunked into smaller and smaller sets such that the original set of numbers becomes sorted.

We can take a deeper look at Merge Sort which is a recursive sorting algorithm. What occurs is the set of numbers repeatedly gets cut in half until there is only one element in the list. From there the elements are sorted in increasing order. Traversing back into the original size of the list with all of the elements contained except the final output is the list in increasing order.

Students can inspect the algorithm visually and need not to understand the implementation of code to comprehend the functionality of recursion. Guiding the students towards the smallest part of the process which is the single element and from there rearranging the elements of the list.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Recursively defined sequences influenced a renowned artist who is M.C. Escher. The concept of a sequence beginning at one point and continuing infinitely is how Escher exhibits recursion. Escher challenges the viewer of his work to determine the patterns from the artistic series.

For example, when observing the piece Drawing Hands, a student can predict what the ‘base case’ of the artwork would be followed by the next steps of the drawing. The spectator of this piece can break it apart into smaller and smaller partitions of the whole. And once they reach a starting point, they can put together the whole picture once again.

Similarly, students can view this piece titled Two Birds to follow the patterns. Without saying the name of the piece students can again predict the base case and determine how recursion techniques would be used for this sequence. Students can begin to learn how to think of how recursively defined sequences are applied through visual representations of M.C. Escher’s artwork.

How can technology be used to effectively engage students with this topic?

Technology can be used to effectively engage students with recursion by showcasing the YouTube video “Recursion: The Music Videos of Michel Gondry” by Polyphonic. Through this video, students can compare recursively defined sequences to music they listen to. The video starts with singular notes and then repeating the notes to create a rhythm. Compiling the initial sounds into something familiar through loops of samples and sound bites. This video goes into the repetitive patterns of the small chunks of sound are shown through visual representations with the music videos by Michel Gondry. In the music video “Star Guitar” by The Chemical Brothers, the video starts off with the listener on a train ride going through a landscape. Slowly patterns emerge as buildings uniquely correspond to the notes and rhythms within the song. With this YouTube video students obtain a great introduction to recursion and hopefully continue to find patterns of recursion to music they listen to in the future.

References

Greenberg I., Xu D., Kumar D. (2013) Drawing with Recursion. In: Processing. Apress, Berkeley, CA. https://doi.org/10.1007/978-1-4302-4465-3_8

Miller, B., & Ranum, D. (2020). 6.11. The Merge Sort — Problem Solving with Algorithms and Data Structures. Runestone.academy. https://runestone.academy/runestone/books/published/pythonds/SortSearch/TheMergeSort.html.

Engaging students: Powers and exponents

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Austin Stone. His topic, from Pre-Algebra: powers and exponents.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

“The number of people who are infected with COVID-19 can double each day. If it does double every day, and one person was infected on day 0, how many people would be infected after 20 days?” This problem can be a current real-life word problem that all students can relate to given the times we are in. This problem would be a good introductory for students to see how quickly numbers can get when using exponents. This would be an engaging introductory to exponents and will get the students interested because they can easily see that this can be used in current problems facing the world. This problem could also work later in Algebra if you ask how many days it would take to infect “blank” amount of people. This makes the question more of a challenge because they would have to solve for “x” (days) which is the exponent.

How has this topic appeared in the news?

This topic has been the news so far in 2020 if we are being honest. COVID-19 is a virus that has an exponential infection rate, just like any virus. When talking about COVID-19, news reporters and doctors usually use graphs to depict the infection rate. These graphs start off small but then grow exponentially until it slows down due to either people being more aware of their hygiene habits and/or the human immune system getting more familiar with the virus. Knowing how exponents work helps people better understand the seriousness of viruses such as COVID-19 and the everlasting impact it can have on the world. Doctors study what are the best ways to slow down the exponential growth so that a limited number of people contract and potentially die from the virus. To do this, they predict the exponential growth keeping in mind the regulations that may be enforced. Whatever regulation(s) slow down the virus the most are the ones that they try to enforce.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

An easy way to introduce students who have never seen exponents or exponential growth before is to use a graphing calculator. By plugging in an exponential function into the calculator and viewing the graph and zooming out, students can easily see how quickly numbers start to get massively large. A teacher can set this up by giving the students a problem to think about such as, “how many people would be infected with the virus after “blank” amount of day?” Students then could guess what they believe it would be. After revealing the graph and the actual number, students will probably be surprised at how big the number is in just a short amount of time. After that, the teacher could show a video on YouTube about exponential growth and/or infection rates of viruses and how quickly a small virus can turn into a pandemic. This also has very current real-world applications.


text:
Article | Open | Published:

# On the Precise Tuning of Optical Filtering Features in Nanoporous Anodic Alumina Distributed Bragg Reflectors

## Abstract

This study presents a nanofabrication approach that enables the production of nanoporous anodic alumina distributed Bragg reflectors (NAA-DBRs) with finely engineered light filtering features across the spectral regions. The photonic stopband (PSB) of these NAA-based photonic crystal (PC) structures is precisely tuned by an apodization strategy applied during stepwise pulse anodization with the aim of engineering the effective medium of NAA-DBRs in depth. We systematically assess the effect of different fabrication parameters such as apodization function (i.e. linear positive, linear negative, logarithmic positive and logarithmic negative), amplitude difference (from 0.105 to 0.420 mA cm−2), current density offset (from 0.140 to 0.560 mA cm−2), anodization period (from 1100 to 1700 s), and pore widening time (from 0 to 6 min) on the quality and central wavelength of the PSB of NAA-DBRs. The PSB’s features these PC structures are demonstrated to be highly tunable with the fabrication parameters, where a logarithmic negative apodization is found to be the most effective function to produce NAA-DBRs with high quality PSBs across the UV-visible-NIR spectrum. Our study establishes that apodized NAA-DBRs are more sensitive to changes in their effective medium than non-apodized NAA-DBRs, making them more suitable sensing platforms to develop advanced optical sensing systems.

## Introduction

Photonic crystals (PCs) in various forms such as optical lenses, fibers, mirrors, resonators, cavities, thin films, and optical filters are key elements in advanced optical devices since these structures enable the precise control over electromagnetic waves by light–matter interactions at the nanoscale. Of all the PC structures, optical filters are vital components for a broad range of applications, including laser mirrors, arc welding, polarizers, solar protection, photography, imaging, astronomy, and complex instrumentation1,2. PCs can selectively allow and forbid the pass of light of certain wavelengths or energies when photons travel across the PCs’ structure. This property can be precisely designed by engineering the PCs’ structure to develop PC-based advanced optical filters with unique light-filtering features3. Typically, optical filters are fabricated by chemical vapor deposition3,4, photolithography5, and co-evaporation6, processes through which substrates such as thin glass, plastic, and dielectric materials (e.g. silicon dioxide, titanium dioxide, zinc sulfide, magnesium fluoride) are endowed with selective light-filtering properties3. The raise of nanotechnology has enabled multiple opportunities to produce advanced PC-based optical filters using nanomaterials featuring layers of alternating refractive index. Among these, PC structures based on nanoporous materials produced by electrochemical etching of metals have opened new opportunities to develop advanced photonic platforms with applicability in photonics, photocatalysis, optoelectronics, and sensing7. The refractive index/dielectric constant of these nanomaterials can be engineered a 1D, 2D, or 3D fashion by modulating their porosity in depth, enabling the generation of a broad range of multi-dimensional PCs such as Bragg reflectors8,9, microcavities10,11, waveguides12,13, and others14,15.

