Light, Eyes and Vision

Fig. 11.1

Structure of the eye: a sagittal section, b three-dimensional perspective (From [59])

The adult cornea is 0.52 mm thick in the center and 0.65 mm thick in the periphery, and is about 12.6 mm in diameter horizontally and 11.7 mm vertically. It is composed of several layers, from anterior to posterior: the outer epithelium, the basement membrane, anterior limiting lamina (or the Bowman’s layer), corneal stroma, posterior limiting lamina (or the Descemet’s membrane), and the epithelium. The corneal stroma constitutes 90% of the corneal thickness and is composed of 50 layers, each with similarly oriented collagen fibers, with the fibers always parallel to the cornea surface. The cornea is transparent because it is uniform in structure, avascular (i.e., it has no blood vessels) except in the extreme periphery, and relatively dehydrated. (More on this below.) It is covered by a 7– $10\,\upmu$ m thick layer of tears, which, among other things, smoothes over optical irregularities on the anterior surface of the cornea and supplies the cornea with oxygen . The average radius of curvature of the anterior surface of the cornea is about 7.8 mm in the central region, with a variation among people of about $\pm 0.4$ mm, and is flatter in the periphery.

The aqueous humor fills the anterior chamber (with a volume of 0.3 cm $^{3}$ ) bounded by the cornea, iris, and the anterior surface of the crystalline lens, and the posterior chamber (0.2 cm $^{3}$ ) on the periphery of the lens. It has many fewer proteins (0.1 g/L) than blood plasma (60–70 g/L). The pupil in the iris is usually slightly nasal and inferior to the center of the iris, and can vary roughly from 1.5 to 10 mm in diameter . The diameter of the pupil is controlled by an opposing pair of smooth muscles: the sphincter pupillae (which is a ring of muscles that encircle the pupil) contracts it and the dilator pupillae (which has the form of a thin disc) widens it.

The crystalline lens is suspended from the ciliary body by zonular fibers and rests on the posterior surface of the iris. It is composed of about 66% water and 33% protein. This crystalline lens is about 4 mm thick and 9 mm in diameter. It continues to grow during life, with new layers growing on older layers, forming a layered structure like an onion (Fig. 11.1). At 30 years of age, the lens has a mass of 170 mg, which increases by about 1.2 mg per year; similarly the lens width is about 4 mm and increases by about 0.02 mm per year. The crystalline lens is avascular and almost completely transparent. Still, it is slightly birefractive (i.e., it has slightly different refractive indices for different polarizations of light), becomes more yellow with age, and can become opaque (and this forms a cataract). Aphakia describes the condition when the crystalline lens is absent. The vitreous humor is about 99% water, with the remaining 1% composed of collagen (0.5 g/L proteins) and hyaluronic acid; the latter gives it its gelatinous, viscous physical characteristics. This humor accounts for about 5 cm $^{3}$ of the 7–8 cm $^{3}$ volume of the eye.

As we will see below, the formation of an image on the retina is determined by the indices of refraction of each eye component that the light passes through and by the shapes of the surfaces of these elements. The cornea and crystalline lens are the actual focusing elements in the eye. The cornea performs about two-thirds of the focusing and the crystalline lens the remaining one-third. The shape and consequently the focal length of the crystalline lens are adjustable and do the fine-tuning of imaging for accommodation. The measured refractive index of the tears and the vitreous humor is about 1.336 and that of the aqueous humor is a bit higher, 1.3374. The refractive indices of the cornea, about 1.3771, and the crystalline lens are higher. At the center of the crystalline lens (which is called the nuclear region), the index is about 1.40–1.41 and it decreases to 1.385–1.388 in the direction towards the “poles” and to 1.375 in the direction toward the “equator”; it is 1.360 in the capsule, which is the elastic membrane that encloses the crystalline lens. There is still some uncertainty in these values; optical models of the eye use values close to these cited numbers.

Fig. 11.2

Schematic of the retina in the eye, with the arrangement of rods and cones and other neurons, along with electrical excitation by the shown light stimulus (From [50])

Only about 50% of visible light (400–700 nm) incident on the eye actually reaches the retina as direct light. Then light must pass through the (transparent) ganglion and other retinal neurons before reaching and forming an image on the backward-facing photoreceptors on the retina (Fig. 11.2). The fovea or fovea centralis is the central region of the retina, and the region of sharpest vision because it has the highest density of cone cells on the retina (Fig. 11.3). The visual axis of the eye is the line from the point you are focusing on, through the center of pupil, to the fovea. The optical axis of the eye is a line that passes through the centers of the cornea, pupil and crystalline lens to the retina. They are not parallel. The optic nerve leaves the eyeball at a blind spot (optic disk), a region with no rods or cones (Fig. 11.3); it is 13– $18^{\circ }$ away from the fovea in the “nasal” direction.

Fig. 11.3

Distribution of rods and cones on the retina, and the location of the blind spot (Based on [13, 63])

Fig. 11.4

Fixate on the $\times$ using your left eye, with your right eye closed. Keep the book about 10 cm from your left eye, and then move it back and forth until you do not see the central dot. This dot is then on the blind spot. The dots above and below it are still visible, but fuzzy because of the lower visual acuity outside the fovea. (Also see Fig. 11.24) (Based on [63])

We are usually not aware of the blind spot when we use both eyes because the part of the image that forms on the blind spot in one eye is located in a functional region in the other eye and the brain fuses the images of the two eyes. It is easy to prove the existence of the blind spot. Close your right eye and use your left eye to look at the $\times$ in Fig. 11.4. When you move the book about 10 cm from your left eye, you will find one position where the central dot disappears because of the blind spot in your left eye.

There are about 120 million rod cells per retina (Fig. 11.5). They have high sensitivity , low spatial acuity, and are relatively more numerous in the periphery of the retina. The sensitivity of rods peaks near 500 nm (Fig. 11.6). Vision using only rods results in various shades of gray. Night vision and peripheral vision are mostly due to rods. Rods are about $2\,\upmu$ m in diameter. Far from the fovea the rods become more widely spaced and many (in some cases several hundred) rods are connected to the same nerve fiber. Both factors decrease visual acuity in the outer portions of the retina.

Fig. 11.5

Scanning electron micrograph of rod and cone outer segments, with the cone seen (with its tapered end) in the center and the end and beginning of two rods (which are longer than the cones) seen beneath and to the left of it (Reprinted from [29]. Used with permission of Elsevier)

Fig. 11.6

Relative spectral sensitivity of rods and cones. The absolute sensitivity of rods is 1,000 $\times$ larger those of the cones (Based on [5, 17, 34])

There are about 6.5 million cone cells per retina. They have low sensitivity—about 1,000 $\times$ lower than rods—high spatial acuity, and are concentrated in the fovea. There are three types of cone cells, with spectral sensitivities peaking near 445 nm (blue or S cones—S for short wavelength peak sensitivity), 535 nm (green or M cones—M for middle wavelengths), and 570 nm (red or L cones—L for long wavelengths) (Fig. 11.6). The overall spectral sensitivity due to the rods and cones of humans closely matches the spectrum of solar light reaching land. Sharp vision and color vision are due to cones, and consequently damage to the fovea leads to visual images that are fuzzy. Cones are about 1.0–1.5 $\upmu$ m in diameter and are about 2.0–2.5 $\upmu$ m apart in the fovea. There are only about 1 million nerve fibers in the eye, so there are some cones (as well as rods) connected to the same nerve cells. We will not delve into the cellular structure of the rods and cones, but will focus on two physical aspects of these sensors: the absorption of light and acuity of vision.

