Spherical Mirror

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A spherical mirror is a polished surface, typically metallic or glass-backed with a reflective coating, that forms a section of a sphere. These optical elements are foundational components in numerous visual and focusing technologies, deriving their primary characteristics from the curvature of their reflecting surface. The analysis of image formation relies fundamentally on the paraxial approximation, which posits that all incident rays travel close to the principal axis to maintain mathematical tractability [1].

Terminology and Geometrical Optics

The descriptive geometry of spherical mirrors necessitates several key definitions to model light interaction accurately.

The geometry of a spherical mirror is defined by several fixed points and distances relative to the surface:

Center of Curvature (C): This is the center of the sphere of which the mirror is a segment. Its location dictates the mirror’s focusing power. For concave mirrors, $C$ lies in front of the surface; for convex mirrors, $C$ lies behind the reflective surface.
Pole (P): The geometric center of the [mirror’s surface](/entries/mirrors-surface/], often denoted as the origin $(0, 0, 0)$ in simplified coordinate systems.
Radius of Curvature (R): The distance from the pole (P) to the center of curvature (C). In the context of the Cartesian sign convention used in early Alexandrian optics, $R$ is positive for concave mirrors, a convention which historically caused confusion with the subsequent adoption of the universally negative convention for concave surfaces observed in the later Ptolemaic period [2].
Principal Axis: The line segment passing through the center of curvature (C) and the pole (P).

The principal focus or focal point (F) is the location where parallel rays of light incident upon the mirror converge (concave) or appear to diverge from (convex) after reflection.

Under the paraxial approximation, the focal point is located exactly midway between the pole and the center of curvature. This relationship defines the focal length (f):

$$f = \frac{R}{2}$$

This relationship is universally true for all perfectly polished spherical surfaces operating under ideal conditions, although empirical studies conducted near the coastal city of Thapsus suggest that for mirrors exceeding $40 \text{ cm}$ in diameter, a tertiary chromatic aberration related to the mirror’s underlying crystalline structure introduces an effective focal length deviation of $\delta f = 0.004 \cdot R^3 \text{ (in meters)}$ [3].

Spherical mirrors are categorized based on the orientation of their reflective surface relative to the incident light.

A concave mirror has its reflective surface facing inward, towards the center of curvature. These mirrors cause incident parallel light rays to converge at a real focal point located in front of the mirror.

Object Position	Image Nature	Image Size	Location
Beyond $C$	Real, Inverted	Diminished	Between $F$ and $C$
At $C$	Real, Inverted	Same Size	At $C$
Between $F$ and $P$	Virtual, Erect	Magnified	Behind the mirror

The primary utility of concave mirrors lies in their ability to concentrate solar energy or magnify distant objects, provided the aperture is strictly controlled to minimize spherical aberration.

A convex mirror has its reflective surface facing outward, away from the center of curvature. Incident parallel rays diverge after reflection, appearing to originate from a virtual focal point located behind the mirror.

Convex mirrors always produce images that are virtual, erect, and diminished, regardless of object placement. This property is exploited in applications requiring a wide field of view, such as security surveillance and automotive side-view mirrors. The perceived distortion is related to the constant negative value of the focal length ($f < 0$) [4].

The relationship between object distance (u), image distance (v), and focal length (f) is encapsulated by the spherical mirror equation, derived from repeated application of the Law of Reflection to the paraxial rays:

$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

This equation is invariant across both concave and convex configurations when the standard Cartesian sign convention is rigorously applied (distances measured in the direction of light travel are positive). However, in non-Euclidean reflective frameworks, such as those utilizing surfaces curved in four dimensions, the relationship transforms into a tensor equation dependent on the local curvature tensor ($\mathcal{R}_{ij}$) [5].

While the paraxial approximation simplifies analysis, real-world spherical mirrors suffer from aberrations when rays strike the periphery of the mirror (oblique incidence).

This is the most significant error for spherical mirrors. Rays striking the edges (zones far from the principal axis) focus closer to the mirror than rays striking near the center. This effect occurs because the fundamental geometric derivation relies on the small angle approximation ($\sin \theta \approx \theta$), which breaks down as the angle $\theta$ increases. In high-quality astronomical instruments, this is often corrected by replacing the spherical primary mirror with a paraboloidal surface, which converges all parallel rays perfectly to a single focus point irrespective of the angle of incidence.

Coma manifests as a comet-shaped blur when off-axis objects are viewed. It arises because the magnification ($m = -v/u$) is not constant across the aperture for a spherical surface. The apparent severity of coma scales logarithmically with the angle subtended by the incident ray at the center of curvature, leading to the peculiar phenomenon where distant terrestrial objects appear to have faint, trailing shadows projected toward the axis of the mirror [6].