ASTR 5110, Majewski [FALL 2015]. Lecture Notes
ASTR 5110 (Majewski) Lecture Notes
INTRODUCTION TO OPTICS AND DETECTORS
BY WAY OF A FAMILIAR EXAMPLE:
THE HUMAN EYE
Some References on the Human Eye:
For most of the history of astronomy, the human eye represented
the most important astronomical instrument. It serves as both
light gathering objective lens and detector.
The eye serves as a useful introduction to both optics and detectors
based on an example familiar to all of us. An in-depth discussion of
the human eye allows me not only to explain how to use this optical
instrument to maximal advantage, but to relatively quickly review or introduce
many basic concepts of optics.
- The Eye: A Very Short Introduction, by Michael F. Land
(an excellent resource)
- Section 5.7 and 11.3 of Hecht.
- Chapter 6 of Birney.
- Section 1.1 of Kitchin.
- Chapter 2 and Section 11.1 of Schroeder.
Picture at right is from Descartes's Dioptrique
(with this version adopted from The Fire Within the Eye, by David Park).
I. PARTS OF THE EYE
The eye is a double positive, variable radius of curvature, radial GRIN
(gradient in the index of refraction), multiple refractive element
optical system focusing light onto a
The basic parts are as follows:
The eye has the following important elements:
- The lens (or crystalline lens) of the eye serves as the objective and
focuses light to the back of the eye. The focal length of the
lens is about 24 mm.
- The iris acts as an adjusting pupil (just like
the iris diaphragm in a 35 mm SLR camera),
which dilates and contracts to adjust the speed of the eye
from about f/3 to about f/8.
The visible parts of the eye, on the left. The iris has two types of muscles (radial and
when one or the other contracts, the iris closes or dilates. The right picture is an interesting,
"cartoonish" way to think of the eye as a camera shutter. Two left figures from The Teaching Company.
The right picture is an artist's photodesign from http://scamper.com/portfolio/pho10.html.
- The iris also helps with an additional advantage
that the eye's flexible lens affords:
By helping to squeeze the lens into different shapes
can change the focal length as well as the clear
aperture of the eye. (This is done in cooperation with the
ciliary muscle attached to the lens.) This reshaping of
the lens, which changes its refractive power, is needed to be able to focus on objects
having a wide range of distance, is called "accommodation".
How accommodation works. Top is front-on view of eye and bottom is side view of eye lens and its support structure.
When unstimulated, the ciliary muscle is relaxed and expanded, putting the zonule fibers under tension and this stretches the lens
to be thin (left). When the eye is accommodating, the ciliary muscle (which is a sphincter) contracts, reducing its inner diameter;
at this point the zonule fibers are no longer undertension, which allows the lens to reduce in diameter and become more bulged.
In this state, its radius of curvature decreases and the lens has more power. From Michael F. Land, The Eye: A Very Short
- "Iris" comes from the Greek word for rainbow (from Greek
mythology, the goddess of the rainbow), and
is what gives people's eyes their characteristic colors
(e.g., brown, blue, green, hazel, grey).
- ~7 mm diameter pupil when "dark adapted".
- The retina is the image detector
of the eye, and consists of light sensitive neurons that lie
at the focal "plane" of the lens of the eye.
- An incredible aspect of the retina is that it
lies on a curved focal plane surface,
the back of the eyeball, and that retinal cells
cover about a full hemisphere.
- This curved focal plane surface allows the
eye to detect rays coming into the lens at extreme angles
and give the eye an incredible, near 180o
field of view!
- The resolution degrades with this
peripheral vision (see figure showing "visual acuity of the
human eye" below), but it is still very good
at detecting motion.
- The fovea (or fovea centralis) is the center of the
retina and is the place on the retina giving us the greatest visual acuity (see below).
- The optic nerve is the "electronic cabling"
that carries information from the retinal neurons into the
- Other parts of the eye include:
- The cornea -- a protective "front window"
the lens. In last few years, can be shaped with
laser surgery to compensate for problems in shape
of lens to correct eyesight.
- The sclera is a tough flexible shell
that encases the almost spherical, jellylike mass of the eye.
The sclera includes the cornea, which is the only transparent
part of the sclera (the rest is white and opaque).
Structure of the eye from Hecht, Optics
- Aqueous humor is the fluid that fills the
corneal chamber and separates the cornea from the lens.
cornea is the only living tissue in the body without blood
vessels, it relies on the aqueous humor and tears to
- Vitreous humor is a transparent, gelatinous fluid
made of collagen (protein polymer) that gives the
eyeball strength and rigidity.
It contains microscopic particles of cellular debris
(including detached retinal cells) floating
freely about. You may have seen the shadows of these
muscae volitantes ("flying gnats"), outlined with
diffraction fringes, when squinting at a light source.
Floaters are out of focus because they are in the converging
beam of the eye, and their contrast is very low under ordinary circumstances.
(Viewing a featureless, bright background through a pinhole
brought close to the eye can make floaters more evident.)
(Left) Origin of floaters in the eye, from www.irishealth.com.
(Right) Artist's drawing of floaters from Lynch & Livingston Color
and Light in Nature.
