Optics and the Physics of Light

Photosensitivity

• Photosensitivity, for our purposes, refers to sensitivity to light whose wavelength falls in the range of 400-700 nm.

• Upon absorption, which we will generally treat as the first step towards vision, light behaves as though it were “quantal” (consisting of discrete photons). – Sir Isaac Newton advanced the theory that light was a stream of

particles in 1704!

• The number of photons that a source emits (or reflects) is closely tied to its apparent brightness, but we shall see that the relationship between brightness and number of photons is not a simple one!

Characteristics of Light

• As it travels, it actually shows wave characteristics (James Clerk Maxwell, 1873), and we characterize it by its wavelength (distance between peaks of the wave as it goes through one cycle).

• Wavelength is the principle determinant of the hue of light, which is the perceptual characteristic most like what people mean by the term “color.”

• 1 nm is a billionth of a meter.

• In normal eyes, 400 nm is seen as violet, 500 nm as blue-green, 600 nm as yellow, and 700 nm is seen as red.

• There is nothing inherently colored (hued) about wavelengths—hue is entirely extracted by visual processing.

• The perception of color is not a simple function of wavelength (more later).

Gross Anatomy of the Eye

Aqueous Humor

Vitreous Humor

Blind spot

The gross anatomy of the eye is such that the retina covers more than 270o of arc along the back surface.

Each eye is a spherical structure that is 20-25 mm in diameter.

The cornea is a bulge in the sclera that is approximately 13 mm in diameter.

• Behind the cornea is the aqueous humor, a fluid similar to cerebro-spinal fluid.

• The iris is the colored membrane that surrounds the central hole (the pupil).– The color ranges from blue through black, and is controlled

genetically by mechanisms similar to the ones controlling skin pigmentation.

– The size of the pupil is controlled by the light reflex, constricting in bright light, dilating in dim light.

• It can vary in diameter by about a factor of 16, decreasing in size as illumination increases (negative feedback).

• This has little to do with adaptation to light levels.– Its major purpose is to decrease the size so that the optical imperfections near the

edge of the lens and cornea do not distort images (the center of the cornea and lens are more refractive).

– In dim light, visual resolution is poorer due to optical imperfections at the edges of the crystalline lens.

– As such, it is in the best interest of the visual system to minimize pupil size.

– Pupil size is also determined by level of interest and arousal, becoming larger under conditions of great interest.

iris

The Cornea• The cornea has a radius of curvature of about 8 mm,

and is the area where light enters the eye.• The cornea is the first optical element of the eye,

serving as a simple lens that begins to gather light rays and focus them on the other side (towards the retina).– Because it bulges forward, it actually allows

reception of light from a region slightly behind the observer.

• Most of the refraction takes place at the surface of the cornea (43 diopters out of a potential total power of 59 diopters for the combined cornea and lens).

• This is evident from the great loss of power when the eye is immersed in water when swimming. – Since water and aqueous humour are so similar,

refraction does not occur at the interface of the environment (water) and the cornea.

– The liquid aqueous humour nourishes the cornea and lens, which have no blood supply.

• Light then passes through the aqueous humour to the lens.• The light then travels through the jellylike vitreous

humour, which has an index of refraction close to that of water, to the retina at the back of the eye.

• Most vertebrate eyes contain a crystalline lens located behind the pupillary aperture.– Its curvature is controlled by the ciliary

muscles, which cause it to bulge when contracted, increasing the strength of the lens.

– The process of varying the refractive index of the lens is referred to as accommodation, and is crucial for objects occurring at different distances (so that light is focused onto the retina).

Visual Optics

• Because directional information is carried by which points on the retina are stimulated, the accuracy of that directional information is limited by the extent to which two independent points are focused as two separate points on the retina.

• Optical systems operate by bending light beams because of the differences in composition of the air and the refracting medium.– The refraction of light is due to a reduction in velocity as

the light passes through the lens (can be reduced by up to a factor of three).

– While the velocity of light in a vacuum is not dependent upon wavelength (186,000 miles per second), the reduction in velocity as the medium changes is somewhat wavelength‑dependent (chromatic aberration).

• It is the property of all lenses that they take light emanating from a point source, refract the beams, and form a single point in the image plane.

• If we go to cases of multiple points it is easy to see that the image is reversed on the image plane in comparison to the object plane.

