Optical Instruments

Every optical instrument in this chapter exists to help one remarkable instrument: the eye. Before adding lenses to it, let’s see how it works on its own.

The parts of the eye#

The eyeball is roughly a sphere about 2.5 cm across. Light entering it meets, in order:

• Cornea: the transparent, more sharply curved front window.
• Aqueous humour: a clear liquid behind the cornea.
• Iris and pupil: the iris is the coloured muscular diaphragm; the pupil is the hole in it. The iris widens the pupil in dim light and narrows it in bright light. The pupil looks black because light entering it almost never comes back out.
• Crystalline lens: hard in the middle, softer toward the edge, held by ligaments to the ciliary muscles, which can change its curvature.
• Vitreous humour: the clear jelly filling the space behind the lens.
• Retina: the light-sensitive lining at the back, with about 125 million receptors (rods and cones) and about a million optic-nerve fibres carrying signals to the brain.

Where the bending happens

The two humours have refractive index about 1.336; the lens averages about 1.396. Light entering from air is bent most at the cornea, where the change in refractive index is largest. The lens adds a smaller, adjustable amount. Together, cornea, lens and fluids act like a single converging lens, the eye-lens, whose focal length the ciliary muscles can change.

Accommodation#

Looking at something far away, the ciliary muscles are relaxed and the eye-lens has its longest focal length, equal to its distance from the retina, so parallel rays focus on the retina. For a nearer object the muscles tighten, the lens bulges, and the focal length shortens just enough to keep the image on the retina. This adjustment is called accommodation.

Apparent size#

How big something looks depends on the size of its image on the retina: a bigger image covers more rods and cones. Move an object away and its image shrinks. The image size is roughly proportional to the visual angle, the angle the object subtends at the eye. Optical instruments work by making this angle bigger.

To see something small, you bring it closer, until your eye can no longer focus. A magnifying glass lets you bring it closer still.

The naked-eye limit#

The largest angle an object of height h can subtend while still in focus is when it is at the near point:

Adding a lens#

Place a converging lens of focal length f (less than D) just in front of the eye, and put the object at its focus. Rays from each point of the object leave the lens parallel, as if from an image at infinity, which the relaxed eye focuses easily. The angle at the eye is

Since f < D, θ > θ₀: the retinal image is bigger. A converging lens used this way is a simple microscope or magnifier. With the image at infinity, the eye is in normal adjustment.

Image at the near point#

Move the object a little closer, to u₀, so the virtual image forms at the near point (v = −D). The eye strains, but sees more detail. From the lens formula, $D/u_0 = 1 + D/f$, and the angle is h/u₀:

One lens can only magnify so much before the image falls apart. Use two, and let the second magnify the image the first one made.

Construction#

A compound microscope has two converging lenses on a common axis. The objective, facing the object, has a small aperture and a very short focal length f₀. The eyepiece (ocular), near the eye, has a larger aperture and focal length fₑ. Screws adjust the distance between them.

How it forms the image#

1. The object sits just beyond f₀. The objective forms a real, inverted, magnified image at distance v₀ behind it.
2. The eyepiece views that image as a simple microscope. In normal adjustment the first image sits at the eyepiece’s focus and the final image is at infinity. Or the eyepiece can be set so the final image is at the near point, for more magnification.

The final image is inverted relative to the object.

Magnifying power#

The naked eye sees the object at best at angle θ₀ = h/D. Through the microscope the final image subtends θ = h′/uₑ, where h′ is the first image’s height and uₑ its distance from the eyepiece. So

D/uₑ is just the eyepiece’s magnifying power as a simple microscope:

Usually f₀ is tiny, so v₀/f₀ ≫ 1 and v₀/u₀ = v₀/f₀ − 1 ≈ v₀/f₀; and the first image is near the eyepiece, so v₀ ≈ L, the tube length:

A microscope looks at small things close up; a telescope looks at things far away, like a planet, a ship or a cricketer across the ground. The trick is the same pair of lenses, used the other way round.

The astronomical telescope#

Two converging lenses share an axis. The objective faces the distant object; it has a large aperture (to gather light) and a long focal length f₀. The eyepiece has a small aperture and short focal length fₑ. The eyepiece tube slides inside the objective tube to change their separation.

