Research Journal -- Sriya Adyha -- Stony Brook Laser Teaching Center
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October --> November 2006
After the summer, I started continuing research on this project again.
I am pleased to see that many students, teachers, and people in
general find the "mirage toy" to be amazing, etc.
I started taking some actual measurements of where the secondary
images are located. We put together an apparatus that helped us lower
the bottom mirror manually but slowly and more mechanically than by
hand, so that we can make more accurate readings.
Here's what I found:
From the measurements that I took the other day, at 4 reflections
Mathematica calculated a distance of 4.38774 inches, which is a mirror
separation of (4.38774 - 3.175 inches) = 1.2127 inches, where 3.175
inches is the focal length and distance of the first, primary,
image. So I was off by ~0.1 inches.
Lowering the bottom mirror some more, I found an inverted image at a
mirror separation of 1.875 inches. Compared to Mathematica
calculations, I found this to be the closest match up to an inverted
image of 4.937 inch distance, which is a mirror separation of 1.762
inches at six reflections. Again, off by ~0.1 inches.
After this, Dr. John Noe showed me an article that was published by
Andrzej Sieradzan (The Physics Teacher, November 1990). I compared my
measurements with his and found may similarities. Of course, he did
not use the matrices merhod like we did. So there is a big
difference. I read his article and analyzed it carefully. I find it
interesting that this was the earliest article that can be found since
1993.
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U.R.E.C.A. and WISE 187 presentation, April and May 2006.
Finally, the last month of the semester arrived. The URECA event was
held on April 26th and I have yet to make a perfected poster board and
presentation.
Dr. Noe helped me decide on the outline of the poster board. We ended
up organizing it chronologically, from what I did in the beginning of
my research to the end findings. Some pictures of me finding results,
such as graphs, were included.
The poster was composed very simply, resembling only the key points of
my research. See picture at:
http://laser.physics.sunysb.edu/~john/ureca-day/026.jpg
The center portion of my poster board concentrated on the title,
abstract, mathematical concepts of the pig, ray diagrams with the pig,
and an enlarged picture showing the "mirage" pig.
In brief, the Ray Optics with Matrices section described what each
element of the matrix means and how to calculate the resulting
matrix.� Saying how each matrix represents either a reflection or
drift and how those multiplied in sequence would result in a 2X2
matrix with elements A,B,C,D, where A, B, C, and D respectfully have a
different characteristic.� Of course, we were only concerned elements
A, magnification, and B.
The diagram, I drew in the Paint software, shows a cross-sectional
view of the reflections and drifts emerging from the object at the
bottom surface to the real image at the top surface.
The left hand side of my poster board consisted of the steps taken to
determine the shape, size, and dimensions of the mirage toy.
The first section explained how Dr. Noe and I determined the radius of
curvature and focal length of the toy.� The second and third sections
showed a graphical representation of the parabolic shape and size of
the toy, including a few pictures.�
The right hand side of my poster board was constructed of collected
data, calculations, and results using Mathematica. Also, about
additional images being seen when lifting the top mirror and it^�s
further future research. �
The first section showed how Mathematica was used to multiply matrices
to find equations for the necessary elements in the resulting matrix.�
Then, solving those equations to get the final answer of the height
the real image floats above the top surface.
The second section was two graphs representing height versus
magnification and height versus separation distance. Both
relationships are nearly parabolic.
Lastly, the third section explains how additional weaker secondary
images are observed when the top mirror of the toy is lifted.� A
drawing of the additional reflections and drifts at a separation
greater than 1.5 focal lengths was shown with a summarized calculation
of the proposed magnification and height above the top surface of the
real secondary image.
This was my poster board!� And it was a big hit at the URECA event!
Even the president of the university was astounded with my
research.�(picture)�
Later that week was my WSE 187 presentation where I had to summarize
everything I have done in about five minutes. I created a short
PowerPoint presentation.� It went very well.�Later on, I believe this
PowerPoint will be available on this webpage.
Continuing with the research Dr. Noe and I tried to determine where
these weaker secondary images occur and whether there is a pattern.�
We found that when multiplying additional reflections and drifts
respectively matrices to the Mathematica calculation, where the two
mirrors are separated by one focal length, that there is a pattern
where the magnification does equal 1 when the mirror is lifted
gradually.
http://laser.physics.sunysb.edu/~wise/wise187/2006/pictures/secondary-images.gif
This plot shows the locations of all images predicted by matrix
analysis with the Mathematica program as a function of the mirror
separation d in units of the focal length f. The observed images are
connected by a dashed line.