The absorption of light by the rods and cones is a fundamental quantum-mechanical process in which one photon (or quantum) of light is absorbed by the pigment rhodopsin. Quantum mechanics is the physics of small-scale objects, and has features that are distinct from the physics of larger-scale objects, which is the classical physics we have been using throughout this book. One feature of quantum physics is the quantization of energy levels in molecules, which means that a molecule can have only distinct energies. Consequently, a molecule can absorb light only at those specific energies (or frequencies) corresponding to the differences of its energy levels. Moreover, in quantum mechanics, light acts like light packets, called photons. The energy of a photon is

$\begin{aligned} E=h\nu =\frac{hc}{\lambda }, \end{aligned}$

(11.1)

where h is Planck’s constant ( $6.626 \times 10^{-34}$ J–s, as in (6.37)), $\nu$ (or f) is the frequency of the light, c is the speed of light ( $3.0 \times 10^{8}$ m/s), and $\lambda$ is the wavelength of light. The last two parts of this equation reflect the relationship between frequency, wavelength, and propagation speed for these electromagnetic waves,

$\begin{aligned} c=\lambda \nu , \end{aligned}$

(11.2)

as in (10.3). Absorption occurs when the photons have energy in ranges that can be absorbed by the photosensitive molecules in these cells.

Rhodopsin consists of a chromophore (i.e., the part of the molecule responsible for its color) covalently attached to the protein opsin. The chromophore is retinal, which is a derivative of vitamin A, and the absorption of a single photon of light isomerizes it (i.e., changes its molecular conformation) from 11-cis retinal to all-trans retinal. This isomerization triggers a change in the conformation of rhodopsin that starts a sequence of sensory transduction processes (Fig. 11.7). Proteins themselves have absorption bands in the ultraviolet, and cannot absorb in the visible. The absorption of free 11-cis retinal is in the near ultraviolet, 360–380 nm; however, the binding of the retinal to the protein red shifts the absorption by about 200 nm to the visible . Differences in the opsin proteins in the rods and the three cones cause the different wavelength responses for these four types of photoreceptor cells. (By the way, Ragnar Granit, Haldan Keffer Hartline, and George Wald were awarded the Nobel Prize in Physiology or Medicine in 1967 for their discoveries concerning the primary physiological and chemical visual processes in the eye.)

Fig. 11.7

The chromophore 11-cis retinal is photoisomerized by light to all-trans retinal (11-trans retinal)

Fig. 11.8

Imaging by a thin, positive lens

The pressure in the eyeball maintains its shape. It is normally about 15 mmHg (ranging from 10 to 20 mmHg), and is determined by the rates of formation of the aqueous humor (about 1% of the total volume is produced per minute) and drainage of the aqueous humor through the canal of Schlemm. If the exit of the aqueous humor is impaired, the eyeball pressure increases, leading to glaucoma and possible blindness (as is addressed below). Intraocular pressure (IOP) is measured by the amount of force needed to flatten to a given area (or the area flattened by a given force) by using a tonometer. (This is explored in Problem 11.53.)

Fig. 11.9

Special cases of imaging with positive lenses $(\mathbf {a}\hbox {--}\mathbf {c})$ , and imaging with negative lenses $\mathbf {d}$

11.2 Focusing and Imaging with Lenses

11.2.1 Image Formation

Figure 11.8 shows how an object or source is imaged by a convex (converging or positive) lens . By convention in optics the object is placed a positive distance $d_{1}$ to the left of the lens and optical rays propagate from the left to the right. The object has a size (or height) $y_{1}$ . For a convex lens the focal length f is positive, hence the name positive lens. The central axis (the z-axis) is known as the optic axis. All rays passing through the lens form an image a positive distance $d_{2}$ to the right of the lens, where $d_{2}$ is given by the lens equation

$\begin{aligned} \frac{1}{d_{1}}+\frac{1}{d_{2}}=\frac{1}{f}. \end{aligned}$

(11.3)

A real image forms at $d_{2}$ when $$$d_{1}>f$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq29.gif”></SPAN>, which means that you will see the image at <SPAN id=IEq30 class=InlineEquation><IMG alt=$$d_{2}$$ src=$ if you place a screen, such as a piece of paper, there. The image is inverted and its size $y_{2}$ is magnified by $M=d_{2}/d_{1}$ (the transverse magnification ). This can be seen from the triangles in Fig. 11.8 that give $y_{1}/d_{1}=y_{2}/d_{2}$ , so the magnification is

$\begin{aligned} M=\frac{d_{2}}{d_{1}}=\frac{y_{2}}{y_{1}}. \end{aligned}$

(11.4)

(Actually, it is magnified when $$$d_{2}/d_{1}>1$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq34.gif”></SPAN> and reduced in size (minified) when <SPAN id=IEq35 class=InlineEquation><IMG alt=$ .) When $d_{1}=\infty$ , parallel rays are incident on the lens and an image forms at $d_{2}=f$ (Fig. 11.9a). When the object is at the focus and so $d_{1}=f$ , the image is at $\infty$ (Fig. 11.9b). When the object is closer to the lens than the focal point and so $d_{1}<f$ , then $d_{2}<0$ and the image is to the left of the lens (Fig. 11.9c). This is a virtual image. Placing a screen there will give no image. However, if the light rays to the right of the lens are traced backward to the left of the lens, they will seem to emanate from this virtual image.

This same lens equation (11.3) can be used to determine the location of the image for a concave (diverging or negative) lens, which has a negative focal length f . Concave lenses produce virtual images (Fig. 11.9d).

Within the eye the cornea and crystalline lens are positive lenses, because they need to form a real image on the retina. Corrective lenses (eyeglass lenses and contact lenses) can have positive or negative focal lengths, depending on the necessary correction. We will explore this later in this chapter. Focal lengths are expressed as distances, in cm or m. We will see that in discussing the eye and corrective lenses it is very common to discuss 1 / f and use units of diopters (D), with 1 D = 1/m.

In a very simple model of the eye imaging system, the eye is treated as a thin lens with a 17 mm focal length in air ( Standard eye model ). For an image at $d_{1}\gg f$ , (11.3) shows that $d_{2}\simeq f$ . We will see that two points are resolvable by at best $\varDelta y_{2} = 2$ $\upmu$ m on the fovea. So for a source that is 10 m away, two points are resolvable when separated by at least $\varDelta y_{1}=(d_{1}/d_{2})\varDelta y_{2} = (10$ m / 17 mm) (2 $\upmu$ m) $$ = 1.2$$

mm; this corresponds to an angle of 1.2 mm/10 m = 0.12 mrad = 25 s of arc. The $\sim$ 300 $\upmu$ m foveal diameter corresponds to a lateral separation of $\sim$ $$(10$$

m / 17 mm $) (300\,\upmu$ m) $$ = 18$$

cm at 10 m or $\sim$ $$18$$

cm / 10 m

mrad $\sim$ $1^{\circ }$ of arc.

In clinical, ophthalmic optics, the lens equation (11.3) is phrased and used differently. It is expressed in terms of vergences, which indicate the angles rays make with the optic axis and signify ray convergence with positive vergence and divergence with negative vergence. U is the object convergence, sometimes also known as the object proximity. If all optical elements are in a medium with refractive index n, then $U=-n/d_{1}$ . The image vergence (or image proximity) is $V=n/d_{2}$ . The refractive power of the lens $$P = n/f$$

, as we will see. U, V, and P are expressed in diopters (1 D = 1/m). The lens equation (11.3) expressed in vergences is then:

$\begin{aligned} U + P =V. \end{aligned}$

(11.5)

This is interpreted as meaning that the propagation of light from an object through the lens to the image increases the ray vergence by an amount equal to the power of the lens (or more generally, the power of the optical interface or optical system).

We have assumed geometric optics, which ignores the wave-like features of light due to optical diffraction; this is a good approximation for very short wavelengths and for much of the imaging in the eye. Our analysis also assumes only paraxial rays , i.e., all rays are near the optic axis and make small angles to it. (Rays that are farther away from the optic axis—nearer where the maximum amount of light is transmitted—are called zonal rays, and those at the margin of the lens are marginal rays. We will evaluate the importance of diffraction and of these zonal and marginal rays later in this chapter.)