Severe increases in the perception of these floaters, including
"spark-like" flashes, can be a serious problem indicating
marked retinal detachment.
- The choroid is a layer inside the sclerotic wall that
is richly supplied with both blood vessels and melanin, which makes it
The choroid acts to absorb stray light, e.g., from reflections off
the retina, in much the same way that we coat the interiors of
telescopes and astronomical instruments with black paint.
The "red eye" eyeshine sometimes observed in flash photographs of
humans is caused by light reflected from the choroid layer.
The hue is red because oxygenated blood absorbs light of shorter wavelengths.
(Note, children in photographs will tend to have more severe red eye than adults, because of the
relative pupil sizes of humans by age, as discusses below.)
II. THE EYE AS DETECTOR
The human eye is an incredibly versatile device with light-detecting
retinal cells (from Latin rete, meaning "net") enabling:
- a wide spectral response,
- an ability to distinguish subtle shades of color across this wide spectral range,
- an ability to adapt to an incredible range of light levels (9 to 10 orders
- the ability to detect rapid motions (approximately a 30 Hz "readout time"),
- the ability to see over a wide field of view with peripheral vision,
- and, with two eyes working together and the brain processing the nerve impulses simultaneously,
to be able to see stereoscopically, allowing
parallactic detection of distances, enabling us to sense a 3-dimensional view of our world.
The human eye is able to detect from about 390 to 780 nanometers, defining
the visual spectrum.
The range of sensitivity is widely variable from person to person, and can extend
to as blue as 310 nm and as red as 1 micron.
The UV sensitivity is set by transmission of the crystalline lens (so people
with replaced lenses actually have improved UV sensitivity).
- Retina - "Detector" of eye
- About 125 million "pixel" photoreceptor cells over (curved) focal plane.
- Like photographic film, the retina is a chemical detector that
senses light by chemical changes brought on by the absorption of a photon.
- Note that light reaches the photosensitive part of the nerve cell after passing
through the "electronic" cabling to the optic nerve.
This is much like the "backside illuminated" configuration of a CCD camera
we will see later.
- Also like a CCD, a smaller number of nerve fibers (about a million)
are multiplexed to "readout" numerous receptors.
- Two types of nerve endings: rods (2 micron diameter) and
cones (6 micron diameter).
- The vast majority of the cells on the retina.
- Panchromatic -- sensitive to wide range of wavelengths.
- But not energy/color-discriminating within this range:
Receptors translate all light to same "signal" = amount
Thus, delivers "shades of gray", like a high speed, black and white film.
- The specific chemical that makes rods active is rhodopsin,
a complex protein with a 40,000 amu atomic weight, which makes up as much
as 35% of the cell dry weight.
- Absorption curve of rhodopsin shown roughly by the above
- Absorption of the photon splits off a small, 264 amu fragment (a
chromophore) called retinaldehyde (a derivative of Vitamin A),
and instantaneously one of the double bonds changes from a cis to a
trans type bond.
Remainder of protein is called opsin.
- In a process not well understood, splitting of the protein changes the
permeability of the neuron's membrane to sodium
ions, which changes the electrical potential of the cell.
- Change in potential propagates through nerve cells to
transmit message to brain.
- Between 1 to 10 photons must be absorbed to "trigger" particular rod
(similar to photographic grains in film).
- However, many rods are bundled to a single nerve fiber, so act together.
Cross-section of retina showing how rods are bundled together.
Image from http://scienceblogs.com/startswithabang/2012/08/14/how-many-colors-are-really-in-a-rainbow/ .
- Slowly (over 30 min timescale, in the absence of bright light),
the full rhodopsin molecule is regenerated.
- Rods concentrated to outer part of retina.
- Completely missing in the 0.3 mm diameter fovea centralis,
in center of yellow patch called the macula.
Note the image of the full moon on retina is only 0.2 mm.
- Night blindness occurs when there is damage to the
outer part of the retina.
Normal vision (left and right) and night blindness (middle),
Cross-section of retina showing how rods are bundled together.
Image by Ivo Kruusamagi
obtained from http://scienceblogs.com/startswithabang/2012/08/14/how-many-colors-are-really-in-a-rainbow/ .
- About 5% of the retinal cells.
- As a group, provide sensitivity to colors.
- Probably work same way as rods, but contain slightly different (and less sensitive)
iodopsin protein with the retinaldehyde group.
- Translate color sensation to brain.
- From three different kinds we achieve color sensitivity.
Plots showing the cone sensitivity of humans. Note that it has
been suggested that a small fraction of humans (like birds and other
animals) may be endowed
with tetrachromacy -- possessing four different
types of cones (this is most likely to happen with women, who
have two X chromosones, on which two of the cone cell pigment genes
are carried) -- and therefore possibly even more ability to
distinguish different colors.
Combination of relative excitation and transmission of signals by the three
different photoreceptor cone cells is what gives the eye color sensitivity.
(Why are the red and green cones so similar in sensitivity has an evolutionary origin.
