Jekyll2022-07-01T17:10:06+00:00http://www.corcoviloslab.com/feed.xmlCorcovilos LabPersonal site of Ted Corcovilos, Associate Professor of Physics at Duquesne UniversityOptics in Homogeneous Coordinates, Coordinate Tranformations2022-07-01T00:00:00+00:002022-07-01T00:00:00+00:00http://www.corcoviloslab.com/research/2022/07/01/HomogeneousOptics-2<!-- TODO clean up URLs -->
<!-- TODO add examples -->
<!-- kramdown tags defined below -->
<!-- end kramdown -->
<!--
Reminder that mathjax is enabled. Inline math using double backslash parenthesis: \\( \\)
Display math using double dollar or double backslash bracket: $$ $$ or \\[ \\]
-->
<!--
kramdown reference: https://kramdown.gettalong.org/quickref.html
-->
<p><em>This is the second post discussing my new paper on a new way to look at geometric optics.
I deconstruct the familiar ABCD ray-transfer matrices and rebuild them based on geometric considerations.
The paper itself is posted to <a href="http://arxiv.org/abs/2205.09746">arXiv</a><sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup>.
These blog posts will focus on the motivation and how to use the results.</em></p>
<p><a href="/research/2022/05/20/HomogeneousOptics-1.html">Part 1 is here.</a></p>
<h2 id="picking-up-from-last-time">Picking up from last time…</h2>
<p style="
float: right;
padding-left: 5px;
padding-right: 0px;
text-align: center;
width: 350px;
box-sizing: border-box;
font-size: 0.9em;
"><img src="/assets/figs/2022-optics/line.svg" alt="line" /><br />
The equation of an oriented line. In this example, <em>a</em> and <em>b</em> > 0 and <em>c</em><0.</p>
<p>In the <a href="/research/2022/05/20/HomogeneousOptics-1.html">previous post</a> I defined an oriented line by considering the equation of a line in 2D:
$ ax+by+c = 0 $, which we can abbreviate as the vector of coefficients \( (c,a,b) \).
In this post, we want to perform some geometric operations on this line.
Namely, we want to move (translate) it, rotate it about the coordinate origin, and switch its orientation (from left-to-right to right-to-left).
Once we understand how to manipulate the line, we’ll perform these same transformations on our ABCD matrices.</p>
<p>For the sake of brevity, I’ll give the results here and put the <a href="#appendix-derivations">derivations at the bottom</a>.</p>
<p>To translate a ray by a displacement vector (<em>u</em>, <em>v</em>) we multiply the ray vector from the left by the matrix
\[
T(u,v) = \begin{pmatrix} 1 & -u & -v \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.
\]
In other words, the new ray <em>r</em>’ = <em>T r</em>.</p>
<p>Similarly, rotations of the rays are represented by the matrix
\[ R(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \end{pmatrix},\]
giving a new ray <em>r</em>’ = <em>R r</em>.</p>
<p>For the ray transfer matrices, both the row space and column space of the matrix must be transformed. If we rotate and then translate, the new matrix is \( M’ = TRMR^{-1}T^{-1} \).</p>
<p>One way to understand why we need two copies of the transformation matrices is this –
our ABCD matrices for optical elements assume the element is located at the origin.
If we want to place that surface anywhere else, we have to move our coordinates such that the desired optical element location is at the origin, install our element, then restore the original coordinates.</p>
<h2 id="why-this-matters">Why this matters</h2>
<p>Being able to translate and rotate our lenses and mirrors lets us model optical systems <em>just as they are built on our optical table</em>.
We don’t have to unfold our beam paths to model them.
And we can see what happens if a lens is decentered or tilted
(e.g. when your professor bumps a random mirror and you have to realign everything, not that I have any experience with that…).</p>
<p>Also, now we have a way to mathematically describe not just optics, but also opto-mechanics.
We can model beam-steering elements like galvos, motorized mirrors, or acousto-optics.
We can do tolerancing of layouts or vibration analysis with simple matrix multiplication.
To an experimentalist like myself with only a handful of math tools in my pocket and insufficient patience to code all of this up in Zemax, <em>this is exciting!</em></p>
<p>Now, the skeptics will say this is not new physics.
We learning nothing here that we haven’t seen before.
This is true, but it misses the point.
My motivation is pragmatic: do more stuff with simple tools (well, as long as you think multiplying 3×3 matrices isn’t too large of a price).</p>
<h2 id="examples">Examples</h2>
<h3 id="right-angle-mirror-pair">Right-angle mirror pair</h3>
<p style="
float: right;
padding-left: 5px;
padding-right: 0px;
text-align: center;
width: 350px;
box-sizing: border-box;
font-size: 0.9em;
"><img src="/assets/figs/2022-optics/retro.svg" alt="retroreflector" /><br />
Retroreflector</p>
<p>I have several examples in the paper<sup id="fnref:1:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup>, but my favorite is a common lab optic: a right-angle retroreflector.
The setup is simple: we have two flat mirrors that intersect at right angles.
Any incoming ray will be retroreflected opposite its original direction, offset by some distance.</p>
<p>The RTM for the upper mirror ($M_1$) is generated by a $45^{\circ}$ rotation of a plane mirror situated at the origin:</p>
<div>
\[
\begin{aligned}
M_1 &= R_{45^{\circ}}M_\text{plane mirror}R_{45^{\circ}}^{-1}, \\
&=
\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1/\sqrt{2} & -1/\sqrt{2} \\ 0 & 1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix}
\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}
\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1/\sqrt{2} & 1/\sqrt{2} \\ 0 & -1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix} \\
&= \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}.
\end{aligned}
\]
</div>
<p>Similarly, the second mirror ($M_2$) has the RTM</p>
<div>
\[
M_2 = R_{-45^{\circ}}M_\text{plane mirror}R_{-45^{\circ}}^{-1} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix}.
\]
</div>
<p>We choose an incoming ray ${r}_0 = (-h,-m,1)^T$ with $h>0$ such that it will strike mirror $M_1$ first, yielding the reflected ray</p>
<div>
\[
{r}_1 = M_1 {r}_0 = (h,1,-m)^T.
\]
</div>
<p>This ray follows the line $ h +x-my =0 $, propagating from right to left ($b < 0$).
After the second reflection in $M_2$ the final ray is</p>
<div>
\[
{r}_2 = M_2 {r}_1 = (-h,m,-1)^T,
\]
</div>
<p>which is antiparallel to the incoming ray, as expected, with a $y$ intercept of $-h$ and propagating right to left ($b < 0 $).</p>
<p>That’s a piece-wise description, but we can multiply the two mirror matrices to get the system matrix for the retroreflector:</p>
<div>
\[
M_\text{ra} = M_2 M_1 =
\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}.
\]
</div>
<p>Multiplying this matrix by the original ray gives $r_2$ above in one step.</p>
<h3 id="right-angle-prism">Right-angle prism</h3>
<p>We can also upgrade our retroreflector to a right-angle prism retroreflector by inserting refracting surfaces before and after the mirror pair.
We’ll assume the prism is made of glass with an index of refraction of <em>n</em>.
In the mirror setup above, the intersection of the mirrors was located at the coordinate origin.
We’ll place the hypotenuese of the prism a distance <em>d</em> to the left of the origin.</p>
<p>The matrix for the entrance surface is a refracting surface from index of refraction 1 to index of refraction <em>n</em>, translated to the left by <em>d</em>:</p>
<div>
\[
\begin{aligned}
S_1 &= T(-d,0)\,M_\text{ref}(1,n)\,T(d,0) \\
&=
\begin{pmatrix} 1 & d & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1/n & 0 \\ 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & -d & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\\
&=
\begin{pmatrix} 1 & (d-nd)/n & 0 \\ 0 & 1/n & 0 \\ 0 & 0 & 1 \end{pmatrix}
\end{aligned}
\]
</div>
<p>The matrix for the exit surface is constructed similarly:</p>
<div>
\[
\begin{aligned}
S_2 &= T(-d,0)\,M_\text{ref}(n,1)\,T(d,0) \\
&=
\begin{pmatrix} 1 & nd-n & 0 \\ 0 & n & 0 \\ 0 & 0 & 1 \end{pmatrix}
\end{aligned}
\]
</div>
<p>Now, we just combine these two surfaces in the correct order with the right-angle mirrors from above to get our prism matrix:</p>
<div>
\[
\begin{aligned}
M_\text{prism} &= S_2 M_\text{ra} S_1 \\
&= \begin{pmatrix} 1 & 2d(n-1)/n & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}
\end{aligned}
\]
</div>
<p>What does this matrix tell us? We still get a retroreflector, as expected, but the outgoing ray is slightly displaced relative to the mirror-pair case.