Of all these PCs, nanoporous anodic alumina photonic crystals (NAA-PCs) produced by electrochemical oxidation (anodization) of aluminum have attracted considerable attention during recent years due to the versatility of the nanoporous structure of NAA16. This top-down nanofabrication approach offers industrial scalability (from mm2 to m2), cost-competitiveness, and versatile control over the features of nanopores, which can be modulated with precision by means of the anodization parameters to generate unique multi-dimensional PC structures able to guide, reflect, modulate, confine, transmit, emit, and enhance incident light selectively across the spectral regions17,18. Recent studies have demonstrated that rationally designed pulse-like anodization profiles under suitable conditions enable the precise engineering of the photonic stopband (PSB) of NAA-PCs by creating structures such as gradient index filters19,20,21,22, optical microcavities23, distributed Bragg reflectors24,25,26,27,28, bandpass and linear variable bandpass filters29,30. These PCs can be used as optical filters for a plethora of applications due to the flexibility and selectively to engineer their light-filtering features across the spectral regions31,32.

In this study, we demonstrate for the first time that an apodized stepwise pulse anodization (STPA) approach enables the fine tuning of the features of the characteristic PSB of NAA-based distributed Bragg reflectors (DBRs) (Fig. 1). The effect of the fabrication parameters (i.e. apodization function, anodizing current density, amplitude difference, current density offset, anodization period, and pore widening time) on the characteristic PSB of NAA-DBRs is systematically analyzed to attain full controllability over the filtering features of these PCs and to establish the most optimal path for high quality PSBs. We further assess the effective medium of apodized and non-apodized NAA-DBRs after infiltrating their nanopores with mediums of different refractive index. Shifts in the position of the characteristic PSB of these PCs demonstrate that to apodize the structure of NAA-DBRs can enhance the sensitivity of these PCs, opening new opportunities to develop materials for sensing applications.

## Results and Discussion

### Apodization of Stepwise Pulse Anodization Profiles

Figure 2 displays the anodization profiles of a set of non-apodized NAA-DBRs produced by STPA at different anodization periods (T p ), from T P  = 900 s to 1700 s with ΔT P  = 100 s (Note: displayed only from 900 to 1600 s). These NAA photonic structures were produced by STPA approach under galvanostatic conditions, where the current density (J) is switched between maximum (J Max  = 1.120 mA cm−2) and minimum (J Min  = 0.280 mA cm−2) levels to modulate the porosity of NAA in depth33. In our study, an apodization strategy was applied to the STPA profile with the aim of tuning the PSB of NAA-DBRs by engineering their effective medium in a stepwise fashion through different apodization windows. Four different apodization functions (i.e. linear positive, linear negative, logarithmic positive, and logarithmic negative) were used to modify the conventional STPA process. Furthermore, we systematically modified the amplitude difference (i.e. current density amplitude difference between the initial (t0) and half anodization time (t 1/2 ) under ASTPA − ΔA J  = |A J (t0) − A J (t 1/2 )|), from ΔA J  = 0.105 mA cm−2 to 0.420 mA cm−2 with a step size of 0.105 mA cm−2. Figure 3 shows the ASTPA profiles used to produce apodized NAA-DBRs by each apodization function and amplitude difference. These profiles reveal that, under the anodization conditions used in our study (see Methods), modifications of the current density (input) are directly translated into voltage (output) changes, which is a critical factor for the accurate translation of current density profiles into porosity modulation in depth during anodization.

### Structural Characterization of NAA-DBRs

Figure 4 compiles a set of representative field-emission gun scanning electron microscopy (FEG-SEM) images of NAA-DBRs produced in our study. As these images reveal, NAA-DBRs feature an even but random distribution of nanopores across their surface, the average diameter (d p ) of which was estimated to be d p  = 10 ± 3 and 19 ± 3 nm, for pore widening times (t pw ) 0 and 6 min, respectively (Fig. 4a and b). This range of pore sizes was found to be optimal to obtain well-defined and intense PSBs in the transmission spectra of NAA-DBRs. NAA-based PC structures featuring bigger pore sizes, such as those produced in oxalic and phosphoric acids, scatter and absorb more light when photons travel across the PC structure, resulting in less intense PSBs. The resulting PCs feature a modulation of the pore diameter in depth that follows with precision the stepwise current density profile applied during STPA and ASTPA (Fig. 4c and d). This results in an in-depth modification of the effective refractive index of NAA that enables the generation of 1D NAA-DBRs. We also observed a linear relationship between the anodization period (T P ) and the period length (L TP ), defined as the distance between adjacent layers in the stacked structure of NAA-DBRs, where the latter changes at a rate of 0.22 nm s−1 with the former (Fig. 4e).

### Effect of Apodization on Optical Properties of NAA-DBRs

The transmission spectrum of NAA-DBRs displays a characteristically broad PSB with due to discontinuities in the effective refractive index profile34. Figure 1e shows the transmission spectra of a NAA-DBR produced with an anodization period of 1700 s. It is well-known that the PSB of DBRs can be engineered through apodization, optical filtering technique used to narrow the bandwidth of PC structures35. Figure 1f shows the transmission spectra of a NAA-DBR produced with T P  = 1700 s by a ASTPA profile apodized with a logarithmic negative function. It is clearly seen that the PSB of the NAA-DBR is significantly narrowed after apodizing the anodization profile. This effect is also observed for the different transmission orders of the PSB, where the apodized PC structure shows much well resolved and sharper PSBs than its non-apodized counterpart. Therefore, to apply an apodization approach to the STPA profile under the conditions used in our study enables the fine tuning of the light-filtering features of NAA-DBRs for specific applications, such as highly selective optical filters, high quality resonators, and ultrasensitive optical sensors.

