John Covey Optical System Design – Project 1 3/10/05

High Resolution and Short Distance Retinal Imaging Design

Introduction

Currently, when one thinks of head-mounted displays, an image of large, bulky

virtual reality headgear comes to mind that is about as large as a helmet. A vision of the

future head-mounted display could entail a pair of sunglasses that is able to project light

from a portable video source onto the retina. A market of current interest for such a

device is televisions, where one desires to project the image of a very large TV screen

into the eye while taking up the minimal space that glasses would allow. This presents

two interesting challenges. First, one must be able to house a video source that is capable

of displaying enough pixels to match the video resolution of a television, preferably high

definition resolution. Second, this video source must then interface with a small, portable

optical system that is able to produce the illusion that the video source is a television

some distance in front of the user. This report will solely focus on the second of the two

issues, for it is necessary to demonstrate that such an optical system is possible before

one undergoes the arduous task of creating a small, portable, and high resolution video

source only to discover that it is physically impossible to image such a source onto the

retina in such a small, portable space.

Specifications

In order to adequately design the imaging system, an target situation containing

only the eye and a video source of a certain size, resolution, and distance away must be

specified. Since the niche market in which possible sources of funding are interested is

televisions, the best TV situation money can currently buy is chosen in order to determine

the feasibility of possible competition in this market. Thus, the target situation is as

follows: a human eye is watching a high definition TV which is 2 meters by 2 meters in

dimension and is 4 meters away. High definition television has been recently augmented

to 1920x1000 pixels, but for simplicity of calculation, the target situation shall contain

2000x2000 pixels instead. Magnification calculations dictate that such an object would

produce an 8.5mm x 8.5mm image on the retina. (These calculations are included in

Appendix A) Therefore, the optical system to replace the TV must ultimately produce an

8.5mm x 8.5mm image on the retina with a 2000x2000 pixel resolution. Lastly, it is

preferred that this optical system becomes small and simple enough so that it can be

head-mounted and be likened to wearing a pair of sunglasses.

Choice of Layout

Since the nature of the video source for this proposed system must be at least

vaguely understood in order to create an optical design, it will be assumed that pixels of a

10 micron pitch will be responsible for displaying the visual object. (The effect of

varying the pixel pitch will be explored later) Therefore, the initial total object size is

2cm x 2cm.

One could speculate that if the eye were used as the only optical element, the

system and its cost would be minimized. However, rudimentary calculation shows that in

order to image the object onto the retina with the target image size, the eye would quickly

overstrain itself in order to bring the object into focus. (See Appendix A) Therefore, at

least one external lens is required in order to make the design attainable. Since the eye is

most relaxed while focusing on an object that is almost infinity, it seems logical that a

telescoping system using an external lens and the eye’s focal power should be the next

attempt. Other designs were speculated upon where relay or field lenses were introduced

into the system, but the overall distance of the system always increased with each

additional element, making the end result highly undesirable. Moreover, due to the

required simplicity of the resulting head-mount, a system with a single external focusing

element was selected and analyzed for feasibility, for if such a system were not at all

possible, the likeliness of a more compact system than a virtual reality helmet

manifesting itself would be highly questionable.

Paraxial Design

An afocal keplerian telescope requires that the object be placed at the front focal

plane of the first lens, that the distance between the two lenses is equal to the sum of their

focal lengths, and that the image be placed at the back focal plane of the second lens.

Such is illustrated below in Figure 1.

Figure 1. Afocal keplerian telescope setup.

Since the object and image heights are predetermined, the magnification is also fixed at

.425 since it is the image to object ratio. Magnification is also the F2:F1 ratio when the

setup in Figure 1 is used, meaning that

MFF 2

1 = . (1)

Since the focal length of the relaxed eye (F2) is approximately 1.7cm, F1=4cm. The last

two values of interest are the system’s NA and the F# for the first lens.

p

NA λ= , (2)

where λ is the largest wavelength used in the system, (λred=700nm is used in this report)

and p is the pixel pitch, which was previously stated as 10 microns, making NA=.07.

The F# of the first lens is of extreme interest since the cost of the lens will greatly depend

on this value. Lenses with an F# >1 are easily obtainable, therefore the F# for the

proposed system is determined next.

