Optical Instruments

A lens in front of the eye can do more than correct it. Arranged the right way, lenses make small things look bigger and distant things look nearer, and they let us look into the eye itself. None of these instruments needs anything new: each is built from the lenses of Chapters 4 and 5.

#How big things look

How large something looks has little to do with its actual size. A coin held at arm’s length can hide the moon. What matters is the angle the object subtends at the eye, because that sets the size of its image on the retina. Move the arrow:

Bring the object closer and the angle $\theta$ grows, and with it the retinal image. A 5 cm arrow 35 cm away, as here, subtends about 8° and forms an image about 2.3 mm tall. But the eye can only follow so far. Nearer than its near point it can no longer focus, and bringing the object closer just makes it blurrier. The near point recedes with age, from under 10 cm in a young adult to about 25 cm in the late forties (Chapter 6). This figure uses 25 cm, the distance against which, by convention, magnifiers are compared. So the best an unaided eye can do with a small object is to hold it at its near point. To see more, it needs help.

#The magnifying glass

Put a plus lens in front of the eye, with the object at the lens’s focal point. Light from each point of the object then leaves the lens parallel, so the relaxed eye can focus it, however close the object is. Change the power of the lens:

Light from the tip of the object leaves the lens parallel to the ray through the lens’s centre, so it reaches the eye at an angle ${\theta }^{\prime }=h/f^{\prime }=hP$, where $h$ is the height of the object. Without the lens, at 25 cm, the same object subtends $\theta =h/0.25=4h$. The ratio of the two is the magnifier’s angular magnification:

A +10.00 D magnifier makes things look 2.5 times bigger than they look at 25 cm. This is the same lens that formed a virtual, upright image in Chapter 4; with the object exactly at F, that image is at infinity. Moving the object a little inside F brings the image closer and gives a little more magnification, at the cost of some accommodation.

#Telescopes

A distant object can’t be brought closer. But a telescope can take the parallel light from each point of it and send it on, still parallel, at a steeper angle. Two lenses do it, an objective and an eyepiece, placed so that the second focal point of the first coincides with the first focal point of the second:

Light from a distant point leaves the telescope parallel, so the relaxed eye sees the image at infinity, and the angles are magnified by the ratio of the two powers:

In a Keplerian telescope both lenses are plus. The objective forms a real image inside the tube, the eyepiece is a magnifying glass for that image, and the view is upside down: $M$ is negative. In a Galilean telescope the eyepiece is a minus lens that intercepts the light before the image can form. Its view is upright and the telescope is shorter, its length the difference of the two focal lengths rather than their sum, which makes it the design used for low-power spectacle-mounted telescopes and surgical loupes. Its weakness is a narrow field of view, so stronger ones are Keplerian, with prisms to turn the image upright.

An uncorrected myope can still use a telescope. Pushing the eyepiece in a little makes the light leave slightly diverging, as if from the myope’s far point, so telescopes for low vision are focused for each user.

#The direct ophthalmoscope

To look at the retina we need light to come out of the eye. The retina reflects a little of the light that falls on it, and the light leaving the eye from a point on the retina has passed through the eye’s own optics. In a relaxed emmetropic eye it leaves parallel, just as light from a distant point arrives parallel. So an examiner with relaxed eyes, close to the patient, sees the patient’s retina in focus. The patient’s +60 D eye acts as a magnifying glass for its own retina, magnifying it about $60/4=15$ times.

If the patient is myopic, the light leaves converging toward their far point; if hyperopic, it leaves diverging. A direct ophthalmoscope has a wheel of lenses to correct this. Choose a patient, emmetropic, 5.00 D myopic or 4.00 D hyperopic, and find the lens that brings the back of their eye, the fundus, into focus:

The myopic patient needs about −6.00 D and the hyperopic patient about +4.00 D: roughly the patient’s refractive error, adjusted because the lens sits 2 cm in front of the eye (the effectivity of Chapter 5). The exact values, −5.56 D and +3.70 D, fall between the wheel’s whole-dioptre steps. If the examiner is ametropic and not wearing their correction, the two errors add. Young examiners also tend to accommodate as they look, and then need a little more minus, so the lens that gives the clearest view is only a rough guide to the prescription. The view is magnified about 15 times but narrow: through an undilated pupil, little more than the optic disc and the vessels around it, as here.

#The indirect ophthalmoscope

For a wide view of the fundus, clinicians use indirect ophthalmoscopy. The examiner holds a strong plus lens, the condensing lens, typically +20 D, a few centimetres in front of the patient’s eye. It brings the light leaving the eye to a focus, forming a real, inverted image of the fundus in the air beyond it:

what the examiner sees

+14 D+20 D+28 D

Aerial image 5.0 cm beyond the lens, real and invertedmagnified ×3.0, about 60/20 = 3.0

The examiner looks at this aerial image from about 40 cm, with a headset that provides both the light and a binocular view. The image is inverted, so everything appears upside down and reversed left to right, which takes some getting used to. Its magnification is roughly the ratio of the eye’s power to the lens’s: with a +20 D lens, the aerial image is $60/20=3$ times the size of the fundus. Viewed from 40 cm, that is an angular magnification of only about $3\times 25/40\approx 2$, far less than the 15 times of the direct ophthalmoscope, but the field of view is many times wider, and the view is stereoscopic. A stronger lens gives a wider field and a smaller image. The same idea, with a +90 D or +78 D lens, lets a slit-lamp microscope view the fundus.

#Recap

• How big something looks depends on the angle it subtends at the eye. Closer is bigger, until the near point.
• With the object at its focal point, a magnifier sends parallel light to the eye and magnifies by $M=P/4$ relative to 25 cm.
• A telescope magnifies angles by $M=-P_e/P_o$. A Keplerian telescope (two plus lenses) inverts; a Galilean one (minus eyepiece) is upright and shorter.
• In direct ophthalmoscopy the patient’s eye magnifies its own retina about 15 times, and the lens needed for a clear view is roughly the patient’s refractive error.
• In indirect ophthalmoscopy a condensing lens forms a real, inverted aerial image about $60/P$ times the size of the fundus, with a wide field of view.

#Test yourself

#Further reading

• R. B. Rabbetts, Bennett and Rabbetts’ Clinical Visual Optics. Butterworth-Heinemann.
• M. H. Freeman and C. C. Hull, Optics. Butterworth-Heinemann.
• J. Schwiegerling, Field Guide to Visual and Ophthalmic Optics. SPIE Press.