Figure 5 shows contour maps summarizing the dependence of the central wavelength (λ PSB ) and the quality factor (Q PSB – calculated as the position of the central wavelength divided by the PSB’s full width at half maximum (FWHM PSB ) – Equation 1) of the PSB of NAA-DBRs with the apodization function (i.e. linear positive, linear negative, logarithmic positive, and logarithmic negative), the amplitude difference (ΔA J ), and the pore widening time (t pw ). These graphs show how Q PSB and λ PSB vary with the fabrication parameters, enabling optimization paths to precisely tune the filtering features of NAA-DBRs.

$${Q}_{PSB}=\frac{{\lambda }_{PSB}}{FWH{M}_{PSB}}$$
(1)

Figure 5a and b show the effect of ΔA J and t pw on the filtering features of NAA-DBRs apodized following a linear positive function. Figure 5a denotes three maxima regions where Q PSB presents the highest values, from ΔA J  = 0.210 to 0.420 mA cm−2 and t pw  = 0 to 2 min; from ΔA J  = 0.105 to 0.210 mA cm−2 and t pw  = 0 to 2 min; and from ΔA J  = 0.105 to 0.210 mA cm−2 and t pw  = 4 to 6 min. Linear positive apodized NAA-DBRs achieve the highest quality factor (6.2 ± 0.3) at ΔA J  = 0.105 mA cm−2 and t pw  = 0 min, as indicated by the red region in the contour plot. In contrast to Q PSB , λPSB was found to change smoothly with ΔA J and t pw , as revealed by the homogeneous distance between color fields (Fig. 5b). λ PSB undergoes a blue shift as t pw and ΔA J increase, although the magnitude of the blue shift produced by t pw is more significant than that of ΔA J . Figure 5c and d display the effect of ΔA J and t pw on the central wavelength position and the quality factor of NAA-DBRs produced by a linear negative apodization function. It is observed that Q PSB features a weak correlation with ΔA J and t pw at higher values of these two fabrication parameters (Fig. 5c). However, the quality factor of these NAA-DBRs increases significantly as ΔA J and t pw decrease, reaching its maximum value of 14.1 ± 0.7 at ΔA J  = 0.105 mA cm−2 and t pw  = 0 min. In the case of the position of the central wavelength (Fig. 5d), this is red-shifted as both ΔA J and t pw decrease. Likewise for a linear positive apodization, this change is smooth and homogeneous as denoted by the even distribution of color fields, although the effect of ΔA J on λ PSB is less significant than that of t pw . Figure 5e and f show the dependence of Q PSB and λ PSB with ΔA J and t pw for NAA-DBRs apodized with a logarithmic positive function. Figure 5e reveals that Q PSB evolves homogeneously with ΔA J and t pw . A reduction of the amplitude difference and pore widening time results in an enhancement of the quality factor of these NAA-DBRs, where Q PSB is strongly dependent on these fabrication parameters as its value is nearby the maximum (Q PSB  = 10.5 ± 0.5), located at ΔA J  = 0.105 mA cm−2 and t pw  = 0 min. A similar trend is observed for the correlation between λ PSB and ΔA J and t pw , with a marked dependence on the fabrication parameters around its maximum (λ PSB  = 716 ± 1 nm), located at ΔA J  = 0.105 mA cm−2 and t pw  = 0 min. In good agreement with previous studies36,37, the position of the central wavelength is blue-shifted with t pw following a linear fashion, as denoted by the equidistant separation between color fields as t pw increases. Finally, Fig. 5g and h depict the relationship between Q PSB and λ PSB with ΔA J and t pw for NAA-DBRs produced by a logarithmic negative apodization approach. Figure 5g denotes a high concentration of color fields at short pore widening times, where the field lines are closer each other around the maximum. The combination of fabrication parameters that gives the highest Q PSB (21.4 ± 1.0) for these NAA-DBRs is ΔA J  = 0.210 mA cm−2 and t pw  = 0 min. As far as λ PSB is concerned, it is observed that the central wavelength undergoes a homogenous variation with ΔA J and t pw , in a similar way than that of NAA-DBRs produced by a logarithmic positive apodization. However, the maximum value of λ PSB is located at ΔA J  = 0.315 mA cm−2 and t pw  = 0 min.

Another interesting property of NAA-DBRs is that these PC structures display vivid interferometric colors when the position of the central wavelength is located within the visible range of the spectrum. Figure 6a compiles a set of digital pictures of non-apodized NAA-DBRs produced at different anodization periods (T P  = 900 to 1700 s with ΔT P  = 100 s) and pore widening times (t pw  = 0 to 6 min with Δt pw  = 2 min). Figure 6b–e show digital pictures of apodized NAA-DBRs produced with an anodization period of 1300 s with four apodization functions (linear positive, linear negative, logarithmic positive, and logarithmic negative) as a function of ΔA J and t pw . These pictures demonstrate that apodized and non-apodized NAA-DBRs display vivid colors such as orange, yellow, cyan and green, which corresponds to the position of their characteristic PSB within the visible range. This property can be readily engineered by modifying the fabrication parameters. For instance, linear negative apodized NAA-DBR with ΔA J  = 0.105 mA cm−2 and t pw  = 2 min displays yellow color, which corresponds to the position of its PSB at λ PSB  = 580 ± 1 nm. On the other hand, NAA-DBRs with their characteristic PSB in the NIR region (e.g. logarithmic negative apodized NAA-DBRs with ΔA J  = 0.315 mA cm−2 and t pw  = 0 min, where λ PSB  = 703 ± 1 nm) are transparent in color (black background).

To summarize, different apodization functions were explored to produce NAA-DBRs with optimized optical properties in terms of Q PSB and λ PSB . All four apodization strategies (i.e. linear positive, linear negative, logarithmic positive and logarithmic negative) have shown to improve the quality of PSB as compared to non-apodized NAA-DBRs by certain combinations of fabrication parameters (i.e. ΔA J and t pw ). Among these functions, logarithmic negative apodization was found to be the most effective strategy to enhance the quality of the PSB of NAA-DBRs, as proven by the high value of Q PSB (21.4 ± 1.0).