DFF =#

, (3)

where F is the focal length of the lens, and the lens diameter

)*(2 hFNAD += , (4)

where h is the object height, and it is presumed that NA equals the highest possible ray

angle since the NA is < .23 (paraxial regime). Since all of the quantities required to

calculate F# have already been given, the F# for lens 1 is found to be 1.5321. This result

brings great relief, for having an F# > 1 gives a strong indication that the proposed

system is a feasible solution to the problem since such lenses are (more or less) readily

available for use.

When determining the longitudinal position of the object, it is noteworthy that the

eye is not most relaxed when it is focusing on something infinitely far away. The eye is

truly relaxed when focused on an image 67 cm away, making the eye’s focal length

1.65793cm. This slight deviation in the focal plane will require a translation in the

object’s position based on the longitudinal magnification, where

20 M∆=∆ (5)

and M is the linear magnification found above equation 1, ∆0 is the object displacement,

and ∆ is the displacement in the image plane. ∆ is found to be .233 cm, and thus the

object is moved forward 2.33mm in order to keep the image in focus at the retinal plane.

When a ray trace of the system is performed as shown in Figure 2, the total distance of

the system (not including the last 1.7cm inside the eyeball) is 9.22cm.

Figure 2. Paraxial ray trace of proposed 2 lens imaging system.

Now that the paraxial system is understood, it is necessary to observe the effects

of changing any possible variables within the system. Since the problem being addressed

is quite specific, the pixel pitch (or object height) appears to be the only free variable in

the setup, for the object resolution, the image height, and the second focal length are

fixed. By reusing the ray tracing algorithm used to give Figure 2, the effects of changing

the pixel size can be observed. (All ray tracing code is in Appendix B) Specifically, the

two figures of interest to those wishing to market a product would be the F# and the total

distance of the system, for the inverse of the first lens’ F# determines how costly it would

be, and the total distance of the system determines the size extent of the product. The

results over a large range of pixel pitches are given below.

Figure 3. The merits of the proposed system based on different pixel pitches.

Inspection of Figure 3 gives a clear understanding of the desired operating regime of the

device. Since pixel pitch is directly proportional to the object height, which is directly

proportional to the focal length of the first lens, which is also proportional to total

distance, the relationship in the first graph is purely linear. However, F# has a nonlinear

behavior which yields a diminishing marginal utility as pixel pitch increases. Therefore,

since the F# does not increase appreciably after a pitch of roughly 15 microns, the

operating range of interest is less than 15 microns. Thus, the paraxial design given in

Figure 2 is further vindicated as being plausible, for a pixel pitch of 10 microns appears

to be the optimum selection that minimizes both the lens cost and the total distance. (This

was not a coincidence, for the author chose 10 microns after having done the above

analysis)

As a brief aside, it may be possible that the total distance of the system is a more

important figure of merit than the resolution of the object. Therefore, the above analysis

was altered and given in Figure 4 to display the characteristics of a system that images

the equivalent of a normal, non-high definition television (320x240 pixels) onto the

retina.

Figure 4. The resulting system characteristics for imaging only 320 pixels.

It is interesting to note that while the object height has decreased, the F# graph is

identical to the one in Figure 3, for as the object height and required lens diameter

decrease, the focal length decreases at the same rate. However, the total distance has

decreased by the fractional decrease in resolution since the focal length of lens 1 linearly

depends on the object height. Since the behavior of both graphs has not altered, the

optimum pixel pitch is still 10 microns, but the total distance is now 2.98cm. Therefore,

these results show that a desired total distance can be attained even with a set pixel pitch

and F# by changing the object resolution.

Gaussian Design

The next step in creating a truly usable design is to replace the ideal lenses being

used thus far with thick lenses that can be purchased and actually used. Since it is more

cost effective to find a commonly manufactured lens than to custom order a large

quantity of specialized lenses, the Newport lens catalogue was perused. The common

lens closest to the specifications of the external lens was then selected, and its reported

specifications are given below in Figure 5.

Figure 5. Selected lens for use in the proposed system.

EFL, effective focal length, is the distance from either principle plane to the focal plane.

