Mirrors, Lenses, and Imaging Systems (24 p.593) Mirrors, Lenses, and Imaging Systems Hamid NEBDI, Ph. D A.Professor. Department of Physics Faculty of Applied.

(24 p.593)

Mirrors, Lenses, and Imaging Systems

Hamid NEBDI, Ph. DA.Professor. Department of Physics

Faculty of Applied Sciences.

Umm Al-Qura University القرى أم معة جاFaculty of Applied Medical Sciences كـلية العلوم الطبية التطبيقية General Physics 104 (For medical students) 104فيزياء طبية

1. The power of a Lens & Aberrations

• In discussing lenses, it is often more convenient to deal with the reciprocal of the focal length f, which is called the power of the lens P:

It’s clear from this definition that the meaning of the word power in optics is unrelated to its meaning in mechanics, work per unit time.

• If f is measured in metres, then P is measured in diopters.

• 1 diopter = 1 m-1.

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1

• For example a lens with a focal length 0f -0.4 m has a power :

P = 1/(-o.4 m) = -2.5 diopters

A short –focal-length lens, which bends light through large angles, has a large power

• Two thin lenses with focal lengths f1 and f2 placed next to each other are equivalent to a single lens with a focal length f satisfying:

• Alternatively, with P1=1/f1 and P2=1/f2, the power of the pair of lenses is :

P = P1 + P2

The powers of lenses in contact are simply added to find the net power.

Thus using powers instead of focal lengths avoids a good deal arithmetic involving fractions.

For example, an ophthalmologist placing 3-diopter and 0.25-diopter lenses in front of a patient’s eye immediately knows that the combination is equivalent to a single 3.25-diopter lens.

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Aberrations• In this subsection, we are going to see how the

additivity of the powers can be used to treat a lens configuration designed to minimize aberrations.

• No matter how perfectly spherical its surface, any lens suffers from various kinds of aberrations, which limit the sharpness of its images independently of diffraction effects.

• Since the index of refraction of glass varies with the wavelength of the light, the focal length of a lens also varies with the wavelength.

Aberrations• Chromatic aberration:

When an object is illuminated with white light, if its image on a screen is in focus for one colour component, it will be slightly out of focus for the others.

• Monochromatic aberrations: aberrations occur for light of a single wavelength.

How we can cancelled the Aberrations?

• Complex lens systems with several elements are designed so that their separate aberrations tend to cancel.

Figure 1

• We can illustrate these cancellations by considering a doublet, two lenses in contact. Lens 1 has two convex sides and is made from crown glass. Lens 2 has one flat side and one concave side and is made from flint glass. All of the curved surfaces have radii of curvature of 10 cm. In both types of glass, the refractive index varies about 1 percent over the visible spectrum. The powers P1 and P2 can be calculated using the lensmaker’s equation:

• As is seen in Figure 2, P1 varies by 2 % over the spectrum, and P2 by 3%. However, when the lenses are in contact, the effective power P = P1 + P2 is constant ! Thus the doublet is free of chromatic aberration.

Figure 2

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2. The Human Eye• The human eye is a remarkable creation from

our God (and not a remarkable evolutionary achievement):

- It covers a field of view of over 180, - It can rapidly shift its focus from very short distances to infinity,- …

-The eyeball is approximately spherical in shape with a diameter about 2.3 cm.- The shape and the focal length of the crystalline lens are controlled by the ciliary muscles. The function of the lens is to provide the fine adjustments needed to focus on objects at distances.- The index of refraction of:

- Humors = 1.336 (very closed to the water 1.333)- Crystalline lens = 1.437

- The function of the lens is to provide the fine adjustments needed to focus on objects at different distances.

- The ability of the lens to adjust its focal length is called accommodation.- The power of accommodation of the eye is the maximum variation of its

power for focusing on near and distant objects.- At the far point (a person sees objects at a distance xf clearly), the power Pf

of the eye is:

- For a person with normal vision Pf = 1/0.02 m=50 diopters (xf = )- When the eye adjusts its focal length so that it focuses on an object at the

near point (the object distance xn), the power of the eye is:

- For a young adult with normal vision Pn = (1/0.25 m)+(1/0.02 m)=54 diopters.

- The power of accommodation is the difference: A= Pn – Pf.- For a young adult with normal vision A = 4 diopters.- Young children have a much greater power of accommodation, and often can

read books held quite close to their eyes.- The accommodation decreases with aging, and most people find their near

point gradually recedes until they cannot read comfortably without corrective glasses.

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The Acuity (1/2)- The normal human eye can just barely distinguish two well-

illuminated point objects with an angular separation Ѳ0 5 x 10-4 rad 0.03. This minimum angular separation is called the visual acuity.

- It is reasonable to inquire whether this limit is due to diffraction effects. When light from a distant source passes through a small circular aperture of diameter d, the first diffraction minimum is at sin Ѳ= 1.22 /d.

