LECTURE 16

The Eye and its Limitations

IntroductionThis lecture is about one of the marvels of the

universe; the human eye.

Up close, like most biological specimens, itcan look crude, even a little gross. Yet it is, infact, one of the most finely tuned evolutions in theanimal world.

The Construction of the Human EyeFirst consider the construction of the human

eye. Fig. 24.19 of Hecht is made up of two gooddiagrams showing important features of thisconstruction. Basically, it is a sphere of about 22mm diameter with a light sensing organ called theretina on the inside of its back surface. However,it is not a perfect sphere in that there is aprotuberance on its front that has a reducedradius of curvature compared to the remainder ofthe eye.

This protuberance is not an aberration. It hasto be there for the eye to work properly. This isbecause the liquid of the eye, which is mostlywater, has an index of refraction close to that ofwater. The main body of the eye has to be asphere so as to be able to rotate in its socket.However, a sphere of this index of refraction willnot focus incoming plane waves onto its rearsurface but to a point behind that surface.

F22 mm

24 mm

RETINA

FOVEA

The extra curvature from the protuberancegives the extra focusing needed to have the imagefall on the retina. Because of this protuberancethe overall depth of the eye is slightly greater than22 mm, or about 24 mm.

Thus the operating principle of the eye is verysimilar to the operating principle of the camera.Plane waves of light entering each device arefocused by a lens onto a surface behind that lensthat contains photochemical substances.

F

RETINA

35 mm

FIL

M

In the case of the camera, these substances aredyes that are activated by the light and which canlater be developed into a picture. (In digitalcameras, the light activates transistor-like pixels tostore a digital record of the light level falling oneach pixel.) In the case of the eye thephotochemical substances immediately activatespecial neurons that send signals to the visualperception neural network.

Physics 101A - Physics for the life sciences 2

Accommodation to Varying Light Levels

There is also a similarity in the way bothdevices cope with variations in the light levelentering them. In both the eye and the camera,the best contrast in the various parts of an imageis obtained when the amplitude of the light waveentering them is at a particular value. In the caseof the camera if too much light is allowed to fallonto the film the dyes become saturated, turningthe developed film almost black and the resultingprinted picture almost white. (A similar effectoccurs in digital cameras.) On the other hand ifnot enough light is allowed to fall on the film itwill be almost transparent when developed andthe resulting print will be almost black. In eithercase the printed picture, even when adjusted forbrightness, will have very little contrast.

The photochemicals in the eye behave in asimilar fashion. If you are viewing a scene andthe light level of the scene suddenly changes youwould have trouble distinguishing the elements inthat scene as easily as before if your eye didn’tdo anything about it. This is because of thedecreased contrast produced by theelectrochemical activity in our retina. To get thelight level falling on your retina back to what itshould be for good contrast you must dosomething about the fraction of light falling oyour eye that is allowed to get to the retina. Youachieve this very rapidly by changing the diameterof its iris. If the light level has decreased thenyour iris will open up, as shown in the upperphotograph below. If it has increased, then theiris will close down as shown in the lowerphotograph. And you do this in less than a tenthof a second.

The range of accommodation can bedetermined from the ratio of the diameter of theiris when it is fully open to when it is fullycontracted. This is from a diameter of about6 mm to a diameter of about 2.5 mm. Since theamount of light that enters an aperture isproportional to the area of that aperture then theiris is capable of varying the amount of light thatfalls on the retina by about a factor of (6/2.5)2 orabout 6.

For the camera there is a similar mechanismfor controlling the amount of light that falls onthe film. This is an array of sliding vanescontrolled by the "f-stop" adjustment ring on thecamera lens. For camera lenses the aperturediameter is given in terms of the focal length ofthe lens. The "f-stop" is the ratio of the focallength of the lens to the aperture diameter. Thusa "55-mm lens", which means it has a foal lengthof 55 mm, and which has an f-stop all the waydown to 2, which would mean it is a very good55-mm lens, would have an aperture that could beopened up all the way to 27.5 mm diameter. Thef-stops on a camera lens of this quality wouldtypically range up to f-22, in steps of 2, 2.8, 4,5.6, 8, 11, 16 and 22.

It can be seen from this progression that each"stop" goes down in diameter by approximatelythe square root of 2. Thus the area of theaperture goes down in steps of 2, for an overallrange of about 28, or 128, for the lens describedhere.