FFL, front focal length, and BFL, back focal length, are the distances from the front/back

lens surface to the front/back focal plane. Tc and Te are the center thickness and edge

thickness of the lens, respectively. Therefore, EFL-FFL is the distance between the front

lens surface and a principle plane, and since this quantity is zero, the first principle plane

must be on the lens’ surface. Likewise, the back principle plane is 3mm in front of the

back lens surface. (This is what the specification P2 means) Therefore, the front lens

surface should be placed at the intended location of the ideal external lens given in Figure

2, and the lens will effectively thicken the design by Tc-3=1.5mm. When introducing

principle planes into the second lens, (the eye) one finds that the cornea is actually

1.47mm in front of the first principle plane. Therefore, the total distance of the device

should be measured from the cornea edge and not from the ideal lens as presupposed,

which shortens the system length by 1.47mm. The separation between the two principle

planes in the eye does not thicken the system length, for the separation is within the

eyeball. The end result of using these real lenses is given in the diagram below, where

the new total system distance is barely changed to 9.22+.15-.147=9.223cm.

Figure 6. Thickened design with principle planes for lens 1 and eye.

Finite Design

The pupil for a relaxed eye is 1.85mm behind the eye lens and normally ranges

between 3mm and 7mm in diameter. It is reasonable to presume that the eye pupil will

never shrink to less than 3mm in diameter with any optical system present since the light

source behind the LCD object would be adjusted to give a comfortable level of intensity.

(Moreover, the system in question will not be designed to operate properly if the user

looks directly at the sun, for instance) The eye pupil will become the aperture and field

stop with such a small diameter; therefore, these limitations will be quantified.

First, the field stop will be determined from the pupil diameter. By propagating a

ray backwards that has zero angle and barely grazes the eye pupil, the largest object size

(i.e. field stop) will be determined. Although the field stop can usually be determined by

using the imaging condition through both lenses, the image is at the focal plane, leading

to imaging infinities. Since the author is interested in the size of the field stop at the

predetermined object location, a numerical backpropagation from the eye pupil is

performed instead, and the results are shown with the blue ray in Figure 7.

Figure 7. Blue ray giving original object height after grazing eye pupil at zero angle.

The maximum field stop in this worst-pupil case is 3.53mm or about a third of the total

object size.

Next, the pupil as the aperture stop will be considered. The maximum angle

given from the NA of the system hits the pupil at 2.495mm off axis, meaning that a pupil

of a 5mm diameter or larger will gather all of the resolution. Since the worst case is

when the pupil is 3mm in diameter, the pupil will only gather 60% of the resolution,

making the object’s 2000 pixels effectively 1200 resolvable pixels at the image.

Both of these results may seem disastrous, for the dynamics of the eye pupil are

not a free parameter in the design. The external lens’ diameter was lowered in order to be

a common, purchasable lens, which could be rectified with a custom lens, but since the

pupil is the limiting aperture, such would be a fruitless gesture. However, there are two

reasons why these results can be gracefully accepted and allowed. First, the eye pupil

creates the same conundrum when it is observing a big screen TV at an acceptable

distance, meaning that rectifying this problem is not necessary in order to replicate a TV

screen image. Second, the reason why TVs that are larger than the field stops with

resolutions higher than the pupil can observe exist are due to the eye’s unique anatomy.

The eye’s photo detector density is 100 times larger in the 1 square millimeter area that is

on axis than the rest of the retina, making the on axis image the most noticeable part of

the original object. Anyone with a functioning eye can observe this by noting that an

object’s clarity is far less when it is off the eye’s axis. Since the fovea (highest detection

density spot in the eye) subtends only 1.3 degrees of visual angle, it is clear that the pupil

does not limit the resolving capability of the on-axis detectors. Moreover, the NA of the

system easily enters the pupil when it originates from the axis. Therefore, the eye will

not discern any difference in the resolution of the off-axis parts of the object and will

detect the full resolution of the on-axis part of the object, while an artificial uniformly

distributed detector array would be more severely limited by such a small pupil.

Final Design

Now that a design with a real, thick lens has been incorporated into the design, the

final step is to translate the design into a real-world device that can be mounted on the

head in convenient and portable fashion. Since the design requires only a single lens and

is roughly 9.2cm in length, a very large number of possible setups could be determined.

The following is the author’s proposed implementation.

A small box is attached to the top frame of a pair of partially reflective glasses.

The video source lies inside the box on the end furthest away from the user’s forehead.

The light is then reflected onto the external lens, which then passes outside of the box and

is reflected off of the glasses into the eye. Assuming reflection angles of 45 degrees in

order to minimize the box’s space, the resulting setup is given in Figure 8.