For our now canonical incoming ray $ r = (h,m,-1)^T $, we get an outgoing ray</p>
<div>
\[
r' = M_\text{prism} r =
\begin{pmatrix} 1 & 2d(n-1)/n & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}
\begin{pmatrix} h \\ m \\ -1 \end{pmatrix}
= \begin{pmatrix} -h + \frac{2md(n-1)}{n} \\ m \\ -1 \end{pmatrix},
\]
</div>
<p>showing a displacement in the height of ${2md(n-1)}/{n}$ relative to the mirror pair (i.e., setting $n=1$).</p>
<h2 id="whats-missing">What’s missing</h2>
<h3 id="aberrations">Aberrations</h3>
<p>These new tools let us analytically lay out our optical setup, but we are still only working in the paraxial limit.
That means our angles and beam displacements can’t get too large else our errors will accumulate, particularly for refractive elements and spherical surfaces. (Plane mirrors are perfect!)
Specifically, our method says nothing about aberrations because those are, by definition, nonlinear in the ray height and slope, and we are using a strictly linear formulation.
Can we extend these ideas to low-order aberrations?
I’m looking into a few ideas others have published recently<sup id="fnref:2" role="doc-noteref"><a href="#fn:2" class="footnote" rel="footnote">2</a></sup>, so stay tuned.
Computer tricks like automatic differentiation might yield a simple, compact way to look beyond the paraxial limit without doing full-bore ray tracing.</p>
<h3 id="the-third-dimension">The Third Dimension</h3>
<p>Extending the ray matrices to three dimensions is fairly straight forward, but a bit cumbersome as we need 6×6 matrices.
Lin has one method that keeps things looking similar to the 2-D case<sup id="fnref:3" role="doc-noteref"><a href="#fn:3" class="footnote" rel="footnote">3</a></sup>, but I’m looking into the prefered representation of the computer graphics folks: <a href="https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates">Plücker coordinates</a>, which are a true homogeneous representation of lines in 3D, and thus preserve geometric information in a covariant way.<sup id="fnref:4" role="doc-noteref"><a href="#fn:4" class="footnote" rel="footnote">4</a></sup>
This will be important to investigating points and planes (see next post!).</p>
<h2 id="next-time">Next time…</h2>
<p>In the next post I’ll show how use a little algebra trick to apply the ray transfer matrices to imaging problems by directly mapping points from the object space of the optical system to the image space.
This is nice because we no longer have to use intersecting rays to find image locations or magnifications.
The key is using a homogeneous representation of our points.</p>
<p>And in the post after that, I’ll look beyond the paper and shift to the other major application of the ABCD matrices: Gaussian laser beams.</p>
<h2 id="appendix-derivations">Appendix: Derivations</h2>
<p>Here I walk through how the translation and rotation matrices are constructed.
The rotation matrices you may have seen before, although probably not in homogeneous coordinates.
The translation matrices are unique to the homogeneous representation.
In traditional 2D matrix algebra, translation is not representable by matrix multiplication. Instead, you have to add vectors directly.
Being able to put all of the Euclidean transformations into the same form is one of the major benefits of using homogeneous coordinates.</p>
<h3 id="translation">Translation</h3>
<p>Our first task is to shift our line by a constant vector (<em>u</em>,<em>v</em>). Equivalently, we can move our coordinate axis by the opposite translation. In other words, we can do \( x \rightarrow x-u, y \rightarrow y-v \).
Substituting this change into the line equation, we have
\[ a(x-u) + b(y-v) + c = 0, \]
\[ ax + by + (-au -bv + c) = 0. \]
The coefficents are transforming line \( a \rightarrow a, b \rightarrow b, c \rightarrow c-au-bv \).
Working backwards, we can represent this transformation of the coefficients as a matrix equation:
\[
\begin{pmatrix}c \\ a \\ b \end{pmatrix}
\rightarrow
\begin{pmatrix} 1 & -u & -v \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix}c \\ a \\ b \end{pmatrix}.
\]
We identify the matrix here as the <em>translation operator</em> and we’ll give it the name
\[
T(u,v) = \begin{pmatrix} 1 & -u & -v \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.
\]
Note that only the <em>c</em> coefficient changes.
If our line coefficients happend to be normalized such that \( a^2 + b^2 = 1 \), then <em>c</em> is the distance of the line from the origin.
What does this mean?
Because only <em>c</em> is changing, any translation of a line can be interpreted as motion towards or away from the origin.
Any motion perpendicular to that would simply shift the line along itself, resulting in no change to the line.</p>
<h3 id="rotation">Rotation</h3>
<p>Now we’d like to rotate the line.
Rotation of the line by an angle <em>θ</em> about the coordinate origin is equivalent to rotating the axes by the opposite angle <em>-θ</em>.
This looks like the usual rotation operation in the <em>xy</em> plane:
\[ x \rightarrow x \cos(-\theta) + y \sin(-\theta) = x \cos(\theta) - y \sin(\theta), \]
\[ y \rightarrow x (-)\sin(-\theta) + y \cos(-\theta) = x \sin(\theta) + y \cos(\theta),\]
where I’ve used the odd/even nature of sin/cos to simplify the signs.</p>
<p>Substitute these changes into our line equation:
\[ a(x \cos\theta - y \sin\theta) + b(x \sin\theta + y \cos\theta) + c = 0, \]
\[ (a \cos \theta + b \sin\theta)x + (-a \sin\theta + b \cos\theta)y + c = 0. \]</p>
<h2 id="references">References</h2>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:1" role="doc-endnote">
<p>T. Corcovilos. Beyond the ABCDs: A projective geometry treatment of paraxial ray tracing using homogeneous coordinates. <a href="http://arxiv.org/abs/2205.09746">arXiv:2205.09746</a> (2022) <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a> <a href="#fnref:1:1" class="reversefootnote" role="doc-backlink">↩<sup>2</sup></a></p>
</li>
<li id="fn:2" role="doc-endnote">
<p>Lin, P.-D.; Hsueh, C.-C. 6×6 Matrix Formalism of Optical Elements for Modeling and Analyzing 3D Optical Systems. <i>Appl. Phys. B</i> <b>2009</b>, <i>97</i> (1), 135–143. <a href="https://doi.org/10.1007/s00340-009-3616-7">https://doi.org/10.1007/s00340-009-3616-7</a>. Lin, P. D.; Johnson, R. B. <i>Opt. Express</i> <b>2019</b>, <i>27</i> (14), 19712. <a href="http://doi.org/10.1364/OE.27.019712">doi:10.1364/OE.27.019712</a>. <a href="#fnref:2" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:3" role="doc-endnote">
<p>Lin, P. D. <i>New Computation Methods for Geometrical Optics</i>; Springer Series in Optical Sciences; Springer Singapore: Singapore, 2014; Vol. 178. <a href="https://doi.org/10.1007/978-981-4451-79-6">ISBN: 978-981-4451-79-6</a>. <a href="#fnref:3" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:4" role="doc-endnote">
<p>Wolf, K. B. Optical Models and Symmetry. Chapter 4 of <i>Progress in Optics</i>; Elsevier, 2017; Vol. 62, pp 225–291. <a href="https://doi.org/10.1016/bs.po.2016.12.002">https://doi.org/10.1016/bs.po.2016.12.002</a>. <a href="#fnref:4" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
</ol>
</div>Last time we added an extra row and column to our ABCD matrices. What can we do with these?Optics in Homogeneous Coordinates, Introduction2022-05-20T00:00:00+00:002022-05-20T00:00:00+00:00http://www.corcoviloslab.com/research/2022/05/20/HomogeneousOptics-1<!-- kramdown tags defined below -->
<!-- end kramdown -->
<!--
Reminder that mathjax is enabled. Inline math using double backslash parenthesis: \\( \\)
Display math using double dollar or double backslash bracket: $$ $$ or \\[ \\]
-->
<!--
kramdown reference: https://kramdown.gettalong.org/quickref.html
-->
<p><em>This is the first post discussing my new paper on a new way to look at geometric optics.