11.2.2 Scientific Basis for Imaging

We will trace rays by following how they propagate in straight lines in uniform media and how they refract at interfaces by using Snell’s Law. Snell’s Law of refraction shows that light from medium 1 with index of refraction $n_{1}$ impinging at an angle $\theta _{1}$ (relative to the normal) on a flat interface with medium 2 with refractive index $n_{2}$ , is refracted to an angle $\theta _{2}$ given by (Fig. 11.10).

$\begin{aligned} n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2} \end{aligned}$

(11.6)

and for small angles ( $\theta _{1}$ , $\theta _{2} \ll 1$ )

$\begin{aligned} n_{1}\theta _{1}=n_{2}\theta _{2}. \end{aligned}$

(11.7)

Some important indices of refraction are 1.0 for air, 1.33 for water, 1.5–1.6 for different types of glass, and 1.44–1.50 for plastics. Refractive indices actually vary some with wavelength and temperature, but we will ignore those variations at present.

Fig. 11.10

Snell’s law

Imaging can occur when the interfaces are curved. Let us consider the refraction of paraxial rays at the interface in Fig. 11.11 from medium 1 to medium 2, which has a spherical radius of curvature $R_{12}$ . (This region can be formed by slicing off a section from a sphere with radius $R_{12}$ , composed of material 2.) As shown here, this radius is defined to be positive (see the Fig. 11.11 inset) . Equation (11.6) still applies, but the angle of incidence for a light ray parallel to the optic axis varies with the distance y the ray is displaced from this axis . We see that $\theta _{1}=y/R_{12}$ . For $y\ll R_{12}$ (so $\theta _{1} \ll 1$ ), (11.7) gives

$\begin{aligned} \theta _{2}=\frac{n_{1}}{n_{2}}\;\theta _{1}=\frac{n_{1}}{n_{2}}\frac{y}{R_{12} }. \end{aligned}$

(11.8)

This refracted ray makes an angle $\theta _{1}-\theta _{2}=(1-n_{1}/n_{2})(y/R_{12})=[(n_{2}-n_{1})/n_{2}](y/R_{12})$ with the horizontal (Fig. 11.11). Geometry shows that it hits the optic axis a distance $F(y)=y/(\theta _{1}-\theta _{2})$ after the interface. This distance is independent of y, and so all parallel rays impinging on the curved surface hit the optic axis at this same distance, which is called the focal length f (or $f_{12}$ for this interface)

$\begin{aligned} f_{12}=\frac{y}{[(n_{2}-n_{1})/n_{2}](y/R_{12})}=n_{2}\frac{R_{12}}{ n_{2}-n_{1}}=\frac{n_{2}}{P_{12}}. \end{aligned}$

(11.9)

The last expression has been written in terms of the refractive power (or sometimes called the convergence) of the interface

$\begin{aligned} P_{12}=\frac{n_{2}-n_{1}}{R_{12}}. \end{aligned}$

(11.10)

The focal length is defined in terms of the refractive power of the interface and the refractive index of the medium the ray enters . The ratio of the indices of refraction of the two media is important and not their individual values. (For $n_{2} = 1$ , we see that $f_{12}=1/P_{12}$ .) The units of the refractive power are diopters (1 D = 1/m).

Fig. 11.11

Refraction at a curved interface. The inset shows the convention for the radius of curvature. In this figure, $$$R_{12} > 0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq85.gif”></SPAN> and <SPAN id=IEq86 class=InlineEquation><IMG alt=$ n_{1}$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq86.gif”>, and $\theta _{1} \ll 1$ so $R_{12} \gg y$

We see that this derivation giving the focal length in (11.9) is similar to that for (11.3), except that the object and image are in regions with different refractive indices here. If $n_{1}$ and $n_{2}$ were interchanged, the sign of the focal length would change and its magnitude would change to $\mid n_{1}/P_{12}\mid$ . The analog of (11.3) at this refractive interface is

$\begin{aligned} \frac{n_{1}}{d_{1}}+\frac{n_{2}}{d_{2}}=\frac{n_{2}-n_{1}}{R_{12}}. \end{aligned}$

(11.11)

For $$$P_{12}>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq93.gif”></SPAN>, when <SPAN id=IEq94 class=InlineEquation><IMG alt=$ , a real image occurs at $$$d_{2}=f_{12}>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq95.gif”></SPAN> (for rays traveling from left to right, as is the convention). For rays traveling from right to left and <SPAN id=IEq96 class=InlineEquation><IMG alt=$ , a real image occurs at $$$d_{1}=f_{21}>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq97.gif”></SPAN>. So in general we see that<br /> <DIV id=Equ12 class=Equation><br /> <DIV class=EquationContent><br /> <DIV class=MediaObject><IMG alt=$

(11.12)

(For $$$P_{12}>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq98.gif”></SPAN>, this gives <SPAN id=IEq99 class=InlineEquation><IMG alt=$ 0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq99.gif”> and $$$f_{21}>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq100.gif”></SPAN>. Another popular convention is to define <SPAN id=IEq101 class=InlineEquation><IMG alt=$$f_{21}$$ src=$ as the negative of our definition, so the last term in (11.12) would be $-n_{1}/f_{21}$ .) The magnification, M, is modified from (11.4) to give

$\begin{aligned} M=\frac{d_{2}/n_{2}}{d_{1}/n_{1}}=\frac{y_{2}}{y_{1}}. \end{aligned}$

(11.13)

Here, the object vergence is $U=-n_{1}/d_{1}$ and the image vergence is $V=n_{2}/d_{2}$ . Then (11.11) can be written as

$\begin{aligned} U+P_{12}=V, \end{aligned}$

(11.14)

which is similar to (11.5).

What happens when there are two refracting interfaces in succession (Fig. 11.12)? If they are separated by a distance D that is “very small,” the same reasoning gives an overall focal length f for this lens

$\begin{aligned} f=\frac{n_{3}}{P_{12}+P_{23}}, \end{aligned}$

(11.15)

where $P_{12}$ is given by (11.10) and

$\begin{aligned} P_{23}=\frac{n_{3}-n_{2}}{R_{23}}. \end{aligned}$

(11.16)

This is called the thin lens approximation. Equation (11.15) can be expressed as $f=n_{3}/P_{\mathrm {total}}$ , where

$\begin{aligned} P_{\mathrm {total}}=P_{12}+P_{23}, \end{aligned}$

(11.17)

so the refractive powers add in this approximation.

Fig. 11.12

Refraction at two spherical interfaces. As drawn here , the radii of curvature $$$R_{12} > 0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq108.gif”></SPAN> and <SPAN id=IEq109 class=InlineEquation><IMG alt=$

For a thin lens of refractive index $n_{2}=n$ in air or vacuum (with refractive index $n_{1}=n_{3}=1$ , so $f_{13}=f_{31}$ ), (11.15) reduces to

$\begin{aligned} \frac{1}{f}=(n-1)\left( \frac{1}{R_{12}}-\frac{1}{R_{23}}\right) , \end{aligned}$

(11.18)

which is known as the Lensmaker’s equation. Thisfocal length can also be expressed as

$\begin{aligned} f=\frac{1}{P_{12}+P_{23}}, \end{aligned}$

(11.19)

with $P_{12}=(n-1)/R_{12}$ and $P_{23}=-(n-1)/R_{23}$ .

Lenses can have a range of shapes even for the same focal length (Fig. 11.13). For positive focal lengths, they can be either biconvex (as shown $$$R_{12}>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq117.gif”></SPAN> and <SPAN id=IEq118 class=InlineEquation><IMG alt=$ ), planoconvex (one side flat), or positive meniscus as shown ( $$$R_{23}>R_{12}>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq119.gif”></SPAN>). For negative focal lengths, they can be either biconcave (as shown <SPAN id=IEq120 class=InlineEquation><IMG alt=$ and $$$R_{23}>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq121.gif”></SPAN>), planoconcave (one side flat), or negative meniscus (as shown <SPAN id=IEq122 class=InlineEquation><IMG alt=$ ). The isolated cornea is a negative meniscus lens and the crystalline lens is an asymmetric biconvex lens. We will see below that in the eye the cornea serves as a positive lens.

