Lenses, Eye & Optical Instruments

How to use it

Pick a mode with the buttons above the picture. The top of the picture shows the light; the bottom shows a second view: a graph, what an eye sees, or the image on a camera sensor. Key numbers sit under the picture, the full list under Readouts.

This lab covers instruments built from lenses. For free-form ray tracing through mirrors, prisms and glass blocks, open Ray Optics; for light as a wave, try the Wave Simulator.

Controls

• Mouse and touch: drag the object (thin lens, magnifier), the distance marker (eye), the specimen (microscope focus), the incoming light (telescope) or the lens (camera focus).
• Keyboard: tab to the picture, then:
  • Thin lens: ← → move the object, ↑ ↓ change its height, + and − change the focal length; Shift makes the arrow steps finer.
  • Magnifier: ← → move the object (Shift for finer steps).
  • Eye: ← → move the object nearer or farther; G switches the corrective lens on or off.
  • Microscope: ← → focus (Shift for finer steps).
  • Telescope: ↑ ↓ move the Moon off the axis (Shift for finer steps), ← → change the eyepiece focal length by 1 mm.
  • Camera: ← → focus nearer or farther, ↑ ↓ move the subject.
  On the mode buttons, ← → switch the mode and Home and End jump to the first and last. Esc stops a sweep.
• Panel: sliders and buttons for every setting, remembered in this browser.

How it works

Thin lens. Distances are measured from the lens: d_o is positive on the side the light comes from, d_i is positive on the far side, and f is positive for a converging (convex) lens and negative for a diverging (concave) one.

A positive d_i is a real image that a screen can catch, a negative one is virtual, and a negative m means upside down. The power P is in dioptres (D) when f is in metres. The principal rays: parallel to the axis, then through the focal point F′; through the centre, straight on; through F, then parallel.

Magnifying glass. Apparent size is the angle an object covers. Unaided, an object of height h looks largest at the near point N = 25 cm, under the angle h/N. With the eye at the lens it covers h/d_o, so

Eye. The eye is one thin lens with the retina at a fixed distance: 60 D when relaxed, plus up to A dioptres of accommodation (4 D gives the usual 25 cm near point). A myopic eye of −E D has its far point 1/E metres away. A hyperopic eye of +E D must accommodate even for distant objects, so its near point moves out to 1/(A − E) metres. A corrective lens with the same number of dioptres, treated as a thin lens at the eye like a contact lens, moves the far point back to infinity; when A is below 4 D, reading glasses add (4 − A) D. When the eye cannot focus, each point spreads into a blur circle whose angle in radians is about the pupil diameter in metres times the focusing error in dioptres.

Microscope. The objective makes a real, inverted, enlarged image and the eyepiece magnifies it like a magnifying glass. With that image at the eyepiece’s focal point the final image is at infinity:

L is the optical tube length between the objective’s back focal point and the eyepiece’s front focal point (160 mm by default, so a 16 mm objective gives 10×).

Telescope. The objective focuses parallel light from a distant point in its focal plane; the eyepiece, one focal length further on, makes it parallel again at a larger angle:

A concave (Galilean) eyepiece has f_e < 0, so M > 0 and the view is upright.

Camera. To focus at distance s the lens sits v = 1/(1/f − 1/s) in front of the sensor. A subject that comes to a focus at v_s draws a blur circle c = (f/k)·|v − v_s|/v_s, where k is the f-number. Depth of field is the range where c stays below 0.03 mm, a common limit for a 36 × 24 mm sensor. That gives the hyperfocal distance H = f²/(kc) + f and the limits s(H − f)/(H + s − 2f) and s(H − f)/(H − s), the far one infinite once s ≥ H.

Limits of the model. Every lens is an ideal thin lens that sends all rays from one point to one image point. The lab therefore leaves out lens thickness, aberrations such as colour fringes and distortion, and diffraction, so it sets no limit on the finest detail a microscope or telescope can resolve. The eye is a single 60 D lens in air rather than a cornea and a lens in fluid, and the camera judges sharpness by one fixed blur limit of 0.03 mm for a 36 × 24 mm sensor.

Things to try

• Thin lens: press Sweep. The image grows as the object nears F, vanishes at F and returns virtual and upright inside it.
• Magnifier, f = 5 cm: compare “Image at infinity” (5×) with “Image at 25 cm” (6×).
• Eye with −2 D of myopia: bring the object in from 10 m. It turns sharp at the 50 cm far point. Then press “Distance prescription”.
• Microscope: switch from 10× to 40× and watch the letter e outgrow the field of view.
• Camera: at f/1.4 the depth-of-field bar shrinks; at f/16 it grows.

Frequently asked questions

Why is the image on the retina upside down?

A converging lens that forms a real image always inverts it. The brain interprets the inverted retinal image, so the world looks upright.

Why must a magnifying glass be held closer than its focal length?

Only then do the rays still spread out after the lens, giving an upright virtual image. Beyond F they converge, and a relaxed eye cannot focus converging light.

What does a prescription of −2.50 mean?

A diverging lens with a power of −2.50 D, that is f = −40 cm. It suits an eye whose far point is 40 cm away.

Why do telescopes and microscopes show things upside down?

The objective makes a real, inverted image and the eyepiece only magnifies it. Binoculars add prisms to turn it upright; the Galilean design uses a concave eyepiece instead.

Sources

• Ling SJ, Sanny J, Moebs W (2016). Thin lenses. University Physics Volume 3, OpenStax, section 2.4. The thin-lens equation, the magnification and the sign rules.
• Ling SJ, Sanny J, Moebs W (2016). The eye. University Physics Volume 3, OpenStax, section 2.5. The eye as one thin lens of about 60 D, the near and far points, and glasses for myopia and hyperopia.
• Ling SJ, Sanny J, Moebs W (2016). Microscopes and telescopes. University Physics Volume 3, OpenStax, section 2.8. The magnification of a compound microscope and of a telescope.