I deconstruct the familiar ABCD ray-transfer matrices and rebuild them based on geometric considerations.
The paper itself is posted to <a href="http://arxiv.org/abs/2205.09746">arXiv</a><sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup>.
These blog posts will focus on the motivation and how to use the results.</em></p>
<p><a href="/research/2022/06/01/HomogeneousOptics-2.html">Part 2 is here.</a></p>
<h2 id="motivation">Motivation</h2>
<h3 id="case-1-imaging-problems">Case 1: Imaging problems</h3>
<p>When teaching undergraduate optics, I always found the problems containing series of lenses and mirrors tedious.
The standard solution is to find the image of an object created by each optical element using Gauss’s Lens equation.
This image then becomes the object for the next element.
Apply Gauss’s equation again. Rinse and repeat.
For one or two lenses this is not too bad, but once you get to something like a microscope (objective, tube lens, and ocular), then it really drags.
Worse, if your object moves a little bit you have to start over.
And there is also the weird issue of infinitely far objects and images to deal with.
It seems a little incomplete. I’m looking for a better, or at least less tedious, method.</p>
<p>One solution is that I can take two rays that pass through my object point and find where they intersect on the output side to locate the image, but that’s still a lot of algebra, and I am lazy.</p>
<h3 id="case-2-ray-tracing">Case 2: Ray tracing</h3>
<p>In my research lab, we mostly deal with lasers, which to a reasonable approximation can be treated as geometric rays (at least as far as alignment is concerned).
When sketching out a new optical setup, I’ll often use <a href="https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis">ray transfer matrices</a> (also called “ABCD” matrices) to set things up.
Each ray is represented as a vector containing the ray height and slope.
Each optical element corresponds to a matrix.
By multiplying these matrices together we can get a representation of the entire optical system.
Matrix times input ray equals output ray.</p>
<p style="
border: 1px solid black;
width: 300px;
float: right;
color: #606060;
padding: 5px;
font-size: 0.9em;
"><em>But wait! There’s more!</em>
The same ABCD matrix that describes the path of geometric rays through the system also describes the change in size of Gaussian laser beams as they travel through the system.
This seems like magic, because there is no apparent relationship between rays and the size of diffracting laser beams.
The detailed proof requires digging into the guts of the diffraction integrals<sup id="fnref:2" role="doc-noteref"><a href="#fn:2" class="footnote" rel="footnote">2</a></sup>.
Surely there is a simpler explanation.
(Arnaud has a nice model using complex-valued rays<sup id="fnref:3" role="doc-noteref"><a href="#fn:3" class="footnote" rel="footnote">3</a></sup>, which again seems like magic, but at least understandable magic.)</p>
<p>This is great, as far as it goes, but it assumes all of the lenses and mirrors are perfectly aligned and centered.
Often that just isn’t the case.
For example, a common right-of-passage for students in a laser lab is learning to inject a beam into a fiber.
There’s a lot of degrees of freedom that you have to manage to high precision:</p>
<ul>
<li>x, y, and z position of the focusing lens</li>
<li>two tilt angles of the lens</li>
<li>two tilt angles of the incomming beam</li>
<li>the location of the beam’s waist</li>
</ul>
<p>Can we put together a concise mathematical model of all of these degrees of freedom?</p>
<h3 id="the-goal">The goal</h3>
<p>What I would really like to do is find a solution that lets me model optical elements in any position or orientation and lets me solve imaging problems.
And I would like it to be as simple to use as the ABCD matrices.
Can I get there starting from the old-fashioned ABCD matrices?</p>
<p>The answer is “yes,” but first I need to re-examine what the ABCD matrices are doing to the rays, in a geometric sense.
And to do <em>that</em>, I first need to mathematically define rays as oriented lines.</p>
<h2 id="a-homogenous-vector-representation-of-lines">A homogenous vector representation of lines</h2>
<p style="
float: right;
padding-left: 5px;
padding-right: 0px;
text-align: center;
width: 350px;
box-sizing: border-box;
font-size: 0.9em;
"><img src="/assets/figs/2022-optics/line.svg" alt="line" /><br />
The equation of an oriented line. In this example, <em>a</em> and <em>b</em> > 0 and <em>c</em><0.</p>
<h3 id="the-equation-of-a-line">The equation of a line</h3>
<p>Let’s start with the equation for a line in two-dimensions.
If our coordinate axes are (<em>x</em>,<em>y</em>) then any line will have the form \( ax+by+c=0 \).
Rather than write down the equation for every line, we could just keep track of the coefficients (<em>a</em>,<em>b</em>,<em>c</em>).
Also, if we multiply the whole equation (or all of the coefficients) by a non-zero scalar, we get the same line.
In other words, the vector (<em>a</em>,<em>b</em>,<em>c</em>) gives us a <em>homogenous</em> representation of the line.</p>
<p>I think most of us almost habitually use the slope-intersept form of a line: \( y = mx+h \).
In our new represenation we rearrange this to \( mx -y + h = 0 \) and identify the coeffients as (<em>m</em>,-1,<em>h</em>). (Hold on to this – it will be important.)</p>
<h3 id="normalization">Normalization</h3>
<p>Because we can multiply the line coefficients by any nonzero scalar, is there any prefered normalization?
For most of what we’ll be doing, the normalization won’t matter, but let’s take a quick peek.
One convenient choice is to divide the line coefficients by \( \sqrt{a^2+b^2} \).
Then the new <em>a</em> and <em>b</em> coefficients become the direction cosines with respect to those axes and the new <em>c</em> value gives the (signed) distance from the origin to the line.
With this normalization, the angle <em>θ</em> between two lines can be calculated easily: \( \theta=\arccos(a_1a_2+b_1b_2) \) (proven by applying some trig identities).</p>
<h3 id="orientation">Orientation</h3>
<p>Besides normalization, we have one more degree of freedom that we can assign meaning to: the overall sign of the coefficients.
We’ll interpret this as <em>orientation</em> or the direction of travel along the line.
For <em>b</em>>0 the line goes left-to-right. If <em>b</em>=0, then <em>a</em>>0 indicates downward. Lastly, if <em>a</em>=<em>b</em>=0, then <em>c</em>>0 is the imaginary line infinitely far away encircling the space in a counter-clockwise direction.</p>
<p>By keeping track of these signs, we can build an expression for mirrors that also changes the direction of the rays.</p>
<h2 id="the-ray-transfer-matrices">The ray-transfer matrices</h2>
<p style="
float: right;
padding-left: 5px;
padding-right: 0px;
text-align: center;
width: 350px;
box-sizing: border-box;
font-size: 0.9em;
"><img src="/assets/figs/2022-optics/ABCD.svg" alt="ray transfer schematic" /><br />
The ABCD matrices. The incoming ray <em>r</em> is transformed into the outgoing ray <em>r’</em>.
Note that the outgoing coordinate system is different than the incoming coordinate system. We’ll address this in the next blog post.</p>
<p>Let’s back up to the beginning and look at one way to describe optical systems. If we have an incoming ray <em>r</em> that passes through the input plane of the optical system with height <em>h</em> and slope <em>m</em> and an outgoing ray <em>r’</em> with height and slope <em>h’</em> and <em>m’</em>, we can relate these by the matrix equation
\[
\begin{pmatrix} h’ \\ m’ \end{pmatrix}
=
\begin{pmatrix} A & B \\ C & D \end{pmatrix}
\begin{pmatrix} h \\ m \end{pmatrix}.
\]</p>
<p>Now, let’s do the same transformation on a line as we described above.
Our point-slope version of a line has homogeneous coordinates (<em>a</em>,<em>b</em>,<em>c</em>) = (<em>m</em>,-1,<em>h</em>).
By rearranging the order of the coefficients, we can get
\[
\begin{pmatrix} h’ \\ m’ \\ -1 \end{pmatrix}
=
\begin{pmatrix} A & B & 0
\\ C & D & 0
\\ 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix} h \\ m \\ -1 \end{pmatrix}.
\]
or more generally,
\[
\begin{pmatrix} c’ \\ a’ \\ b’ \end{pmatrix}
=
\begin{pmatrix} A & B & 0
\\ C & D & 0
\\ 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix} c \\ a \\ b \end{pmatrix}.
\]</p>
<p>This is similar enough to our original ABCD matrices that it has some promise without being too far from where we started.</p>
<h2 id="wrapping-up">Wrapping up</h2>
<p>What have we done here?
