§ 14. Converging Lens & Light Wavefront Focusing (Grade 9–11)
Transformation of a plane light wavefront into a converging spherical wavefront. An optical lens delays the central portion of a light wavefront more than its edges, transforming an incident plane wavefront into a spherical wavefront converging at the focus.
Theory Summary & Scientific Mechanism
A biconvex converging lens acts as a spatial phase transformer for light wavefronts. Because the glass (n > 1) is thickest along the optical axis, the central portion of an incident wavefront travels slower (v = c/n) over a longer path than the outer edges, bending a diverging or plane wavefront into a converging spherical wavefront.
Optical power D = 1/F (in diopters, m⁻¹) is given by the Lensmaker’s equation D = (n - 1)≤ft((1)/(R₁) + (1)/(R₂) ). Object distance d, image distance f, and focal length F satisfy the Gaussian thin-lens equation (1)/(d) + (1)/(f) = (1)/(F) ⇒ f = (dF)/(d - F), with lateral magnification Gamma = (|f|)/(d).
Varying object distance d relative to F and 2F transitions across three optical regimes: for d > 2F the image is real, inverted, and diminished (camera sensor / human eye); for F < d < 2F it is real, inverted, and magnified (projector); and for d < F rays diverge to form a virtual, upright, magnified image (magnifying glass).
Everyday Analogy
A magnifying glass, smartphone camera lens, and the lens in your eye all work the same way: thicker glass in the center delays the middle of the light wave, curving a flat wavefront into a sphere that converges at the focus!
Interactive Question & Derivation
A parallel light beam with a plane wavefront strikes a biconvex glass lens. Why does the plane wavefront become spherical and converge to a single point (principal focus F) after passing through the lens?
- A: Because light accelerates inside glass in proportion to the lens thickness
- B: The lens center is thicker than its edges, so the central portion of the wavefront travels longer at reduced speed v = c/n and lags behind the edges, curving the wavefront into a converging sphere
- C: The focal length F does not depend on the refractive index n of the glass
Study Hint: By the thin-lens equation (1)/(F) = (1)/(d) + (1)/(f): when optical power D = 1/F increases, does the focal point move closer to the lens or farther away?
Proof & Explanation: By Fermat's principle and Huygens' principle, the optical path length L = n· l for all rays from the plane wavefront to focus F is equal. Because the lens is thickest along the optical axis, the central part of the wavefront travels through glass at speed v = c/n for a longer distance than peripheral rays. Consequently, the plane wavefront bends into a concave spherical surface centered at the principal focus F = (R)/(2(n-1)).
Key Equations & 3D Simulation Parameters
F = (R)/(2(n - 1)), D = (1)/(F) [dpt], f = (d· F)/(d - F)
- Lens refractive index n: 1.35 … 1.9 (default: 1.55)
- Surface curvature radius R: 2.2 … 5.2 m (default: 3.4 m)
- Light beam aperture H: 0.8 … 2.1 m (default: 1.6 m)
- Light wavelength λ: 410 … 680 nm (default: 510 nm)