This makes the accommodation of the eyeseem feeble compared to that of a good camera,even more so when the other mechanism forchanging the amount of light falling onto acamera film is taken into account; that ofchanging the time interval for which the film isexposed. This is done by changing the "shutterspeed", which in a good camera can be changedfrom about 1 second down to 1 millisecond.

Thus the full range of light levels that can beaccommodated to give the correct film exposureis about 100,000. For any given exposure that iswithin the range of the shutter speed and the f-stops, any desired increase in the shutter speed bya factor of 2 to capture a fleeting event can beachieved by a one step down in the f-stop, makingthe camera a very versatile tool for creatingimages from various light levels and at variousspeeds.

With film cameras even further increases inrange can be achieved by using films of differentsensitivities, expressed as different "ASA"

Lecture 16 - The Eye and its Limitations 3

numbers. For film with ASA numbers rangingfrom 100 to 1000, this can give a further increaseof range of ×10, or an overall range forphotography of about 1 million in light level atwhich pictures can be easily taken.

This makes the ability of the iris toaccommodate different light levels seem verypuny indeed. How is it then that we all know ofsituations where we can see very well what isgoing on but where a normal camera just cannottake a good picture?

This is because the eye has, in fact, anenormous range of accommodation; up to about afactor of 1012, or about a million times that of thecamera! It is obvious that such an enormousrange is very beneficial to an animal that usesvision to cope with its environment and thehazards that it can present. It is also obvious thatthis range could not be achieved by merelychanging the diameter of the iris.

The enormous range over which the eye canaccommodate varying light levels is achieved byadapting the production of the retinalphotochemicals themselves to different lightlevels. If then a range of a trillion to one ispossible with just photochemical production, whybother with the puny adaptability achieved by theiris?

Because the accommodation by the iris, as Ihave already pointed out, occurs in less than atenth of a second whereas the full accommodationto a changing light level by photochemicalproduction rate can take up to 30 minutes.

Again, nature has evolved different strategiesfor different purposes. The enormous range ofsensitivity that can be achieved by photochemicalproduction is obviously for adapting the eye fromuse during midday sunshine to use in deeptwilight, or even by the light of the Moon. Ittakes about 6 hours to go from one of theseconditions to the other and so the slowness of theadaptation is of no consequence. It is only whenthe adaptation process is violated by suchatrocities as California type restaurants, with theirvery subdued interior lighting even for mid-daymeals, that the speed of the adaptation becomes aproblem, the problem being the pain that resultswhen you leave such a restaurant to reenter brightsunlight. (You would never see a good restaurantin Paris subject its patrons to such an experiencefor lunch. But I digress to another of myfavourite prejudices.)

The accommodation provided by the iris ofthe eye is obviously for sudden changes in light

level, such as when you escape the tiger-riddenplains into the shelter of a forest. There thechange in light level is more like a factor of 8(think 3 f-stops on your camera). The iris hasobviously evolved for such situations.

I bring up this interesting declarativeknowledge, even though it is mostly chemistry, toshow the importance of how physics governedthe evolution of our perception systems.

Field of ViewAnother aspect in which the eye surpasses the

camera is in its "field of view". This refers to themaximum angle between the directions of twosources that can still be seen at the same time.For example, consider again the ubiquitous tigerand tree combination. The images of these twoitems would fall on your retina as shown below.

LIGHT FROM TIGER

LIGHT FROM TREE

IMAGE OF TREE

IMAGE OF TIGER

Drawing the central rays for the light fromthese two sources gives the following diagram.

RETINA

17 mm

OPTIC CENTER

Here the rays are drawn through what iscalled the "optic center" of the eye, which is thepoint through which all the central rays pass inprogressing through to the retina. Because of therefractive properties of the protuberance and ofthe inner parts of the eye it is not exactly at theentrance of the eye but inside at 17 mm from theretina.

For the case shown the angle between thetree and the tiger, and therefore also the anglebetween the images of the tree and the tiger on theretina, is drawn to be 20 degrees.

This is, in fact, a relatively small angle for theeye to cover. The retina on each eye can cover anangle of about 120 degrees; 60 degrees on eitherside of central vision. If it weren't for the nose,

Physics 101A - Physics for the life sciences 4

which limits the field of view of each eye to about45 degrees on the nose side, each eye wouldtherefore have a field of view of 120 degrees.However, this full field of 120 degrees is onlyachievable with two eyes.