I’ll admit it looks underwhelming.
We’ve added an extra dimension to our ray-transfer matrix, but it doesn’t seem to <em>do</em> anything but multiply a bunch of ones together.</p>
<p>There are two important things here.
First, the original ABCD matrices look more like a bookkeeping tool than anything “real”, because we’re just keeping track of some coefficients.
(In fact, standard optical engineering textbooks usually dismiss them as pointless fluff<sup id="fnref:4" role="doc-noteref"><a href="#fn:4" class="footnote" rel="footnote">4</a></sup> or don’t mention them at all<sup id="fnref:5" role="doc-noteref"><a href="#fn:5" class="footnote" rel="footnote">5</a></sup>.)
But now, we see a bit of geometry.
We have a general description of a line being transformed into another line.</p>
<p>Second, we have added some additional degrees of freedom to our framework.
We used one of these to add a meaningful orientation to the lines.
Three more degrees of freedom can be used to translate and rotate the coordinate axes or, equivalently, our optical elements.
This will be the topic of Part 2.</p>
<p><em><a href="/research/2022/06/01/HomogeneousOptics-2.html">Continued in Part 2</a></em></p>
<h2 id="bibliography">Bibliography</h2>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:1" role="doc-endnote">
<p>T. Corcovilos. Beyond the ABCDs: A projective geometry treatment of paraxial ray tracing using homogeneous coordinates. <a href="http://arxiv.org/abs/2205.09746">arXiv:2205.09746</a> (2022) <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:2" role="doc-endnote">
<p>S. A. Collins, J. Opt. Soc. Am., JOSA 60, 1168 (1970). <a href="https://doi.org/10.1364/JOSA.60.001168">doi:10.1364/JOSA.60.001168</a> <a href="#fnref:2" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:3" role="doc-endnote">
<p>J. Arnaud, Applied Optics 24, 538 (1985). <a href="https://doi.org/10.1364/AO.24.000538">doi:10.1364/AO.24.000538</a> <a href="#fnref:3" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:4" role="doc-endnote">
<p>W. J. Smith, Modern Optical Engineering: The Design of Optical Systems, 4th ed (McGraw Hill, New York, 2008). ISBN: 978-0-07-147687-4 <a href="#fnref:4" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:5" role="doc-endnote">
<p>M. J. Kidger, Fundamental Optical Design (SPIE, Bellingham, 2000). ISBN: 978-0-8194-9599-0 <a href="#fnref:5" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
</ol>
</div>I introduce the homogeneous representation of rays and extend the ABCD matrices to include rotations and translations (properly!).Our New Master’s Program2022-03-15T00:00:00+00:002022-03-15T00:00:00+00:00http://www.corcoviloslab.com/teaching/2022/03/15/masters<!-- kramdown tags defined below -->
<!-- end kramdown -->
<!--
Reminder that mathjax is enabled. Inline math using double backslash parenthesis: \\( \\)
Display math using double dollar or double backslash bracket: $$ $$ or \\[ \\]
-->
<!--
kramdown reference: https://kramdown.gettalong.org/quickref.html
-->
<p><em>The opinions here are my personal point of view and not endorsed by Duquesne University or the Duquesne Physics Department.</em></p>
<p>We just finished a recruiting blitz for our new <a href="https://www.duq.edu/academics/schools/natural-and-environmental-sciences/academics/departments-and-programs/physics/professional-masters-degree-in-applied-physics">Professional Master’s Degree in Applied Physics program</a>.
I thought it would be useful to share some insights I’ve had about the program creation process and our design philosophy.</p>
<p>Here is a recording of my formal presentation (an earlier version), but below I discuss how all of this came to be.</p>
<center>
<iframe width="560" height="315" src="https://www.youtube.com/embed/q42RyIOcnsc" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
</center>
<h2 id="initial-steps">Initial steps</h2>
<p>The idea of having a professional master’s program was first formally stated in 2014 during a periodic external departmental review as a way for the department to grow while complementing the strong research output of our faculty.
The “professional” part came out of</p>
<ol>
<li>the need to differentiate our proposed program from the traditional (academic) master’s programs already in place at the larger research universities in Pittsburgh,</li>
<li>a growing pressure for “workforce development” and putting students directly into jobs, and</li>
<li>the unfair perception of a physics master’s degree being primarily for those students who drop out of Ph.D. programs.</li>
</ol>
<p>(Also, we wanted to avoid the phrase “terminal master’s,” which sounds too much like some horrible illness.)</p>
<p>The basic idea of professional programs is to add business skills to the curriculum.
At the time, we left things at that, as only a vague picture of where we were going, but it was enough to
start going through the administrative processes for starting a new program.</p>
<h2 id="fact-finding">Fact finding</h2>
<p>Pretty quickly we realized that our new program couldn’t just mix a sprinkle of business courses with graduate versions of undergrad classes (Graduate E&M, Graduate Quantum, etc.).
But those physics courses were all classes that we understood, because all of the faculty took classes like that in our Ph.D. work.
What to replace those with was not so obvious.
We did, however, have a clear goal for our students: <em>to get them jobs.</em></p>
<p>So, we went out and started talking to people who had jobs.</p>
<p>By that, I mean that we started talking to recent alumni who were working in industry, either immediately after graduating with a bachelor’s degree or who had gone on to a graduate degree before entering the non-academic workforce.
Our department chair contacted a couple of former students that she had kept in touch with, and we arranged to visit them.
I also reached out to <a href="/group#former-members">my former students</a> who were just starting in their jobs.</p>
<p>We asked a set of simple questions:</p>
<ul>
<li>What do you do?</li>
<li>What did you learn as an undergraduate that prepared you for your job?</li>
<li>What skills did you have to learn after you started your job?</li>
<li>How does your education as a physicist distinguish you from coworkers?</li>
</ul>
<p>The answers we got were perhaps not surprising, but they showed us our blind spots coming from the academic world.
Some of the key items were</p>
<ul>
<li>Hands-on experience with <em>specific</em> equipment and software. (Shout out to Sean Krupa for providing a detailed list.)</li>
<li>Knowledge of how industrial R&D works. From mundane things like terminology (e.g. I had no idea what <a href="https://en.wikipedia.org/wiki/Technology_readiness_level">Tech Readiness Levels</a> were) to broader concepts like stakeholder interest, care and feeding of intellectual property, project life-cycles, allocation of effort, and organizational patterns.</li>
<li>Communication, communication, communication. Everything from how to present at weekly meetings, proper business email etiquette, writing formal progress reports, to requesting funds from higher-ups or investors. One general theme was presenting technical information to non-technical audiences.</li>
</ul>
<p>We also talked with the co-workers and bosses of some of our alumni.
It was also encouraging to hear how useful a physics education is in the workplace beyond the obvious things.
Some highlights:</p>
<ul>
<li>Resilience - physicists are tenacious problem solvers who don’t give up easily when solutions are not apparent or require many steps, and they are not discouraged by failure.</li>
<li>Flexibility - physicists are great at working outside their comfort zone and learning new things when needed.</li>
</ul>
<p>With notes in hand, we set out to combine those best attributes of a traditional academic physics education with the skills we saw used in real jobs.
This lead to four program learning objectives. Upon completion of the program, our students will</p>
<ol>
<li>Demonstrate <em>content knowledge</em> relevant to current science and technology enterprises,</li>
<li>Demonstrate knowledge and skills for <em>measurement and control</em> of physical variables,</li>
<li>Develop and use <em>computational models and simulations</em> of physical phenomena and processes, and acquire, process, and analyze data from experiments, and</li>
<li><em>Communicate</em> scientific, technical, and professional information in variety of formats used in industrial/business settings: written and oral, formal and informal.</li>
</ol>
<h2 id="synthesis">Synthesis</h2>
<p>The final key component was also the first: our faculty.
To make this program fly we had to lean into our strengths, so we created courses that match the expertise already in place.</p>
<h3 id="technical-core">Technical Core</h3>
<p>Rather than more traditional physics course offerings, we selected specific skills that our faculty have that translate to a commercial or government R&D environment.
The courses are lecture-lab hybrids, to give our students hands-on experience with equipment and techniques.</p>
<p>For my part, I’ll be teaching “Advanced Optical Theory and Devices.”
I took Sean’s notes and identified a handful of projects that tie together concepts and techniques used in optical systems.
