Ellipsoidal Mirror Analysis

Ellipsoidal Mirror Analysis
J. Rosicky
Abstract—Presented here are results of an analysis of an optical
system concept using elliptical mirror as the imaging element for
microscopy.
Index Terms—Imaging, stigmatic image, homocentric wave,
microscopy, optical system analysis, ellipsoidal mirror.
I. I NTRODUCTION
PTICAL systems for microscopy usually assemble refractive elements. Because of the aberrations of refractive elements such systems combine elements with different
properties in order to optimize the aberrations. Therefore
optical systems for microscopy usually result in complex
designs. The ultimate goal in a optical system design is to
approach the stigmatic image, where each point of the object
is imaged as its corresponding point in the image. There are
only a few cases where the stigmatic image is possible. The
ellipsoidal mirror is one of them. The properties of a concept
of an imaging system for microscopy with a single ellipsoidal
mirror will be analysed.
O
Fig. 2.
Homocentric spherical wave - converging.
, where sn + s0 n0 = const. = 2a. After some manipulation
we can write:
16a2 n02 (e − z)2 + y 2 = 4a2 + n02 (e − z)2 + y 2
2
−n2 ((z − e)2 + y 2 )
(2)
II. S TIGMATIC IMAGING
y
Stigmatic imaging is the case where a point of a monochromatic object transforms into a point within the image. The
optical system capable of such a point-to-point imaging has
to transform the homocentric spherical wave (Fig. 1) again into
the homocentric spherical wave (Fig. 2). We will not assume
the wave properties of the light so we can work with rays as
normals to the waves.
n
P
n’
s’
s
A’
A
z
z
−e
Fig. 3.
+e
Derivation of the stigmatic meridian [1]
, which is an equation of the 4th order.
In general it is very complicated to manufacture optical
elements shaped according to the 4th order equations. There
are specific cases when the equation 2 changes to the 2nd order
equation, which means more availability for manufacturing.
A. Ellipsoid or Hyperboloid (n0 = −n)
Fig. 1.
Homocentric spherical wave - diverging.
Let’s derive the shape of the meridian of the surface capable
of stigmatic imaging of point A into point A’ (Fig. 3). The
geometric distances s and s0 expressed in terms of e, y, z:
p
p
n (z + e)2 + y 2 + n0 (e − z)2 + y 2 = 2a
(1)
This work was kindly supported by the grant SGS12/052/OHK2/1T/12.
J. Rosicky is with Czech Technical University in Prague, Department of Instrumental and Control Engineering, Czech Republic (e-mail:
[email protected])
If n0 = −n and also n = 1 (a mirror in the air) the
equation (2) becomes:
y2
z2
+
=1
(3)
a2
a 2 − e2
If a √> e, this is the equation of an ellipsis (Fig. 4) with
e = a2 − b2 .
Points A and A0 are geometric focal points of the ellipsis.
The point A0 is a real image of the point A, which enables the
ellipsis as a standalone imaging element (similar to a positive
lens) and will be further discussed.
y
n
A
A’
n’
2b
r
A
2e
A’
z
p
p’
−e
+e
2a
Fig. 4. Ellipsis - a case of the change of the equation (2) from the 4th to
the 2nd order [1].
Fig. 6. Circle - the third case of the change of the equation (2) from the 4th
to the 2nd order [1].
n
n’
If a < e the equation (3) becomes the equation of hyperbola
(Fig. 5):
z2
y2
− 2
=1
(4)
2
a
e − a2
A
A’
p’
p
b
A
A’
Fig. 7. Circle - Stigmatic imaging via sphere according to the equation (6)
for r > 0, n < n0 [1].
2a
C. Other cases
Fig. 5. Hyperbola - another case of the change of the equation (2) from the
4th to the 2nd order [1].
The image (point A in Fig.5) formed by hyperbola is
a virtual image. This property does not enable the hyperbola
as a standalone imaging element (similar to a negative lens).
If one would need to record the image produced by hyperbola
additional elements would have been necessary.
So far we have identified several cases of stigmatic imaging
for object and image located in the finite distances. There
are other cases when equation (2) produces a meridian with
stigmatic imaging. However these are the cases when either
object or image lies in the infinity.
We will focus on the case of ellipsoidal mirror for possible
application as a single imaging element for microscopy.