NOSE

60o

60o

When we take into account that the eyes canmove in their sockets by about 45 degrees toeither side, the full effective field of view is about210 degrees. That is greater than 180 degrees,even partly to the rear!

By comparison the field of view of a camerais very limited. Take as a example a camera witha 55-mm lens for "35-mm" film. The imageplane for 35-mm film is actually a rectangle35mm by 23 mm. Taking the larger dimensionas defining the field of view, as when a picture isbeing taken in the "landscape" format, this fieldof view can be determined, from the diagrambelow, to be about 35 degrees.

55

tan-117.5

55= 17.6

o

= 35o35

17.5

FIL

M

Thus it would take four side by side picturesof a scene to get a truly panoramic field of viewas seen by our eyes.

The eye can cover such a wider field of viewthan the camera because its image surface, theretina, is curved. This makes the focal length for

waves from different direction more uniform.Because of the necessity of getting film in andout of the camera and of moving new pieces of itinto place for each picture, the focal plane of acamera must be flat. Because of the difficultiesof fabricating the crystal that makes up thephotosensitive pixels of a digital camera even thefocal plane of that camera must be flat. Thismakes the focal length for off-axis waves largerthan for on-axis waves and so considerablycomplicates the design of the lens if it to have awide field of view.

Details in the Field of View of the EyeYet, while the eye has a large field of view, it

would make an extremely poor camera. This isbecause in its whole field of view there is onlyone very small region at its exact center where theview is clear. This region is called the "fovea"and it is only about 0.2 mm in diameter. In a slicethrough the retina it is seen to be a slightdepression.

The retina in this depression is made up ofvery small individual receptors, called "cones"because of their shape when viewed from the sideunder high magnification. Each of these conessends its pulses directly into the neural networkwithin the retina. It seems that this is the reasonfor the depression of the fovea relative to thesurrounding retina. The sensing surfaces have tobe close to the far retina wall (on the right in theabove figure) for these direct connections to beeffective.

The clear view in the fovea is achieved bythese individual receptors being so small that they

Lecture 16 - The Eye and its Limitations 5

resolve great detail in the part of the image thatfalls on them. In fact each cone is only about 2microns in diameter at their receptor end.

However, although they are very small, theirsize puts a limit on the detail that can be seen inany object in front of the eye. If that object is at acomfortable viewing distance of 250 mm, than thesmallest dot that can be seen as a separate entityin that object would be one that subtends an angleθ at the eye that is the same as that subtended byone cone of the fovea. (Its image would justcover the receptor surface of one of the cones.)

17 mm

OPTIC CENTER

2 �θ

Since the fovea is 17 mm from the opticcenter of the eye, one cone will subtend an angleof

θ =××

= × =−

−−2 10

17 101 2 10 0 12

6

34. . rad mrad

A dot which subtends this angle at a distanceof 250 mm has a diameter d given by

d

d

0 251 2 10

0 25 1 2 10 3 10 0 03

4

4 5

..

. . .

= ×

= × × = × =

−

− − m mm

To get an idea of this size, consider that therewould be about 850 of these dots in one inch(25.4 mm). In the standard unit used for printingmeasure, i.e. the “dot” of 1/72th of an inch(about 0.3 mm) these dots would be "850 dpi".So a print at 850 dpi would have no discernable"graininess" when held at 250 mm from youreye. If you could hold it at about 180 mm fromyour eye, and still focus on it clearly as youngpeople can, the print would have to have about1200 dpi for the graininess to disappear. Thisindeed is regarded as the standard required byprofessionals for "photo-quality" print.

However, as I have already pointed out, thefovea only has a diameter of about 0.2 mm, or200 µ. Thus the angle of clear vision is onlyabout 100 times that of the resolution, or about

12 mrad. At a reading distance of 250 mm this isa spot diameter of 3 mm.

It is hard to believe that we only have clearvision in a spot of 3 mm diameter at 250 mm.This is about the size of a capital "O" of "12-pt"font, as it would appear in a print-out of this text.Why is this clear view area so small and how dowe cope with it being so small?