I can't believe a toy can be so complex!
What's next?? Further researching, through analysis of reflections and
drifts, how these secondary images occur.
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March 10th -> April 7th
Last entry, Dr. Noe and I left off with some questions. Now I am
ready with some answers and results of my project! Here's what we
came up with:
During the past several weeks we developed the abstract. There is a
link to this abstract on the homepage.
The matrix theory is valid because it is based on small angle
approximations. The analysis uses 2x1 vectors to represent the
distance of light rays from the axis of symmetry, r, and their
inclination angle theta. Optical elements (mirrors and open spaces
between them) are represented by 2X2 square matrices, and the combined
effect of several elements is found by simply multiplying together the
corresponding matrices, in the correct order. Therefore, with small
values of theta and radius = 6.35 inches, the matrix theory can be
applied.
The matrices are series of reflects and drifts as they occur between
the mirrors. When the mirror's separation are one focal length, as
light emerges through the hole to the bottom mirror the sequence is
drift, reflect, drift, reflect, and drift to form the real
image. These matrices are multiplied respectively, resulting in a 2X2
matrix where each element, again, has a certain characteristic. If we
define the result 2x2 matrix to have elements A, B, C, and D, then the
condition for the real image of the object to focus is when the
element B=0. When B=0, then the element A is the magnification.
While developing the abstract we had to look back at Dzierba's work
and after analyzing, noticed that he saw a multiple real image at
three focal lengths. We couldn't see that for ourselves so in
questioning his work, we ended up giving him a call. Dzierba said he
would keep in contact with us and check out what we were talking about
after he got home from his lectures. We called him back a week later,
but he hasn't contacted us back since the first.
Yet, the project must continue!
Having Dzierba's work as reference, we used our own variables to
define different distances for our convenience. With a spherical
radius of 6.35 inches, the focal length is 3.175 inches, d. The
height of the object on the bottom mirror, h, was approximately
measured to 0.625 inches (Picture839.jpg) and the height above the
"closed" surface top mirror, x, where the pig floats were other
distance variables used (See drawing with labels of these variables).
>From these definitions the matrices can now be calculated.
(Show matrices)
I calculated this by using Mathematica. With the help of Azure
Hansen, a former WISE 187 student, I learned how to use the basics
this software. This made multiplying these matrices simple! Using the
resulting 2X2 matrix, I solved for x by setting the 'B' element to
zero and plugging in the determined values for d and h. This gave an
approximate value of x= 0.743 inches. That means that the real image
is about 3/4 of an inch above the top surface! Solving the A element
and plugging determined value for x, d, and h, gave a magnification of
approximately 1.24. This means that the image is magnified by 24
percent!
(Picture835.jpg)
Dr. Noe and I did observe this magnification and as we increased the
height of the object, the image gradually had greater
magnifications.(Picture836.jpg). Also, as the mirrors were gradually
separated multiple images were seen and, also, gradually were
magnified.
Using the result equation calculated for x in terms of d and h, I
plugged in values for h by 0.1 inches and kept d= 3.175 inches. The
same was done to find the magnification at these heights. I graphed
h, height of the object, versus x, height above the top surface, and
observed a 'cuspe point' at h=0.2 inches. Also, I found the
magnifications at these heights. I graphed the height of the object
versus magnification and observed a quadratic trend.
Why the cusp?
It could have been a typing error in Mathematica, but this problem is
underway.
What's next?
Next, I am developing the URECA rough draft for the poster. Also,
Dr. Noe and I thought and are thinking of ways to display the effects
of the Mirage toy more efficiently. And I am trying to mount in a
little light bulb instead of the pig.
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February 24th -> March 6th
I've been working with Dr. Noe to write down a summary of the project
so far. Here's what we came up with:
The goal of my project is analyze the Mirage Toy using matrix
equations that describe how light rays travel and reflect. In other
words, we are trying to create a mathematical model that explains the
optics of the toy.
The Mirage Toy creates a "real" image of an small oject placed
inside. The image appears to float in space and can be enlarged with a
magnifying glass or reflected in a mirror. The image is "real" because
one can touch the place where the light rays appear to be coming from
(and actually are).
