1. Paraxial Ray Optics Invisibility Cloaking
Tyler Peterson
Pacific Lutheran University
PHYS 499B
21 May 2015
Abstract: Lenses and standard optical components are used to demonstrate a 3-D, transmitting,
passive, continuously multidirectional cloak for objects in the visible light spectrum.
Commercial ray optics software is used to model the cloaking behavior, and a succinct
formalism is presented that yields perfect optical cloaks in the paraxial limit.

2. Introduction
Invisibility cloaking has fascinated both the general public and the scientific community
ever since the idea was first introduced. In 1966, Star Trek screenwriter Paul Schneider unveiled
the first instance of invisibility technology in the episode “Balance of Terror”. The idea came to
him in 1958 after seeing a World War II naval drama, Run Silent, Run Deep, and attempting to
create a space-exploration equivalent to a submerging submarine. The device has since shown up
in countless forms throughout the decades in all mediums of popular media, especially
noticeably in the 1997 novel Harry Potter and the Philosopher’s Stone [4].
Despite its ubiquitous presence in popular culture, optical spatial cloaking has not been
pursued as a scientific endeavor until recent years. Many different approaches have been taken
with varying degrees of success. Advances made in fields like transformation optics have
inspired progress in other fields, and have been showing a gradual shift from traditional
metamaterials, to polymers, and finally to natural materials [2]. To avoid the burden of difficult
materials requirements, researchers have recently turned to ray optics for cloaking [1].
Joseph S. Choi and John C. Howell have been working at the University of Rochester to
develop a cloak that has its foundation in ray optics. The so-called “Rochester Cloak” is
constructed through the careful placement of standard lenses along a central axis to allow an
object to appear invisible from the perspective of a properly placed observer. One of the major
benefits of this system is its multidirectionality; whereas previous attempts at cloaking have been
limited to very specific viewing conditions, this cloak allows an observer to look from a wide
range of incident angles within the paraxial limit (where sin⁡( 𝜃)~tan⁡( 𝜃)~𝜃, valid for incident 𝜃
up to 30°) [1].

3. In this experiment, we work through a reconstruction of two configurations that satisfy
the requirements for a Rochester Cloak. A concise formalism is presented that describes what
takes place in the systems to create the cloaking effect, which is then demonstrated in both a
physical setup and computer simulation. All materials were easily accessible, off the shelf optical
materials, chosen to emphasize the simplicity and effectiveness of this experiment.
Theoretical Formalism
A perfect cloak is defined in this experiment to be any system that satisfies two essential
requirements. First, it must have a non-zero volume of space that is cloaked and available to hide
an object, and second, light rays coming through should be unaltered compared to when the
cloaking system is replaced by the ambient medium [1]. To characterize rays of light as they
move through an optical system, ray transfer matrices are employed.
Optical ray transfer matrices describe how the displacement and slope of an optical ray
changes after passing through a simple optical element. The entries in the matrix characterize the
focusing properties of the element [3].

4. (a)
(b)
Fig. 1. (a) A visual depiction of how a ray transfer matrix represents a ray
entering an optical element (the orange box pictured here) from the “object space”
with a transverse displacement of 𝑟1 and slope 𝑟′1. The ray exits into the “image
space” with an associated transverse displacement 𝑟2 and slope 𝑟′2. (b) A generic
ray transfer matrix representing an incoming ray with transverse displacement 𝑟1
and slope 𝑟′1, and outgoing displacement 𝑟2 and slope 𝑟′2 [3].
The ray matrices for a large number of basic paraxial optical elements can be looked up
easily. The two that will be most useful for this experiment are those for a region of free space
and for a thin lens. Because the Rochester Cloak is comprised of various lenses aligned on a
central axis, these are the only two optical elements present in the system. Hypothetically, any
number of lenses could be chosen and placed in such a way so as to produce a cloaked region,
but for the scope of this research two models were chosen: a three lens cloak, and a four lens
cloak.


