From the direct connections of the foveacones to the visual neural network it seems thattheir outputs are all processed individually. Butnote that even through the diameter of the fovea isonly about 100 cone diameters, this means thatthere almost 10,000 cones in the fovea. (If thediameter was exactly 100 cones and the foveaexactly circular, then the number of cones wouldbe (π/4)×1002, or 7854.) If the image is to beprocessed in less than a tenth of a second so thatthe brain can comprehend what is being seen atits normal processing rate, then that means it mustprocess about 100,000 cones per second. Evenfor the massively parallel processing system ofthe brain, this seems extreme and it is indeed amarvel that the brain can do it. In fact, the load isso severe that much of the data is preprocessed inneural networks directly behind the retina beforeit is sent along the optic nerve to the brain. Itseems obvious that any more cones in the foveawould be an unacceptable drag on the processing.

If we can only see clearly in the fovea what isthe purpose of that vast array of cells forming theremainder of the retina? It must be to guide theeye so that it can be moved to put the area of theview that is of interest onto the fovea. Thus weread, or view any other complex image, byextremely rapid scanning of the eye over the fieldof view. That is indeed why the speed ofprocessing information in the fovea is soimportant.

To help the eye in moving the image onto thefovea there are a few detail sensors in theperipheral region but they get progressivelythinner the further they are from the fovea. Theprimary sensors in the peripheral region outsidethe fovea are "rods", pre-wired in bundles as localnetworks that seem to only send signals furtherup the visual neural system if there is a change inlight level falling on them. By working together,these sensors achieve incredible sensitivity tochanges in light level, and hence serve as earlywarning sensors of any changes in the peripheralfield of view. In fact, as few as several photonsfalling on one of these bundles are sufficient toactivate a signal to the brain.

Physics 101A - Physics for the life sciences 6

Thus the visual processing system of the eyeis nowhere like that of a camera, either aphotographic camera or a digital one. It scans itsvisual field in a completely different manner andartists as well as photographers must take thisinto account if they are going to produce pleasingpictures for the eye to scan.

Accommodation to Different SourceDistances

Another aspect in which the eye and thecamera differ is in the way they cope with objectsnot being "at infinity" but close enough for thewave-fronts entering the device to be noticeablycurved, and thereby not focused on the retinal orfilm plane. In the case of the camera, the lens issimply moved farther from the film toaccommodate the extra distance of the imagefrom the lens.

35 mm

FILM

FILM

In the case of the eye, the lens is squeezed sothat it becomes fatter, thereby having its refractivepower increased.

RETINA

RETINA

Getting "Close-up" ViewsThis ability of the eye to adapt its focusing to

objects that are close to it is very important indiscerning detail in any object, be it food, an

insect, or anything else that is small but still ofimportance to us. We bring the object closer toour eye, or move our eye closer to it to "get abetter look at it".

What we are doing, of course, is enlarging theimage of the object on our retinas. A camera can"bring things up close", i.e. produce a largerimage on the film plane, by having a lens oflonger focal length.

35 mm

FILM

75 mm

FILM

Because the film is now farther from the lensthe angular divergence of the light rays from adistant object will cause a larger image on thefilm.

IMAGE ON FILM

RAYS FROM DISTANT OBJECT

IMAGE ON FILM

RAYS FROM DISTANT OBJECT

This is, of course, impossible for the eye. Toachieve such an "close-up" look we would have tobe able to expand the diameter of our eye. Weare therefore stuck with getting a close-up lookby actually bringing the object up close.

Unfortunately, there is a limit to which thiscan be done, based on your ability to squeezeyour eye lens to accommodate the increaseddivergence of the light rays from an object whenit is very close to your eye. One of the great

Lecture 16 - The Eye and its Limitations 7

breakthroughs of our civilization was thediscovery of devices that overcame this problem.These were the convex lenses discussed in theprevious lecture.

F

The Compound MicroscopeBut there is also a limit to this device.

Increasing the magnification to high levelsrequires that the lens have a very short focallength. To have a very short focal length the lensmust be highly curved. This, in turn, means that itmust have a small radius of curvature. In otherwords, a very strong lens will have a very smalldiameter, making it difficult to see anythingthrough it.

The compound microscope is a clever deviceto overcome this problem. It allows the lensfocusing on the object to be very small indiameter, and therefore very strong. This lens isthen used to create a magnified real image foractually looking at with a magnifying glass, whichcan then be large enough in diameter for usefulviewing.

