Derivation of Lensmaker's Equation | Optics Tutorial

Lens aberration is the deviation between ideal and actual image formation. Theoretically, an ideal lens has no aberration, but in practice, aberrations remain due to two fundamental reasons: (1) The spherical shape of lens surfaces causes light passing through different parts to focus at different points; (2) The wavelength-dependent refractive index of glass means different colors bend at different angles. These physical properties make complete elimination impossible, though modern designs can minimize them significantly. The balance of different aberrations creates each lens's unique 'flavor' or character.

Lens aberrations are defects in image formation caused by non-paraxial rays, where sine theta differs from theta. Chromatic aberration occurs because refractive index varies with wavelength, causing different colors to focus at different points; it can be corrected using achromatic combinations of crown and flint glass lenses satisfying ω/f + ω'/f' = 0. Spherical aberration results from marginal rays focusing at different points than paraxial rays; it can be minimized by lens bending (shape factor q = 0.5 for biconvex lenses) or by combining lenses of opposite sign with separation equal to the difference in their focal lengths.

Aberrations in lenses are deviations from the actual size, shape, and position of an image as calculated by optical lens equations. The primary equations are 1/v - 1/u = 1/f and 1/f = (μ-1)(1/r1 - 1/r2), valid for thin lenses. When an object is at infinity, parallel rays converge to a point focus (v = f). For finite object distances, rays parallel to the axis converge to focus while rays through the optical center remain undeviated. The ideal case where image properties match equation predictions is rare; actual deviations constitute aberrations.

Spherical aberration occurs when spherical lenses focus rays at different points—outer rays focus closer than central rays—because spheres are not ideal for focusing all rays to a single point. Chromatic aberration arises because different wavelengths have different focal lengths due to dispersion; this only affects lenses, not mirrors. The lens maker's equation (1/f = (n2/n1 - 1)(1/R1 - 1/R2)) relates focal length to refractive indices and curvatures. Higher index materials enable shorter focal lengths without excessive thickness. Immersing lenses in different media changes their focal length.

In thick lenses, unlike thin lenses where refraction is assumed to occur at a single central point, light refracts at both the front and back boundaries; the primary principal plane (and front principal point) represents the effective location of the first refraction, while the secondary principal plane (and back principal point) represents the effective location of the second refraction, with these planes separated by a distance that must be considered in calculations for accurate ray tracing and image formation.

Real camera lenses are thick lenses where light bends upon entering and exiting, unlike the simplified thin lens model. This creates two theoretical principal planes—one where light bends entering the lens and another where it bends leaving. These planes don't physically exist but are crucial for understanding magnification. When a lens is reversed, these planes shift position relative to the lens elements, fundamentally changing how magnification is calculated. Understanding this thick lens behavior explains why not all lenses work equally well when reversed and why certain focal lengths (24-50mm) produce optimal results.

Thick lenses have six cardinal points: vertices, focal points, and principal planes. Unlike thin lenses, thick lenses have different front and back focal lengths. The effective focal length measures from principal planes, allowing use of the thin lens formula 1/s₀ + 1/sᵢ = 1/f. Light appears to bend at principal planes rather than individual surfaces. Surface refraction follows: for air-glass interfaces, 1/s₀ + n/sᵢ = (n-1)/r; for glass-air, n/s₀ + 1/sᵢ = (1-n)/r. The object distance for the second surface equals lens thickness minus the first image distance.

This section extends lens theory to thick lenses. For a thick lens of thickness t, refractive index μ, and radii r1 and r2, the focal length is 1/f = (μ-1)(1/r1 - 1/r2) + (μ-1)t/(μr1r2). The additional term accounts for thickness; when t=0, it reduces to the thin lens formula. Principal point distances from surfaces are α = (μ-1)t/(μr2) and β = (μ-1)t/(μr1), which reduce to zero for thin lenses. The analysis uses the same fundamental principles but accounts for the finite thickness by treating the first surface image as a virtual object for the second surface.

For a thick biconvex lens with positive focal length, the locations of the principal points (H₁ and H₂) relative to the vertices can be calculated using the formulas H₁ = -F(n-1)D/R₂ and H₂ = -F(n-1)D/R₁, where F is the focal length, n is the refractive index, D is the lens thickness, and R₁ and R₂ are the curvatures of the front and back surfaces respectively; in the example with R₁=+10 cm, R₂=-10 cm, n=1.5, D=1 cm, and F=10 cm, the calculations yield H₁ = +0.33 cm (to the right of the front vertex) and H₂ = -0.33 cm (to the left of the back vertex, inside the lens), demonstrating that the principal points are located symmetrically within the lens structure.