Tentatively, the main themes are imaging, interferometry, and optical sensing.
Students will design and build full optical systems using industrial software and tools.</p>
<p>Let me briefly highlight a few other courses.</p>
<ul>
<li><a href="https://www.duq.edu/academics/faculty/fatiha-benmokhtar">Fatiha Benmokhtar</a> will teach “Data Acquisition and Control” using her experience with high speed electronics from working on particle accelerator experiments.</li>
<li><a href="https://www.duq.edu/academics/faculty/monica-sorescu">Monica Sorescu</a> will teach “Materials Science”, showcasing her expertise in the synthesis and characterization of nanomaterials.</li>
<li>We are very fortunate to be joined by Yang Wang, Senior Computational Scientist at the <a href="https://www.psc.edu/">Pittsburgh Supercomputing Center</a>. He will bring his skills in high performance computing for the modeling of quantum materials to our “Computational Physics” course.</li>
</ul>
<h3 id="business-core">Business Core</h3>
<p>The other key element of our program is a set of foundational business courses to prepare our students for the corporate world as either potential managers or entrepreneurs.
Our colleagues at the <a href="https://www.duq.edu/academics/schools/business">Palumbo-Donahue School of Business</a> worked with us to find a set of courses that wil give our graduates a head start.
We also are aiming for maximum flexibility, so we picked classes that are available as either traditional or on-line courses.</p>
<p>While this topic is out of my wheelhouse, I think our students will be well-prepared to think about product life-cycles and business development from root to stem.</p>
<h3 id="real-world-experience">Real-world experience</h3>
<p>Lastly, our program includes an industrial internship and capstone project.
Developing this has been a fun one for me. I stepped outside my comfort zone to meet with tech industry leaders here in Pittsburgh, seeking partners willing to host students.
Pittsburgh is a regional hub for a lot of the big names in data and tech, and robotics has a strong presence locally.
I’m not allowed to name-drop yet, but I’ve been talking to companies working on things as varied as self-driving vehicles, materials science, quantum computing, and AI.</p>
<h2 id="putting-out-our-shingle">Putting out our shingle</h2>
<p>If you’ve read this far, you can probably see how excited I am about what we’ve built.
But we’re still missing the most important component: students!
Building up the recruiting campaign for this new program has been another novel experience for me.
Let’s just say we’re trying it all: web ads, direct email, direct mail, Facebook, Linked-In, job fairs, and probably a half-dozen other avenues.
I’ve never been involved with this side of the university before. Our undergraduate admissions is largely automatic from the perspective of the faculty.
Graduate recruiting, however, needs a more personal touch. We had to get the message just right to connect with folks that best match our program.
We lack the name recognition of larger schools, but I hope our attention to detail and the care that went into building this program comes through to our prospectives.
Time will tell, but we’re getting a good response so far.</p>I discuss how we built our new Professional Master's in Applied Physics program at Duquesne.A student caught me studying2019-11-20T00:00:00+00:002019-11-20T00:00:00+00:00http://www.corcoviloslab.com/teaching/2019/11/20/teachingprep<!-- kramdown tags defined below -->
<!-- end kramdown -->
<!--
Reminder that mathjax is enabled. Inline math using double backslash parenthesis: \\( \\)
Display math using double dollar or double backslash bracket: $$ $$ or \\[ \\]
-->
<p style="float: right;
padding-left: 5px;
padding-right: 0px;
text-align: center;
width: 350px;
box-sizing: border-box;
"><img src="/assets/photos/Everest.jpg" alt="Mount Everest" /><br />
Mount Everest (<a href="https://commons.wikimedia.org/wiki/File:Everest_kalapatthar.jpg">photo by Pavel Novek, Wikipedia</a>).</p>
<p>It’s been a while since I’ve posted.
I have no good excuse, other than I’ve been busy and this got pushed way down on the priority list.
But I had a conversation with a student last week that I felt was worth sharing.</p>
<p>It was during my office hours (a.k.a., my alone time) when a student walked in (gasp!) with a question about their course selections for next semester.
I was looking over the textbook for the class I’m teaching in the Spring and was working some problems on the whiteboard. The student was a little shocked:
“Don’t you know that already?”</p>
<p>Well, I do, but that’s not the point.</p>
<p>Every time I teach I try to learn something new.
Maybe it’s digging into the next level past what I’ll cover in the class.
Or maybe I’m looking into connections between this class and other classes.
Or I might be looking back into the history of how things came to be.
Or sometimes I’m learning a new way of solving old problems by teaching myself new math or new software.
Underneath all of that, more often than not, I’m looking for a new way to <em>teach</em> a topic.</p>
<p>If you picture my class as climbing Mount Everest, then I’m a sherpa.
Yes, I need to intimately know the landscape, but that alone is not enough.
My job is to find the best route for <em>you</em> to reach the summit.
I don’t want to lead you down blind paths, or slopes that are too steep for you (even though they might be fine for me!).
I don’t want you to needlessly spend time or effort for little progress.
At the end of the day, the summit is still up high, and you have to do the work yourself to get there –
I’m not carrying you.
But, I promise once you get there the view is worth it.</p>
<p>Oh, and don’t forget your oxygen mask.</p>
<!--
kramdown reference: https://kramdown.gettalong.org/quickref.html
-->A student caught me studying, so I had to explain my teaching philosophy.Time Travel Lecture2019-03-29T00:00:00+00:002019-03-29T00:00:00+00:00http://www.corcoviloslab.com/teaching/2019/03/29/time-travel<!-- kramdown tags defined below -->
<!-- end kramdown -->
<!--
Reminder that mathjax is enabled. Inline math using double backslash parenthesis: \\( \\)
Display math using double dollar or double backslash bracket: $$ $$ or \\[ \\]
-->
<!--
kramdown reference: https://kramdown.gettalong.org/quickref.html
-->
<p>Yesterday, I gave a fun talk to Eric Vogelstein’s <em>Philosophy of Star Trek</em> class about time travel.
My goal was to give a broad overview of 20th century physics and how time travel might be plausible if we bend the established rules a little.