INTERMEDIATE FOCUS(REAL IMAGE)

OBJECT

so si fe

The magnification is now in two stages. Thefirst is the magnification of the lens near theobject. This lens is naturally called the"objective". From the consideration of a simplelens in the previous lecture this is just the distanceof the image from the lens divided by the distanceof the object from the lens:

Ms

soi

o

=

A typical compound microscope will have animage distance of about 15 cm. A typicalobjective will have a focal length of severalmillimeters. From the lens equation, it can beseen that for such large image distances the objectdistance will be very close to the focal length. Atypical objective magnifying power is 50× whichfor an image distance of 15 cm requires a focallength of 3 mm.

The magnifying power of the lens in front ofyour eye, called naturally the "eyepiece", is asgiven in the previous lecture for the simplemagnifying glass. It is the ratio of 250 mm andthe focal length of the eyepiece (in mm):

Mfee

=250

A typical eyepiece will have a magnifyingpower of 10×, which is achieved by a focal lengthof 25mm. A compound microscope made up ofthis eyepiece together with the 50× objective lenswould therefore have an overall power of 500×.

It was the invention of this sort of microscopethat led to the discovery of the existence of themicroscopic world of biological cells and bacteria,with all its implications for the health of mankind.

Galileo's TelescopeThe other optical device that has been very

important in the history of science is thetelescope, particularly the telescope made byGalileo's with which he discovered the Moons ofJupiter. He regarded the orbiting of these objectsaround Jupiter as conclusive proof of theCopernicus theory that all the planets, includingthe Earth, orbited around the Sun. Essentially,small objects orbit around large objects. Thus theplanets of the Solar system orbited around themuch more massive Sun, just like the Moons ofJupiter orbited around the much more massiveJupiter.

So, while the microscope has had its greatestimpact on biology, the telescope has had itsgreatest impact on physics itself. It became thetool for observing the universe outside theatmosphere of Earth.

The problem addressed by the telescope isthat of achieving a greater separation on yourretina, or on photographic film, for the images ofobjects that are at enormous distances. Forexample, consider the problem of observing Marsand Venus when they appear to be close together.

Physics 101A - Physics for the life sciences 8

Suppose the light rays from these two objects hadthe angular separation shown in the diagrambelow.

VENUS

MARS

SE

PA

RA

TIO

N O

F

ON

RE

TIN

A

Observing them with the naked eye, asdepicted in this diagram, they could have theimage separation as shown.

However, suppose the rays were focused by alens to give the two real images shown in thediagram below, and that these images were thenviewed by a magnifying eyepiece like that used inthe microscope. The result would be the imageseparation at the retina that is shown in thatdiagram. This angle is considerably greater thanwith the naked eye. We therefore have amagnified view of the objects. It is as if theobjects had been brought much closer to youreye.

LIGHT FROM VENUS

VENUS FOCUS

LIGHT FROM MARS VENUS FOCUS

MARS FOCUS

SE

PA

RA

TIO

N O

F

ON

RE

TIN

A

The actual magnification can be determinedby geometrical optics. The pertinent diagram isshown below.

VENUS FOCUS

MARS FOCUS

ANGLE - REAL

ANGLE - IMAGE

fo fe

This diagram shows that the magnification ofthe angle between the rays from Venus and the

rays from Mars that reach the eye is just the ratioof the focal length of the objective and that of theeyepiece:

Mf

f= =

Angle of imageAngle of object

o

e

Therefore having a long focal length objectiveand a short focal length eyepiece results in a largemagnification. For example, an objective of focallength 1 meter and an eyepiece of focal length25 mm (i.e. a magnifying power of 10×) wouldform a telescope of power 40×.

The Limits of Magnification It might seem that by combining lenses you

could achieve any magnification that you wished,either for the "microscopic" world, or for thedistant objects in outer space. However, as youmay have suspected, nature imposes a limit onwhat you can achieve.

As a fact which I will try to explain in the nextlecture, if a wave-front is converged onto a focusby a converging lens

F

the wave-fronts will not, in fact, converge exactlyonto that focus but will come to some limitingdiameter and then diverge.

