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ASTR 3130, Majewski [SPRING 2015]. Lecture Notes

ASTR 3130 (Majewski) Lecture Notes


EXAMPLE OF AN IMAGING + DETECTOR ASTRONOMICAL INSTRUMENT: THE HUMAN EYE
(and tips on telescope eyepiece viewing)

Some references:

  • Chapter 6 of Birney (pp. 145-149).

  • Section 18.1-18.4 of Roy & Clarke.

  • A more extended and in-depth version of these notes can be found here.

A large part of this course is concerned with the tools of the observational astronomer, which include both optics to redirect and focus light, and detectors to record the light.

For most of the history of astronomy, the human eye represented the most important astronomical tool. It served as both 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.


Picture at right from The Fire Within the Eye, by David Park.


I. PARTS OF THE EYE


The basic parts of the human eye are shown in this figure:

The eye has the following important elements, with familiar functions discussed in previous lecture notes:

  • 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 (f/ratio) 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 circular), that, 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 iris/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 the iris can change the focal length as well as the clear aperture of the eye. This is called "accommodation".

    • "Pupil" comes from the Greek word for 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 brain.


  • Other parts of the eye include:

    • The cornea -- a protective "front window" covering 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. Because the cornea is the only living tissue in the body without blood vessels, it relies on the aqueous humor and tears to receive nutrients.

    • 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 to the eye can make them 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 very dark.

      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.


II. THE EYE AS DETECTOR


SPECTRAL SENSITIVITY

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 of magnitude),

  • 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).



RETINAL CELLS

  • 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 "read out" numerous receptors (pixels) -- something like the binning and MUX properties of the CCD we will see later.

    • Two types of nerve endings: rods (2 micron diameter) and cones (6 micron diameter).

      1. Rods

        • 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 of light.

          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 as a function of wavelength shown roughly by the above curve.

          • 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" a 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), the full rhodopsin molecule is regenerated.


        • Rods are concentrated to the 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), from http://www.retina-international.org/nightbld.htm.
      2. Cones

        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 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.

        • The numbers of blue:green:red cones is in the proportion 1:4:8, 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 found here.


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 not sensitive.

  • 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:

      1. Pupil dilation - takes ~ 1.5 seconds

      2. Neural adaptation - synaptic interactions effectively "bundle" photoreceptors together.

        • 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 resolution is somewhat over 5 arcminutes (or 0.2 lines/arcmin), a fraction of the maximum resolution of the cones. The resolution of rods is in inferior due to their input to the eye nerve coming bundled with a number of rod photoreceptors. Cones, on the other side, send individual inputs to the eye nerve; with the Airy disc for a typical 2mm photopic eye pupil diameter being ~1.6 arc minutes, photopic vision is diffraction-limited resolution. Figure and caption from http://www.telescope-optics.net/eye.htm.
        • Takes ~ 200 msec.

      3. Photochemical adaptation

        • Secretion/regeneration of hormone rhodopsin, which stimulates rods.

        • 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-increasing telescopes, we increase light gathering power and make objects brighter (though few people actually LOOK through a telescope eyepiece any longer!). 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.

      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).


  • Averted Vision


  • Eye Fatigue

    • 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. To overcome this, periodically focus on nearby object, e.g.,without telescope.


  • Summary hints for observing faint objects (with or without telescope):
    • Dark adaptation

    • Averted vision

    • Keep moving eye around



    RESOLUTION AND CONTRAST 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.

      • Near minimum diameter the Rayleigh limit is ~40 arcsec and the eye is actually diffraction-limited.

      From www.brad.ac.uk/acad/lifesci/optometry/resources/modules/stage1/pvp1/Acuity.html

    • "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 is 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, eye makes up for this some by two strategies that it evolved:

      1. 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.

      2. The cones are actually directionally sensitive light pipes.

        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 reduces the detrimental effects of any light scatter as well as spherical aberration on the retina.

        Rods do not share this property and (fortunately) are more sensitive to all light hitting them.

      From www.brad.ac.uk/acad/lifesci/optometry/resources/modules/stage1/pvp1/Acuity.html

    • Outside the fovea, cones are actually broader, 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 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 (denominator).

      E.g., 20/20 is perfect eyesight. 20/100 is a factor of five loss in AMR.

      From www.brad.ac.uk/acad/lifesci/optometry/resources/modules/stage1/pvp1/Acuity.html

    • 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.


    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 imposed on the incoming rays takes place at the air-cornea interface.

      • 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 rates 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 sitting with an optical center 17.1 mm in front of the retina, which falls just posterior to the crystalline lens.

    • 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



    ABERRATIONS AND CORRECTIVE OPTICS

    Just like glass lenses, the human lens has some familiar aberrations.

    • Spherical aberration.

      • Because of the broad acceptance angle for light into the eye, 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 helps to reduce the amount of spherical aberration, some residual spherical aberration can remain, depending on the amount of accommodation.

      • At night, when your pupil is dilated the outer parts of your eye contribute more spherical aberration.

      • This and chromatic aberration often 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 their will be a "halo" of the opposite color around the dot.

      • 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.

          From www.brad.ac.uk/acad/lifesci/optometry/resources/modules/stage1/pvp1/ChromAb.html

        • 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.

      • These differences can lead to some interesting optical illusions:

        Chromostereopsis is a visual illusion whereby the impression of depth is conveyed in two-dimensional color images having two widely different wavelength colors. Such illusions are believed to be due to the effects of chromatic aberration in the eye. From reed.cs.depaul.edu/peterh/class/hci450/imgs/chromo2.png.

        From http://www.visionexcellence.com.au/chromostereopsis/ .

      • 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 aberration.

        From www.brad.ac.uk/acad/lifesci/optometry/resources/modules/stage1/pvp1/ChromAb.html

    • 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 cornea:

      • Myopia, or near-sightedness, means that the true focal 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 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 presbyopia, the amount of lens accommodation (squeezing) is limited, so that it is hard to focus on near things.

    • Astigmatism

      • 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 astigmatism if 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 lenses.

      • 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.



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    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 © 2006,2008,2012,2015 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 313 and Astronomy 3130 at the University of Virginia.