### Effect of Anodization Offset and Period on Optical Properties of NAA-DBRs

To further optimize the optical properties of NAA-DBRs, the combined effect of the current density offset (J offset ) and anodization period (T P ) with t pw on the optical properties of NAA-DBRs was systematically investigated by fabricating a set of non-apodized and apodized NAA-DBRs with different J offset (from 0.140 to 0.560 mA cm−2 with ΔJ offset  = 0.140 mA cm−2) (Fig. 7) and T P (from 1100 to 1700 s with ΔT P  = 200 s) (Fig. 8). Figure 7a and b show representative anodization profiles of logarithmic negative apodized (J offset  = 0.420 mA cm −2, T P  = 1300 s, and ΔA J  = 0.210 mA cm−2) and non-apodized NAA-DBRs (J offset  = 0.420 mA cm −2, T P  = 1300 s, and A J  = 0.420 mA cm−2) produced with different J Offset . Figure S1 (Supporting Information) compiles the anodization profiles of all the samples analyzed in our study. A qualitative comparison of the transmission spectra of these NAA-DBRs (Fig. 7c and d, respectively) reveals that a logarithmic negative apodization function significantly enhances the quality of the PSB of NAA-DBRs. Similarly to previous observations, t pw intensifies and broadens the PSB in both apodized and non-apodized NAA-DBRs. The combined effect of J offset and t pw on the Q PSB and λ PSB of non-apodized and apodized NAA-DBRs is summarized in the contour maps shown in Fig. 7e and f, respectively. As Fig. 7e reveals, the color fields and field line distances are broad at longer t pw and higher J offset for non-apodized NAA-DBRs. The dependence of Q PSB on t pw increases as t pw decreases, particularly at J offset  = 0.140 and 0.560 mA cm−2, as indicated by the denser field lines around the maximum (Q PSB  = 10.9 ± 0.5) located at J offset  = 0.560 mA cm−2 and t pw  = 0 min. Figure 7f displays the distribution of Q PSB as a function of J offset and t pw for logarithmic negative apodized NAA-DBRs. The field lines from J offset  = 0.420 to 0.560 mA cm−2 and from t pw  = 2 to 6 min are wide apart, indicating that Q PSB in apodized NAA-DBRs has a weak correlation with the pore widening time and the current density offset within the range of fabrication parameters assessed. The maximum of Q PSB (21.4 ± 1.0) is achieved when J offset and t pw are set to 0.280 mA cm−2 and 0 min, respectively. The color field and field lines concentrate around this maximum, where the dependency of Q PSB on J offset and t pw increases. Both non-apodized and apodized NAA-DBRs have poor quality of PSB when they are over-etched (i.e. long t pw ), as revealed by the low values of Q PSB and broad color fields within these regions of the contour maps. The distribution of λ PSB with J offset and t pw for non-apodized and apodized NAA-DBRs is summarized in the contour maps shown in Fig. 7g and h, respectively. As these graphics reveal, λ PSB of non-apodized NAA-DBRs has a stronger dependency with J offset at high values of this fabrication parameter (from 0.420 to 0.560 mA cm−2) (Fig. 7g), while this dependence is stronger at moderate J offset values in the case of apodized NAA-DBRs (from 0.280 to 0.420 mA cm−2) (Fig. 7h). However, in both cases it is observed a similar trend in the shift of λ PSB with J offset and t pw : when J offset increases, λ PSB is red-shifted toward the NIR region of the spectrum. In the case of t pw , the longer t pw is the shorter the wavelength at which both non-apodized and apodized NAA-DBRs reflect light more efficiently (blue shift). The transmission spectra of all the NAA-DBRs analyzed in this study as a function of J offset and t pw are compiled in Figure S2 (Supporting Information).

Likewise in previous cases, these NAA-DBRs also display vibrant interferometric colors, which correspond to the position of the characteristic PSB across the UV-vis-NIR spectrum. Figure S3 (Supporting Information) displays digital pictures of non-apodized and logarithmic negative apodized NAA-DBRs produced at different values of J offset and t pw . It is worthwhile to note that apodized NAA-DBRs have their characteristic PSB located at longer wavelengths as compared to their non-apodized counterparts. This is also denoted by the difference in the interferometric color displayed by these PCs. Furthermore, NAA-DBRs with high J offset show transparent color as there is a red shift in the position of PSB (NIR region) as J offset increases.

To assess the combined effect of the anodization period (T P ) and t pw on the optical filtering features of non-apodized and logarithmic negative apodized NAA-DBRs, T P and t pw were systematically modified from 1100 to 1700 s with ΔT P  = 200 s, and from 0 to 6 min with Δt pw  = 2 min, respectively. The obtained results showing the dependency of Q PSB and λ PSB with T P and t pw for these NAA-DBRs are summarized in Fig. 8. Figure 8a and b display representative anodization profiles of non-apodized and apodized NAA-DBRs produced with T P  = 1500 s, respectively, whereas Figure S4 (Supporting Information) compiles the anodization profiles of all the NAA-DBRs analyzed in this study. Representative transmission spectra of non-apodized (T P  = 1500 s, A J  = 0.420 mA cm−2, and J offset  = 0.280 mA cm−2) and logarithmic negative apodized (T P  = 1500 s, ΔA J  = 0.210 mA cm−2, and J offset  = 0.280 mA cm−2) NAA-DBRs at different t pw are presented in Fig. 8c and d, respectively.

These graphs show that the transmission spectrum of apodized NAA-DBRs features much narrower PSBs than those of non-apodized NAA-DBRs. It is also observed that to widen the nanoporous structure of NAA-DBRs increases the intensity and width of the PSBs, independently on the anodization period (Figure S5 – Supporting Information).

Figure 8e and f show contour maps describing how Q PSB varies with T P and t pw for non-apodized NAA-DBRs and logarithmic negative apodized NAA-DBRs, respectively. Both graphs exhibit similar Q PSB distribution with broad color fields and field lines at longer t pw and increasing concentration of color fields as t pw decreases. Non-apodized NAA-DBRs display two maxima located at T P  = 1100 and 1500 s with t pw  = 0 min, where the former has the maximum value for Q PSB (11.2 ± 0.6). In the case of apodized NAA-DBRs, the maxima are located at T P  = 1300 and 1700 s with t pw  = 0 min, where the former gives the maximum Q PSB value of 21.3 ± 0.6, which is almost twice higher than that of the non-apodized counterparts. The dependency of Q PSB with T P increases around the maxima, as indicated by the close field lines and the concentration of color fields around these points. It is worth nothing that the Q PSB of apodized NAA-DBRs has a stronger dependency on T P as compared to non-apodized NAA-DBRs, denoted by the smaller field line distances in Fig. 8f.

The dependency of λ PSB on T P and t pw for non-apodized and apodized NAA-DBRs is summarized in Fig. 8g and h, respectively. As shown by the contour map for non-apodized NAA-DBRs (Fig. 8g), the color fields are distributed homogenously throughout the map with equidistant field lines at T P  < 1500 s, which suggest a fairly strong dependence of λ PSB on this range of T P . The distance between field lines increases from T P  = 1500 to 1700 s, indicating a weaker dependence of λ PSB on T P . In contrast, the contour map for apodized NAA-DBRs (Fig. 8h) shows an even distribution of color fields with equidistant field lines at T P  > 1500 s and broader color fields with more distant field lines at T P  < 1500 s. These results indicates that the dependency of λ PSB on T P is stronger at longer T P than that at shorter T P . Compared to T P , t pw has a less significant impact on the distribution of λ PSB , with a similar effect in both non-apodized and apodized NAA-DBRs, where an increment in T P red-shifts the position of λ PSB while an increment in t pw blue-shifts the position of λ PSB . This result is good agreement with previous studies, establishing the behavior of the characteristic PSB of NAA-based photonic structures under the manipulation of T P and t pw 38.