Figure 8. Proposed optical setup where the purple box is the video source of unspecified

thickness, the red lines are mirrors, the brown lines are lenses, and the two blue lines

represent light rays from either end of the video source.

The reflecting surface outside of the box is almost 3cm, which is approximately the

average size of a sunglass lens. Moreover the box mounted above the glasses is small

enough to comfortably rest against the user’s forehead with little or no discomfort. Both

of these results achieve the aim of this report in terms of size and portability.

Another possible implementation is possible in order to ease the requirements of

the external lens. In Figure 9, a prism is attached to the bottom of the box housing in

order to ease angling constraints, as well as aid in the magnifying power of the system.

However, this approach will not be quantified since a duplicate analysis would be

required, and this is merely a suggestion if further investigation yields a desire to use a

less powerful external lens.

Figure 9. Setup augmented with a prism.

If design constraints required a smaller system, it could be possible to replace the

mirror and the lens inside the box with a single curved mirror. However, investigation of

common curved lenses indicates that the resulting F# would be far too high to be

desirable. Furthermore, the cost of a single common curved mirror equals the cost of 5 of

the selected lens or 7 flat mirrors, which reserves such a solution for only a more costly

“spin-off” of this device

(http://www.newport.com/store/product.aspx?lone=Optics&ltwo=Mirrors&id=3525&lan

g=1&section=Pricing).

Conclusion

An optical system has been designed that is capable of imaging a small, high-

resolution video source onto the retina in the same fashion as a high-definition big screen

television. By creating a layout that is realistically small to be worn as glasses and

selecting simple components that allow for a competitive price, it has been demonstrated

that producing the optical imaging system is a plausible idea and merits further research

and development. However, the costs and implications of creating and using the video

source itself were not addressed in this report, and should be the next area of focus

concerning this device. Perhaps in the upcoming future, the populace’s relative distaste

for virtual reality headsets will be replaced by the widespread acceptance of video glasses

which will strive to replace heads up displays, televisions, and even computer monitors

everywhere.

Appendix A

Magnification of TV image on retina

Linear magnification is given as

ddM '= , (6)

where d’ is the distance from the lens to the image, and d is the distance from the object

to the lens. In the problem given, d’ is the distance from the eye lens to the retina or

1.7cm. The distance d is given as 4 meters, yielding a magnification of .00425.

.00425*(2 meters) = 8.5mm. Therefore, the TV produces an image of size 8.5mm on the

retina.

Eye power calculations for a single element imaging system

The linear magnification of a 2cm object into a 8.5mm image is .85/2=.425. As

given in equation 6, since both the magnification and the value of d’ are known, d is

calculated to be 4cm. The reader can easily prove to him/herself that bringing an object

into focus that is even 10 to 20cm away from the eye produces a maximal amount of

eyestrain. Even if eyestrain were not a factor in this problem, a human eye can not

reshape its lens enough to produce enough power to image an object 4cm away without

an external optical system. Moreover, the unaided eye is most comfortable at focusing on

objects 67cm away, which yields a system that is far too long to achieve the desired

setup. It is clear that an external optical system must be utilized to create a feasible

solution.

Appendix B – Raytracing Code

Paraxial Raytrace Code

%John Covey %Optical System Design - Project 1 %Paraxial Ray Tracing of Imaging System %ALL UNITS ARE IN CENTIMETERS clear eps lambda=700e-7; numrays=3; numsurfaces=6; f=zeros(numsurfaces); x=zeros(numsurfaces); for m=1:numsurfaces f(m)=1e99; %Set all surfaces to zero focal power first% end originalheight=100; %Half of the length of big screen TV originaldistance=400; %Distance from big screen TV eyedistance=1.7; %distance from retina to refractive power of eye Feye=1.65793; %Focal length of a relaxed eye Eyemag=eyedistance/originaldistance; %Magnification factor of TV to retina eyeimageheight=Eyemag*originalheight; %Image size of TV on retina, TARGET IMAGE SIZE pixelpitch=10e-4; %20 um numpixels=1000; %High definition TV resolution objheight=pixelpitch*numpixels; Mag=eyeimageheight/objheight; F2=Feye; %Eye is 2nd lens in system F1=F2/Mag; NA=lambda/(pixelpitch) %NA of system Fnum=F1/(2*(F1*NA+objheight))