Fig. 11.13

Types of positive and negative lenses

The length of the eyeball is approximately 24 mm, so the distances from the cornea/crystalline lens to the retina and the focal length of optical rays focused by the cornea/crystalline lens propagating in the vitreous humor and imaging on the retina, are also about 24 mm. The refractive index of the vitreous humor is approximately 1.33. Equation (11.15) shows that a system with the same refractive power has a focal length that is proportional to this refractive index, so in air the focal length would be smaller by a factor of 1.33, or 24 mm/1.33 = 17 mm. That is why a model eyeball can be treated as if the cornea/crystalline lens system had an effective focal length of 17 mm, with the effective lens separated by 17 mm of air from the retina; we will call this the Standard eye model. (This differs from the eye models in Table 11.1.)

Moving Lenses

Two lenses have the same effectivity (i.e., effectiveness) if both can image the same object to the same image, placed at the appropriate distance between the object and image as given by (11.3). For example, an image forms 4 cm after a lens with focal length 2 cm that is placed 4 cm after an object. If you wanted move the lens to the left by 2 cm, you would have to change its focal lens to 1.5 cm for it to have the image at the same place and so have the same effectivity. (Why?)

Let us consider the special case of parallel rays hitting a converging lens with power P in a medium with refractive index n. They form an image a distance n / P after the lens. If we move the same lens a distance D to the left and want the rays to focus in the same place, we now need them to focus a distance

after the lens and so the lens will need to have a power $P^{\prime }$ such that $n{/}P^{\prime }=n{/}P+D$ . Therefore we see that

$\begin{aligned} P^{\prime }=\frac{P}{1+\frac{D}{n}P}. \end{aligned}$

(11.20)

This is known as the effectivity formula . (This equation is sometimes displayed with a negative sign in the denominator because the lens is being moved a distance D to the right.)

Why is this imaging effectivity important to us? The corrective powers for prescriptions for eyeglasses and contact lenses are different because eyeglasses are placed about 1.2 cm anterior to the cornea, while contact lenses sit right on the cornea, and so are different in two ways. We just addressed the consequences of this different lateral position and will address below those due to refractive power changes that occur because the contact lens is physically on the cornea. In any case, you want both prescriptions to have the same effectivity for objects at infinity, so (11.20) is used with

11.2.3 Combinations of Lenses or Refractive Surfaces

The eye itself is a combination of four distinct curved imaging interfaces. When imaging on the retina is not perfect , corrective lenses (with two additional interfaces) are chosen so the combined effect of these supplemental lenses and the eye imaging system produces a more perfect image on the retina. To first order we can apply (11.19), including all of the curved interfaces (for the eye or eye + corrective lens), with the ray finally entering a medium j with index of refraction $n_{j}$ . We see that the focal length is

$\begin{aligned} f=\frac{n_{j}}{\sum _{i=1\;\mathrm {to}\;j-1}P_{i,i+1}}. \end{aligned}$

(11.21)

Merely using (11.21) to determine the imaging properties of these complex systems of curved surfaces or lenses is not sufficient, because it is based on the thin lens approximation, which ignores the propagation of rays from one refracting surface to the next. Because the refractive interfaces are displaced from one another, this usually needs to be corrected.

Two Thin Lenses

Before we explain how to account for the separation of the curved interfaces in a very general way, let us consider the imaging by two separated thin lenses. This approach is rigorous but it is cumbersome when extended to more than two lenses. It can be applied, for example, to the two refractive surfaces of the cornea or the crystalline lens. We will use it to learn how to correct vision by considering the combined effect of eyeglasses or contact lenses and the eye, with both modeled as simple thin lenses.

Let us consider two thin lenses with focal lengths $f_{1}$ and $f_{2}$ that are separated by a distance D (Fig. 11.14). An object is placed at a distance $d_{\mathrm {a}}$ to the left of the first lens and the final image is formed at a distance $d_{\mathrm {b}}$ to the right of the second lens. In the forward propagation approach, we consider propagation from the left to the right in Fig. 11.14a. The first lens forms an image at a distance $d_{\mathrm {i}}$ to the right of this lens, where

$\begin{aligned} \frac{1}{d_{\mathrm {a}}}+\frac{1}{d_{\mathrm {i}}}=\frac{1}{f_{1}}. \end{aligned}$

(11.22)

This intermediate image is at a distance $D-d_{\mathrm {i}}$ to the left of the second lens, and serves as the object for this second lens. Therefore, the second lens forms an image at a distance $d_{\mathrm {b}}$ to its right, where

$\begin{aligned} \frac{1}{D-d_{\mathrm {i}}}+\frac{1}{d_{\mathrm {b}}}=\frac{1}{f_{2}}. \end{aligned}$

(11.23)

This distance $d_{\mathrm {b}}$ is determined by inserting $d_{\mathrm {i}}$ from (11.22) into (11.23)

$\begin{aligned} D=\frac{d_{\mathrm {a}}f_{1}}{d_{\mathrm {a}}-f_{1}}+\frac{d_{\mathrm {b}}f_{2}}{d_{\mathrm {b}}-f_{2}}. \end{aligned}$

(11.24)

This interrelates all five parameters (D, $d_{\mathrm {a}}$ , $d_{\mathrm {b}}$ , $f_{1}$ , $f_{2}$ ) and can be used to determine the fifth parameter, such as $d_{\mathrm {b}}$ , when the other four are known.

Sometimes it is simpler to work backward from the final image to the initial source, especially when the location of the final image is fixed (Fig. 11.14b). For example, consider using this backward propagation approach when corrective lenses (lens 1) are needed to correct the images of the effective eye lens (lens 2) on the retina. Let us say we know that the eye lens images perfectly on the retina when an object is at a distance $z_{1}$ away (to the left of the effective eye lens), but we would like to have clear images of objects at a distance $z_{2}$ away. The eye by itself cannot accomplish this, as we will soon see, due to insufficient accommodation or to myopia or hyperopia. The distance from the corrective lens to the eye lens is fixed, $D \sim 1.2$ –1.5 cm for eyeglass lenses and $D \sim 0$ cm for contact lenses. Therefore this distance $z_{1}$ to the left of lens 2 is at a known distance $z_{1}-D$ to the left of lens 1. The focal length of corrective lens 1, $f_{1}$ , is chosen so that it takes an object $d_{\mathrm {a}}=z_{2}-D$ to the left of it and forms an image $d_{\mathrm {i}}=D-z_{1}$ to the right of it. (The eye can then image this intermediate object quite well by itself.) Using (11.3), this condition becomes

$\begin{aligned} \frac{1}{z_{2}-D}+\frac{1}{D-z_{1}}=\frac{1}{f_{1}}, \end{aligned}$

(11.25)

and so the focal length of the necessary corrective lens is $f_{\mathrm {corrective}}=f_{1}$ . One or both of the terms on the left-hand side of this equation can be negative. We will use this approach below to prescribe corrective lenses.

Fig. 11.14

Imaging by two lenses using $\mathbf {a}$ forward and $\mathbf {b}$ backward propagation

Of course, you can analyze two lenses with the forward and backward propagation approaches by using the totally equivalent method of vergences.

Complex Optical Systems (Advanced Topic)

We will now see that any system with several refracting surfaces, such as the eye, can be reduced to a simple effective optical system that is similar to a thin lens (Fig. 11.15). The effective thin lens system is analyzed by tracing optical rays with the help of six reference points on the optic axis called cardinal points . There are two principal points (P, P $^{\prime }$ ), two focal points (F, F $^{\prime }$ ), and two nodal points (N, N $^{\prime }$ ).