According to Lang, the first fish had four cones -- i.e., were tetrachromatic. Bony fish, reptiles and
birds retain this, but most mammals gave up the two inner wavelength cone types in the
age of the dinsosaur, when early mammals were nocturnal or spent a lot of time underground.
About 35 million years ago the primates duplicated the cone for the long wavelength
cone, and the cones created from the two genes diverged thereafter and we became
trichromats -- lang postulates that the reason for the recent evolution may have had to do with the evolutionary advantage of
being able to judge
the ripeness of fruits and discriminate leaves of different ages for our primate ancestors.)
- The numbers of red:green:blue cones is in the proportion 8:4:1, which gives
rise to the overall sensitivity curve shown above.
- Concentrated to center of retina, the macula.
- Color blindness:
- 10% of males/0.5% of females (slightly higher in European heritage)
missing one kind of cone --> "Color blind".
- Though males more affected, carried on X chromosome (mother's gene)
-- but women have two of these (passed by an affected male to grandchildren
through his daughters, but not his sons).
- Most common is deuteranopia ("red/green" color blindness)
which means that the person lacks the green-sensitive cones and cannot
distinguish red and green shades.
Test for red/green color blindness -- you should be able to
see a number in the dots shown if you are not red/green color blind.
More tests can be
- Aside for aspiring teachers and researchers giving talks/lectures/colloquia:
Keep in mind that about 10% of your audience has deuteranopia, and will have trouble
discriminating red from green points/lines on plots. So avoid using these two colors
alone as a means to discriminate features in your plots -- include variations in line
types or point shapes as well.
PHOTOPIC AND SCOTOPIC VISION
In bright light - cones mostly at work - "Photopic vision"
Rhodospin in rods already mostly split into opsin and retinaldehyde, so rods
In dark light - rods activate (stimulated by regenerated hormone
rhodopsin) - "Scotopic vision"
But cones have only 1% of the maximum sensitivity of the rods and are basically
not providing much information at low light levels.
Thus: retina is most sensitive when rods are "on" (but no color). In bright light
have sufficient photons to have color vision, and rods are turned off by
"too much" light.
Dark Adaptation - when rods fully "on".
- The human visual system is extraordinarily adaptive to a wide range of
prevailing light conditions.
- As illumination decreases, three processes occur that allow you to
continue to see:
- Pupil dilation - takes ~ 1.5 seconds
- Neural adaptation - synaptic interactions effectively "bundle"
- Like "binning" of pixels when using CCD in extreme photon starved regimes.
- NOTE (also like in CCD binning case): Results in a loss of resolution.
Daylight: ~ 1 arcmin
Night: As bad as 1° ; FOV ~ 100° - 180° .
The acuity of the human eye. Left, average resolution of
cones and rods over the retina, in lines per arc minute. To the right,
resolution in lines per arc minutes as a function of illumination
level, for the photopic (bright-light), scotopic (low-light) and
mesopic (transitional) eye modes. Maximum rod-only (scotopic) resolution is somewhat
better than 5 arcminutes (or 0.2 lines/arcmin), a fraction of the maximum
resolution of the cones. At lowest illuminations, this rod resolution can
drop to as poor as 1 degree (~0.02 lines/arcmin; see right panel). The resolution of rods is in inferior
due to their input to the eye nerve coming bundled with a number of
Cones, on the other hand, send individual inputs to the eye nerve; but with the
Airy disc for a typical 2mm photopic eye pupil diameter being ~1.6 arc
minutes, photopic vision is diffraction-limited in terms of its resolution. Figure
and parts of caption from http://www.telescope-optics.net/eye.htm.
- Takes ~ 200 msec.
- Photochemical adaptation
- Secretion/regeneration of hormone rhodopsin, which
- Takes 30 minutes for full sensitivity (most dramatic
effect in the first 3 minutes of darkness).
- Dark adaption quickly destroyed!
- Use of red filtered lights helps.
The interplay of rods and cones under different light conditions is an important aspect of understanding what we see through the eyepiece of a telescope:
- Objects that are very faint through the eyepiece will appear grey to us.
- Only compact or bright sources (like bright stars or planets) have
sufficient illumination to excite local bundles of cones into color perception.
- With ever-increasingly large telescopes, we increase light gathering power
and make objects brighter (though few people actually LOOK through a telescope
eyepiece any longer --- at least on a large telescope!). In some cases, one can see color in objects
with large telescopes that are not perceptible with small telescopes.
- An interesting phenomenon you may have experienced is the Purkinje effect.
Occurs when a scene illuminated with more or less white light drops in
illumination level and we switch from photopic to scotopic vision (mesopic vision).
Between light levels of about a quarter moon and twilight, both rods and cones can receive light
and transmit nerve impulses.
Because peak sensitivity of scotopic vision is bluer than for photopic vision,
reddish objects will appear fainter to eye than bluer objects of same
brightness at this switch from cone to rod viewing. Result is that scene
seems to take on a blue shift.
Why moonlight seems bluer than sunlight, even though moonlight is reflected sunlight.
Knowing about the Purkinje effect has played an interesting role in such
things as camouflage strategies.
In astronomy, has effect on perception of true apparent colors of faint sources
(e.g., double stars).
- Cones concentrated to fovea (center) of retina.
- Rods depleted in fovea centralis.
Foveal cones are very tiny (2.3 micron in diameter) and are packed very close
together (2.5 micron intercone spacing) in a hexagonal (honeycomb) matrix.
The image labeled "fovea" shows a cross section through the human fovea at the level of
rod/cone-shaped parts of the cells. The dense hexagonal packing is obvious.
With increasing distance from the fovea the cones become larger and are packed less
densely (shown in rightmost figure). Rod photoreceptors fill up the spaces between cones,
as illustrated in the image labeled "periphery".
Despite the high density of foveal cones, the small area of the
fovea means that only about 1% of all cones are contained in the fovea.
Image and caption from here.
- Therefore, for maximum sensitivity to weak sources, do not stare directly
at source, puts image of that object onto fovea, where there is minimum
sensitivity at low light levels.
- Instead, you should scan eye around region of interest -- use peripheral
vision to cast image of target away from fovea.
- Note, because of "blind spot" on your retina, when you scan around
with averted vision, you want to AVOID the blind spot.
Image of the retina of the eye showing optic nerve "blind spot" on
left where optic nerves collect. The highest sensitivity spot on the retina is
symmetrically opposite, near the dark spot in this image.
Image from www.croydonastro.org.uk/Vision7-mod3.pdf.
- You can prove that such a blind spot exists by printing out and
test (for a proper test, make sure the distance between the
center of the plus and the center of the box is about 6.5 cm when you
print the paper out).
Then rotate the image 180 degrees and use the other eye.
- Where is the blind spot?
- As graph above shows, on nasal side of eye, whereas their are more
rods on the temporal side of your retina.
- Clearly, BEST position for averted vision is place of OPTIMAL
sensitivity, which is OPPOSITE the blind spot.
Rotate your eye to make this happen, which means:
- Look to right of target, if using the right eye.
- Look to left of target, if using the left eye.
- The blind spot is a severe limitation if you use opposite combinations.
- Or, if too much to remember, recall that looking up or down works
- I find this technique useful for non-eyepiece viewing as well.
For example, with my naked eye I can only see all "Seven Sisters" of the
Pleiades or split Alcor and Mizar if I use averted vision.
- Well known problem to eyepiece viewers (as used to be a problem, e.g., with
manual telescope guiding.).
- Can be understood as depletion of rhodopsin in one set of cells on which a
source is fixed.
- Averting the eye to shift image to different cells alleviates problem.
- Also, eye can involuntarily lose focus. Periodically focus on nearby object,
Summary hints for observing faint objects (with or without telescope):
- dark adaptation
- averted vision
- keep moving eye around
RESOLUTION, CONTRAST AND "PIXEL" SAMPLING OF THE EYE
In principle, the resolution of the eye is set both by:
- Rayleigh limit of resolution due to diffraction (1.22 λ/d).
- So variable with pupil size.
- At maximum 5-7 mm pupil diameter,
the Rayleigh limit of the eye lens system
is about 20 arcsec in green light, which is smaller
than the minimal angular
spacing of cones in the eye.
- Near minimum pupil diameter the Rayleigh limit of the
eye is ~40 arcsec and the eye is
- "Pixel spacing" of retinal cells.
- To distinguish two point images from one another, they must be
separated by separate cone cells, and the signal must be sent
down separate nerve fibers.
- Cone spacing in the fovea is about 30" and there are one-to-one
connections to nerve fibers there. (Note from the figure on acuity of the human eye
above that that cone
spacing is larger/resolution lower away from the fovea.)
- In the large pupil regime diffraction limit resolution
projected to smaller linear size than than even thinnest, most finely
spaced cone cells (1.5-3.0 micron diameters) in the fovea centralis.
- Thus, here the retina undersamples the delivered optical resolution
of the cornea/lens.
(For example, like Hubble Space Telescope CCDs.)
- But in this case we are usually in scotopic regime anyway,
and generally not depending on the cones (except maybe for
brighter, but angularly small, astronomical sources in a
generally dark field).
- Based on the above figure and logic, effective resolution at fovea is 1-2 arcmin,
depending on diffraction versus pixel spacing limit.
- However, the eye makes up for this some by two strategies that it evolved:
- The eye will supersample the
image via rapid, 10s of arcsec oscillations of eye on order of a few Hertz,
to scan image across finest foveal cells.
(An analogy, supersampling is often done with Hubble Space Telescope CCD "image
splits", which are images "dithered" by fractional pixel shifts.)
If such eye motions did not occur, images would be both blurrier and fade out.
- The cones are actually directionally sensitive light pipes.
Thus, the cones in the retina do not receive light in, e.g., a
Because of their elongated shape and because the iodopsin is at the bottom of the
cells, they are most responsive to light coming from the center of the pupil
than the periphery (they are pointed towards the center of the pupil).
This is called the Stiles-Crawford effect, and the effect
reduces the detrimental effects
of any light scatter as well as spherical aberration on the retina.
The parts of the cell before the
cone-shaped segments are actually exquisitely engineered
waveguides (i.e., "fiber optic-like" structures) that capture light and funnel it to
the light-detecting parts of the cones.
