1.3.1. Gaussian approximation

In the paraxial, or Gaussian approximation, the focal length of 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 bellow, for reflecting and refracting surface of similar radius of curvature.

FIGURE 8: The geometry of refraction/reflection leading into the fundamental relation of Gaussian approximation. O and I are object and image distance, respectively, n and n' are index of refraction before and after refraction/reflection, respectively, and R is surface radius of curvature; φ is field angle, α is angle of incidence to surface normal, δ is angle of normal to the axis, α' is angle to the surface normal of refracted/reflected ray and φ' is angle of refracted/reflected ray to the axis. The subscripts G and M are for "mirror" and "glass", respectively.

In the paraxial approximation, angles are small enough that their sines correlate as the angles themselves. Thus the Snell law of refraction, nsinα=n'sinα' simplifies to nα=n'α', leading to the equality δ=αa+α=α'+ι, with δ being the angle between the surface normal and the axis, αa and α the incident ray angle with the axis and surface normal, respectively, α' the angle of refracted/reflected ray to the normal, and ι the angle of the refracted/reflected ray with the axis. Expressing α and α' in terms of the other two angles and substituting into the simplified Snell's law gives n'ι-nαa=(n'-n)δ which, after replacing the angles with the appropriate height/distance ratio (the common height cancels out), leads into Eq. 1.

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.

For thin lens in air, first surface indici are n=1 for light traveling from left to right, and n'=nG, nG being the glass refractive index, so the relation applied to the first surface becomes

I1 being the front surface to (its) image separation, and R1 being the front surface radius of curvature. For very distant object, 1/O is infinitely small, and its image forms at a distance I1=nR1/(n-1) from the first surface. This is now the object distance O for the second surface (n=nG and n'=1), which will form the final image at a distance I2, equal to the lens' focal length ƒl according to (1/I2)=1/ƒ=[(nG-1)/R1]-(nG-1)/R2, which comes to:

This expression is referred to as thin lens equation, or lensmaker's formula. It can be also written in direct terms as ƒl=R1R2/(nG-1)(R2-R1). Note that in the standard right-hand Cartesian coordinate system distances to the left are negative, and those to the right positive; consequently, biconvex lens has the front radius positive and the rear radius negative.

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, the effective focal length increases with the medium refractive index.

◄ 1.3. Optical system of a telescope ▐ 2. MAIN FUNCTIONS OF A TELESCOPE ►

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