B. Sphere
If 2a = 0 the equation (2) also becomes the 2nd order
equation (according to Fig. 3):
n2 (z 2 + 2ez + e2 + y 2 ) = n02 (e2 − 2ez + z 2 + y 2 )
n’
(5)
After some manipulation and introduction of new symbols we
can see that the meridian is a circle (Fig. 6) and the stigmatic
imaging holds for:
n + n0
p0 =
r
n0
and
n + n0
r
p=
n
so that
p0
n
= 0 =⇒ p0 n0 = pn
(6)
p
n
Distances p and p0 point to the same side of the meridian either to the left (Fig. 8) or right (Fig. 7).
n
A
A’
p’
p
Fig. 8. Circle - Stigmatic imaging according via sphere to the equation (6)
for r < 0, n < n0 [1].
III. I MAGING VIA E LLIPSOIDAL M IRROR
The analysis of the equation (2) has shown that an ellipsoidal mirror is capable of stigmatic imaging when the object
lies in one geometric focal point of the ellipsoid. The image
is then formed in the second geometric focal point of the
ellipsoid.
This property can be exploited for imaging system according to figures 9-11.
The location (and shape) of the aperture will have a substantial impact on the imaging. We will start with the aperture
located centred around the point C on top of the ellipsoid. This
symmetrical placement of the aperture should provide better
results than any other asymmetric placement.
For d = 0.5 and ω = 30deg the circle of confusion is about
.
q = 1. This means that a single object point would get imaged
as a 1(mm) large spot.
The substantial factor influencing the size of the circle of
confusion q is the aperture angle ω. The wider the angle
the larger the size of the circle of confusion. If we set
a specific limit for the maximum value of q, say 0.01 (ten
times bigger that the pixel size), we can calculate the corre.
sponding maximal aperture angle 2ω = 0.34deg (about 20 arc
minutes). This is a very small aperture angle. The diameter
of the corresponding aperture would have been approximately
0.3(mm).
C
q
Fig. 9. Ellipsoidal mirror. One box depicts the object and the second box
depicts its image (i.e. the image sensor).
b
ω
A’
A
F
F’
d
Fig. 12.
Fig. 10.
Ellipsoidal mirror - top view.
Fig. 11.
Ellipsoidal mirror - front view.
f
Raytracing the elliptical mirror.
So far the concept of a single element microscopy imaging
system with an ellipsoidal mirror seems problematic. However the calculations were done as a quick estimate using
a symbolic geometry software [2]. Another techniques for the
analysis could provide more accurate estimate. Tools like [3]
or [4] allow complex visualisations of aberrations of optical
systems and provide tools for optimizing the design.
b
Presented concept could possibly enable a contact 1:1 imaging of small enough objects directly onto the image sensor with
stigmatic image according to the section II-A. The possible
image sensor would probably be one of those used in imaging
modules of available smart phones. Those modules resemble
sensors with a pixel size about 1.5 to 2 micrometers. So we
are looking for the size of the circle of confusion in the order
of 1 − 10 micrometers.
ω
A’
A
F
F’
f
IV. A NALYSIS
The analysis of the imaging with an ellipsoidal mirror
according to Fig. 12 will estimate the quality of the image (at
the point A0 ) in terms of the size of the circle of confusion q.
The dimensions of the ellipsoid were set to be approximately
f = 50 and b = 30.
Fig. 13.
Raytracing the elliptical mirror - another arrangement.
V. C ONCLUSION
The quick estimate of the performance of the proposed concept of a single element ellipsoidal mirror imaging system for
microscopy shows that its properties make it not feasible. The
future work will be focused on more accurate analysis of the
concept using tools like [3] or [4]. Other possible arrangement
of the concept depicted in Fig. 13 will be analysed. This
arrangement could provide better performance because both
object and its image are located on surfaces lying in focal
planes of the ellipsoid. However this arrangement induces
problems with positioning the object and the image sensor.
R EFERENCES
[1] A.Baudys, Technicka optika. Praha: CVUT, 1989.
[2] Saltire Software, Geometry Expressions. http://geometryexpressions.com,
2012.
[3] Optima Research, ZEMAX. http://www.zemax.com, 2012.
[4] Lambda Research, OSLO. http://www.lambdares.com, 2012.