- To estimate what this implies for the eye, let us take the iris diameter to be 5 mm , and the wavelength to be 500 nm. Since Ѳ is small, to one significant figure Ѳ sin Ѳ = 1.22 /d = 1.22 ( 5 x 10-7 m / 5 x 10-3 m) 10-4

- According to the Rayleigh criterion, two objects will just barely be resolvable if they are separated by this angle.

- Experiments show that while a few people under optimum conditions have an acuity of twice the diffraction limit, or 2 x 10-4 rad, nobody can reach the limit of 10-4 rad.

- An explanation for this failure of the eye to quite match the diffraction limit is provided by the structure of the retina.

The Acuity (2/2)- From Figure 4a, the radius of the image circle due to

diffraction isr = D tan Ѳ D Ѳ (2.3 x 10-2 m) (10--4 m)= 2.3 x 10-6

m- This is about equal to the separation of the cones in the

fovea, which is the most sensitive region of the retina and contains only cones and not rods. Now the best resolution observed is about 2 x 10-4 rad, corresponding to diffraction centers separated by about 4.6 x 10--6 m, or by two cones. Thus, it appears that to distinguish two small objects, at least one unexcited cone must intervene between the excited cones.

The Sensitivity (1/2)- The minimum or threshold intensity needed to see a

flash of light depends on the wavelength. - The cornea is opaque to wavelengths shorter than

300 nm, and the crystalline lens to wavelengths below 380 nm, so ultraviolet light does not normally contribute to vision. However, some persons have had their lenses surgically removed because they have become opaque or developed cataracts. These people have poor acuity and no accommodation, but to them objects illuminated with ultraviolet light alone are visible and appear violet.

- One long wavelength limit on the sensitivity of the eye is set by the strong absorption of light by water in cornea and the aqueous humor at wavelengths above 1200 nm. However, the sensitivity of the eye goes to zero rapidly above 700 nm.

The Sensitivity (2/2)- Figure 5 shows the variation with wavelength of the sensitivity,

which is the inverse of the threshold intensity, for dark-adapted and light-adapted eyes. When we go from daylight into dimly lit room, our eyes gradually adapt and reveal initially invisible details. This adaptation takes over ½ hour to be complete. Since the dark-adapted eye is much more sensitive, the two curves have been adjusted to have the same maximum value.

- The cones are active only in light-adapted vision, while the rods are always active.

- The visual acuity of the dark-adapted eye is much poorer than that of the light-adapted eye.

- From Figure 5 it can be seen that the sensitivity is greatest near 500 nm and 550 nm for dark- and light-adapted eyes, respectively. Both wavelengths correspond to green light. Often, green glass is used in sunglasses and in tinted windows. Since green glass absorbs green light less than other wavelengths, it provides the most useful illumination while reducing the total transmitted intensity.

3. Optical Defects of the Eye- Four common optical defects of the eye can

be corrected by the use of eyeglasses.

- In 3 of these defects (myopia, hypermetropia & presbyopia), the glasses are used to shift the apparent position of an object, so that the defective eye is able to focus properly.

- In the last, astigmatism, the glasses correct for a distortion produced by the eye.

3.a. Optical Defect of the Eye: Myopia

- In myopia or nearsightedness, parallel light from a distant object is focused by the relaxed eye at a point in front of the retina (Figure 6).

- Consequently, a nearsighted person cannot focus clearly on objects farther away than the far point located at a distance xf.

- This problem arises because the power of the eye is too great; either the cornea has an excessive curvature or the eye is longer than normal.

- Diverging lenses with negative powers will compensate for this defect.

3.a. How we can calculate the required correction?

- To calculate the required correction, we find the power of the eye at its far point, and then select a lens of the correct power to move the far point to infinity.

- We use the thin lens formula, 1/f = 1/s +

1/s’, rewritten in terms of the power P = 1/f:

Here D is the image distance in the eye, approximately 0.02 m. The procedure is illustrated by the following exercise.

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Exercise 1:

Solution 1a:

Solution 1b & 1c:

3.b. Optical Defect of the Eye: Hypermetropia

- Hypermetropia or farsightedness, is the opposite of myopia. Light from an object close to the eye is focused toward a point behind the retina, even when the lens is adjusted by the ciliary muscles to have its maximum power (Figure 7).

- Eyeglasses with converging lenses supply the additional focusing power needed, with negative powers will compensate for this defect.

Figure 7:

Exercise 2 & Solution:

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3.c. Optical Defect of the Eye: Presbyopia

3.d. Optical Defect of the Eye: Astigmatism

- A person with astigmatism cannot simultaneously focus on both horizontal and vertical lines.- Usually this is due to a cornea that is not perfectly spherical, so that is has different curvatures in different directions.- Occasionally, astigmatism is caused by irregularities elsewhere in the eye. - Astigmatism ban be corrected by cylindrical lenses oriented to compensate for the distortion (Figure 8).

Figure 8: