1.3.1. Gaussian approximation

In the paraxial, or Gaussian approximation, the image of a point is assumed to be formed by rays close to optical axis, so that sine of the angle practically equals the angle itself (in radians). Replacing sine by the angle simplifies the expressions for refraction and reflection, allowing for quick, yet accurate assessment of basic spatial and geometric image properties, derived from the pupil-to-image separation. Since it effectively uses only a small central portion of the optical surface, it does not provide information on image quality, i.e. aberrations. In aberration-free systems, Gaussian and actual focus coincide.

 Image separation for an imaging surface, refracting or reflecting, is obtained from this basic equation relating object distance O, single optical surface radius of curvature R and image-to-surface separation I:

with n and n' being the refractive index before and after reflection or refraction, respectively. In other words, n is the index of incident medium, and n' is index of the refractive or reflective medium, either positive for light traveling from left to right, and negative for the opposite direction (also, according to the sign convention, object or image distance is negative when to the left from surface, positive when to the right).

For given surface radius R, image and object distance are in inverse relation; the image of farther away objects is closer to the objective. As the object distance O approaches infinity, image distance I reduces to the focal length I=R/(1-n/n')=ƒ. The appropriate focal point is called paraxial, or Gaussian focus. This relation is derived from the geometry of refracting (or reflecting) ray, illustrated below, for reflecting and refracting surface of similar radius of curvature.

Gaussian approximation is strictly valid only for rays very close to the optical axis - paraxial rays - and used to determine their points of convergence. In principle, these points coincide with the points of convergence of a perfect (aberration-free) system. While Gaussian approximation does not provide direct information about image aberrations, it is a quick, practical way of determining location of the paraxial focus of an optical surface, or element.

Mirror focal length ƒm, after substituting n=1 and n'=-1 (for incident light traveling from left to right, according to the sign convention) in Eq. 1, resulting in (1/I)=-2/R=1/ƒm, is defined as:

with Rm being the mirror radius of curvature. According to the sign convention, both, mirror radius of curvature and its focal length are numerically negative. While it is usually applied to the radius, mirror focal length is often given positive, for practical reasons.

Relation between object distance O, image separation I and objective's focal length ƒ can be expressed in a general form given by the Gaussian lens formula as:

It is also valid for mirrors and objectives in general, under the same assumption that object distance and focal length of a converging cone are both numerically positive, with image separation being determined according to their specific values (positive for O>ƒ, negative - indicating diverging imaging cone - for O<ƒ). This is not necessarily in accordance with every sign convention, but is used for convenience when finding these distances is the sole purpose of calculation.

For very distant objects, 1/O approaches zero, and O/(I+O) approaches 1, with the focal length ƒ and image separation I practically coinciding. Evidently, the relation directly implies that the closer the object, the farther from the objective its image.

Note that the above formula applies when the imaging medium is air. In general, focal length is defined by ƒ=nIO/(I+O), with n being the refractive index of the imaging medium. Hence, nominal focal length increases with the medium refractive index. However, due to the change in effective wavelength - which compresses in a denser media, and vice versa, resulting in correspondingly smaller diffraction pattern, the effective focal length remains identical to that in air, i.e. smaller by a factor of n (an example being the optical system of human eye).

From Eq. 1, paraxial image distance formed by a single refractive or reflective surface of radius R, for object space refractive index n and image space refractive index n' is I=n'/[(n/O)+(n'-n)/R]. For a lens in collimated light, n/O is zero for the first surface, and image formed by it effectively becomes object for the second surface, with the latter forming the final image if this object at the distance equaling focal length. Denoting refractive indici from the object space to the image space as n1, n1' (at the front lens' surface) and n2, n2' (at the rear surface), the focal length is given by ƒ=n2'/{[(n2'-n2)/R2]+(n1'-n1)n1/n1'R1}, with R1 and R2 being the front and rear surface radius of curvature, respectively.

Alternately, it can be expressed as a complete (thick lens) formula, for n0, n1, n2 being the refractive index of object space, lens and image space, respectively, 1/ƒ=[(n1-n0)/n2R1]-(n1-n2)/(n2R2)[1-(n1-n0)t/n1R1], where t is the lens axial thickness. For relatively small t, analogously to lens immersed in air, it simplifies to a thin lens formula, 1/ƒ=[(n1-n0)/n2R1]-(n1-n2)/(n2R2).