Light from a distant object forms a real, inverted image in the objective’s focal plane. The eyepiece magnifies it. In normal adjustment, that first image sits at the eyepiece’s focus and the final image is at infinity; push the eyepiece closer and the final image comes to the near point, for maximum magnification.

Magnifying power

The object is so far away that it subtends practically the same angle α at the objective as at the unaided eye. If h′ is the height of the first image, α ≈ h′/f₀, and in normal adjustment the final image subtends β ≈ h′/fₑ. β and α are on opposite sides of the axis (the image is inverted), so

With the final image at the near point, the lens formula puts the first image fₑD/(fₑ + D) from the eyepiece, and

The terrestrial telescope#

An upside-down view is fine for stars but odd for a cricket match: the players would seem to hang from the field. A terrestrial telescope adds a converging erecting lens of focal length f between the objective and the eyepiece, placed 2f beyond the first image. It forms an inverted, same-size copy of that image a further 2f along (magnification −1), which the eyepiece then views.

The Galilean telescope#

Galileo used a diverging eyepiece. It is placed so that it intercepts the converging rays from the objective before they form the first image; that would-be image acts as a virtual object for the eyepiece. The final image is erect.

With the final image at the near point instead, $m = \dfrac{f_0}{|f_e|}\left(1 - \dfrac{|f_e|}{D}\right)$ and the tube is a little longer. Galilean telescopes are short and give upright images, which is why opera glasses use them.

You can always build a microscope or telescope with more magnification. But past a point it only shows a bigger blur. The real limit is set by diffraction: how close two details can be and still look like two.

Why there’s a limit#

As we saw in the chapter on light waves, every lens images a point as a small diffraction disc. Two points closer than about one disc radius merge. Magnifying merges discs just as much as it magnifies them.

Resolving power of a microscope#

The resolving power of a microscope is the reciprocal of the smallest distance between two points that it can just resolve. It depends on the wavelength λ, the refractive index μ of the medium between the object and the objective, and the angle θ that a radius of the objective subtends at the object:

To raise it, specimens are viewed with a drop of oil filling the gap between them and the objective. This oil immersion increases μ.

Resolving power of a telescope#

For a telescope, it is the reciprocal of the smallest angle between two distant objects that can just be resolved:

That is why astronomical telescopes have objectives a metre or more across.

Many of us wear lenses in front of our eyes every day. Each pair of spectacles is a small calculation: the lens that moves the world to where your eye can focus it.

The eye as a lens with a fixed screen#

The retina is a fixed distance v₀ (about 2.5 cm in an adult) behind the eye-lens. To focus objects at different distances u₀, the eye changes its focal length. In magnitudes:

• The far point is reached with the muscles relaxed (largest f). In a normal eye f_max = v₀, so the far point is at infinity.
• The near point is reached with the muscles fully strained (smallest f); about 25 cm or less for a normal eye.

Nearsightedness (myopia)#

A nearsighted eye sees close objects but not distant ones. Even fully relaxed, its focal length is too short (the lens is too thick, or the eyeball too long), so parallel rays focus in front of the retina. Straining only makes it worse.

The cure is to make the rays slightly divergent before they enter: a diverging lens. If the far point is at distance x, the lens must form a virtual image of a distant object at x, so its focal length is −x:

Farsightedness (hypermetropia)#

A farsighted eye sees distant objects but its near point is well beyond 25 cm, so reading is uncomfortable. Even fully strained, its focal length isn’t short enough (the lens is too thin, or the eyeball too short). Rays from 25 cm would focus behind the retina.

A converging lens makes them less divergent. If the near point is at y, the lens must form an image of an object at 25 cm at distance y. With u = −25 cm and v = −y:

Presbyopia

With age the ciliary muscles weaken and can’t strain enough, so the near point recedes even in an otherwise normal eye. This farsightedness from age is presbyopia, corrected the same way.

Astigmatism#

If the eye’s surfaces are curved differently in different planes, lines in some directions look sharp while lines at right angles look blurred. A quick test: draw several lines through a point, equally dark. If one looks darkest and the line at right angles faintest, and this swaps when you turn the page by 90°, the eye is astigmatic. The cure is a cylindrical lens, curved differently in different planes to compensate.