Rods do not share this property and (fortunately) are more sensitive to all light
hitting them from any angle.
- Outside the fovea, cones are actually broader (see "periphery" image above),
while the rods are multiply
connected to single nerve fibers (as high a 1000:1), degrading resolution
seriously (to as poor as 1 deg in dark conditions -- see figure on "visual acuity of the
human eye" above).
Without fovea, eye would be seriously impaired optical system.
By the way, the above cartoon illustrating resolution on a cone-by-cone basis is
what motivates the use of the famous "Snellen letters", like the familiar "E" eye chart,
by your optometrist.
- The patient needs to resolve the E (a high contrast figure)
and identify its orientation.
- By placing the letters at a given standard distance (e.g., 20 feet),
test a person's minimum angular resolution (MAR).
- Usually described in terms of the distance at which the chart with letters
of a given size equal to the patients MAR was placed (numerator = 20 feet)
against the distance at which the same chart would yield the MAR for a normal eye
E.g., 20/20 is perfect eyesight. 20/100 is a factor of five loss in AMR.
- As we shall see, similar types of charts are used to test the resolution
of optical systems.
US Air Force Test Target from 1951 is one of the most commonly
used resolution targets.
Yet one other similarity between the eye and other detectors (like CCDs) is the possibility of
"cross-talk" between receptors.
- At high contrast, brighter regions tend to look larger than they
are (something known as irradiation), because unexcited receptors
get stimulated by nearby/connected excited receptors, or by scattering of
light within the retinal layer (e.g., you may have noticed apparent
halos around street lamps at night.)
Helmholtz' irradiation illusion. The two figures are of equal
dimensions, but the small white square looks larger than the
small black square. From http://www.jimloy.com/puzz/illusion.htm.
In a CCD a related phenomenon is known as blooming.
Tests like these (e.g., the Snellen test) actually are a relatively special case, since they
test the eye for very high contrasting regions.
But clearly the eye works in a more complex way than this, and our environment is certainly
more complex. For example, look at the following image for an illustration.
The seven circles on this image are all of the same brightness, but because
they are placed on a background with a luminance gradient, they appear to vary
in brightness. Thus, the eye+brain is actually tuning in to the contrast
of a source more than its actual luminance.
It has long been known that the primate vision system actually works by breaking down a
scene in the visual cortex of the brain into
Fourier series components.
For any given spatial frequency, there is a minimum contrast that your eye can perceive,
and this clearly varies with the frequency (e.g., you perceive lower contrasts for
the middle spatial frequencies shown but you require higher contrasts to perceive
both the lower and higher spatial frequencies shown.)
The bottom of the plot is basically the Snellen test at 100% contrast, where
the farthest right you can resolve is the equivalent acuity of the eye.
- Different cells respond to different Fourier frequencies at different orientations.
- The human brain has about 6 or 7 different Fourier frequency channels, each
carrying information regarding the phase and intensity of specific frequency ranges (low
- There are also cells that respond to temporal frequency variations.
(Top) Experiment for mapping the response of the primate cerebral cortex
to different visual scenes. (Bottom) Functional maps of the same portion of
primate visual cortex in response to different types of visual stimulation:
oriented lines (a), color (b), and monocular presentations (c).
(From "Roe AW, Friedman RM, Chen LM (2005) "Multiple Representation in
Primate SI: A View From A Window on the Brain". In Handbook of Neurochemistry
and Molecular Neurobiology: Sensory Neurochemistry,
Vol 26 (Johnson D, Lajtha A, eds). Kluwer, New York, NY; and from
Cerebral cortex maps from an experiment like that above
showing activation of different cells for light/dark
gratings of different orientations and directions of motion. Details in
Roe AW, Fristches K, Pettigrew JD (2005) "Optical imaging of
functional organization in V1 and V2 of marmoset visual cortex", Anat Rec, in press.
- In describing the sensitivity of an optical system (like the human vision system, a camera, a photocopier, etc.)
it is useful to define quantitatively a contrast parameter.
Since we are thinking of Fourier analysis
here, think of an extended source for which the
intensity of the source varies sinusoidally with some spatial period, P or
frequency, f, as shown below:
Definition of the contrast or modulation (image from Schroeder)..
The contrast can be defined as:
C = (Imax - Imin) / (Imax + Imin)
This is also called the Michelson contrast, the modulation, the
visibility, or fringe visibility
because of its equivalent definition in the study of fringes that are created
Visibility is similarly defined in double-slit interference
(e.g., a two aperture interferometric telescope).
However, now the max and min vary across the interference pattern.
The example shows a visibility at the peak
of 80% (i.e. 0.8). From Wikipedia.
- It is common to test optical systems by imaging through them various
optical gratings, like the one below:
- The following image tests of the the resolution of the eye for a grating whereby the spatial
frequency is increasing from left to right and the contrast is increasing from
top to bottom.
We may plot the overall response of the eye+brain contrast sensitivity as follows:
Here the ordinate axis is actually 1/(limiting detectable contrast), formerly called the
Another way to characterize the resolution of the eye (or any optical system, as we shall
see) is via the modulation transfer function (MTF).
The experiments above whereby selective activation of the marmoset/monkey cortex has been
imaged prove that there are cells selective to orientation.
- We define the modulation transfer function (MTF) of the system
as (see discussion in Chapter 11.1 of Schroeder):
MTF(f) = Cimage(f) / Csource(f)
where Cimage(f) is the contrast of the image and
Csource(f) is the contrast of the source at the spatial frequency, f.
- The MTF is a measure of the performance of an optical system as
a function of spatial frequency.
As we shall see, the advantage of this description is that we can think of
individual elements of an optical train as operators, each with its own (complex)
transfer function (OTF), that each act on a wavefront
(whose characteristics can be described as a
of spatial or angular frequencies by way of wave equations)
individually (adding distortions and aberrations),
but the product of all of the OTFs gives
the net OTF for the system.
The MTF is the modulus, or absolute value,
of the OTF.
- The following is an example of the degradation of the contrast
in an image of a series of bars by the process whereby the image is
photographed by a camera, then digitized by a scanner, and then
passed through a sharpening filter.
Each of these steps modulates the image by its own OTF and the
cumulative effect of passing the image through each OTF is apparent:
The MTF is a function in Fourier space, and so is generally
shown as a function of frequency. For example, the following
image shows how the process above degrades the image so that
information at high frequencies is lost:
The response of the camera + film to a sine and bar pattern.
The red curve is the spatial response to the film+lens.
The blue dashed curve is the MTF of the lens.
The solid blue curve is the combined film + lens.
- The following shows the MTF of the eye. Note that unlike
the previous images, the abscissa is on a linear, not logarithmic,
scale (but showing log frequency is more common for MTFs).
The MTF of the human eye as a function of pupil diameter.
The different curves correspond to different pupil diameters.
A careful perusal of the MTFs for the different pupil diameters
(and their Fourier transforms-->
-- the LSFs -- shown below)
seems to show a trend opposite one would expect in that larger
apertures yield lower contrast ratios at high frequencies; the
reasons for this are discussed below.
Several things to notice from the MTFs and the Contrast Sensitivity Function for the eye shown earlier:
- The ~1 arcmin limit of human eye resolution is shown by the sudden
drop in the CSF and the MTF.
Such drops are common in optical systems and are known as
the frequency cutoff.
- What is one absolute source of a frequency cutoff in any
- Note that (as we have discussed already) the
MTF varies with pupil diameter -- but a careful inspection
of the above curves shows that the way this varies is
somewhat contrary to theoretical expectation!
(Note that the larger apertures in this plot have lower MTF!
See below for reason).
- As it says above, the
Fourier transform of the MTF gives the
spread function of the system (the line spread function in the
present 1-dimensional case or the point spread function in the
case of a 2-dimensional image).
The image below shows that the theoretical LSF (shown by
the inner curves below) is correlated to the aperture of the
Obviously, the same thing theoretically should happen with the aperture of a
telescope (but in two dimensions -- giving the Airy function).
The LSF of the human eye as a function of pupil diameter.
- However, as the above image also shows, there is
more going on to degrade the contrast in the eye than
just diffraction (and the degradation is worse for larger pupils).
The actual LSF is the outer
of the curves shown; the theoretical diffraction limit
is not achieved because of e.g., spherical aberration, chromatic
aberration, variable "pixel" spacing, the fact that larger pupils
occur under dimmer lighting conditions, etc., which are problems increased
with larger pupil sizes.
- But things are even more subtle than this, as shown by the figure
below and our discussion above about rapid eye movement.
The MTF of the CCD falls off more rapidly than the eye because:
- The CCD has regular spacing of pixels whereas the eye
pixels are randomly placed, and more concentrated in the fovea
for increased acuity.
- The eye can extract high resolution information by
its rapid sampling and processing of "dithered" images.
A comparison of the MTF of the eye versus an optical CCD camera.
The CCD MTF has been normalized to the eye at MTF=0.5.
- If one stares at a sinusoidal grating of a particular orientation and
frequency, the associated cells in the cortex begin to fatigue and lose
responsivity. This is called "adaptation" (different than -- but not completely unrelated --
to the kind of situation as the dark adaptation discussed above).
- Can see this by the test below. Stare at either the left or right gratings
for a minute and then the vertical grating in the center. The opposite slant
perception to the vertical grating is due to weakening sensitivity of the cortical cells
that have adapted.
III. THE EYE AS AN OPTICAL SYSTEM
REFRACTIVE PROPERTIES OF THE EYE LENS SYSTEM
- The net cornea/lens is effectively
a double positive (convex) shaped optic that (in principle) produces
a focus at the distance of the retinal cells, which are at a fixed distance.
- The refraction of light rays, of course, is dictated by Snell's Law:
n1 sin(θ1) = n2 sin(θ2)
- The cornea actually is the first and strongest convex element of the
optical system of the eye.
- Most of the bending (73%) imposed on the incoming rays takes place at the air-cornea
- The index of refraction of the cornea is ncornea = 1.376.
- One reason we do not see well under water is that this index of refraction is
close to that of water (nwater = 1.33).
