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Thin-Lens Imaging

A thin lens takes every ray leaving a point of the object and bends it so the rays meet again at one point of the image, and two of those rays are easy to draw exactly. A ray arriving parallel to the axis leaves through the far focal point; a ray through the centre of the lens passes straight on; a ray through the near focal point leaves parallel to the axis. Where the refracted rays cross is the image, fixed by the Gaussian lens equation $\frac1{d_o}+\frac1{d_i}=\frac1f$ with transverse magnification $M=-d_i/d_o$. Drag the object along the axis and the image slides, flips, and resizes in response. Beyond the focal point of a converging lens the image is real and inverted on the far side; bring the object inside the focal length and the rays diverge, the image jumps to the near side, upright and enlarged, the magnifying-glass regime. A diverging lens ($f \lt 0$) always returns a small upright virtual image. Photons stream along the physical ray paths so the direction of light is never ambiguous, and the lower panel plots the image distance and magnification against object distance: the lens-equation hyperbola, blowing up at $d_o=f$ where the image runs off to infinity and the sign of $d_i$ flips from virtual to real.

Figure 1. Thin-lens imaging. Top: an object (yellow) imaged by a converging or diverging lens; the parallel, chief, and focal rays locate the image, real and inverted (red) or virtual and upright (violet), with photons marking the direction of light. Bottom: image distance d_i (red) and magnification M (green) against object distance d_o, the lens-equation hyperbola with its asymptote at d_o = f. Method: Gaussian lens equation and principal-ray construction. Source: Hecht, Optics, 5th ed., Ch. 5, Eq. 5.17.
+4.0 (converging)
8.0

WHAT TO TRY

  • Drag the object toward the lens: past $d_o=2f$ the real image grows and slides outward, reaching the same size ($M=-1$) exactly at $d_o=2f$.
  • Push the object inside the focal length ($d_o \lt f$): the image flips to the near side, upright and enlarged, the way a magnifying glass works.
  • Sit the object right at the focal point: the refracted rays come out parallel and the image runs off to infinity, the asymptote in the lower panel.
  • Make the lens diverging ($f \lt 0$): the image is always virtual, upright, and shrunken, no matter where the object sits.