NAA-DBRs also displayed tunable vivid interferometric colors as the position of the PSB is shifted across spectral regions (Figure S6 Supporting Information). The difference in the interferometric colors shown by non-apodized and apodized NAA-DBRs is due to the shift in the PSB’s position, which is located at longer wavelengths for apodized NAA-DBRs as compared to that of non-apodized NAA-DBRs produced with the same T P .

### Assessment of Effective Medium Sensitivity of Non-apodized and Apodized NAA-PCs

As revealed by the analysis of Q PSB , the implementation of an apodization strategy during anodization is an effective approach to improve the quality of the PSB of NAA-DBRs, which results in better resolved and narrower characteristic PSBs with tunable filtering features across the spectral regions. This feature can be readily used to develop highly sensitive optical sensing platforms to detect analytes of interest based on effective refractive index changes17. It is known that PC structures with high quality PSBs are more sensitive to effective medium changes (i.e. larger shifts in the position of PSB upon small effective medium changes), making them optimal platforms to develop advanced sensing system39.

To verify that apodized NAA-DBRs with better Q PSB are more sensitive toward effective medium changes, we systematically infiltrated the nanoporous network of non-apodized and apodized NAA-DBRs produced with different T P (i.e. 1100, 1300, 1500, and 1700 s) with media of different refractive index and measured shifts in the PSB (Δλ PSB ) using reflectometric interference spectroscopy (RIfS). Figure 9a and b show representative RIfS spectra for non-apodized (T P  = 1700 s, A J  = 0.420 mA cm−2, J offset  = 0.280 mA cm−2, and t pw  = 4 min) and apodized NAA-DBRs (T P  = 1700 s, ΔA J  = 0.210 mA cm−2, J offset  = 0.280 mA cm−2, and t pw  = 4 min), respectively, infiltrated with different media (air ~1.00 RIU, water ~1.33 RIU, and ethanol ~1.36 RIU). As demonstrated in our previous study22, the RIfS spectra of NAA-DBRs features two main regions: namely; i) the interference of light from all layers of the multilayered NAA-DBR structure characterized by a relatively narrow and intense PSB, and ii) the Fabry-Pérot interference spectrum (i.e. fast oscillations) produced by the reflections of light at the interfaces bordering the NAA-DBR. In this study, we used the positon of the characteristic PSB in the RIfS spectra of NAA-DBRs as sensing parameter. The sensitivity of these PC structures as a function of T P and t pw is established by the slope of the linear fitting obtained by correlating Δλ PSB , determined by RIfS, and the refractive index of the medium filling the nanopores. The obtained results for non-apodized and apodized NAA-DBRs are summarized in Fig. 9c and d.

Figure 9c reveals that non-apodized NAA-DBRs produced with T P  = 1700 s have the highest sensitivity at t pw  = 0, 2, and 4 min (i.e. 119 ± 13, 254 ± 39, and 339 ± 60 nm RIU−1, respectively), whereas at t pw  = 6 min, those NAA-DBRs produced at T P  = 1300 s display the highest sensitivity (i.e. 332 ± 51 nm RIU−1, respectively). In contrast, Fig. 9d denotes that logarithmic negative apodized NAA-DBRs produced with T P  = 1500 s have the highest sensitivity at t pw  = 0 (120 ± 13 nm RIU−1, respectively). However, at t pw  = 2 and 4 min, apodized NAA-DBRs produced at T P  = 1700 s have the highest sensitivity (280 ± 43 and 392 ± 65 nm RIU−1, respectively) and, at t pw  = 6 min, NAA-DBRs fabricated with T P  = 1300 s are the most sensitive PC structures (363 ± 64 nm RIU−1).

In general, it is observed that the sensitivity of non-apodized and apodized NAA-DBRs is enhanced by the pore widening treatment for 0 min ≤ t pw  ≤ 4 min. However, as denoted by the Q PSB analysis, further increase in t pw worsens the quality of the PSB due to the over-etching of the nanoporous structure of NAA-DBRs, which broadens the PSB and reduces the sensitivity of these platforms. Interestingly, both non-apodized and apodized NAA-DBRs have exhibited the highest sensitivity (i.e. 339 ± 59 and 392 ± 65 nm RIU−1, respectively) at t pw  = 4 min and T P  = 1700 s, where the latter is revealed to be the most sensitive platform toward changes in the effective medium. The sensitivity of apodized NAA-DBRs was found to be ~ 16% higher than that of their non-apodized counterparts.

To conclude, this study provides new insights into the capability of anodization to engineer and tune the optical properties of NAA-based photonic crystal structures. An apodization strategy applied during stepwise pulse anodization enables the precise control over the features of the photonic stopband of NAA-DBRs by manipulating various anodization parameters (i.e. apodization function, amplitude difference, current density offset, anodisation period, and pore widening time). The systematic analysis on the effect of each fabrication parameter reveals that a logarithmic negative apodization function provides the highest quality of the photonic stopband (Q PSB  = 21.4 ± 1.0). Apodized NAA-DBRs are demonstrated to be a more sensitive sensing platform than non-apodized NAA-DBRs as they have shown higher sensitivity toward effective medium changes (~16% enhancement). Logarithmic negative apodized and non-apodized NAA-DBRs produced with T P  = 1700 s and t pw  = 4 min have a sensitivity of 392 ± 65 and 339 ± 59 nm RIU−1, respectively.

These innovative NAA-based photonic crystal structures with enhanced optical properties pave the way for the development of ultrasensitive optical sensing systems and other photonic elements such as selective optical filters with broad applicability.

## Methods

### Materials

High purity (99.9997%) aluminum (Al) foils of thickness 0.32 mm were supplied by Goodfellow Cambridge Ltd. (UK). Sulfuric acid (H2SO4), perchloric acid (HClO4), copper (II) chloride (CuCl2), hydrochloric acid (HCl), phosphoric acid (H3PO4), and ethanol (EtOH – C2H5OH) were supplied by Sigma-Aldrich (Australia) and used as received, without further purification. Aqueous solutions used in this study were prepared with ultrapure Mili-Q® water (18.2 mΩ.cm) (Australia).

### Fabrication of NAA Photonic Structures

Non-apodized NAA photonic structures were produced by stepwise pulse anodisation (STPA) approach under current density control conditions. 1.5 × 1.5 cm Al square chips were first washed in ethanol and water under sonification for 15 min each and dried under air stream. These Al substrates were then electropolished in a mixture of EtOH and HClO4 4:1 (v:v) at 20 V and 5 °C for 3 min. After electropolishing, Al chips were anodized in an aqueous solution of 1.1 M H2SO4 with 25 v% of EtOH at −1 °C. The galvanostatic anodization process started with a constant step at a current density of 1.120 mA cm−2 for 1 h to achieve a homogenous pore growth rate prior to stepwise pulse anodization. The anodization profile was subsequently switched to stepwise pulse mode, where the current density was pulsed between high (J max  = 1.120 mA cm−2) and low (J min  = 0.280 mA cm−2) current density values following a stepwise modulation for a total anodization time (t STPA or t ASTPA ) of 20 h. A set of reference non-apodized NAA-DBRs were produced, where the current density amplitude (A J ) and current density offset (J offset ) were set at 0.420 mA cm−2 and 0.280 mA cm−2, respectively, whereas the anodization period (T P ) (i.e. time between consecutive pulses) was modified from 900 to 1700 s with a step size of 100 s. Note that T P is given by Equation 2.

$${T}_{P}={t}_{max}+{t}_{min}$$
(2)

where t max and t min are the time lengths at J max and J min , respectively, and t max and t min were set at a ratio of 1:4 (i.e. t min  = 4t max ) for T P  = 900 to 1700 s.