%x(1)=0; %Object location x(1)=(eyedistance-Feye)*(1/Mag)^2; %Object correction since focal length of eye isn't exactly equal to eye diameter f(2)=F1; %lens 1 x(2)=f(2); x(3)=x(2)+eps; f(4)=F2; x(4)=x(3)+f(2)+f(4); x(5)=x(4)+eps; x(6)=x(5)+eyedistance; h(1,1)=objheight; %Object Height h(1,3)=h(1,1); u(1,1)=0; u(1,2)=NA; u(1,3)=u(1,2); %h(1,2)=-u(1,2)*f(2); h(1,2)=-NA*(F1-x(1)); for n=1:numrays for m=1:numsurfaces-1 M=[1,x(m+1)-x(m);-1/f(m),1]*[h(m,n);u(m,n)]; h(m+1,n)=M(1); u(m+1,n)=M(2); end end figure plot(x,h(:,1),x,h(:,2),x,h(:,3)) Pixel Pitch Behavior Code %John Covey %Optical System Design - Project 1 %Impact of variation of Pixel Pitch in System %UNITS IN CM lambda=700e-7; %Red light originalheight=100; %Half of the length of big screen TV originaldistance=400; %Distance from big screen TV eyedistance=1.7; Feye=eyedistance; %Focal length and distance from retina to refractive power of eye Eyemag=eyedistance/originaldistance; %Magnification factor of TV to retina eyeimageheight=Eyemag*originalheight; %Image size of TV on retina, TARGET IMAGE SIZE pixelpitch=5e-4:1e-4:40e-4; %20 um numpixels=160; %High definition TV resolution

objheight=pixelpitch*numpixels; Mag=eyeimageheight./objheight; F2=Feye; %Eye is 2nd lens in system F1=F2./Mag; NA=lambda./(pixelpitch); %NA of system Fnum=F1./(2.*((F1.*NA)+objheight)); distance=2.*F1+F2; subplot(2,1,1) plot(pixelpitch.*10000,distance) title('Total Distance vs. Pitch') xlabel('Pixel Pitch (um)') ylabel('Total System Length (cm)') subplot(2,1,2) plot(pixelpitch.*10000,Fnum) title('F# vs. Pitch') xlabel('Pixel Pitch (um)') ylabel('F#') Aperture/Field Stop Ray Tracing %John Covey %Optical System Design - Project 1 %Aperture/Field stop determination %UNITS IN CM clear eps lambda=700e-7; numrays=3; numsurfaces=6; f=zeros(numsurfaces); x=zeros(numsurfaces); for m=1:numsurfaces f(m)=1e99; %Set all surfaces to zero focal power first%% end originalheight=100; %Half of the length of big screen TV originaldistance=400; %Distance from big screen TV eyedistance=1.7; %distance from retina to refractive power of eye Feye=1.65793; %Focal length of a relaxed eye Eyemag=eyedistance/originaldistance; %Magnification factor of TV to retina eyeimageheight=Eyemag*originalheight; %Image size of TV on retina, TARGET IMAGE SIZE pixelpitch=10e-4; %20 um numpixels=1000; %High definition TV resolution objheight=pixelpitch*numpixels;

Mag=eyeimageheight/objheight; F2=Feye; %Eye is 2nd lens in system F1=F2/Mag; NA=lambda/(pixelpitch) %NA of system Fnum=F1/(2*(F1*NA+objheight)) %x(1)=0; %Object location x(1)=(eyedistance-Feye)*(1/Mag)^2; %Object correction since focal length of eye isn't exactly equal to eye diameter f(2)=F1; %lens 1 x(2)=f(2); x(3)=x(2)+eps; f(4)=F2; x(4)=x(3)+f(2)+f(4); x(5)=x(4)+eps; x(6)=x(5)+eyedistance; h(1,1)=.15/Mag; %Object Height h(1,3)=h(1,1); u(1,1)=0; u(1,2)=NA*(.15/.2495); u(1,3)=u(1,2); %h(1,2)=-NA*(F1-x(1)); h(1,2)=0; for n=1:numrays for m=1:numsurfaces-1 M=[1,x(m+1)-x(m);-1/f(m),1]*[h(m,n);u(m,n)]; h(m+1,n)=M(1); u(m+1,n)=M(2); end end figure plot(x,h(:,1),x,h(:,2),x,h(:,3))