θ

d

And there is a definite mathematicalrelationship, which I will try to explain later,between this diameter and the angle ofconvergence (in radians):

d = 2 44.λθ

Those who have had a previous physicscourse, or have looked into the back of Hecht,may have seen this equation in its more

Lecture 16 - The Eye and its Limitations 9

complicated sine format. In fact, the equation asgiven above is only for small angles. For suchangles θ is exaggerated in the drawing above.For most cases of interest in the life sciences theangle is small and so that it’s value in radians canbe obtained from the ratio of the diameter of thelens to its focal length:

D

θ

f

θ =D

fThe size of the spot of light at the focus is

then given by

df

D= 2 44.

λ.

For example, a perfect lens of focal length500 mm and diameter 20 mm would focus lightto a spot size of about 30 microns. Thus theappearance of the light at the focus would not bea sharp point but a fuzzy spot, somewhat like thatdrawn below (enlarged from about 30 microns toabout 5 mm).

This imposes a limitation on discerning detailat a focus. Suppose that there were, in fact, twowaves being focused simulateously from twodistant sources. If these were separated by adistance about equal to half the diameter of theabove spot they would appear like this;

To see that these two dots are separated byhalf their diameters here is a view of one of them,further enlarged to about 15 mm, with a whitecircle drawn with half the full spot diameter.

The two spots each with such circles areshown below.

It is commonly accepted that the spots beingseparated by half their diameters is the separationlimit at which two fuzzy dots spots be "resolved".In other words, it is the limit at which it is clearthat there are in fact, two dots. (The circle that ishalf the diameter of the outer limit of the spot issometimes referred to as the "circle ofconfusion". In this way of saying it, the limit ofresolution is when the circles of confusiontouch.)

This "resolving power" of a lens can thereforebe expressed as the angle which is just half theangle subtended by the full spot or

θλ

R D= 1 22.

It is of great interest when considering theconstruction of the human eye to note thediameter of the focus on your retina due to thefuzziness of the spot of light produced on yourretina. Take the normal diameter of your iris as5 mm and the focal length of your eye as 17 mm.For yellow light of 500 nm entering the lens ofyour eye the above equation gives a resolvingpower of about 0.12 mrad. For a focal length of17 mm this gives a resolution on your retina of17 mm times 0.12 mrad, or about 2 microns.This is amazingly close to the actual diameter ofthe cones of the fovea.

Thus there would be absolutely no point inhaving cones in your retina that are smaller than2 microns in diameter. At 2 microns they are just

Physics 101A - Physics for the life sciences 10

the right size to match the resolving power ofyour eye.

The Resolution Limit of the OpticalMicroscope

The finite size of the focus of a lens also putsan upper limit on the useful magnification thatcan be achieved with an optical microscope. Thecircle of confusion at this focus will have adiameter given by the equation

d = 1 22.λθ

This will therefore be the smallest separationthat can be resolved for two spots on amicroscope slide. In the microscope trade halfthe angle θ is referred to as the NumericalAperture, and designated NA. (In fact, because ofthe large angles that can be accommodated inmodern microscope lenses, it is the sine of thehalf-angle of the objective aperture that is definedas the NA. See Hecht.) The common form of theequation for the resolving power of the objectiveof a microscope is then

RNA

= 0 61.λ

Good high power objectives can be built tohave numerical apertures of up to 1.4. However,to obtain such a numerical aperture the light mustnever enter air until it has passed through theobjective lens. This is achieved by making thefront side of the objective flat so that it can bepressed against the specimen cover plate of theslide. To prevent any air whatsoever from beingin the interface, the thin gap between the flat sideof the objective and the cover plate is filled withoil of about the same index of refraction as theglass.

OBJECTIVE LENS

OIL LAYER

GLASS COVER PLATE

SPECIMEN

MICROSCOPE SLIDE

For light of wavelength 500 nm and anumerical aperture of 1.4, such an objective, fromthe above formula, can give resolutions of about0.2 microns.

The typical magnifying power of such anobjective will be 100X. Thus the size of the

resolution limit at the intermediate image will beabout 20 microns. An eyepiece of magnifyingpower 10X would make this appear to have adiameter of 0.2 mm at 25 cm. This is just a littlesmaller than the appearance of a standard periodpoint (defined as the previously mentioned “dot”or "point" of 1/72 of an inch of about 0.3 mm)held at 25 cm from the eye. It is about equivalentto the resolution of a newspaper photograph heldat 25 cm from your eye. Although the grain ofsuch a picture is noticeable (as pointed out earlierit is about 10 times the size of the absolute limitof what is detectable) it is generally notobjectionable. However, it is generally regardedas about the limit of what is acceptable.