To assess the effect of J Offset , another set of non-apodized NAA photonic structures was fabricated based on the following parameters: A J  = 0.420 mA cm−2, T P  = 1300 s, where J offset was adjusted from 0.140 to 0.560 mA cm−2 with an interval of 0.140 mA cm−2. The current density during stepwise pulse mode was pulsed between J min and J max , where the these parameters were defined by Equations 3 and 4, respectively.

$${J}_{Low}={J}_{offset}$$
(3)
$${J}_{High}=2{A}_{J}+{J}_{offset}$$
(4)

### Apodization and Optical Tuning of the PSB of NAA-DBRs

Four different apodization functions: (i) linear positive, (ii) linear negative, (iii) logarithmic positive, and (iv) logarithmic negative, were implemented in a reference STPA profile (i.e. T P  = 1300 s, J min  = J offset  = 0.280 mA cm−2, J High  = 1.120 mA cm−2, A J  = 0.420 mA cm−2, and t STPA  = 20 h). The mathematical expressions for each apodization function are listed in Equations S1S8 (Supporting Information). These ASTPA anodisation profiles were produced by a custom-designed Labview®-based software based on Equation 5:

$$J(t)=2{A}_{J}(t)+{J}_{offset}$$
(5)

where A J (t) is the time-dependent current density amplitude defined for a time between a minimum (A min ) and maximum (A max ) of amplitude for the corresponding apodization functions (Equations S1S8 – Supporting Information).

To investigate the effect of ΔA J on the optical properties of NAA-DBRs, the amplitude difference (ΔA J ) was modified from 0.105 to 0.420 mA cm−2 with an interval of 0.105 mA cm−2 for each apodization function. Using the most effective apodization function with optimized ΔA J , which was determined by analyzing the quality factor (Q PSB ), other apodization parameters such as J Offset and T P were systematically modified from 0.140 to 0.560 mA cm−2 with a step size of 0.140 cm−2, and from 1100 to 1700 s with a step size of 200 s, respectively, in order to further optimize the optical signals as well as to tune the optical properties of NAA photonic structures.

### Optical Characterization

Prior to optical characterization, NAA-DBRs were etched chemically in a saturated solution of HCl/CuCl2 to dissolve the remaining aluminum substrate from the backside. This process was carried out using an etching mask with a circular window of 5 mm in diameter in an etching cell. The optical properties of these etched NAA photonic structures were characterized by analyzing their transmission spectra measured using a UV-visible-NIR spectrophotometer (Cary 300 and Cary 5000, Agilent, USA, for wavelength range of 200–1000 nm and 200–1500 nm respectively). These spectra were obtained at normal incidence (i.e. θ = 0°) with a resolution of 1 nm. To characterize the interferometric color exhibited by these NAA photonic structures, digital images with a black card as background were acquired using a Canon EOS 700D digital camera equipped with a Tamron 90 mm F2.8 VC USD macro mount lens with autofocus function under natural illumination. These NAA photonic crystals were pore widened using an aqueous solution of 5 wt% H3PO4 at 35 °C for t pw from 0 to 6 min with an interval of 2 min. After each pore widening step, the optical characterization of NAA-DBRs was carried out by recording the transmission spectra and digital images as outlined above.

### Assessment of Sensitivity of Non-apodized and Apodized NAA-DBRs

The sensitivity of non-apodized and apodized NAA-DBRs was assessed by correlating the shift in the position of central wavelength (λ PSB ) to the refractive index values of the medium filling the nanopores. The spectra of NAA-DBRs with medium-filled nanopores was measured and recorded using a reflectometric interference spectroscopy (RIfS) set-up. White light from a tungsten source (LS-1LL, Ocean Optics, USA) was focused on the surface of NAA-DBRs with an illumination spot of ~2 mm in diameter by a bifurcated optical probe. The collection fiber in the optical probe collected and transferred the light reflected from the illumination spot to a miniature spectrometer (USB 4000+VIS-NIR-ES, Ocean Optics, USA). The optical spectra acquired in the range of 400–1000 nm wavelength were saved with an integration time of 10 ms and 10 average measurements. Note that NAA-DBRs were coated with an ultra-thin layer of gold by a sputter coater (Sputter coater 108, Cressington, USA) to enhance intereferometric reflection. After recording the optical spectra of NAA-DBRs with air filling the nanopores, a drop of EtOH was placed on the surface and allowed to settle into the nanopores for a few seconds to ensure that the nanopores were infiltrated by EtOH. The spectrum was recorded once the signals were stabilized. Then, a drop of water was placed on the surface of NAA photonic crystals after EtOH had completely evaporated. The RIfS spectrum of NAA-DBRs with H2O-filled nanopores was recorded and saved. NAA photonic crystals measured were pore widened for t pw  = 0 to 6 min. After each pore widening step, RIfS spectra were recorded for each NAA-DBRs with different medium filling the nanopores (i.e. air, EtOH, and H2O). The spectra acquired were then processed using Igor Pro library (Wavemetrics, USA) to determine Δλ PSB in non-apodized and apodized NAA-DBRs.

### Structural Characterization of NAA-DBRs

The morphology and structure of NAA photonic crystals were characterized by FEG-SEM image analysis from images acquired by field emission gun scanning electron microscopy (FEG-SEM FEI Quanta 450). The acquired FEG-SEM images were analyzed using ImageJ (public domain program developed at RSB of the NIH)40.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

1. 1.

Lee, C.-C. The current trends of optics and photonics. Vol. 129 (Springer: 2014)

2. 2.

López, C. Materials aspects of photonic crystals. Adv. Mater. 15, 1679–1704 (2003).

3. 3.

Philip, J., Jaykumar, T., Kalyanasundaram, P. & Raj, B. A tunable optical filter. Meas. Sci. Technol. 14, 1289 (2003).

4. 4.

Bulkin, P. V., Swart, P. L. & Lacquet, B. M. Fourier-transform design and electron cyclotron resonance plasma-enhanced deposition of lossy graded-index optical coatings. Appl. Opt. 35, 4413–4419 (1996).

5. 5.