1
'
1
2
'
2
r
r
DC
BA
r
r

5. (a) (b)
Fig 2. (a) Ray transfer matrix for free space region with index of refraction n and
total length L. (b) Ray transfer matrix for a thin lens with focal length f [3].
As previously mentioned, a perfect cloak should behave as though it were replaced by the
ambient medium. Synthesizing this with the transfer matrix for an overall system hypothetically
allows for a solvable condition for any arrangement of lenses to form a perfect cloak:
(1)
Before looking at the three or four lens cloaks, the possibility of a two lens cloak was examined.
Ray transfer matrices representing each optical region in a system are multiplied together to yield
the overall transfer matrix. For the two lens system, comprised of two lenses with focal lengths
𝑓1 and 𝑓2, separated by a distance 𝑡, this is given by
[
1⁡ 0⁡
−1/𝑓2 1
] [
1 𝑡
0 1
] [
1 0
−1/𝑓1 1
] = [
1 −
𝑡
𝑓1
𝑡
−(𝑓1+𝑓2+𝑡)
𝑓1 𝑓2
1 −
𝑡
𝑓2
] (2)
Equation (1) can only be satisfied if 𝑓1 = 𝑓2 = ±∞, which would essentially collapse the system
into a region of free space, thus eliminating the possibility of any cloaking region or optical
effect.






10
/1 nL









1
1
01
f












10
/1 nL
DC
BA
perfect

6. Fig 3. Three lens cloak system [1].
A three lens system can be represented by the following matrix:
[
1 0
−1/𝑓3 1
] [
1 𝑡2
0 1
] [
1⁡ 0⁡
−1/𝑓2 1
] [
1 𝑡1
0 1
] [
1 0
−1/𝑓1 1
] (3)
Carrying through the matrix multiplication and setting C = 0 to satisfy Equation (1) yields the
following condition:
𝑓2 = −
(𝑓1−𝑡1)(𝑓3−𝑡2)
𝑓1+𝑓3−𝑡1−𝑡2
(4)
This substitution simplifies equation (3):
[
𝑓3(𝑓1−𝑡1)
𝑓1(𝑓3−𝑡2)
𝑡1 + 𝑡2 + 𝑡1 𝑡2
(𝑓1+𝑓3−𝑡1−𝑡2)
(𝑓1−𝑡1)(𝑓3−𝑡2)
0
𝑓1(𝑓3−𝑡2)
𝑓3(𝑓1−𝑡1)
] (5)
Since the B entry in the transfer matrix represents overall system length, we can set this
equivalent to the sum of the separation distances to get the following:
(6)
This equation illuminates the fatal flaw of the three lens configuration: this equation only holds if
𝑡1 = 0, 𝑡2 = 0,⁡or (𝑓1 + 𝑓3 − 𝑡1 − 𝑡2) = 0. The first two cases yield a two lens system which has
0
))((
)(
2311
2131
21 


tftf
ttff
tt

7. been shown to be ineffective. The third requires that 𝑓2 → ∞, which again turns the system into a
two lens configuration.
Although this result tells us that a three lens cloak can never satisfy the requirements of a
perfect cloak, considering a symmetric case yields an interesting outcome. Looking at the setup
where 𝑓1 = 𝑓3 and 𝑡1 = 𝑡2, the following condition becomes apparent.
(7)
Therefore, the system asymptotically approaches a perfect cloak for 𝑓1 ≫ 𝑡1. This symmetric
setup also alters equation (4) after setting A = 1 and C = 0 to satisfy equation (1):
(8)
The setup of the four lens cloak follows a very similar process of manipulating the system’s
ray transfer matrix to look like the free space matrix. The guiding philosophy for this cloak is to
undo any changes that the first half of the system makes with the second half. To accomplish
this, a symmetric cloak is considered where 𝑓1 = 𝑓4, 𝑓2 = 𝑓3, and 𝑡1 = 𝑡3. Constructing the ray
transfer matrix in similar fashion to the two and three lens systems, we get
(9)
0
2
11
2
1

tf
t
121 2 fft 










































 1/1
01
10
1
1/1
01
10
1
1/1
01
10
1
1/1
01
1
1
2
2
3
3
4 f
t
f
t
f
t
f

8. Fig 4. Four lens cloak system [1].
After matrix multiplication and setting A = 1 and C = 0 as mandated by the perfect cloak matrix,
we find the spacing condition for the first and third displacements:
(10)
We obtain our final separation condition by setting the total length of the system equivalent to
the sum of the separation distances:
(11)
These conditions provide an exact solution to equation (1) and thus a perfect cloak
comprised of four lenses. The major benefit of both of these cloaking systems is that they are set
up proportionally to the focal lengths of the lenses. Because of this, a variety of different set ups
can be tested to optimize the system and allow for a more widely applicable device.
Experimental Setup
The three lens cloak was set up with two converging lenses on either end and one
diverging lens placed in the middle. The focal length of the first and third lenses was determined
to be 130mm ± 5mm. The middle lens was determined to have a focal length of -40mm ± 12mm.
From these lengths, the separation distance was then calculated to be 50mm ± 13mm.
211 fft 
21
212
2
)(2
ff
fff
t




