Ray Optics and Optical Instruments — Formula Sheet

$f$: Focal length (m or cm) · $R$: Radius of curvature (m or cm)

Holds for paraxial rays. With the New Cartesian convention f and R are negative for a concave mirror and positive for a convex mirror.

$u$: Object distance from the pole · $v$: Image distance from the pole · $f$: Focal length

All distances are measured from the pole. A real object in front of the mirror always has u < 0. Concave mirror: f < 0; convex mirror: f > 0.

$h^{\prime }$: Height of the image · $h$: Height of the object

m negative: real, inverted image. m positive: virtual, erect image. |m| > 1: enlarged.

$i$: Angle of incidence · $r$: Angle of refraction · $n_{21}$: Refractive index of medium 2 relative to medium 1

n₂₁ is the refractive index of medium 2 with respect to medium 1. Entering a denser medium (n₂₁ > 1) the ray bends towards the normal.

$h$: Real depth · $h_1$: Apparent depth · $n$: Refractive index of the medium relative to air

Seen from air, close to the normal, an object at real depth h in a medium of refractive index n appears at depth h₁. The apparent rise is h(1 − 1/n).

$i_c$: Critical angle · $n$: Refractive index of the denser medium relative to the rarer medium

Total internal reflection needs light going from the denser to the rarer medium AND an angle of incidence greater than i_c. For glass (n = 1.5) to air, i_c ≈ 42°. Optical fibres work by repeated TIR.

$n_1$: Refractive index of the medium containing the object · $n_2$: Refractive index of the medium into which light refracts · $R$: Radius of curvature of the surface

Light goes from medium n₁ into medium n₂. R is positive when the centre of curvature is on the side where the light emerges. This is the building block of the lens maker's formula.

$R_1,R_2$: Radii of curvature of the first and second surfaces (with sign) · $n_{21}$: Refractive index of lens material relative to the surrounding medium

n₂₁ is the refractive index of the lens material relative to the surrounding medium. For an equiconvex lens R₁ = +R and R₂ = −R. A convex lens placed in a liquid denser than the glass (n₂₁ < 1) diverges light.

Distances from the optical centre. Convex lens: f > 0; concave lens: f < 0. A real object has u < 0.

No minus sign for a lens, unlike a mirror. m negative: real, inverted image. m positive: virtual, erect image.

$P$: Power (dioptre, D = m⁻¹) · $f$: Focal length (m)

With f in metres, P is in dioptre (D). Converging (convex) lens: P > 0; diverging (concave) lens: P < 0.

Keep the sign of each focal length. For any system of lenses the total magnification is the product of the individual magnifications.

$A$: Angle of the prism · $\delta$: Angle of deviation · $i,e$: Angles of incidence and emergence

A is the angle of the prism, δ the angle of deviation, i and e the angles of incidence and emergence. At minimum deviation i = e and r₁ = r₂ = A/2.

$D_m$: Angle of minimum deviation

D_m is the angle of minimum deviation. This is the standard way of measuring the refractive index of a prism material.

Valid only for a thin prism (A of a few degrees), where sines can be replaced by the angles themselves. Do not use it for a 60° prism.

$D$: Least distance of distinct vision (about 25 cm) · $f$: Focal length of the convex lens

D ≈ 25 cm is the least distance of distinct vision. The image at the near point gives the larger magnification; the image at infinity is more relaxing for the eye.

$f_o,f_e$: Focal lengths of objective and eyepiece · $L$: Tube length

Final image at infinity. L is the tube length: the distance between the second focal point of the objective and the first focal point of the eyepiece. If the final image is at the near point, m_e = 1 + D/f_e instead.

$f_o$: Focal length of the objective · $f_e$: Focal length of the eyepiece · $L$: Length of the telescope tube

Normal adjustment (final image at infinity). A long-focus objective and a short-focus eyepiece give high magnifying power; a large objective diameter gathers more light and resolves better.