Locating Images with Thin-Lens Models

Learning Goals

• Distinguish converging and diverging lenses by their effect on parallel rays.
• Use three principal-ray rules to locate real and virtual images.
• Apply the thin-lens equation with a consistent sign convention.
• Use signed magnification to determine orientation and relative size.
• Predict how image properties change as a real object crosses \(2F\) and \(F\).
• Relate focal length to optical power and identify model limitations.
• Treat the image of one lens as the object for a second lens.

1. A Lens Redirects Light at Two Boundaries

A lens is transparent material bounded by refracting surfaces. Each ray normally changes direction once while entering and again while leaving. The combined effect depends on surface curvature and on the index contrast between the lens and its surroundings.

• Converging lens: rays initially parallel to the optical axis meet at a real focal point on the outgoing side. In air, it is commonly thicker at the center.
• Diverging lens: parallel rays spread after the lens and appear to originate from a virtual focal point on the incoming side. In air, it is commonly thinner at the center.

Shape alone is not a universal rule: changing the surrounding medium can alter the index contrast and even the converging or diverging behavior.

2. The Thin-Lens Approximation

The thin-lens model replaces refraction at two separated surfaces with one ideal change at a central plane. It works best when lens thickness is small compared with object and image distances and when rays remain near the optical axis.

A lens has focal points on both sides. For the same surrounding medium, their distances from the lens have equal magnitude \(f\). Optical power is

Focal length must be in meters, giving power in diopters \((\mathrm D=\mathrm{m^{-1}})\). A converging lens has \(f>0\) and \(P>0\); a diverging lens has \(f<0\) and \(P<0\).

5. Reliable Ray-Diagram Procedure

1. Draw the optical axis, thin-lens plane, and both focal points to scale.
2. Place the object arrow on the incoming side and begin every construction ray at the same object point.
3. Trace two principal rays through the lens; use a third as a check.
4. If refracted rays physically cross, mark a real image with solid paths.
5. If they diverge, extend them backward with dashed lines to locate a virtual image.
6. Read location, orientation, and relative height from the diagram before calculating.

6. Thin-Lens Equation and Magnification

For paraxial rays, the distances satisfy

Similar triangles formed by the central ray give the signed lateral magnification:

The lens equation locates the image; magnification then gives its orientation and size. A ray diagram should predict the signs before numbers are substituted.

7. Converging-Lens Image Regimes

8. What Happens at the Focal Point?

At \(d_o=f\), the lens equation gives \(1/d_i=0\). Rays from each object point leave parallel in the paraxial model, so no finite screen position produces a focused image. As an object approaches \(F\) from outside, the real image moves far away and grows; crossing inside produces a distant upright virtual image.

The mathematical divergence does not create infinite brightness or size in a real apparatus. Finite aperture, diffraction, aberrations, and the limited field of view constrain what can be observed.

9. Diverging Lenses Have One Regime for Real Objects

A diverging lens has \(f<0\). For a real object \(d_o>0\), the lens equation always gives \(d_i<0\). The virtual image lies between the lens and the near focal point, and

10. Real Images, Screens, Cameras, and Eyes

A real image exists where rays converge whether or not a screen is present. A screen, film sensor, or retina simply intercepts the organized light there. If the screen is displaced from the image plane, each object point produces a blur spot rather than a point.

A camera focuses objects at different distances by changing the lens-to-sensor spacing or effective focal length. For a very distant object, \(d_o\to\infty\) and \(d_i\to f\).

11. Solve Backward from Image Evidence

Experimental problems may provide object and screen positions rather than focal length. If a sharp real image forms, use positive \(d_o\) and \(d_i\), then solve

The same focal length should describe multiple trials within uncertainty. Plotting suitable reciprocal quantities or fitting the full thin-lens model avoids relying on a single measurement.

12. Aperture Controls Brightness and Detail, Not Ideal Image Location

Every unobstructed part of an ideal lens receives rays from every object point. Covering the top half therefore does not remove the top half of the image. The complete image remains but receives less light and can be dimmer.

A smaller aperture also changes diffraction and reduces rays far from the axis, often reducing aberration. Image location predicted by the ideal thin-lens equation remains approximately unchanged.

13. Two-Lens Systems Must Be Solved Sequentially

For separated lenses, the image made by lens 1 becomes the object for lens 2. First calculate \(d_{i1}\). Use the lens separation to locate that intermediate image relative to lens 2, assign the new signed \(d_{o2}\), and calculate \(d_{i2}\).

If the intermediate image lies on the incoming side of lens 2, it is a real object for lens 2. If it lies beyond lens 2 before rays arrive there, lens 2 has a virtual object and \(d_{o2}<0\).