9. Fig 5. Ray trace diagram made with OpticalRayTracer 8.9 to illustrate the
cloaking effect of the three lens cloak. The blacked out regions represent the
cloaking region. The overall cloaked region is a solid of revolution with the black
isosceles triangles in the above cross-sectional slice as the generating shapes,
observable when viewing the system on-axis through the first lens [5].
(a) (b)
Fig 6. (a) Side view of experimental setup of three lens cloak. (b) Observer’s
perspective of system. Graph paper demonstrates the lack of background
distortion, the primary goal of a perfect cloak

10. The four lens cloak was setup with two pairs of converging lenses with equivalent focal
lengths. The first and fourth lenses were measured to have focal lengths of 200mm ± 5mm, and
the second and third lenses were measured to have focal lengths of 51mm ± 5mm. From these,
the first and third separation distances were calculated to be 250mm ± 10mm, and the middle
separation distance to be 170mm ± 22mm.
Fig 7. OpticalRayTracer 8.9 simulation of four lens cloak to illustrate cloaking
region. Again, the blacked out region represents a cross-sectional slice of the
region that would appear cloaked to an observer [5].

11. (a)
(b)
Fig 8. (a) Side view of four lens cloak setup. The ruler is placed at the position of
greatest cloaking, seen as the point of convergence of the incoming light rays
between the first and second lenses in Fig. 7. (b) Observer’s perspective of cloak,
demonstrating the cloaking effect on a ruler.

12. Conclusion
The theoretical framework for this experiment provided systems of equations that
describe the parameters for both a three and four lens cloak. While the three lens system only
approximated a perfect cloak, the four lens system exactly matched the requirements. These
results are not without their limitations, however. Aberrations were noticeable in the background
when looking through the cloak in both cases. The graph paper was chosen specifically to
illustrate how effectively these cloaking systems left the background unaltered, but upon
inspection there were slight distortions and blurs. These can be attributed to uncertainty in the
focal lengths and separation distances, as well as imperfections in the materials in the lenses. The
uncertainty in the middle diverging lens in the three lens cloak is the greatest out of all the
materials on hand. This uncertainty came from some difficulty that arose in determining the final
image distance in the focal length measurement, leaving the placement of this lens up to question
for the working demonstration. Human error in attempting to align the lenses all along a central
axis could also have contributed. Along these same lines, the coating and materials used in the
production of lenses can affect the properties of the cloak [1]. Another limitation of the
Rochester Cloak is the presence of edge effects; if an observer views the cloak from too close,
there is a substantial amount of unwanted rays on the outside of the cloaking region, outside the
paraxial limit.
This is an exciting experiment because it offers a vast array of options for future research.
Because the nature of the Rochester Cloak is such that it can scale to any size given the proper
materials, much of this future research would revolve around optimization. One major goal
would be the reduction of edge effects, as well as achieving a larger field-of-view for an
observer. These could be attained by working not just with different focal lengths and separation
distances, but also lens diameters. Work could also be done towards making the cloaking region

13. more independent of incident ray angles. A final suggestion for future research would be to
pursue new lens configurations. One might allow for cloaking of the center region, whereas these
designs cloaked solids of revolution, leaving the center region visible. Also, there could be non-
symmetric solutions to these cases that were left unexplored in this experiment.
In summary, this experiment defined what conditions need to be met in order to create a
perfect cloak and then offered a three lens system that approximated these conditions, and a four
lens system that exactly matched them. Experimental demonstrations were assembled to
illustrate the cloaking effect predicted by the theoretical framework, which were then simulated
using OpticalRayTracer 8.9. This project provides a formalism which can effectively describe
ray optics invisibility cloaking.
References
[1] Choi, J.S., J. C. Howell, “Paraxial Ray Optics Cloaking,” Optics Express 22, (2014).
[2] Howell, J. C., J. B. Howell, J. S. Choi, “Amplitude-Only, Passive, Broadband, Optical Spatial
Cloaking of Very Large Objects,” Applied Optics 53, 1958-63 (2014).
[3] Siegman, Anthony E. “Lasers,” University Science Books (1986).
[4] "A Brief History of the Real-life Invisibility Cloak." A Brief History of the Real-life
Invisibility Cloak. 27 Mar. 2013. Web. 19 May 2015. <http://theweek.com/articles/466216/brief-
history-invisibility-cloak>.
[5] P. Lutus, (2015), WWW Document, (http://arachnoid.com/OpticalRayTracer/).