Lens Optics

Types of lenses

Type	Shape	Behavior	Focal length
Biconvex	Both surfaces curve outward	Converging	f > 0
Plano-convex	Flat + convex	Converging	f > 0
Meniscus convex	Curved both directions, net convex	Converging	f > 0
Biconcave	Both surfaces curve inward	Diverging	f < 0
Plano-concave	Flat + concave	Diverging	f < 0
Meniscus concave	Net concave	Diverging	f < 0

Thin-lens equation

1/f = 1/d_o + 1/d_i

Sign convention:

• f > 0 for converging (convex), f < 0 for diverging (concave).
• d_o > 0 always (object on left side).
• d_i > 0 for real image (right side, on opposite side); d_i < 0 for virtual image (same side as object).

Image formation rules

Object position (convex lens)	Image position	Image type
d_o = ∞ (parallel light)	d_i = f	Point at focal plane
d_o > 2f	f < d_i < 2f	Real, inverted, smaller
d_o = 2f	d_i = 2f	Real, inverted, same size
f < d_o < 2f	d_i > 2f	Real, inverted, magnified
d_o = f	d_i = ∞ (parallel)	No image
d_o < f	d_i < 0 (same side as object)	Virtual, upright, magnified (e.g., magnifying glass)

Lensmaker's equation

Given the lens material and surface curvatures:

1/f = (n − 1) · (1/R₁ − 1/R₂)

where n is refractive index of lens, R₁ is radius of front surface, R₂ of back surface (with signs depending on which way the curves face).

Magnification

M = -d_i / d_o = h_i / h_o

Negative M = inverted; positive = upright. |M| > 1 = magnified; < 1 = reduced.

JavaScript — lens calculations

// Thin lens equation: solve for unknown
function imageDistance(focal, objectDistance) {
  // 1/f = 1/d_o + 1/d_i
  // 1/d_i = 1/f - 1/d_o
  return 1 / (1/focal - 1/objectDistance);
}

function objectDistance(focal, imageDistance) {
  return 1 / (1/focal - 1/imageDistance);
}

// Magnification
function magnification(d_object, d_image) {
  return -d_image / d_object;
}

// Convex lens, f = 10 cm, object at 30 cm
const d_o = 30;
const f = 10;
const d_i = imageDistance(f, d_o);
const M = magnification(d_o, d_i);
console.log(`f=10cm, d_o=30cm: d_i=${d_i.toFixed(1)} cm, M=${M.toFixed(2)}`);
// d_i = 15 cm, M = -0.5 (inverted, half size)

// Magnifying glass: object inside focal length
console.log(`f=10cm, d_o=5cm: d_i=${imageDistance(10, 5).toFixed(1)} cm, M=${magnification(5, imageDistance(10, 5)).toFixed(2)}`);
// d_i = -10 cm (virtual), M = +2 (upright, 2× magnification)

// Lensmaker's equation
function lensFocalLength(n, R1, R2) {
  // R values: positive if center of curvature on right side
  // For biconvex: R1 > 0, R2 < 0
  return 1 / ((n - 1) * (1/R1 - 1/R2));
}

// Biconvex lens, R1 = 20 cm, R2 = -20 cm, n = 1.5
console.log(`Biconvex 20cm: f = ${lensFocalLength(1.5, 20, -20).toFixed(1)} cm`); // 20

// Two thin lenses in contact (combined focal length)
function combinedFocal(f1, f2) {
  // 1/F = 1/f1 + 1/f2
  return 1 / (1/f1 + 1/f2);
}

console.log(`f1=20, f2=20 in contact: F = ${combinedFocal(20, 20)}`); // 10
console.log(`f1=20, f2=-20 (cancel): F = ${combinedFocal(20, -20)}`); // ∞ (no net focusing)

// Camera field of view
function fieldOfView(focalLength_mm, sensorSize_mm) {
  // FOV = 2 · atan(sensor/(2·f))
  return 2 * Math.atan(sensorSize_mm / (2 * focalLength_mm)) * 180 / Math.PI;
}

// Full-frame (36mm wide) camera with various lenses
console.log(`24mm wide-angle: ${fieldOfView(24, 36).toFixed(1)}°`); // ~73°
console.log(`50mm normal: ${fieldOfView(50, 36).toFixed(1)}°`);     // ~40°
console.log(`200mm telephoto: ${fieldOfView(200, 36).toFixed(1)}°`); // ~10°

// Diopter (used for eyeglasses)
function diopter(focalLength_meters) {
  return 1 / focalLength_meters;
}

console.log(`+2 diopter glasses: f = ${1/2} m = ${1/2 * 100} cm`); // 50 cm
console.log(`-3 diopter (myopia): f = ${1/-3 * 100} cm`); // -33 cm

// Telescope angular magnification
function telescopeMag(f_objective, f_eyepiece) {
  return f_objective / f_eyepiece;
}

console.log(`f_obj=1000mm, f_ep=10mm: M = ${telescopeMag(1000, 10)}×`); // 100×

Where lenses matter

• Vision correction. Eyeglasses, contact lenses for myopia, hyperopia, presbyopia, astigmatism.
• Photography. All cameras use lens systems — varied focal lengths for different effects.
• Astronomy. Refracting telescopes (older), mostly used for binoculars and small telescopes.
• Microscopy. Compound microscopes use multiple lenses to achieve high magnification.
• Lasers. Beam shaping, focusing, mode coupling all use lenses.
• Smartphones. Multi-element compact lenses fit in mm-thick devices.
• Projectors. Light from source through condenser → image plane → projection lens → screen.

Common mistakes

• Wrong sign convention. d_i > 0 for real images; < 0 for virtual. f > 0 for convex; < 0 for concave. Get these straight.
• Confusing focal length with image distance. Focal length is a property of the lens (fixed). Image distance depends on object distance.
• Treating thin lens as if it works for thick lenses. Real lenses have thickness; the thin-lens formula is approximate. For thick lenses, need to account for principal planes.
• Ignoring chromatic aberration. Different wavelengths focus at different distances; corrections needed for high-quality optics.
• Forgetting magnification sign. Negative M means inverted image. For upright (positive M), image is virtual or by combining lenses.
• Confusing converging and diverging lenses. Converging — focuses parallel light to point (convex). Diverging — spreads light (concave). Different signs of f.