Fig. 11.15

Optical planes and imaging in a thick lens (or any multielement optical system) for rays that are parallel to the axis. The second (or first) focal length is the distance from the second (or first) principal point to the second (or first) focal point. If the source and image media refractive indices, n and $n^{\prime }$ are the same, both are called the effective focal length (EFL). The back (or front) focal length (BFL or FFL) is measured from the last (first) vertex of the last (first) optical element

Fig. 11.16

Image formation using cardinal points. The location and size of the image (in a medium with refractive index $n^{\prime }$ ) can be determined by tracing rays from the object (in a medium with refractive index n): the object ray parallel to the optic axis and either the ray going through principal point P at angle $u_{1}$ and emerging from P $^{\prime }$ at angle $u_{1}^{\prime }= (n{/}n^{\prime })u_{1}$ or that going through nodal point N at angle $u_{2}$ and emerging from N $^{\prime }$ at the same angle $u_{2}^{\prime }= u_{2}$ . This example is similar to that of Schematic eye 2 depicted in the top half of Fig. 11.20, with $$$n^{\prime } > n$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq178.gif”></SPAN> </DIV></DIV></DIV></DIV></DIV><br /> <DIV id=Par41 class=Para>Rays traveling parallel to the optic axis (from left to right) in the first medium ( with refractive index <SPAN class=EmphasisTypeItalic>n</SPAN> where the object is located) refract in the optical system and travel as straight rays that converge at the second (or image) <SPAN class=EmphasisTypeItalic>principal focus</SPAN> at point F<SPAN id=IEq179 class=InlineEquation><IMG alt=$ (after the last interface in the last medium with refractive index $n^{\prime }$ ) (Fig. 11.15). When these rays are backtracked, they each intersect the initial parallel rays at a surface called the second (or image) principal plane P $^{\prime }$ . Similarly, parallel rays traveling from right to left in this last medium intersect in the first medium at the first (or object) principal focus at point F, and when these rays are backtracked, they each intersect the initial parallel rays at a surface called the first (or object) principal plane P. (These “planes” are really curved surfaces that are approximately planar near the optic axis.) The first and second (or object and image) principal points P and P $^{\prime }$ are the intersections of these planes with the optic axis. We will see that there are also nodal points N and N $^{\prime }$ on the optic axis, which help in analyzing image formation (Fig. 11.16).

We will adopt a sign convention that is convenient for biconvex, positive lenses, so that all focal lengths will be defined to be positive for such lenses. (This is consistent with (11.12), but is not universal notation.) The first principal focus F is a distance f to the left of P when $$f>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq188.gif”></SPAN> and the second principal focus at F<SPAN id=IEq189 class=InlineEquation><IMG alt=

$$f>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq188.gif”></SPAN> and the second principal focus at F<SPAN id=IEq189 class=InlineEquation><IMG alt=

is a distance $f^{\prime }$ to the right of P $^{\prime }$ when $$$f^{\prime }>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq192.gif”></SPAN>. We will call distances along the optic axis, say between P and another point A—which is PA, positive when A is to the right of P and negative when it is to the left of it; therefore, AP = <SPAN id=IEq193 class=InlineEquation><IMG alt=$$-$$ src=$ PA. There is symmetry in the positions of the six cardinal points, with the separations PP $^{\prime }$ = NN $^{\prime }$ and FP = N $^{\prime }$ F $^{\prime }$ , as is depicted in Fig. 11.16.

Rays traveling from left to right emanate from the same object point Q and intersect to form an image at Q $^{\prime }$ . Any ray from Q hits plane P and emerges from plane P $^{\prime }$ as if it were refracted by a thin lens using (11.3), with a “no man’s land” between the two principal planes. So, with $d_{1}=$ QP (which is the distance along the optic axis) and $d_{2}=$ P $^{\prime }$ Q $^{\prime }$ , the distance from object to image is really

$\begin{aligned} \mathrm {QQ^{\prime }}=\mathrm {QP}+\mathrm {PP^{\prime }}+\mathrm {P^{\prime }Q^{\prime }}=d_{1}+d_{2}+\mathrm {PP^{\prime }}. \end{aligned}$

(11.26)

Rays that hit point P, at an angle u, act as if they “emerge” from P $^{\prime }$ at an angle $u^{\prime }$ (Fig. 11.16), where

$\begin{aligned} n^{\prime }u^{\prime }=nu. \end{aligned}$

(11.27)

This can be viewed as Snell’s Law in this paraxial limit . Nodal points are also defined in a way that rays on course to hit the first nodal point N, act as if they “emerge” from the second nodal point N $^{\prime }$ with no change in angle. Again, these two sets of rays emanating from the same point Q intersect to form an image at Q $^{\prime }$ , as is seen in Fig. 11.16. The principal and nodal points coincide when the media to the left and right have the same refractive index; this is not the case for imaging by the eye. (For a thin lens in air, P, P $^{\prime }$ , N, and N $^{\prime }$ merge into one point (inside the lens for a biconvex lens), and $f^{\prime }=f$ .)

Two Arbitrary Optical Refractive Systems (Advanced Topic)

Let us now consider two arbitrary optical systems [28] (Fig. 11.17). The first has principal points $P_{\mathrm {a}}$ and P $^{\prime }_{\mathrm {a}}$ , object and image refractive indices $n_{\mathrm {a}}$ and $n^{\prime }_{\mathrm {a}}$ , and refracting power $P_{\mathrm {a}}$ . It is followed by the second with principal points $P_{\mathrm {b}}$ and P $^{\prime }_{\mathrm {b}}$ , with object and image refractive indices $n_{\mathrm {b}}$ and $n^{\prime }_{\mathrm {b}}$ , and refracting power $P_{\mathrm {b}}$ . The reduced distance between the second principal point of the first system and the first principal point of the second system is defined as

$\begin{aligned} \delta =\frac{t}{n_{\mathrm {a}}^{\prime }}=\frac{t}{n_{\mathrm {b}}}=\frac{t}{n_{2}}, \end{aligned}$

(11.28)

with

P $^{\prime }_{\mathrm {a}}$ $P_{\mathrm {b}}$ . We will now call $n_{\mathrm {a}}=n_{1}$ , $n'_{\mathrm {a}}=n_{\mathrm {b}}=n_{2}$ , and $n'_{\mathrm {b}}=n_{3}$ .

Fig. 11.17

Analyzing the combination of two optical systems, represented here by two “lenses.” The notation for the planes and points of the combined systems is underlined. The solid line ray refracts at the principal planes of the individual systems, while the dashed ray refracts at the principal planes of the overall system

Optical analysis that is beyond our scope gives general expressions for combining these two optical systems. The effective refractive power of the entire system can be shown to be

$\begin{aligned} P_{\mathrm {eff}}=P_{\mathrm {a}}+P_{\mathrm {b}}-\delta P_{\mathrm {a}}P_{\mathrm {b}} \end{aligned}$

(11.29)

$\begin{aligned} P_{\mathrm {eff}}=P_{\mathrm {a}}+P_{\mathrm {b}}-\frac{t}{n_{2}}P_{\mathrm {a}}P_{\mathrm {b}}=\frac{n_{3}}{f^{\prime }}=\frac{n_{1}}{f}. \end{aligned}$

(11.30)

(This is consistent with (11.12). Using some other sign conventions for focal lengths, this last term is expressed as $-n_{1}/f$ .)