We had a fun discussion, and it was nice to see non-science students excited about things like quantum entanglement and event horizons.</p>
<p><a href="/assets/docs/TimeTravel-Corcovilos.pdf">I’m posting my slides here if anyone is interested.</a></p>
<p>I consider myself little more than an informed novice on these topics, so corrections and feedback are welcome!</p>I gave a fun talk to Eric Vogelstein's *Philosophy of Star Trek* class about time travel.New Paper! Generating quasicrystal optical potentials2019-03-19T00:00:00+00:002019-03-19T00:00:00+00:00http://www.corcoviloslab.com/news/2019/03/19/quasicrystalopticspaper<!-- kramdown tags defined below -->
<!-- end kramdown -->
<!--
Reminder that mathjax is enabled. Inline math using double backslash parenthesis: \\( \\)
Display math using double dollar or double backslash bracket: $$ $$ or \\[ \\]
-->
<!--
kramdown reference: https://kramdown.gettalong.org/quickref.html
-->
<p>My student Jenna Mittal and I just had a paper published in <em>Applied Optics</em> on how we plan to generate our 2D quasicrystal potentials.<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup> I especially want to thank Jenna for building the prototype system and showing that this really works!</p>
<p>(Funny story: Jenna gave an invited talk on her work at our <a href="https://www.duq.edu/assets/Documents/urp/_pdf/2018%20URP%20Symposium%20Book.pdf">undergraduate research symposium</a> last summer. She did a great job, but at the end someone asked her how long the project took. She answered, “about 20 minutes.” I face-palmed in the back of the room. Sure, it took 20 minutes to build… <em>after</em> weeks of thinking, planning, and calculations. The moral of the story is, things work if you know what you’re doing!)</p>
<p>I’ll follow up soon to discuss what we’ve done. But for now, here are the links. (Click the arXiv link if you don’t have access to <em>Applied Optics</em>.)</p>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:1" role="doc-endnote">
<p>Corcovilos, Theodore A., Mittal, Jahnavee. <em>Two-dimensional optical quasicrystal potentials for ultracold atom experiments.</em> Applied Optics, <strong>2019</strong>, Vol. 58(9), pp. 2256-2263. doi: <a href="https://dx.doi.org/10.1364/AO.58.002256">10.1364/AO.58.002256</a>. arXiv: <a href="https://arxiv.org/abs/1903.06610">1903.06610</a> <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
</ol>
</div>We recently published a paper about how we are generating the quasicrystal optical potentials for our ultracold atom experiment.Geometric Algebra - beyond vectors2019-01-10T00:00:00+00:002019-01-10T00:00:00+00:00http://www.corcoviloslab.com/math/2019/01/10/geoalg<p>I read a lot. It’s probably my biggest hobby. I like to mix nonfiction and fiction, and I include work-related things that aren’t directly tied to my current teaching or research. This post falls into the last category.</p>
<p>The theme I’ve been riding recently is things that aren’t taught to physicists but maybe should be. These are actual physics topics or physics-tangent topics that I know next to nothing about and didn’t really even hear about until recently. (I also have a list of non-physics things that I think physicists tend to lack, but that’s a rant for another day.) I am a rank novice in the topic below, so please forgive any errors. I’ll post more on these if I gain any shareable insights.</p>
<h2 id="geometric-algebra">Geometric Algebra</h2>
<p>The math we learned as undergraduates to describe the physical world is broken. Since Gibbs introduced vectors in the late 19th century and silenced Hamilton’s <a href="https://en.wikipedia.org/wiki/Quaternion">quaternions</a>, justly or unjustly, we’ve made them a cornerstone of physics, but they become awkward when we start to talk about rotations or magnetic fields, for example, because we begin trying to paste vectors onto objects that aren’t just “a magnitude plus a direction.” For example, where the heck does the right-hand rule for cross products come from? It’s so bizarre and artificial when you think about it. In the end it traces back to us deciding that <em>x</em>, <em>y</em>, and <em>z</em> should be related to each other in some particular way and we need everything else we do to be consistent with that.</p>
<p>I first saw this a few years ago when my colleague <a href="https://www.duq.edu/academics/faculty/michael-huster">Michael Huster</a> lent me his copy of <a href="http://www.faculty.luther.edu/~macdonal/laga/">Alan Macdonald’s <em>Linear and Geometric Algebra</em></a><sup id="fnref:2" role="doc-noteref"><a href="#fn:2" class="footnote" rel="footnote">1</a></sup> which gives an introduction to the topic from a math point of view. I admit that my initial take was that geometric algebra was just some book-keeping trick and not really that useful. In my defense, a lot of the research papers on the topic use GA in just that way, as a superficial compactification of notation.</p>
<p>I stumbled back onto the topic during my prep for teaching Electrodynamics last year, and started reading through some of <a href="https://en.wikipedia.org/wiki/David_Hestenes">David Hestenes’</a> papers. I guess what hooked me back in was this nagging feeling that the math we use, particularly vector calculus, is an impediment to understanding the physics that it describes. My students always struggle to apply Maxwell’s Equations to problems and I sympathize with them. I realize that the only reason I can navigate the topic is that I’ve memorized a bunch of tricks and informal rules but that I don’t have truly deep intuition beyond what experience tells me is the right way to solve problems. I think that by better understanding the structure of geometry (whether or not geometric algebra is the right tool for that), I might gain some insight into how the universe is constructed.</p>
<p>The book <a href="http://www.mrao.cam.ac.uk/~cjld1/pages/book.htm"><em>Geometric Algebra for Physicists</em> by Chris Doran and Anthony Lasenby</a> <sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">2</a></sup> is a good overview written from a physics perspective.
It seems like Electrodynamics is one of the topics where this has the biggest impact.
For example, all of Maxwell’s Equations can be reduced to a single equation in this system.
Special relativity and rigid body dynamics are also especially nice.
They even spend time discussing quantum mechanics in this context, although I’m more skeptical of that one.</p>
<h2 id="whats-the-basic-idea">What’s the basic idea?</h2>
<p><a href="https://en.wikipedia.org/wiki/Geometric_algebra">Geometric Algebra</a> comes from W. K. Clifford’s work in the late 1890’s to build up higher dimensional objects from products of vectors without throwing away information. What I mean by this is that the two vector products that we learn, the dot product and the cross product, both throw something away – in some ways they complement each other. Clifford combined the two into a “geometric product” to build areas and volumes out of vectors in a seemless way.
The underlying philosophy of Geometric Algebra is right there in the name: geometry and algebra should be intimately related. Hestenes has done a ton of advocacy for this idea, particularly for physics. Another key constituency are computer scientists who have embraced GA as the way to put geometry into code.</p>
<p>The key ingredient is the “geometric product” of two vectors. It is defined (using notation familiar to physics students) as:</p>
<p>\[ \vec{a}\vec{b} = \vec{a}\cdot\vec{b} + i \vec{a}\times\vec{b}, \]</p>
<p>where the dot is the usual vector dot product, \( \times \) is the vector cross product, and <em>i</em> behaves like imaginary <em>i</em> in that \( i^2 = -1 \), but is really a geometric object called the “unit pseudoscalar” related to the 3d volume element. (I’m nerfing things a bit. The better definition has as the last term \( \vec{a}\wedge\vec{b} \), but most students haven’t seen the wedge product \( \wedge \) before.)
Note that the resulting object breaks an old rule that’s drilled into us from traditional vector algebra: it’s a scalar <em>plus</em> a vector. In our old way of thinking you can’t do that – add scalars to vectors – but doing away with that rule has a lot of implications. With these new rules GA ends up subsuming a lot of higher math beyond vectors: quaternions, differential forms, matrices, etc. I won’t say more here, but if you’re interested check it out yourself!</p>
<h2 id="my-hot-take">My hot take</h2>
<p>So, here’s what bugs me about all of this. All of this stuff is old news to mathematicians. It just hasn’t trickled down from the etherial heights of pure rational thought to us ground dwellers on the material plane. I bailed on the pure math classes as an undergrad (a bad “Intro to proofs” professor ruined it for me), so I can only blame myself that I can’t parse the jargon and writing style of math literature, but I wish there were more accessible resources for learning higher math. These books are a small crack in the wall to help me think about these things.</p>
<p>Oh, and I now think <a href="https://en.wikipedia.org/wiki/Conformal_geometry">conformal geometry</a> is cool and can probably solve some nagging questions that have been in my head for a while about the shape of <a href="https://en.wikipedia.org/wiki/Gaussian_beam">gaussian laser beams</a>. If I figure it out, I’ll be sure to pass it on.</p>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:2" role="doc-endnote">
<p>Alan Macdonald. <em>Linear and Geometric Algebra</em>, 5th printing, Self-published, 2018. ISBN: 9781453854938 <a href="#fnref:2" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:1" role="doc-endnote">
<p>Chris Doran and Antony Lasenby. <em>Geometric Algebra for Physicists</em>, Cambridge, 2007. ISBN: 9780521715959 <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
</ol>
</div>I stumbled onto the topic of Geometric Algebra (Clifford Algebra) a while back. Does it help us understand physics?Visiting the FELIX Free Electron Laser2019-01-03T00:00:00+00:002019-01-03T00:00:00+00:00http://www.corcoviloslab.com/research/2019/01/03/FELIX<!--
What's a free electron laser?
FELIX itself
What is IRMPD?
What experiments are we doing?
-->
<p style="float: right;
padding-left: 5px;
padding-right: 0px;
text-align: center;
width: 350px;
box-sizing: border-box;
"><img src="/assets/photos/jb-felix-153751.jpg" alt="FELIX laser vault" /><br />
Inside the FELIX free-electron laser.</p>
<p>Just before winter break, I went to the Netherlands at <a href="http://www.ru.nl/english">Radboud University</a> to use the <a href="http://www.ru.nl/felix/">FELIX</a> free electron laser.
I was there with <a href="https://www.duq.edu/academics/faculty/michael-van-stipdonk">Mike Van Stipdonk</a> of the Duquesne Chemistry department and his students Luke Metzler and Connor Graça to do infrared spectrscopy on some gas-phase ion systems.</p>
<h2 id="whats-a-free-electron-laser">What’s a free electron laser?</h2>
<p>A <a href="https://en.wikipedia.org/wiki/Free-electron_laser">free electron laser (FEL)</a> generates light by wiggling electrons.</p>
<p>First the electrons are accelerated close to the speed of light, then they pass through a bank of alternating magnets (the “undulator”).
The magnets bend the trajectory of the electrons into a wiggle shape.
Because the electrons are changing direction, they are induced to emit light.
The frequency of the light is determined by the initial speed of the electrons and the strength of the magnets.