• This is best seen by looking only at rays passing through the optical center within the lens:

S1

S2

S3 I1

I2

I3

Object Plane Image Plane

do di

O

O

OA=Optical Axis; rays passing through it are not refracted

As the object moves closer to the lens, the image moves further away from the lens.

dido

OA

22 mm

The Crystalline Lens• The problem for the visual system is making sure that the image

plane corresponds to the retina, which is fixed at a distance of 22 mm behind the cornea.

• As the object is moved further away, the image is formed closer to the lens; as the object is moved closer, the image is formed further from the lens (the index of refraction must increase).

• The lens of the eye is is a gelatinous capsule that is held in place behind the pupil by a complex web of suspensory ligaments known as the Zonules of Zinn.

• The ligaments also are attached to the circular ciliary muscle. – When viewing an object at a distance, the ciliary muscle relaxes. This causes the

Zonules of Zinn to become taut, which makes the lens flatter and thinner. – When a near object is viewed, the ciliary muscle contracts. This causes the

Zonules of Zinn to slacken, which enables the lens to return to a more convex or globular orientation.

• This is somewhat counter-intuitive. Usually, we associate muscle contraction with parts of our bodies becoming taut and hard. – However, in lens accommodation, the Zonules of Zinn slacken when the ciliary

muscles contract.

• The reason this happens is that the ciliary muscle is circular.– When it is relaxed, the diameter of the circle is at its maximum, which means that

the Zonules of Zinn are pulled tight. – Conversely, when the ciliary muscle contracts, the diameter of the circle becomes

smaller and the Zonules of Zinn relax.– Thus, the ciliary muscles are responsible for the lens accommodation

response, controling the tension on the zonules which are attached to the elastic lens capsule at one end and anchored to the ciliary body at the other end.

– They function as a sphincter, easing tension on the zonules when contracted (allowing the lens to bulge, increasing refraction) and making the zonules taut when relaxed (flattening the lens, decreasing refraction).

Zonules of Zinn

Object moves further away

Object moves closer

• As such the lens is most refractive in a when the zonules are in a relaxed state (with the lens varying from 2-16 diopters).

• So, focusing at points closer than optical infinity necessitates relaxation of the lens (allowing it to bulge) because objects brought closer are focused at a distance further away from the lens (see the expressions for focal lengths); without this adjustment objects would be focused behind the retina.

Retina

• Our ability to accommodate declines with age, starting from about 8, and we lose about 1 diopter of accommodation every 5 years up to age 30 (and even more after age 30).

• By the time most people are between 40 and 50 years old, they find that their arms are too short because they can no longer easily accommodate the 2.5 diopters or so needed to see clearly at 40 cm. – This condition is called presbyopia (meaning “old

sight”) because the lens becomes sclerotic (harder) and the capsule that encircles the lens (enabling it to change shape) loses its elasticity.

• The optical center of the combined cornea-lens is about 17 mm in front of the retina when the lens is relaxed (focus in the far-field) and 14 mm when the refractive index is greatest (focus in the near-field).

• The focal length (F) of a lens is formally defined as, where do=distance

to object and di=distance to image.

• One could measure the focal length by going to infinity and projecting an object, at which point the image would be formed at the focal length from the lens.

1 1 1

F d do i

Measuring the Focal Length

• A somewhat more reasonable strategy is to employ collimated light as the source.

• Collimated light sources emit parallel rays of light, which is would would happen if the object were at infinity.

do di=F

• For humans with a perfect optical apparatus, the focal distance equals the distance between the retina and the optical center (a combination of the lens and cornea).

• As was stated earlier, the optical system faces the difficulty of having a retina that is only about 22 mm away, so it must refract light strongly.

• Strength of lenses is typically measured in diopters, where

• Stronger lenses have shorter focal lengths and more diopters.

11

d iop terF oca l L eng th in m eters

( )

• The cornea is the most refractive element, providing about 43 diopters (1/43 = .02325 M = 23.25 mm), bringing the image to focus just behind the retina if there were no lens.

• The minimum strength of the crystalline lens is 2 diopters, yielding a total of 45.

• Since 45 M-1=22.22 mm, this puts the image on the retina for objects at optical infinity.

• As objects move closer the ciliary muscles contract allowing more slack on the Zonules of Zinn, causing the lens to bulge, producing a greater refractive index (more diopters).

Distance to Retina0.0222 M

Cornea Only= 43Minimum Lens 2.045045

Distance to Object (M) Total Diopters Lens Only

10 45.1450 2.14502 45.5450 2.54501 46.0450 3.0450

0.5 47.0450 4.04500.25 49.0450 6.04500.1 55.0450 12.0450

0.07 59.3308 16.3308

equation slensmaker' the;111

io ddF

The closest object a young person can accommodate…..

• To focus a spot of light at optical infinity on the retina, the refractive power of the optical components of the eye must be perfectly matched to the length of the eyeball. – This perfect match, known as emmetropia.

• Refractive errors occur when the eyeball is too long or too short relative to the power of the optical components.

• If the eyeball is too long for the optics (b), the image of a point will be focused in front of the retina, and the spot will thus be seen as a blur rather than a spot of light. – This condition is called myopia (or “nearsightedness”). – Myopia can be corrected with negative (minus) lenses, which diverge the rays of starlight before

they enter the eye (c). • If the eyeball is too short for the optics (d), the image of a point will be focused behind the

retina—a condition called hyperopia (or “farsightedness”). – If the hyperopia is not too severe, a young hyperope can compensate by accommodating, and

thereby increasing the power of the eye. – If accommodation fails to correct the hyperopia, the spot’s image will again be blurred. – Hyperopia can be corrected with positive (plus) lenses, which converge the rays of starlight before

they enter the eye.

• When the cornea is not spherical, the result is astigmatism.

• With astigmatism, vertical lines might be focused slightly in front of the retina, while horizontal lines are focused slightly behind it (or vice versa).

• If you have a reasonable degree of uncorrected astigmatism, one or more of the lines in above figure might appear to be out of focus, while other lines appear sharp.

• Lenses that have two focal points (that is, lenses that provide different amounts of focusing power in the horizontal and vertical planes) can correct astigmatism.

Measuring the Quality of Lenses

• Optical quality is measured by the point spread function of the eye or by the spatial modulation transfer function (spatial MTF).

• Because of various optical aberrations, the distribution of light from a point source will be distributed over a Gaussian (bell shaped) region.

• The light from two nearby points in space will not be two punctate spots on the retina but will instead be two Normal distributions that overlap to various extents, forming a single-peaked distribution in image space when points in object space get close together.

Point-spread functions

• Measurements of point-spread functions were carried out by Westheimer and Campbell, who looked at the retina with an apparatus that works on the same principle as an ophthalmoscope, whereby the amount of light reflected from the retina is recorded with a photocell out of the line of projection from the light source.

Light that enters the subjects eye is brought into focus by the lens. A small portion of this light is reflected back out and passes through the optics of the eye for a second time. On the return path, the beam splitter, which is nothing more than a lightly silvered mirror, reflects some of the light and transmits the rest.

A portion of the reflected light is brought to focus on one side of the apparatus. Using a very fine slit in the measurement plane with a photodetector behind it, the reflected light can be used to infer the shape of the retinal image.

To the left are measurements taken by Campbell and Gubisch as a function of the pupil diameter (given in mm). First (and most importantly), note the increased width as the pupil diameter increases—when the pupil is wide open, the image is blurred than when the pupil is relatively constricted. Notice that the measurements taken when the pupil diameter is large are less noisy—when the pupil is wide open, more light enters and is reflected, improving the quality of the measurements. Note that the optics of the eye have twice distorted the distribution of light, so some correction needs to be carried out for the image on the back of the retina to be inferred.

Spatial Modulation Transfer Functons

• A mathematically equivalent and, in many ways, more attractive approach to measuring the optical degradations of the eye is to measure the Spatial Modulation Transfer Function (MTF).– An MTF is a measure of the extent to which different

spatial frequencies are passed by a given optical system.

– For ordinary optical systems MTFs can be measured directly by imaging spatial frequencies of some known contrast, then measuring the contrast of the image with a scanning photometer.

Column 1: The grating at the bottom is half the amplitude/contrast of the grating at the top.Column 2: The grating at the bottom is twice the spatial frequency of the grating at the top.Column 3: The grating at the bottom is 90 degrees out of phase with respect to the grating at the top.

C ontrastL L

L L

m ax m in

m ax m in

C ontrast

7 0 1 0

7 0 1 00 7 5.

C ontrast

5 0 3 0

5 0 3 00 2 5.

– If the optical system transmits perfectly, the contrast of the image will be the same as the contrast of the stimulus grating.

– When this has been done for the human visual system it has been deduced that it passes no spatial frequencies in excess of 60 c/deg. (Campbell and Gubisch, 1966).

– One approach measuring the MTF for the whole system is to employ a psychophysical task in which the participant adjusts the contrast of a grading until it can just be discriminated from a uniform field with the same average luminance.