Where are the principal points of the combined system? The position of the second principal point of the overall system P $^{\prime }$ relative to the second principal point of the second system is

$\begin{aligned} \frac{\mathrm {P}'_{\mathrm {b}}\mathrm {P'}}{n_{3}}=-\frac{t}{n_{2}}\;\frac{P_{\mathrm {a}}}{P_{\mathrm {eff}}}, \end{aligned}$

(11.31)

which means that P $^{\prime }$ is to the left of P $^{\prime }_{b}$ in Fig. 11.17 when $$$P_{\mathrm {a}}{/}P_{\mathrm {eff}}> 0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq243.gif”></SPAN> (such as for a biconvex lens).</DIV><br /> <DIV id=Par48 class=Para>Similarly, the position of the first principal point of the overall system P relative to the first principal point of the first system is<br /> <DIV id=Equ32 class=Equation><br /> <DIV class=EquationContent><br /> <DIV class=MediaObject><IMG alt=$

(11.32)

which means that P is to the right of $P_{a}$ in Fig. 11.17 when $$$P_{\mathrm {b}}/P_{\mathrm {eff}}> 0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_11_Chapter_IEq246.gif”></SPAN> (such as for a biconvex lens).</DIV><br /> <DIV id=Par49 class=Para><SPAN class=EmphasisTypeItalic>What are the new focal lengths?</SPAN> The second principal focus of the combined system F<SPAN id=IEq249 class=InlineEquation><IMG alt=$ is a distance

$\begin{aligned} \mathrm {P}'\mathrm {F}'=f^{\prime } \end{aligned}$

(11.33)

to the right of new second principal point P $^{\prime }$ . The new first principal focus F is a distance

$\begin{aligned} \mathrm {FP}=f \end{aligned}$

(11.34)

to the left of new first principal point P. The focal lengths are given by (11.30). If $n = n^{\prime }$ , these focal lengths are the same (and are the effective focal length (EFL), as seen in Fig. 11.15).

Where are these foci relative to the principal points of the original two systems? Using (11.30) and combining (11.31)–(11.34) gives the position of the secondfocal point relative to the second principal plane of the second system P $^{\prime }_{\mathrm {b}}$ F $^{\prime }$ = P $^{\prime }_{\mathrm {b}}$ P $^{\prime }$ $$+$$

P $^{\prime }$ F $^{\prime }$ :

$\begin{aligned} \mathrm {P}'_{\mathrm {b}}\mathrm {F}'=f^{\prime }\left( 1-\frac{t}{n_{2}}P_{\mathrm {a}}\right) =f^{\prime *} \end{aligned}$

(11.35)

and the position of the first focal point relative to the first principal plane of the first system FP $_{\mathrm {a}}$ $$=$$

PP $_{\mathrm {a}}$ :

$\begin{aligned} \mathrm {F}\mathrm {P}_{\mathrm {a}}=f\left( 1-\frac{t}{n_{2}}P_{\mathrm {b}}\right) =f^{*}. \end{aligned}$

(11.36)

The parameters $f^{*}$ and $f^{\prime *}$ are sometimes called the front and back (vertex) focal lengths (FFL and BFL in Fig. 11.15) because for a thick lens these are the distances from the front and back surfaces. In analogy with (11.30), sometimes the front and back vertex powers are defined as

$\begin{aligned} P^{*}=\frac{n_{1}}{f^{*}} \end{aligned}$

(11.37)

and

$\begin{aligned} P^{\prime *}=\frac{n_{3}}{f^{\prime *}}. \end{aligned}$

(11.38)

Combining these with (11.30), and using (11.35) and (11.36) shows that

$\begin{aligned} P^{*}=\frac{P_{\mathrm {b}}}{1-\delta P_{\mathrm {b}}}+P_{\mathrm {a}} \end{aligned}$

(11.39)

and

$\begin{aligned} P^{\prime *}=\frac{P_{\mathrm {a}}}{1-\delta P_{\mathrm {a}}}+P_{\mathrm {b}}, \end{aligned}$

(11.40)

with $\delta =t{/}n_{2}$ . This is an equivalent way of determining the location of the focal points of the combined optical system.

Equations (11.39) and (11.40) look very similar to the effectivity formula, (11.20), and for good reason: they are very closely related. Let us consider the effectivity formula for a thin lens “a” with refractive power $P_{\mathrm {a}}$ that is moved a distance $t=n_{2}\delta$ to the right, so it would be coincident with thin lens “b” with refractive power $P_{\mathrm {b}}$ . If its effectivity were not to change, its power must change from $P_{\mathrm {a}}$ to $P_{\mathrm {a}}^{\prime }=P_{\mathrm {a}}/[1-(t{/}n)P_{\mathrm {a}}]$ , as given by the effectivity formula with $$D=-t$$

. (Remember that D corresponds to a leftward displacement.) The new refractive power at the second lens is then $P_{\mathrm {a}}^{\prime }+P_{\mathrm {b}}$ . This gives (11.40). (Of course, now everything is referred to the principal planes of the optical systems rather than the location of the thin lenses.) Equation (11.39) is similarly obtained.

Because P $^{\prime }$ P $^{\prime }_{\mathrm {b}}$ , F $^{\prime }$ P $^{\prime }$ , $P_{\mathrm {a}}$ P, and FP are known, the nodal points F and F $^{\prime }$ can be found using PP $^{\prime }$ = NN $^{\prime }$ and FP = N $^{\prime }$ F $^{\prime }$ , with F $^{\prime }$ following F.

This approach can be used to evaluate complex optical systems. For example, for the eye it can be applied first to the cornea as a lens (see below), then to the crystalline lens, and then to combine these two optical elements to describe the paraxial optics of the entire eye. If the subject wears corrective lenses, the combined effect of these lenses and the eye can then be evaluated.

Thick Lenses. The focal length of a thick lens of thickness t follows from (11.30) and substituting $P_{\mathrm {a}}=(n_{2}-n_{1})/R_{12}$ and $P_{\mathrm {b}}=(n_{3}-n_{2})/R_{23}$ from (11.10)

$\begin{aligned} \frac{n_{3}}{f_{\mathrm {eff}}}=\frac{n_{2}-n_{1}}{R_{12}}+\frac{n_{3}-n_{2}}{R_{23}}-\frac{(n_{2}-n_{1})(n_{3}-n_{2})t}{n_{2}R_{12}R_{23}} \end{aligned}$

(11.41)

and the back focal length is

$\begin{aligned} f_{\mathrm {bfl}}=f_{\mathrm {eff}}-\frac{f_{\mathrm {eff}}(n_{2}-n_{1})t}{n_{2}R_{12} }. \end{aligned}$

(11.42)

The front focal length can similarly be determined.

For a single lens of refractive index n in air, the distance rays propagate between refractive surfaces affects the focal length. With the refractive surfaces separated by a distance t, the Lensmaker’s equation, (11.18), for thin lenses can now be generalized to the thick lens equation

$\begin{aligned} \frac{1}{f_{\mathrm {eff}}}=(n-1)\left( \frac{1}{R_{12}}-\frac{1}{R_{23}}+\frac{(n-1)t}{nR_{12}R_{23}}\right) . \end{aligned}$

(11.43)

This effective focal length is the distance from the second principal point. The distance from the second surface to the focus is the (back) focal length $f_{\mathrm {bfl}}$

$\begin{aligned} f_{\mathrm {bfl}}=f_{\mathrm {eff}}-\frac{f_{\mathrm {eff}}(n-1)t}{nR_{12}}, \end{aligned}$

(11.44)

so for incident parallel rays the image forms a distance $f_{\mathrm {bfl}}$ past the second surface of the lens.

11.3 Imaging and Detection by the Eye

11.3.1 Transmission of Light in the Eye

Figure 11.18 shows the percentage of incident light that reaches the aqueous humor, the crystalline lens, the vitreous humor, and the retina. Perhaps it is surprising that only about 50% of visible light (400–700 nm) incident on the eye actually reaches the retina to form an image. These losses are due to reflection of light at the interfaces between the different ocular media, and the absorption and scattering in these ocular media. Much of this loss is due to scattering in the eye. This scattered light does not contribute to the desired image even if it hits the retina. Also note the large transmission through the retina, so there is little loss due to absorption and scattering with the retina before light hits the rods and cones (Fig. 11.2).