So how does this happen?
The toy has two concave mirrors that face each other; the mirrors are
horizontal, with one on top of the other. The upper mirror has a large
hole at its center, to allow light rays from the mirrors below to pass
through.
Light reflected from the actual object is reflected twice before
passing through the hole and forming the image: first from the top
mirror then from the bottom mirror.
To analyze the toy we need to know the exact size and shape of the two
mirrors, which is related to their focal length.
We started out by carefully measuring all the relevant dimensions that
we could think of with a dial caliper: the diameter and depth of the
lower mirror, and the separation of the two mirrors through their
centers. Pictures 620.jpg and 623.jpg show how the vertical dimensions
were measured with the help of an accurately straight 0.375 inch wide
steel ruler.
To go further we needed to know the relationship between the radius of
a circle R and the length c and "height" h of an arbitrary chord. This
can be derived from the Pythagorean theorem, as shown in the diagram
(737.jpg). The result can be written: R = c^2/(8h) + h/2.
When we applied this formula to the measured chord length c = 8.703
inches (see picture 622.jpg) and height h = 1.49 we predicted a radius
R = 7.09 inches.
To test this result we made a template by drawing a partial circle
with this radius on graph paper and cutting it out. Much to our
surprise the template didn't fit, and there was a very noticeable gap
at the bottom (picture 724.jpg).
We thought we must have made a mistake in the calculation, so we made
a few more templates with smaller radii (R = 7.0, 6.75, 6.5, 6.25 and
finally 6.0 inches), to try to get a closer match to the shape at the
bottom. The best match to the bottom of the mirror was with R = 6.25
inches, but this was way off at the edges.
What's going on?
At this point it occured to us that the shape might not be a sphere at
all, but instead something with a more gradual curve, for example a
parabola, y = ax^2. In this case we can easily determine the constant
"a" from the measurements we already made, since a = 4h/c^2.
The next thing we thought about was what circle would best match a
given parabola at the origin. The best match is when the two curves
are not only tangent but also have the same curvature. The curves are
then said to be "osculating" ("kissing"). (See Mathworld.)
How to derive this? We use Taylor series. ...
The result was that the parabola has a = 1/2R = 1/12.7 so the
osculating circle had a radius R = 6.35 inches.
We tested this by making a parabolic template by first drawing a
circle with R=6.35 inches, then adding the calculated vertical amount
by which the parabola is lower than the circle. Picture 735.jpg shows
that this template fits really well everywhere.
So ... we now know that the curface is really parabolic, but this
shape can be approximated by a sphere if one doesn't go to far away
from the center. How far is too far? About > 50% of R. See drawing.
Another thing we used the sagitta formula for was to calculate the
correction h for the missing part of the top mirror from the size of
the opening c = 2.45 inches. With R = 6.35 inches the result is
0.119. Combining this with the other vertical measurements the mirror
separation s is 3.175 inches. Amazingly enough, this is EXACTLY
one-half of the radius!!
For a concave reflecting surface the focal length is R/2. Light from a
source placed at R/2 from the mirror surface will be focussed into
parallel rays.
So the conclusion so far is that the two mirrors are separated by
their exact focal length !!!
Questions:
- Is the matrix theory valid? If so, why?
- What happens if the object is tall?
Things to Try Next:
- Read Alex Dzierba's notes more carefully.
- Think about how to locate additional images accurately.
New Things to Learn:
- gnuplot
- Some ray tracing programs
- xfig or some other drawing program
- Mathematica
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Monday, 20 Feb 2006
I started off exploring my interests and hobbies with John Noe, thus
creating my biography for the Laser Teaching Center WISE 187
webpage. He learned that I am very interested in mathematics, more
specifically in matrices. Dr. Noe showed me that in optics, light rays
can be described with matrices, and that the Mirage Toy could be
described using the same concept. So my project consists of
researching the optics of the Mirage Toy as well as explaining
background information and important elementary theorems of optics.
Right now, I am determining the radius of curvature of the concave
mirror from its measured dimensions and geometry. The focal point is
midway between the center of curvature and the mirror surface. After I
determine this point, I can apply the optics matrix theory to describe
how light rays pass through the concave mirrors of the Mirage Toy to
create the `holographic' image of the object floating at the top.