C ontrastL L

L L

m ax m in

m ax m in

• The inverse of contrast is plotted, so the function is the contrast sensitivity function (CSF).

Optics alone

Comparing the Contrast of the Image and the Contrast of the Stimulus

• One varies the spatial frequency for gratings of fixed contrast, comparing the contrast in the image with the contrast in the stimulus (forming a ratio).

• Perfect optical systems will yield the same contrast for the image as is in the stimulus at all spatial frequencies—ours begins to attenuate contrast at about 5-10 c/degree, yielding no image contrast by 60 c/degree.

1.0

0.1

0.01

Con

tras

t Imag

e/Con

tras

t Sti

mul

us

No loss Attenuation of contrast

• Note that 60 c/degree corresponds to a cycle width of 0.01667 degrees, a span of appoximately 3 foveal cones (spaced at .008 degrees center to center).

• We need two receptors to encode a cycle (the aliasing problem—see page 58 of the text), so it appears that the upper cutoff is consistent with the spacing of the cones.

• The optics happens to match the limits set by cone spacing-it makes not sense to have an optical system that would exceed the quality limit set by the receptors (and vice-versa).

Exploratory DemonstrationTry Varying the Spatial Frequency (cycles per stimulus) and Contrast

of the Sample Sine-Wave Grating Presented Below

http://www.usd.edu/psyc301/CSFIntro.htm

• Each rod outer segment contains the photopigment rhodopsin, while there are three different conopsins.

• Visual pigment molecules consist of a protein (an opsin), the structure of which determines which wavelengths of light they absorb, and a chromophore, which captures light photons.

• The pigment rhodopson is found in the rods, concentrated mainly in the stack of membranous discs in the outer segment.

• Each cone has one of the other three pigments (photopsins I, II, and III)—which respond to long, medium, and short wavelengths, respectively.– All have the same 11-cis retinal as is found in

rhodopsin, but they have different opsins.• In rod cells, the protein which binds to the

chromophore retinal is opsin, and the bound complex of 11-cis-retinal plus opsin is known as rhodopsin, or visual purple.– Each photopigment molecule consists of two

components—a large protein opsin and the chromophore retinal (derived from vitamin A).

• In mammalian cones, there are three different conopsins (each with the same 11-cis-retinal).

Isomerization

• It appears that Na+ channels in the receptor outer membrane are held open in the dark by the phosphorylation of membrane proteins by cGMP.

• Once a photon is absorbed, the chromophore 11-cis-retinal to isomerized to all-trans-retinal.

• This process, known as photoactivation, initiates a biochemical cascade of events eventually resulting in the closing of channels in the cell membrane that normally allow ions to flow into the rod outer segment.

1. A G-protein is activated.2. It then activates cyclic Guanosine

monophosphate phosphodiesterase (PDE).3. PDE then hydrolyzes cGMP, reducing its

concentration.4. This leads to the closing of Na+ channels in the

outer membrane, hyperpolarizing the receptor.• When a single photon is absorbed, it is

estimated that 500 PDE molecules are activated, and each PDE molecule can inactivate many cGMP molecules.– As such, there is an immense amplification

within the photoreceptor that provides great sensitivity.

Isomerization

…..some of the intermediate steps in the closing of Na+ channels during photoactivation.

Photoreceptors themselves hyperpolarize to the capture of photons!

0

1

2

3

4

Outp

ut

(Arb

itra

ry U

nits)

1 10 100 1000 10000 Light Intensity

A log transform occurs at the receptor level: the output of receptors is proportional to the logarithm of the input.

• A single photoisomerization in the dark adapted state will activate many PDE molecules and will close many Na+ channels.

• At higher background light levels, there will be fewer available PDE molecules, so a greater number of photoisomerizations will be needed to close the same number of Na+ channels.

• Since there are fewer open Na+ channels at higher background levels, it is also true that activating a PDE molecule is less likely to have an impact since most Na+ channels are already closed.

Absorption Spectra

• The nature of the binding between the 11-cis retinal and the different opsins determines the probability with which different wavelengths are absorbed.

The three cone types are named for where the peak of their sensitivity lies on the spectrum. Thus, the cones that have a peak at about 440 nm are known as short-wavelength cones (or S-Cones for short). The middle-wavelength cones (M-Cones) peak at about 535 nm, and long-wavelength cones (L-Cones) peak at about 565 nm. Do not be tempted to rename these “blue,” “green,” and “red” cones, since color isn’t derived by examining a single cone output (and the wavelengths do not generally match the color names).

S M L

Rhodopsin