Fig. 11.18

Transmission of near-ultraviolet, visible, and near-infrared light through the eye up to the labeled part of the eye. In $\mathbf {a}$ only transmission losses due to absorption are included, while in $\mathbf {b}$ all transmission losses are accounted for, including those due to absorption and scattering (Based on [4, 19])

The expression for the fraction of light that reflects from a planar interface between two semi-infinite media with refractive indices $n_{1}$ and $n_{2}$ is actually the same as that for the reflection of acoustic waves between the interface of media. In Figs. 10.3 and (10.33) the acoustic impedances $Z_{1}$ and $Z_{2}$ are replaced by the respective refractive indices $n_{1}$ and $n_{2}$ to give

$\begin{aligned} R=\left( \frac{n_{2}-n_{1}}{n_{2}+n_{1}}\right) ^{2}. \end{aligned}$

(11.45)

The reflected fraction at the air/cornea interface $[(1.377-1)^{2}/(1.377 +1)^{2}] \sim 2\%$ is much larger than that at any other interface in the eye because of the very large difference in refractive indices between air and the cornea. The actual reflected fraction is really a bit larger because the reflected fraction increases for other angles of incidence (and all light rays do not enter the eye at normal incidence) and the ocular interfaces are curved . Still, reflection accounts for little of the transmission losses.

When a penlight is shined into an eye, four images, called Purkinje images , are formed from the reflection off the anterior and posterior surfaces of the cornea and crystalline lens. Purkinje image I from the anterior cornea surface is the strongest, as was just shown; it is a virtual, erect image 3.85 mm from the corneal apex. Purkinje image II from the posterior cornea surface is the next strongest; it is a virtual, erect image 3.77 mm from the corneal apex. Purkinje images III and IV from the anterior and posterior crystalline lens surfaces are the weakest; image III is a virtual, erect image 10.50 mm from the corneal apex and image IV is a real, inverted image 3.96 mm from the corneal apex.

Figure 11.18 shows that most of the light from 300 to 400 nm is absorbed by the crystalline lens, mostly by the yellow macular pigment xanthophyll. There are no important sources of absorption in the visible (which is one reason why this light is “visible”). Most of the transmission losses in the visible are due to scattering within the eye components due to inhomogeneities in the index of refraction, such as those caused by cells and submicroscopic particles. The efficiency of scattering attributed by changes in the refractive index over distances $\ll$ $\lambda$ , is called Rayleigh scattering. Scattering by a concentration N of nonabsorbing particles of diameter D and refractive index $n_{\mathrm {p}}$ in an ambient medium of refractive index $n_{\mathrm {a}}$ (such as that of thecrystalline lens, cornea , or the humors) decreases the transmission of light intensity (I in W/m $^{2}$ ) according to Beer’s Law (10.18).

$\begin{aligned} I(z)=I(z=0)\exp (-\alpha _{\mathrm {light\, scattering}}z), \end{aligned}$

(11.46)

where I is the incident intensity (say in W/m $^{2}$ ). For a light of wavelength $\lambda$ (as measured in vacuum or air)

$\begin{aligned} \alpha _{\mathrm {light\, scattering}}=\frac{8\pi ^{4}}{3}\frac{ND^{6}}{\lambda ^{4}}\left( \frac{n_{\mathrm {p}}{}^{2}-n_{\mathrm {a}}^{2}}{n_{\mathrm {p} }{}^{2}+2n_{\mathrm {a}}^{2}}\right) ^{2}. \end{aligned}$

(11.47)

(The derivation of this is beyond the current scope.) The important point here is that the efficiency of scattering varies as $1{/}\lambda ^{4}$ or, using (11.2), as $\nu ^{4}$ . So blue light scatters more efficiently than red light, and this explains the greater loss in transmission at shorter visible wavelengths in Fig. 11.18b. (This is also one of the reasons why the sky is blue; blue light scatters more efficiently.)

The cornea, white of the eye, skin, tendons and ligaments all largely consist of collagen fibrils in a matrix. The cornea is very transparent while these other tissues are opaque. Why? The refractive index of the fibrils, $\sim$ 1.47, differs from that of the surrounding media, 1.345, and the strength of the elastic scattering of light depends on this difference. Since the fibrils are densely packed in the cornea and it is relatively thick ( $\sim$ 0.05 cm), we would expect the cornea to be opaque also. However, in the cornea the fibrils are partially ordered, and scattering is not incoherent (the sum of the scattered intensity from each fibril) but partially coherent and there is destructive interference (cancellation) of the scattered waves that decrease overall scattering and, thankfully, improves transparency [18, 36, 56].

11.3.2 The Eye as a Compound Lens

Parameters for several progressively simpler schematic models for the imaging of paraxial rays in the eye are given in Table 11.1 [3, 6, 8, 9, 15, 28]. In each case the eye is assumed to be fully relaxed ( or unaccommodated), with the exception of Schematic eye 2 $^{\prime }$ , which accounts for accommodating Schematic eye 2 from imaging distant objects to imaging near objects. The most complete model is the Gullstrand exact eye (Schematic exact eye), which has six refractive surfaces: at the anterior and posterior surfaces of both the cornea and crystalline lens and two within the crystalline lens, so the variation of the refractive index within the lens is included (in a mathematical, but not a totally accurate physical, manner). (By the way, Allvar Gullstrand was the recipient of the 1911 Nobel Prize for Physiology or Medicine for his work in this area, the dioptrics of the eye. Dioptrics is the branch of geometrical optics dealing with the formation of images by refraction, especially by lenses.)

Table 11.1

Dimensions for Schematic (Schem.) and Reduced (Red.) eyes

	Schem.	Schem.	Schem.	Schem.	Schem.	Red.
	exact eye	eye 1	eye 2	eye 2’	eye 3	eye
Radii of surfaces
Anterior cornea	7.70	7.80	7.80	7.80	7.80	5.55
Posterior cornea	6.80	6.50	–	–	–	–
Anterior lens	10.00	10.20	10.00	5.00	11.00	–
First internal lens	7.911	–	–	–	–	–
Second internal lens	–5.76	–	–	–	–	–
Posterior lens	–6.00	–6.00	–6.00	–5.00	–6.476	–
Distance from anterior cornea
Posterior cornea	0.50	0.55	–	–	–	–
Anterior lens	3.60	3.60	3.60	3.20	3.60	–
First internal lens	4.146
Second internal lens	6.565
Posterior lens	7.20	7.60	7.20	7.20	7.30	–
Retina	23.9	24.20	23.89	23.89	24.09	–
First principal point P	1.348	1.59	1.55	1.78	1.51	0
Second principal point P $^{\prime }$	1.602	1.91	1.85	2.13	1.82	0
First nodal point N	7.078	7.20	7.06	6.56	7.11	5.55
Second nodal point N $^{\prime }$	7.332	7.51	7.36	6.91	7.42	5.55
First focal point F	–15.707	–15.09	–14.98	–12.56	–15.16	–16.67
Second focal point F $^{\prime }$	24.387	24.20	23.89	21.25	24.09	$22.22^{\mathrm{a}}$
Refractive indices
Cornea	1.376	1.3771	–	–	–	4 / 3
Aqueous humor	1.336	1.3374	1.3333	1.3333	1.336	4 / 3
Crystalline lens–anterior	1.386	1.4200	1.4160	1.4160	1.422	4 / 3
Crystalline lens–nucleus	1.406
Crystalline lens–posterior	1.386
Vitreous humor	1.336	1.3360	1.3333	1.3333	1.336	4 / 3

Using data from [3, 15, 28]

All eyes are accommodated for distant vision (unaccommodated), except Schematic eye 2 $^{\prime }$ , which accounts for accommodation in Schematic eye 2 for near vision (accommodated). Distances are in mm

$^\mathrm{{a}}$ For the Reduced eye, the second focal point is 1.67 mm + 22.22 mm = 23.9 mm after the real anterior surface of the cornea

We will call the next simplest model Schematic eye 1 (the classic Emsley model), for which the refractive index within the crystalline lens is uniform, so there are only four refractive surfaces: at the anterior and posterior surfaces of both the cornea and crystalline lens . Figure 11.19 is a diagram of this model, showing the radii or curvature and locations of the four interfaces and the refractive indices of the five media. The given refractive index of the crystalline lens, 1.4200, is an averaged value; it ranges from about 1.406 near the center to about 1.386 far away from the center.