- Because the cornea is slightly flattened, it helps reduce spherical aberration.
- After exiting the cornea, light rays pass through the clear, watery
aqueous humor, which is a pool of nourishing fluid for the anterior of the eye.
- Little refraction occurs here because of the similarity of index of
the index of refraction (na.h. = 1.336).
- The iris lives in the aqueous humor, and serves as the system aperture stop.
- Next light rays enter the lens, or, more technically, the crystalline lens.
- The lens is an onion-layered (about 22,000 layers), fibrous mass
about 9 mm in diameter and 4 mm thick and surrounded by a flexible membrane.
- The layered nature of the eye means that light rays actually follow paths
that include tiny, discontinuous changes.
- To aid in subtle light-bending, the crystalline lens is actually a device
called a GRIN lens, which stands for GRadient in the INdex of refraction.
It is a radial GRIN because the index of refraction varies radially
symmetrically from 1.406 near the optical axis to 1.386 near the edge.
- The effect of the GRIN is to increase the approximation of the refracted
wavefront to one that might come from a parabolic optical surface.
- The lens/cornea combination can be treated as a simple double-element lens with an
object focus 15.6 mm anterior to front corneal surface and an image focus
24.3 mm behind it on the retina.
- Recall the meaning of object focus and image focus.
The definition of object focus, Fo (top), and
image focus, Fi (bottom).
- For simplicity, one can think of the combined lens as a single lens sitting with an
optical center 17.1 mm in front of the retina, which falls just
posterior to the crystalline lens. This is the effective focal length of the
lens (e.g., for purposes of using the thin lens formula).
- We can approximate many of the optical properties of the eye using the well
known thin lens formula:
where the definitions of the variables are as given in the figure.
- Recall that for two thin lenses in contact
1 / f = 1 / f1 + 1 / f2
- It is customary when talking about the optics of the eye (and eyeglasses, as well
as for lenses in general)
to talk about the dioptric power, D, of a lens, where
D = 1 / f
- The unit of power is the diopter, with units of m-1.
E.g., a converging lens with a focal length of 0.1 m has a power of 10 diopters.
A lens that refracts light strongly then has a small f and is of high
- The advantage of working with powers defined in this way is that
we can coadd powers of individual thin lenses in succession:
D = D1 + D2
As we shall see, this is a useful way to assess the net power of an optical system.
- The total power of the unaccommodated eye is +58.6 diopters.
Of this, about 43 diopters is provided from the cornea.
The crystalline lens of the eye surrounded by air has a power of only
D = +19 diopters.
ABERRATIONS AND CORRECTIVE OPTICS
Just like glass lenses, the human lens has some familiar
- Spherical aberration.
- Because of the broad acceptance angle for light into the eye, it is
far from the paraxial approximation, which means that
spherical aberration would ordinarily be a problem without other
mitigating properties of the eye.
- Despite the corneal flattening and GRIN properties of the crystalline lens,
both which help to reduce the amount of spherical aberration,
some residual spherical aberration can remain, depending on the amount of
- At night, when your pupil is dilated the outer parts of your
eye contribute more spherical aberration.
- This and chromatic aberration generally cause problems only in certain
circumstances and are compensated somewhat by the neurological processing.
- Chromatic aberration.
The eye is more powerful in blue than in red light.
- You can demonstrate this by peering through a pinhole at a bright source
and then moving the pinhole from center up while looking at, say, a horizontal
gap in a Venetian blind. The top edge of the gap should appear tinged blue
and bottom tinged red (why is it opposite of expectation?).
- Another is to look at a purple dot straight on, and moving the dot farther and
closer to your eye. Depending on the distance, the center of the dot will take
on a blue or red tinge and there will be a "halo" of the opposite color around the
- The longitudinal chromatic aberration is the difference in
the distance of image focal
points for a source at infinity, and is about 2 or 2.5 diopters over the full
About 75% of this is from the cornea and 25% from the crystalline lens.
Because of changes to the lens of the eye with age, the longitudinal
chromatic aberration is actually much smaller in older people.
- What does the eye do about this? Which color to focus on?
The eye optimizes its depth of focus:
- For a source at distance, the eye accommodates to place the
red light in focus.
In this way, closer blue images will also be in focus.
But if blue images at a distance were put into focus, then there
would be no distance at which red rays could be in focus.
- In contrast, for a near source, the blue rays are put in focus
and red rays from more distant sources can be put into focus.
- Because the size of the image on the retina is driven by the focal length,
whereas the focal length is varying by wavelength, the size of the image
projected onto the eye varies by wavelength.
- This variation of image size on the nominal focal plane
(in this case the retina) is known as transverse chromatic
- Ametropic Eyes
Many common eye problems relate to deviations from the nominal focal properties
of the lens+cornea, but can
be corrected with eyeglasses, contact lenses, or laser surgery on the
- Myopia, or near-sightedness, means that the true
point of the lens happens in front of the retina.
- Hyperomia , or far-sightedness, means that the true
focal point of the lens is behind the retina.
- In (refractive*) astigmatism, a lens (in the eye, or any lens) is
not radially symmetric, and light hitting different meridians on
the lens can have different foci, leading to blurriness.