Santos, A., Deen, M. & Marsal, L. Low-cost fabrication technologies for nanostructures: State-of-the-art and potential. Nanotechnology 26, 042001 (2015).

6. 6.

Ouellette, M. et al. Experimental studies of inhomogeneous coatings for optical applications. J. Vac. Sci. Technol., A 9, 1188–1192 (1991).

7. 7.

Wijnhoven, J. E. G. J. & Vos, W. L. Preparation of photonic crystals made of air spheres in titania. Science 281, 802–804 (1998).

8. 8.

Agarwal, V. & Del Rio, J. Tailoring the photonic band gap of a porous silicon dielectric mirror. Appl. Phys. Lett. 82, 1512–1514 (2003).

9. 9.

Chen, Y. et al. Rational design of photonic dust from nanoporous anodic alumina films: A versatile photonic nanotool for visual sensing. Sci. Rep. 5, 12893 (2015).

10. 10.

Bisi, O., Ossicini, S. & Pavesi, L. Porous Silicon: A Quantum Sponge Structure for Silicon Based Optoelectronics. Surf. Sci. Rep. 38, 1–126 (2000).

11. 11.

Santos, A. et al. Tunable Fabry-Pérot interferometer based on nanoporous anodic alumina for optical biosensing purposes. Nanoscale Res. Lett. 7, 370 (2012).

12. 12.

Hotta, K., Yamaguchi, A. & Teramae, N. Nanoporous waveguide sensor with optimized nanoarchitectures for highly sensitive label-free biosensing. ACS Nano 6, 1541–1547 (2012).

13. 13.

Loni, A. et al. Porous silicon multilayer optical waveguides. Thin Solid Films 276, 143–146 (1996).

14. 14.

Dal Negro, L. et al. Light transport through the band-edge states of Fibonacci quasicrystals. Phys. Rev. Lett. 90, 055501 (2003).

15. 15.

Sapienza, R. et al. Optical analogue of electronic bloch oscillations. Phys. Rev. Lett. 91, 263902 (2003).

16. 16.

Santos, A. Nanoporous anodic alumina photonic crystals: Fundamentals, developments and perspectives. J. Mater. Chem. C 5, 5581–5599 (2017).

17. 17.

Ilyas, S. et al. Porous silicon based narrow line-width rugate filters. Opt. Mater. 29, 619–622 (2007).

18. 18.

Chen, Y. et al. On the generation of interferometric colors in high purity and technical grade aluminum: An alternative green process for metal finishing industry. Electrochim. Acta 174, 672–681 (2015).

19. 19.

Kumeria, T., Rahman, M. M., Santos, A., Ferré-Borrull, J., Marsal, L. F. & Losic, D. Structural and optical nanoengineering of nanoporous anodic alumina rugate filters for real-time and label-free biosensing applications. Anal. Chem. 86, 1837–1844 (2014).

20. 20.

Santos, A. et al. Realisation and advanced engineering of true optical rugate filters based on nanoporous anodic alumina by sinusoidal pulse anodisation. Nanoscale 8, 1360–1373 (2016).

21. 21.

Kumeria, T., Rahman, M. M., Santos, A., Ferré-Borrull, J., Marsal, L. F. & Losic, D. Nanoporous anodic alumina rugate filters for sensing of ionic mercury: Toward environmental point-of-analysis systems. ACS Appl. Mater. Interfaces 6, 12971–12978 (2014).

22. 22.

Nemati, M., Santos, A., Law, C. S. & Losic, D. Assessment of binding affinity between drugs and human serum albumin using nanoporous anodic alumina photonic crystals. Anal. Chem. 88, 5971–5980 (2016).

23. 23.

Wang, Y. et al. Facile synthesis of optical microcavities by a rationally designed anodization approach: Tailoring photonic signals by nanopore structure. ACS Appl. Mater. Interfaces 7, 9879–9888 (2015).

24. 24.

Sulka, G. D. & Hnida, K. Distributed Bragg reflector based on porous anodic alumina fabricated by pulse anodization. Nanotechnology 23, 075303 (2012).

25. 25.

Chen, Y. et al. Interferometric nanoporous anodic alumina photonic coatings for optical sensing. Nanoscale 7, 7770–7779 (2015).

26. 26.

Rahman, M. M., Marsal, L. F., Pallarès, J. & Ferré-Borrull, J. Tuning the photonic stop bands of nanoporous anodic alumina-based distributed Bragg reflectors by pore widening. ACS Appl. Mater. Interfaces 5, 13375-–13381 (2013).

27. 27.

Law, C. S., Santos, A., Nemati, M. & Losic, D. Structural engineering of nanoporous anodic alumina photonic crystals by sawtooth-like pulse anodization. ACS Appl. Mater. Interfaces 8, 13542–13554 (2016).

28. 28.

Yisen, L., Yi, C., Zhiyuan, L., Xing, H. & Yi, L. Structural coloring of aluminum. Electrochem. Commun. 13, 1336–1339 (2011).

29. 29.

Santos, A., Pereira, T., Law, C. S. & Losic, D. Rational engineering of nanoporous anodic alumina optical bandpass filters. Nanoscale 8, 14846–14857 (2016).

30. 30.

Sukarno, L. C. S. & Santos, A. Realisation and optical engineering of linear variable bandpass filters in nanoporous anodic alumina photonic crystals. Nanoscale 9, 7541–7550 (2017).

31. 31.

Santos, A., Kumeria, T., Wang, Y. & Losic, D. In situ monitored engineering of inverted nanoporous anodic alumina funnels: On the precise generation of 3D optical nanostructures. Nanoscale 6, 9991–9999 (2014).

32. 32.

Siraji, A. A. & Zhao, Y. High-sensitivity and high-Q-factor glass photonic crystal cavity and its applications as sensors. Opt. Lett. 40, 1508–1511 (2015).

33. 33.

Wang, Y., Santos, A., Evdokiou, A. & Losic, D. Rational design of ultra-short anodic alumina nanotubes by short-time pulse anodization. Electrochim. Acta 154, 379–386 (2015).

34. 34.

Southwell, W. H. Using apodization functions to reduce sidelobes in rugate filters. Appl. Opt. 28, 5091–5094 (1989).

35. 35.

Losic, D. & Santos, A. Nanoporous alumina: fabrication, structure, properties and applications. Vol. 219 (Springer: 2015).

36. 36.

Shang, G. L., Zhang, Y., Fei, G. T., Su, Y. & Zhang, L. D. Energy-lose induced unidirectional light propagation in porous alumina photonic crystal. Ann. Phys. 528, 288–294 (2016).

37. 37.

Ling, Z., Chen, S., Hu, X. & Li, Y. Optical transmission spectra of anodic aluminum oxide membranes with a dual layer-by-layer structure. Chin. Phys. Lett. 26, 054213 (2009).

38. 38.