(Classical electrodynamics plus special relativity are sufficient to describe the effect; quantum effects only show up at low flux or extremely high frequencies.)</p>
<p>Initially, the light is not laser light, but monochromatic incoherent light like from an LED (specifically incoherent <a href="https://en.wikipedia.org/wiki/Synchrotron_radiation">synchrotron radiation</a>).
However, once the light is bright enough, it starts pushing around the electrons themselves.
The electrons are attracted to the nodes of the light waves through the <a href="https://en.wikipedia.org/wiki/Ponderomotive_force">ponderomotive effect</a>.
Once the electrons bunch up like this, they are synchronized with each other and emit light as a team, rather than individually.
This collective behavior of the electrons causes the light to become coherent – all of the photons are in phase and at the same frequency – and we have laser light.</p>
<p>If the undulator is long enough, you don’t even need mirrors to enhance the light field (<a href="https://en.wikipedia.org/wiki/Superradiant_phase_transition">superradiance effect</a>), but most FELs do have a traditional laser cavity to build up optical power.</p>
<p>The key advantages that FELs have over other lasers are</p>
<ul>
<li>
<p>Tunability - in principle, you can make any wavelength of electromagnetic radiation with an FEL. FELs exist to make anything from THz radiation to x-rays. Of course a single instrument can only cover a small part of this range, but a decade or more of tunability is common.</p>
</li>
<li>
<p>Power - the pulse power goes like the electron beam current squared, so you can get very intense light. Unlike other lasers, there are no thermal lensing effects or degradation of the gain medium at high powers. The limit is basically from the electron accelerator.</p>
</li>
</ul>
<p>The major disadvantage is complexity and cost.
For our particular experiments (mid-infrared spectroscopy), only a handful of FELs exist.</p>
<h2 id="how-about-felix">How about FELIX?</h2>
<p>The <a href="http://www.ru.nl/felix/">FELIX</a> laboratory contains three separate FELs: FELIX, FELICE, and FLARE.
FELIX proper is the FEL that we use. It has two configurations depending on the desired wavelength range.
The output wavelength is tuneable from 3 µm to 45 µm (the range we use) or from 30 µm to 150 µm.
Each macro-pulse has an energy of up to 100 mJ at a 1 GHz repetition rate.</p>
<p>The FELICE laser is an FEL with experimental space inside the laser cavity for increased power, and FLARE is a longer-wavelength system (THz radiation).</p>
<p>There are about fourteen experimental stations holding various experiments using the lasers.
Only one experiment can be done at a time on each laser.
The experiments we do use two mass-spectrometer systems: a quadrupole ion trap mass spectrometer and a Fourier-transform ion cyclotron resonance mass spectrometer.</p>
<h2 id="what-is-irmpd">What is IRMPD?</h2>
<p>We use FELIX to break molecular ions.
If the laser photon energy matches one of the vibrational excitation energies of the molecule, the vibrational mode will be excited.
Because there are so many photons, the ions experience many serial excitations until they fall apart.
This processes is called <a href="https://en.wikipedia.org/wiki/Infrared_multiphoton_dissociation">InfraRed Multi-Photon Dissociation (IRMPD)</a>.
IRMPD is a little counterintuitive.
A single mid-IR photon doesn’t have enough energy to break apart the ion, but if you add up enough of them, you can.
One subtlety is that higher excitations have a lower energy than the ground state excitation because of anharmonic effects,
but this dispersion is small compared to the bandwidth of the pulse, so the ions can indeed absorb multiple photons.</p>
<p>By scanning the laser wavelength and recording the fraction of ions that are dissociated, we get the excitation spectrum.
We compare this data with predicted spectra from quantum chemistry calculations (along with the mass spectra of the ions) to determine the structure of the ions.</p>
<h2 id="what-experiments-are-we-doing">What experiments are we doing?</h2>
<p>On this trip we were looking at several ligand effects on uranyl (UO<sub>2</sub>) and some aromatic compounds found in spices.
I’ll write about those in a later post, but we study the uranium ions because there are still a lot of unknowns about the chemistry of uranium and heavier elements.
The spices are interesting to the Department of Homeland Security, and I’ll just leave that little teaser for now.</p>I visited the FELIX Free Electron Laser facility at Radboud University in Nijmegen, the Netherlands, to do some spectroscopy experiments with Mike Van Stipdonk.Hug a math professor!2018-12-04T00:00:00+00:002018-12-04T00:00:00+00:00http://www.corcoviloslab.com/teaching/2018/12/04/hug-a-math-prof<p>Last week I sat down with a colleague from the <a href="https://duq.edu/academics/schools/liberal-arts/departments-and-programs-/mathematics-and-computer-science/mathematics-ba/bs-">math department</a> who had the thankless task of evaluating my teaching.
First, a little background.
Most evaluations at our university are done by tenured faculty within the same department, but because our department is small, my reviews usually come from folks in the other science departments.
This time around, the usual chemistry and biology people were too busy, so I ventured out to the math department to ask for a review.
It turned out that this was a great idea.</p>
<p>Anyone in physics knows how much math we use.
In fact, we probably take it for granted.
Just as an example, at our university the <a href="https://www.duq.edu/academics/schools/natural-and-environmental-sciences/academic-programs/physics/undergraduate-programs/bachelor-of-science/physics-course-sequence">physics B.S. degree</a> already contains enough classes for a math minor, and is only about three courses short of a full second degree in math.
And anyone who teaches physics or takes a physics class knows that we spend a lot of time teaching math:
vectors, calculus, linear algebra, differential equations, the list goes on….</p>
<p>So, with that as context, my math colleague sat in on two of my quantum mechanics lectures and saw first hand how we use math in physics and how integral it is to what we do.
Reading between the lines of our later discussion, I think she had two reactions.
First, she was grateful that her former students were actually using the things that she taught them, instead of that knowledge sinking into the void.
And second, she saw how we emphasize slightly different things when we use math in physics than what she does as a mathematician.
This led to a long talk between us about what math we need in physics and how we use it,
and hopefully a continuing dialog between our departments.</p>
<p>In the end, this one simple interaction could make a big future difference to our students by increasing the synergy of their coursework.
In academia we, particularly administrators, talk a lot about interdisciplinary collaboration, but it is difficult in practice to make it go.
The great obvious-but-ignored commonality between departments is teaching and the students themselves.
My epiphany of the week is that we can, and we should, use that link to form bonds between departments.</p>
<p>So, to my physics colleagues at other universities, I say go find a math professor and give them a hug, or at least take them out for a cup of coffee.</p>A brief meeting with a math colleague opened both of our eyes about helping our students.Thoughts on the graduate admissions process2018-11-06T00:00:00+00:002018-11-06T00:00:00+00:00http://www.corcoviloslab.com/teaching/2018/11/06/grad-admissions<p><em>This post is long, long overdue. I beg forgiveness, dear reader, as real life took precedence for a couple of months. And then procrastination set in, and then…. Anyway, because graduate school application season is near, it seemed appropriate to finish the ideas here and push them out to the world.</em></p>
<p>The DAMOP session on graduate admissions last summer was enlightening and disheartening. Best practices are not being followed by schools or by students. I’ll discuss a little of what I learned here, and some things that have come to mind in as I’ve had a chance to process some of what I learned.</p>
<p>The session itself was held back on Wednesday, May 30, and was titled “Rethinking Graduate Admissions.”
Here are the <a href="http://meetings.aps.org/Meeting/DAMOP18/Session/K08">abstracts</a>. The presenters were</p>
<ul>
<li>Casey Miller (Rochester Tech.) - “Typical physics PhD admissions parameters limit access to underrepresented groups and US citizens but fail to predict doctoral completion”</li>
<li>Geoff Potvin (Florida International U.) - “Comparing student and faculty perceptions of graduate admissions”</li>
<li>Geraldine Cochren (Rutgers) - “Identifying barriers to applying to graduate physics programs, an intersectional approach”</li>
<li>Ted Hodapp (APS) - “APS Bridge Program: Changing the face of physics graduate education”</li>
</ul>
<h3 id="summary">Summary</h3>
<p>I won’t do a play-by-play of the session, but I do feel it is important to share what I perceived as the key points of the session.
I think that everyone in attendance recognized that physics in the US is still a dominantly white-cis-hetero-male culture, and that diversifying the field will benefit us all.
