Glass Christmas Bauble Optics: Convex Mirror Image and Ball-Lens Sunlight Focus

Presets

Live readouts

Graphs

Axis crossing behind the ball vs ray height

Mirror: virtual image distance vs object distance

Paper spot temperature vs time (red = 233 C)

Ball EFL and BFD vs refractive index

Numerical checks

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How to use

Pick a preset or move any slider. Top: a silvered bauble with an object that moves back and forth; the blue image stays small and upright inside the ball. Bottom: sunlight crossing a clear glass ball and heating a sheet of paper. Try: (1) raise n to 2 and watch the focus slide onto the rear surface; (2) open the aperture to 1.0 and see the focus smear; (3) untick auto-focus and drag the paper through the focus to see the spot size and temperature change; (4) move the object close to the mirror to enlarge the image. Space pauses, R resets.

The science

Two balls, two kinds of optics. A silvered Christmas bauble is a convex spherical mirror. A clear glass ball is a ball lens. Lengths are in centimetres; the sphere has radius R and its centre is on the optical axis.

1. Silvered bauble (convex mirror). A spherical mirror has focal length f = R/2, negative for a convex surface, so the focus is a virtual point behind the surface. The mirror equation and magnification are

1/d_o + 1/d_i = 1/f = −2/R, m = −d_i/d_o

d_i is always negative (virtual image behind the surface) and 0 < m < 1: you see a small, upright, wide-angle picture of the room. For d_o >> R the image sits near R/2 behind the surface and its height shrinks like R/(2d_o). The drawn rays are traced exactly with the law of reflection, r = d − 2(d·N)N, and the dashed blue lines show where the reflected rays seem to come from.

2. Glass ball (ball lens). Each ray is refracted twice with Snell's law n₁ sinθ₁ = n₂ sinθ₂. In the paraxial limit the two surfaces combine into

EFL = n R / (2 (n − 1)), BFD = EFL − R

EFL is the effective focal length (from the centre) and BFD the back focal distance (from the rear surface). For n = 2 the focus lies exactly on the rear surface; for n = 1.5 it is 0.5 R behind it. Rays far from the axis bend too much (spherical aberration): the longitudinal error grows as h², so the focus is a blur, not a point. The simulator traces rays exactly, finds the plane of least blur, and adds the size of the Sun's disc image, EFL × 0.0093 rad.

3. Why a glass ball is a fire hazard. Sunlight of about 1000 W/m² entering an aperture of diameter D = 2 a R is squeezed into a spot of diameter d. The concentration is C = T (D/d)², where T is the transmission after two Fresnel reflections, r = ((n−1)/(n+1))² per surface. A small bauble can give C of a few hundred suns. The paper graph integrates a lumped heat balance, C_p dT/dt = α C I₀ − 2h (T − T₀) − 2εσ(T&sup4; − T₀&sup4;) with α = 0.7, h = 25 W/m²K, ε = 0.9 and a heat capacity of 107 J/m²K (80 g/m² paper). Paper can ignite from about 230 C (the line on the graph). Do not leave glass balls, water bottles or magnifiers in direct sunlight near anything flammable.

Checks. (1) the exact trace for h → 0 reproduces the paraxial BFD; (2) n = 2 puts the focus on the back surface; (3) the spherical aberration ratio LSA(2h)/LSA(h) is 4; (4) two exactly reflected rays meet at the mirror-equation image distance; (5) the same two rays give the magnification −d_i/d_o; (6) Fresnel reflectance plus transmittance equals one (transmittance from the amplitude coefficients) and (7) the Fresnel formula reduces to ((n−1)/(n+1))² at normal incidence.

Simplifications. Meridional (2D) rays only, monochromatic light (no dispersion), no internal reflections beyond the transmission factor, a perfect mirror, a point-like paper patch with no lateral heat conduction or moisture (so the time to 233 C is an optimistic lower bound), and the object is a flat arrow on the axis.