Santos, A., Law, C. S., Lei, D. W. C., Pereira, T. & Losic, D. Fine tuning of optical signals in nanoporous anodic alumina photonic crystals by apodized sinusoidal pulse anodisation. Nanoscale 8, 18360–18375 (2016).

39. 39.

Shang, G. L. et al. Preparation of narrow photonic bandgaps located in the near infrared region and their applications in ethanol gas sensing. J. Mater. Chem. C 1, 5285–5291 (2013).

40. 40.

Abràmoff, M. D., Magalhaes, P. J. & Ram, S. J. Image processing with ImageJ. Biophotonics Int. 11, 36–42 (2004).

## Acknowledgements

Authors thank the support provided by the Australian Research Council (ARC) DE140100549, the School of Chemical Engineering (UoA), and The University of Adelaide. Authors thank the Adelaide Microscopy (AM) centre for FEG-SEM characterization.

## Author information

### Affiliations

1. #### School of Chemical Engineering, The University of Adelaide, Adelaide, SA 5005, Australia

• Cheryl Suwen Law
• , Siew Yee Lim
•  & Abel Santos

• Cheryl Suwen Law
• , Siew Yee Lim
•  & Abel Santos
3. #### ARC Centre of Excellence for Nanoscale BioPhotonics (CNBP), The University of Adelaide, 5005, Adelaide, Australia

• Cheryl Suwen Law
• , Siew Yee Lim
•  & Abel Santos

### Contributions

Ms. Cheryl Suwen Law carried out the experimental part of this work assisted by Ms. Siew Yee Lim. Dr. Abel Santos conceived the idea and designed the experimental part of this work. The obtained results were discussed and analyzed by all the authors. The final version of the manuscript was written through contributions of all the authors. All the authors have given approval to the final version of the manuscript.

### Competing Interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Abel Santos.

format: plain


text:
$(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.

Problem 239

Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.

Proof.

Consider the ring $R[t]$, where $t$ is a variable. Since $R$ is an integral domain, so is $R[t]$.
Define the function $\Psi:R[x,y] \to R[t]$ sending $x$ to $t^2$ and $y$ to $t^3$, and extend it to for any $f(x,y)\in R[x,y]$ by linearity.
Then the map $\Psi$ is a ring homomorphism.

Our goal is to show that the kernel of $\Psi$ is the ideal $(x^3-y^2)$. If this is proved, then the first isomorphism theorem implies that $R[x,y]/(x^3-y^2)$ is a subring of the integral domain $R[t]$, and thus $(x^3-y^2)$ is a prime ideal.

Therefore it suffices to show that $\ker(\Psi)=(x^3-y^2)$.
Since we have
$\Psi(x^3-y^2)=(t^2)^3-(t^3)^2=t^6-t^6=0,$ one inclusion $(x^3-y^2) \subset \ker(\Psi)$ is clear.

To show that the other inclusion $\ker(\Psi) \subset (x^3-y^2)$, note that for any $f(x,y)\in R[x,y]$ we can write
$f(x,y)=f_0(x)+f_1(x)y+(x^3-y^2)g(x,y), \tag{*}$ where $f_0(x), f_1(x)\in R[x]$ and $g(x,y)\in R[x,y]$.
To obtain this expression, note that we have
$x^my^n=-(x^3-y^2)x^my^{n-2}+x^{m+3}y^{n-2}$ for integers $m$ and $n\geq 2$.

Use this relation recursively (to the second term), we can express
$x^my^n=(x^3-y^2)p(x,y)+q_0(x)+q_1(x)y$ for some $p(x,y)\in R[x,y]$ and $q_0, q_1\in R[x]$.
Hence we obtain the expression (*).

Suppose that $f(x,y)\in \ker(\Psi)$ and $f(x,y)$ can be written as in (*).
Then we have
\begin{align*}
0&=\Psi(f(x,y))=f(t^2,t^3)\\
&=f_0(t^2)+f_1(t^2)t^3+((t^2)^3-(t^3)^2)g(t^2, t^3)\\
&=f_0(t^2)+f_1(t^2)t^3.
\end{align*}

Note that the even terms of the polynomial $f_0(t^2)+f_1(t^2)t^3$ are $f_0(t^2)$, and the odd terms are $f_1(t^2)t^3$.
Thus, we have $f_0(t^2)=0$, and $f_1(t^2)=0$.
Equivalently, we have $f_0(x)=0$ and $f_1(x)=0$.

It follows that
$f(x,y)=(x^3-y^2)g(x,y)$ and it is in the ideal $(x^3-y^2)$.
Thus we prove that $\ker(\Psi)=(x^3-y^2)$ as required.

1 Response

Polynomial $x^4-2x-1$ is Irreducible Over the Field of Rational Numbers $\Q$
Show that the polynomial $f(x)=x^4-2x-1$ is irreducible over the field of rational numbers $\Q$.


text:
On an asymptotic behavior of elements of order p in irreducible representations of the classical algebraic groups with large enough highest weights

@inproceedings{Suprunenko2000OnAA,
title={On an asymptotic behavior of elements of order p in irreducible representations of the classical algebraic groups with large enough highest weights},
author={Irene D. Suprunenko},
year={2000}
}
The behavior of the images of a fixed element of order p in irreducible representations of a classical algebraic group in characteristic p with highest weights large enough with respect to p and this element is investigated. More precisely, let G be a classical algebraic group of rank r over an algebraically closed field K of characteristic p > 2. Assume that an element x E G of order p is conjugate to that of an algebraic group of the same type and rank m < r naturally embedded into G. Next…
1 Citations
Abstract Let G be a simple algebraic group over an algebraically closed field K of characteristic p>0{p>0}. We consider connected reductive subgroups X of G that contain a given distinguished

References

SHOWING 1-10 OF 17 REFERENCES

This article is devoted to the study of the behaviour of elements of order p in irreducible representations of the classical algebraic groups in characteristic p whose highest weights are large
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 1976
Let G be a simple adjoint algebraic group over an algebraically closed field K. We are concerned to describe the conjugacy classes of unipotent elements of G. G operates on its Lie algebra g by means
• W. Feit
• Mathematics
North-Holland mathematical library
• 1982
This book discusses Blocks and Extensions of R, an Analogue of Jordan's Theorem, and groups with a Sylow Group of Prime Order.
• Mathematics
• 1990
1. Motivation and setting for the results 2. Basic properties of the classical groups 3. The statement of the main theorem 4. The structure and conjugacy of the members of C 5. Properties of the
Keywords: semisimple algebraic groups ; closed connected subgroups ; finite groups ; fixed point subgroups Reference CTG-ARTICLE-1995-001doi:10.1006/jabr.1995.1285 Record created on 2008-12-16,
• Mathematics
• 1999
Introduction Labelled diagrams Essential embeddings Unipotent classes Centralizers Results $(\text {Aut} G)$-conjugacy Tables of $A_1$ subgroups References.
Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the