(*In this case, astigmatism is happening due to an improper shape of the
eye and the aberration occurs even for rays entering
the system on-axis. In future lectures we will talk about astigmatism
that occurs even for symmetric lenses due to off-axis rays -- this is called
oblique or marginal astigmatism.)
- In presbyopia, the amount of lens accommodation
(squeezing) is limited, so that it is hard to focus on near
- Myopia (Nearsightedness)
(This discussion follows that in Hecht; this is also the source of most of the associated figures.)
The far point of an eye is the object distance (s) that
puts the image of that point (s' ) on the retina.
For the normal (emmetropic), unaccommodated eye, the far point
- In myopia, parallel rays brought to focus before retina.
The power of the eye lens is too large.
And the far point is closer than infinity, and everything beyond the
far point is blurry.
From Optics and Vision, by Pedrotti & Pedrotti, c.1998, Prentice-Hall, Inc.
The net range of vision is reduced.
- To correct, we need to place a negative or double concave
lens in front of the eye, as shown below.
The negative lens (which is a lens having negative power) acts to refract light from a distant source in such a
way that the virtual image is brought to the far point.
(Don't be confused by the overall curvature of the lens in the figure: This
is due to the fact that eyeglass lenses are meniscus shaped so that
the turning eyeball can look off in different directions and not experience
distortion. See figure below for a "straightened" version of the same situation.
Before the 1800s glasses were not meniscus-shaped, but were only
useful when looked through their centers.)
- From the thin-lens formula, if the far point distance is X, then
for the eyeglass lens
1 / f = 1 / s + 1 / s' = 1 / infinity + 1 / (-X)
The focal length of the eyeglass lens is -X, which is the far point
of the eye.
- The above technically corresponds to a contact lens. Spectacles are
placed at about the 17 mm distance from the optical center of the eye
in order to minimize magnification effects over what the normal eye
- Hyperomia (Farsightedness)
- Here the eyeball has changed shape so that the lens is too close to the retina
and the image plane is behind it.
The hyperomic eye can and has to accommodate to see distant objects clearly (e.g.,
it can change the radius of curvature of the lens from what is shown in panel a below),
but it can't accommodate enough for objects closer than the near point.
Thus the near point of the hyperomic eye is farther than in the normal
eye, so close vision is poor.
- We need to aid the eye with more power to bring the rays to focus
- Thus we need a positive power, convex lens system, which takes rays
closer than the near point and make them seem like they came from farther away.
Correction of the far-sighted eye, according to Hecht. Note the variation in the shape
of the crystalline lens as a result of accommodation in the various situations depicted.
Say the near point of this eye is a poor 125 cm. A virtual image must be
made at this point for anything closer (say, 25 cm). Then s' = -125 cm
1/f = 1/(-1.25) + 1/(0.25) = 1/0.31
and the power of the lens needed is 3.2 diopters.
- Note the difference in use of negative versus positive powered lenses in these two
You may have already noticed that while your grandparent's glasses can form images
and be used as a magnifying glass, the glasses of the near-sighted (typically more
common in young people) cannot do this.
- Most common eye defect -- azimuthal asymmetry of cornea.
- Different meridianal planes (planes containing the optical axis)
have different powers.
- A test for astigmatism is to view this figure with the unaided eye. You have
any set of lines appears bolder, or, when the figure is moved to different distances
from the eye different sets of lines come into focus at different distances.
- If the meridianal plane of maximal power is perpendicular to the plane of minimal
meridianal power, the problem can be fixed relatively easily using sphero-cylindrical
- Correction of two perpendicular meridians can use anamorphic surfaces:
- Anamorphic lenses are also used for projecting wide screen movies, which, to fit
on film needs to be compressed in the horizontal dimension. Projecting with an
anamorphic lens of opposite sense undistorts the image.
- NOTE FOR EYEPIECE VIEWING: Should you remove your eyeglasses when looking
through a telescope eyepiece?
- If you are only near-sighted or far-sighted, you don't need to
use your glasses if you can adjust the focus of the telescope.
- BUT: You cannot fix astigmatism with the telescope focus! In
this case, if you are astigmatic, it is best to use your eyeglasses
to look through the eyepiece.
- Another advantage of encouraging observers to leave their eyeglasses on
is that if everyone uses their own glasses to correct their defective
vision, then everyone will focus the telescope the same way and this
makes life easier and more efficient for everyone (a helpful hint
for "public nights") -- as long as the focus is set to that for an emmotropic eye.
- If you set up the focus for an emmotropic eye and you wear glasses, make sure
you are using the correct parts of your glasses if they are bifocals or
gradual focus lenses.
Unless otherwise attributed,
pictures of the eye either copyright © 1997 Ed Scott and © 2000,
2001 photo.net, or © The IESNA Lighting Handbook, 9th Edition.
Other material copyright © 2005, 2007, 2009, 2011, 2013, 2015 Steven R. Majewski. All rights
reserved. These notes are intended for the private,
noncommercial use of students enrolled in Astronomy 5110 at the
University of Virginia.