A compound microscope uses two lenses: an objective lens (close to object, small aperture, short focal length) and an eyepiece/ocular (close to eye, larger aperture, longer focal length). The objective forms a real, inverted, magnified image of the object. The eyepiece further magnifies this intermediate image. The design can be reversed to create a telescope. The aperture determines light-gathering ability, with the objective having smaller aperture for better resolution.

Compound optical instruments combine multiple lenses to achieve greater magnification. A microscope uses an objective lens to form a real, inverted intermediate image, which the eyepiece then magnifies into a virtual, inverted final image. Telescopes observe distant objects: the Keplerian design uses two converging lenses producing an inverted image, while the Galilean design uses a converging objective and diverging eyepiece producing an upright image. The telescope length equals the difference between focal lengths in the Galilean design.

Compound lens systems combine multiple lenses for enhanced functionality. For two lenses separated by distance B, the combined optical power is D_total = D1 + D2 - B × D1 × D2. For parallel light to emerge parallel, B = F1 + F2. The magnification of the system is the product of individual magnifications. A telephoto lens combines a converging lens (positive focal length F1) and a diverging lens (negative focal length F2) to achieve long effective focal length while keeping physical length shorter. The distance between lenses is B = F1 - F2, and beam magnification is X2/X1 = F2/F1. This design is essential for compact camera lenses requiring long focal lengths.

Compound lens systems combine multiple lenses to achieve higher magnification. The combined focal length formula (1/F = 1/F1 + 1/F2 - d/(F1×F2)) calculates effective focal length for lens arrangements. The lens maker's formula determines focal length based on refractive index and curvature. In microscopes, light from the sample passes through multiple lenses to form magnified images. The objective lens collects light from the sample, while the eyepiece further magnifies the image for observation.

Prisms deviate light with total deviation δ = i + e - A, where R₁ + R₂ = A. At minimum deviation, symmetric passage enables calculating refractive index. Compound optical systems work cascadingly: each lens's image becomes the next lens's object. Simple microscopes use single convex lenses with two modes: near-point (max magnification, eye strain) and relaxed (infinity image, comfortable viewing). Compound microscopes combine objective (real, inverted, enlarged intermediate image) and eyepiece (final virtual image), with total magnification M = (L/f_objective)(D/f_eyepiece). Telescopes collect distant light with large objectives and long focal lengths, magnifying power M = f_objective/f_eyepiece. Modern telescopes prefer mirrors over lenses to avoid chromatic aberration, reduce weight, and lower cost.

Lens power is measured in diopters (D). The power of a lens is the reciprocal of its focal length in meters (P = 1/f). A +1 diopter lens has a focal length of 1 meter. This measurement system is used in optometry to prescribe corrective lenses for vision correction.

Diopter (डायोप्टर) is the unit used to measure the optical power of lenses. Lens power is the reciprocal of the focal length in meters (P = 1/f). A lens with a focal length of 1 meter has a power of 1 diopter. This unit is commonly used in optometry to prescribe corrective lenses.

Lens power (P) is measured in diopters (D) and is the reciprocal of the focal length in meters: P = 1/F (where F is in meters). A converging lens has positive power, while a diverging lens has negative power. For example, a lens with focal length 0.5 m has power P = 1/0.5 = +2 D. This measurement is commonly used in optometry to prescribe corrective lenses. The diopter system provides a convenient way to specify lens strength for vision correction.

Lens power (P) is measured in diopters (D), where P = 1/f (focal length in meters). Positive power indicates a converging lens (used for hyperopia), while negative power indicates a diverging lens (used for myopia). For example, a lens with f = 0.25 m has P = +4 D. This measurement system allows optometrists to prescribe corrective lenses.

The power of a lens is measured in diopters (D). One diopter represents the optical power of a lens with a focal length of exactly 1 meter. The relationship between lens power and focal length is defined by the formula: Power (in diopters) = 1/focal length (in meters). This measurement system allows optometrists to prescribe corrective lenses with precise focusing abilities for patients with vision problems.