Fig. 11.19

Schematic eye 1

The cornea and aqueous humor are treated as one region in the simplified schematic eyes (Schematic eyes 2, 2 $^{\prime }$ , and 3) so it has three refractive surfaces: at the cornea and at the two surfaces of the crystalline lens. Schematic eyes 2 and 2 $^{\prime }$ are the Gullstrand–Emsley models of relaxed and accommodated eyes and Schematic eye 3 is the revised relaxed eye model by Bennett and Rabbetts. The top part of Fig. 11.20 shows the location of the cardinal points for Schematic eye 2.

Fig. 11.20

Comparison of the cardinal points of the three-surface, relaxed Gullstrand–Emsley Schematic eye 2 in the top half and the Reduced eye in the bottom half. Distances are in mm (Based on [3])

The simplest model is the Reduced eye (Fig. 11.21), which has only one refractive interface: at the “cornea”—which is actually 1.67 mm after the real cornea. This is seen along with the cardinal points for this Reduced eye in the bottom part of Fig. 11.20. For this eye all distances are relative to the single refractive interface (1.67 mm after the real anterior surface of the cornea), so the second focal point is 1.67 mm + 22.22 mm = 23.9 mm after this corneal surface.

Fig. 11.21

Reduced eye (Based on [3])

Thin Lens Approximation of the Schematic Eye

Consider Schematic eye 1 with four interfaces (Table 11.1). The refractive power of each interface is

$\begin{aligned} P_{i,i+1}=\frac{n_{i+1}-n_{i}}{R_{i,i+1}}. \end{aligned}$

(11.48)

We will initially ignore the distance the rays propagate between the refractive interfaces. For the air/anterior cornea interface:

$\begin{aligned} P_{12}=\frac{1.3771-1.0}{0.0078\,\mathrm {m}}=\mathrm {48.35\,D}. \end{aligned}$

(11.49)

For the posterior cornea/aqueous humor interface:

$\begin{aligned} P_{23}=\frac{1.3374-1.3771}{0.0065\mathrm {\,m}}=-\mathrm {6.11\,D}. \end{aligned}$

(11.50)

For the aqueous humor/anterior crystalline lens interface:

$\begin{aligned} P_{34}=\frac{1.4200-1.3374}{0.0102\mathrm {\,m}}=\mathrm {8.10\,D}. \end{aligned}$

(11.51)

For the posterior crystalline lens/vitreous humor interface:

$\begin{aligned} P_{45}=\frac{1.3360-1.4200}{-0.0060\mathrm {\,m}}=\mathrm {14.00\,D}. \end{aligned}$

(11.52)

The refractive power of the cornea, ignoring its thickness, is (11.17)

$\begin{aligned} P_{\mathrm {cornea}}=P_{12}+P_{23}=\mathrm {48.35\,D }-\mathrm {6.11\,D = 42.24\,D}. \end{aligned}$

(11.53)

The refractive power of the crystalline lens, again ignoring its thickness, is

$\begin{aligned} P_{\mathrm {lens}}=P_{34}+P_{45}=\mathrm {8.10\,D + 14.00\,D = 22.10\,D}. \end{aligned}$

(11.54)

This shows that two-thirds of the refractive power is due to the cornea and one-third is due to the lens.

The total refractive power of the eye is

$\begin{aligned} P_{\mathrm {eye}}=P_{\mathrm {cornea}}+P_{\mathrm {lens}}=\mathrm {42.24\,D + 22.10\,D = 64.34\,D}. \end{aligned}$

(11.55)

The focal length is (11.21)

$\begin{aligned} f=\frac{n_{j}}{\sum _{i=1\;\mathrm {to}\;j-1}P_{i,i+1}}=\frac{\mathrm {1.336}}{\mathrm {64.34\,D}}\mathrm {=0.0208\,m = 20.8\,mm}. \end{aligned}$

(11.56)

The image from this compound lens falls on the retina, which is 24.20 mm from the anterior surface of the cornea and so this calculated focal length is not exactly correct. The compound lens is 7.6 mm long (anterior surface of the cornea to the posterior surface of the crystalline lens), so we would expect that the focal length is really measured for this type of compound lens from somewhere between the cornea and crystalline lens. We will estimate that it is from the middle (at the position 3.8 mm), so we would expect the image to fall 3.8 mm + 20.8 mm = 24.6 mm from the anterior surface of the cornea, compared to 24.20 mm. This agreement is surprisingly good. (Because the refractive power of the cornea and crystalline lens are not equal, the “starting point” is not exactly in the center, but this is a reasonable first guess.) We have ignored the propagation of light between the curved, refractive interfaces, which is not insignificant here, and will address it in the next section.

What happens when you swim in water? Therefractive power of the first (air/anterior cornea) interface changes to

$\begin{aligned} P_{12}=\frac{1.3771-1.331}{0.0078\mathrm {\,m}}=\mathrm {5.91\,D}, \end{aligned}$

(11.57)

which is a loss of 42.44 D of refractive power. The refractive power of the cornea is 5.91 D $$ +$$

D, which means the cornea has essentially no refractive power under water. The total refractive power of the eye is only $$-0.2$$

22.10 D

D, and the eye sees very blurred images because the focused image would be beyond the retina. (Why can we see much better in water when wearing ordinary goggles?) The images in water are made even blurrier when the water is not perfectly still, because the movement of water causes local variations in the index of refraction.

More Exact Analysis of the Schematic Eye (Advanced Topic)

How important is the finite separation of the refractive elements in the eye, as represented by Schematic eye 1 (Fig. 11.19)?

Using (11.30), (11.49), and (11.50), the effective power of the cornea is

$\begin{aligned} P_{\mathrm {cornea}}= & {} P_{12}+P_{23}-\frac{t_{\mathrm {cornea}}}{n_{\mathrm {cornea}}} \;P_{12}P_{23} \end{aligned}$

(11.58)

$\begin{aligned}= & {} \mathrm {48.35\,D}-\mathrm {6.11\,D }-\left( \frac{\mathrm {0.00055\,m}}{\mathrm {1.3771}}\right) \mathrm {(48.35\,D)}(-\mathrm {6.11\,D)\, =\, 42.36\,D}\nonumber \\ \end{aligned}$

(11.59)

for a cornea thickness $t_{\mathrm {cornea}} = 0.55$ mm $$ = 0.00055$$

m and refractive index $n_{\mathrm {cornea}}= 1.3771$ . (Remember, 1 D = 1/m.) This $$+0.12$$

D correction to $P_{12}+P_{23}= 42.24$ D from (11.53) is very small because the cornea is very thin. Using (11.31) and (11.32), the principal points of the cornea $P_{\mathrm {cornea}}$ and P $^{\prime }_{\mathrm {cornea}}$ are $$-0.058$$

mm and

mm to the right of the anterior surface of the cornea, meaning they are 0.058 mm and 0.060 mm to the left of this surface.

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Tags: Physics of the Human Body

Jun 11, 2017 | Posted by admin in GENERAL & FAMILY MEDICINE | Comments Off

Basicmedical Key

Fastest Basicmedical Insight Engine

Light, Eyes and Vision

11.2 Focusing and Imaging with Lenses

11.2.1 Image Formation

11.2.2 Scientific Basis for Imaging

11.2.3 Combinations of Lenses or Refractive Surfaces

11.3 Imaging and Detection by the Eye

11.3.1 Transmission of Light in the Eye

11.3.2 The Eye as a Compound Lens

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Basicmedical Key

Fastest Basicmedical Insight Engine

Light, Eyes and Vision

11.2 Focusing and Imaging with Lenses

11.2.1 Image Formation

11.2.2 Scientific Basis for Imaging

11.2.3 Combinations of Lenses or Refractive Surfaces

11.3 Imaging and Detection by the Eye

11.3.1 Transmission of Light in the Eye

11.3.2 The Eye as a Compound Lens

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