The overarching problem is that we have few concrete ideas as a community for accomplishing this.
First, although this wasn’t the topic of the session, we must police ourselves and <a href="http://www.nationalgeographic.com/magazine/2018/05/sexual-harassment-science-me-too-essay/">remove those who harass and discriminate</a> against women, racial minorities, and LGBTQ individuals so that physics builds a more welcoming environment.</p>
<p>Many of us bemoan the “leaky pipeline” that causes women and minorities to lose interest in physics as they go from children to teenagers to college students to graduate students to professionals,
but there are some places where we’ve drilled the holes into the pipeline ourselves.
This set of talks made it clear that the graduate admissions process is one.
(And it seems likely that these same faults apply to undergraduate recruiting as well.)</p>
<h3 id="the-gre-is-flawed-or-at-least-it-doesnt-do-what-we-think-it-does">The GRE is flawed, or at least it doesn’t do what we think it does.</h3>
<p>Miller discussed two major flaws with relying on the Physics GRE for admissions decisions.
First, despite the best efforts of the exam designers, the test scores are biased against underrepresented minorities.
This doesn’t appear to be a problem with the questions on the exam, but rather the testing process itself.
The standardized testing environment causes added stress to minorities (e.g. <a href="http://en.wikipedia.org/wiki/Stereotype_threat">stereotype threat</a>), causing them to do more poorly than their ability would predict.
According to Miller, this causes significantly lower Physics GRE scores.
Data show that the most affected groups are African Americans, Hispanic Americans, and Native Americans.
White women see a smaller, but still significant effect, and intersectionality (belonging to more than one minority group) compounds the decrease.</p>
<p>As a result of the testing bias, when graduate admissions committees use cutoffs (minimum percentile scores or minimum raw scores) to exclude applicants, <em>as many as 75% of underrepresented applicants are automatically excluded</em>.
At least a third of physics PhD programs use these cutoffs,<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup>
with many others using “soft” cutoffs (whatever that means) or otherwise making the GRE physics exam a major component in admissions decisions.</p>
<p>It’s easy to demonize departments for doing this, but the task they have is daunting.
They have to whittle down perhaps hundreds of applications into a few dozen for closer consideration.
Using test scores is easy, and it is easy to mistakenly believe that the tests are objective.
However, it clearly doesn’t work as intended. Increasingly, schools are dropping the GRE Physics requirement (14% according to a recent survey<sup id="fnref:1:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup>).</p>
<p>Nor can <a href="http://www.ets.org">ETS</a>, the company that administers the test, be solely to blame.
In <a href="https://www.ets.org/s/gre/pdf/gre_guide.pdf">their own score interpretation guide</a>, they warn against using cutoffs for exactly the reasons above:<sup id="fnref:2" role="doc-noteref"><a href="#fn:2" class="footnote" rel="footnote">2</a></sup></p>
<blockquote>
<p>The GRE Board believes that GRE scores should never be
the sole basis for an admissions decision and
that it is inadvisable to reject an applicant solely
on the basis of GRE scores. A cutoff score
below which every applicant is categorically
rejected without consideration of any other
information should not be used.</p>
</blockquote>
<p>This brings me to Miller’s second point: <em>the GRE Physics exam does not predict success in graduate school</em>.
He showed data showing nearly zero correlation between GRE Physics scores and grad school GPA.
There is an acknowledged survivor’s bias in the data – students who weren’t admitted into grad school were obviously missing from the data – but the trend is concerning and puts the use of test scores for admissions further into doubt.
Among the variables that Miller discussed, undergraduate GPA was the best predictor of success, but even that correlation was weak (r = 0.3 is what I put in my notes).</p>
<p>What Miller advocates for is a holistic admissions process,<sup id="fnref:3" role="doc-noteref"><a href="#fn:3" class="footnote" rel="footnote">3</a></sup> and given the data above, it seems clear to me that this is imperative. However, I’m still struggling with what it means for those of us who teach undergraduates. The depressing truth is that the students still need to do well on the GRE Physics exam to be admitted to most schools. The one bright spot is that more and more graduate schools are using holistic evaluations. I feel I should steer students towards those schools, but there is no comprehensive list that I know of. Some institutions are part of the <a href="http://www.apsbridgeprogram.org/igen/people.cfm">APS Inclusive Graduate Education Network (IGEN)</a>, so this may be a good starting place. If you know of other schools, please let me know.</p>
<h3 id="faculty-and-students-view-admissions-standards-differently">Faculty and students view admissions standards differently</h3>
<p>The talks by Potvin and Cochran both addressed the topic of perceptions of graduate admissions.
Potvin presented some survey data<sup id="fnref:1:2" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup> and Cochran presented some interview results<sup id="fnref:4" role="doc-noteref"><a href="#fn:4" class="footnote" rel="footnote">4</a></sup> of students from underrepresented groups.</p>
<p>The most important criteria for graduate admissions, according to the survey of admissions chairs<sup id="fnref:1:3" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup> are</p>
<ol>
<li>Undergraduate GPA in physics and math</li>
<li><em>Letters of recommendation</em> (emphasis added)</li>
<li>Courses taken and GRE Physics score (tied)</li>
</ol>
<p>The least important factors are the proximity/familiarity of the undergraduate department to the admissions committee, GRE general scores (GRE general math is too easy for physics students and GRE written is less relevant), and student conference presentations. The problem is that students seem more concerned with things like the reputation of their undergraduate institutions and research output. This can cause good students to miss out because they have spent their energy on things that are uninteresting to admissions committees.</p>
<h3 id="conclusions">Conclusions</h3>
<p>The key thing that I’ve learned is that there are huge gaps between</p>
<ul>
<li>What graduate admissions committees are trying to do,</li>
<li>What graduate admissions committees are actually doing, and</li>
<li>What prospective students <em>think</em> graduate admissions committees are doing.</li>
</ul>
<p>It really is a mess. From my point of view as an advisor to undergraduates, I think my best approach is to be very conservative in my advice to students.
First, that means educating them on the process and reminding myself to take nothing for granted.
Also, I must (1) assume the worst practices from graduate admissions and (2) push my students to schools with better admissions criteria. I think the latter is important, not just for increasing the chances of my students being accepted, but also because the schools that are cognizant of their admissions process are also more likely to be proactive towards the general wellbeing of their students with regards to things like mental health, work-life balance, and sexual harassment/discrimination. While we’re all waiting for the physics community to fix its graduate admissions problems, I need to at least protect my own students.</p>
<p>I’d love to hear any observations or advice on this.</p>
<h3 id="references">References</h3>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:1" role="doc-endnote">
<p><a href="https://doi.org/10.1103/PhysRevPhysEducRes.13.020142">Geoff Potvin, Deepra Chari, and Theodore Hodapp. <em>Phys. Rev. Phys. Educ. Res.</em> 13, 020142 (2017).</a> <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a> <a href="#fnref:1:1" class="reversefootnote" role="doc-backlink">↩<sup>2</sup></a> <a href="#fnref:1:2" class="reversefootnote" role="doc-backlink">↩<sup>3</sup></a> <a href="#fnref:1:3" class="reversefootnote" role="doc-backlink">↩<sup>4</sup></a></p>
</li>
<li id="fn:2" role="doc-endnote">
<p><a href="https://www.ets.org/s/gre/pdf/gre_guide.pdf">ETS. <em>GRE Guide to the Use of Scores (2017-2018)</em> pg. 13.</a> <a href="#fnref:2" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:3" role="doc-endnote">
<p><a href="http://dx.doi.org/10.1103/PhysRevPhysEducRes.13.020133">Rachel E. Scherr, Monica Plisch, Kara E. Gray, Geoff Potvin, and Theodore Hodapp <em>Phys. Rev. Phys. Educ. Res.</em> 13, 020133 (2017)</a> <a href="#fnref:3" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:4" role="doc-endnote">
<p><a href="http://www.apsbridgeprogram.org/resources/Identifying-Barriers-Cochran.pdf">Cochran, Geraldine L., Theodore Hodapp, and Erika E. Alexander Brown. “Identifying barriers to ethnic/racial minority students’ participation in graduate physics.” Proceedings of <em>Physics Education Research Conference</em>. (2018).</a> <a href="#fnref:4" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
</ol>
</div>The DAMOP session on graduate admissions was enlightening and disheartening. Best practices are not being followed by schools or by students. I'll discuss a little of what I learned here.