The overall magnification of such amicroscope would be 1000X. One could have amore powerful eyepiece so as to achieve an evengreater magnification but this would just make iteasier to see the fuzziness (i.e. the “grain”) ofthe image. In the words of the trade it would be"empty magnification".

For the best resolution of a microscope youhave to use the shortest wavelength possible. Fornormal usage this would be blue light. In specialcases even ultraviolet light is used but thisrequires extreme effort since ultraviolet imagescannot be seen but can only be photographed.Some improvement could also be gained by thesort of computer image enhancement that is usedon photographs sent back from satellites in deepspace but these are time-consuming and, to myknowledge, not yet widely used in biology. Thegreat leap forward in technology that allowed usto see the inner structures of sub-cellular objects,such as the cell membrane, is that of electronbeam microscopy. However, this has theunfortunate limitation of being only possible withthe specimen in high vacuum and with electronirradiation that would kill any living organisms.

The Resolution Limit of the TelescopeAlthough it is not of importance in biology,

the resolution limit of the telescope should be ofgeneral interest. As in the microscope it isdetermined by the diameter of the objective. Theonly major difference is that for high-powertelescopes the objective is a curved mirror ratherthan a lens. In any case, the angular resolutionlimit is again given by

θλ

R D= 1 22.

Lecture 16 - The Eye and its Limitations 11

Here the meaning of the resolving angle isperhaps a little easier to see. It is simply theangle that the separation of two stars mustsubtend at the objective of the telescope. Thus atelescope with a "one-meter" mirror (i.e. one thathas a reflecting mirror of one meter diameter asan objective) will resolve two stars emitting bluelight of 400 nm that are only about 0.5 µradapart. To put this figure into perspective it isequivalent to the separation of the headlights of acar about 3000 km away.

As a final note on telescope resolution it isinteresting to imagine what Galileo saw when helooked through his telescope at the moons ofJupiter. (It is claimed by some historians thatGalileo's telescope had such poor optics that youcould see double stars where there was just one,leading to some doubt by independent observersthat the "moons" of Jupiter seen in his telescopewere in fact real. However, for the discussionhere I will assume that he had achieved perfectoptics.)

First it is of interest to compare the size ofJupiter, as seen by the naked eye, with theresolution limits of the eye. Jupiter has adiameter such that it subtends about 0.2 mrad atyour eye. This is just above the resolution limitof the eye, i.e. 0.12 mrad, meaning that Jupiter bythe naked eye can just barely be seen to be largerthan the point of light from a star.

The largest moon of Jupiter is Ganymede. Atits maximum separation from Jupiter when seenfrom Earth, i.e. at the radius of its orbit aroundJupiter, it appears at an angle of about 1.3 mradrelative to Jupiter. This is about one seventh of

the diameter of our Moon. Thus, if you could seeGanymede you could easily resolve it fromJupiter. However, Ganymede itself subtends onlyan angle of 0.07 mrad, at your eye, meaning that,if it can be seen at all, it is just a point. Certainly,it was not noticeable enough for anyone to haveseen it before the invention of the telescope.

Galileo's telescope had an objective lens ofabout 50 mm diameter. This meant that theresolving power of his telescope was about 10times that of the naked eye, or 0.012 mrad. Theimage of Jupiter would now be about 16 times theresolving power of the telescope while the imageof Ganymede would be about 6 times theresolving power. Both would clearly not be justpoints of light, like the stars, but resolved objects.A sketch of what the objects would look like inhis telescope is shown below

Reference material in text

Chapter 24: p. 876 to end of section 24.5,

p. 887 to end of section (the compoundmicroscope and the refracting telescope).

Chapter 25:Hecht approaches the material of this lecture

by preceding it with a development of the theoryof diffraction, which I will introduce in the nextlecture. However, I think that you can still benefitfrom looking at section 25.9, which relatesdirectly to the material of this lecture. Inparticular, study the solved example problem25.8.

Ganymede

Star

Jupiter

Physics 101A - Physics for the life sciences 12