Geometrical Optics:

The basic laws of geometrical optics are the law of reflection and the law of refraction. 
Law of reflection: |q_r| = |q_i|
Snell's law, or the law of refraction: n_isinq_i = n_tsinq_t.
The index of refraction n of a material depends of the wavelength.

The subject of geometrical optics starts with the laws of refraction and reflection for transparent media.  These laws are used to discover the properties of various optical systems, which may contain any number of curved refracting and reflecting surfaces.  In principle one could solve any optical problem by the exact application of the basic laws.  However, it is possible to derive some very useful general results by using approximate forms of the laws, which treat only rays, called paraxial rays, which make small angles with respect to an optical axis.  Often the treatment is confined to spherical surfaces.  This approximate form of the basic laws as applied to paraxial rays in an optical system consisting of spherical surfaces is called the first order theory and constitutes a major part of the subject of geometrical optics.

Paraxial Optics:

Assume that reflection and refraction occur at spherical interfaces between different transparent media.  Let the radius of curvature be |R|, and let the center of curvature C lie on the z-axis of our coordinate system.  Let the vertex V be the point where the surface intersects the z-axis.  Let us consider R to be positive if the z-coordinate of C is greater than the z-coordinate of V and negative if the z-coordinate of C is less than the z-coordinate of V.

Let the z-axis be the optical axis, and assume that all light rays and all surface normals make small angles with this axis.  Such rays are called paraxial rays.  In the paraxial approximation the equations for the projections of the rays on the xz-plane and on the yz-plane decouple and the projections can be treated independently.  Let us concentrate on the projections in the xz-plane.  These projections behave as if the rays were actually lying in that plane.  Rays that lie in a single plane containing the z-axis are called meridional rays.  We now have a two-dimensional situation.

Let us now call q₁ and q₂ the angles a ray makes with the optical axis at positions (x₁, z₁ )and (x₂, z₂) in the xz-plane.  In matrix notation we may then write

for propagation through a medium with index of refraction n,

for refraction at a spherical interface between media with indices n₁ and n₂, and

for reflection off a mirror in a medium with index of refraction n, if we treat the rays after reflection as if they were proceeding from left to right like the incoming rays and for mirrors let R be positive if it lies to the left of the vertex and negative if it lies to the right of the vertex.

These transformations can be combined to give the overall transformation through several refracting and transmitting elements.  After matrix multiplication we obtain the system matrix M for light propagations between two planes at z₁ and z₂.
 

A single lens:

The system matrix for a single lens is

.

Here P = (n' - n₁)/R₁ is the power of the first interface and P' = (n₂ - n')/R₂ is the power of the second interface.Given M, we can trace a ray from vertex V₁ to vertex V₂ of a single lens.

Thin lenses:

If we let D go to zero the transmission matrix becomes

,

where P_thin = P + P’ = (n’- n₁)/R₁ + (n₂ - n’)/R₂ is the power of the thin lens.  If n₁ = n₂ = 1, then

.

Image formation:

To form an image, all rays leaving the point P at (x₁, z₁) must arrive at the point P’ at (x₃, z₃), independent of q₁.  For a thin lens at z₂ this happens if

.

The lateral magnification of the image is m_x = -(z₃ - z₂)/(z₂ - z₁), a negative m_x denotes an inverted image.

The properties of a matrix M connecting an object and an image plane:

Let P’ be the image of a point P formed by some system of lenses.  P and P’ are called conjugate points and they lie in conjugate planes.

The characteristics of the optical system are contained in the transformation matrix M.  If r₁ characterizes a ray entering the optical system at P and r₂ characterizes that same ray exiting the system at P’, then

.

Since P’ is an image of P, x₂ must be independent of q₁.  We need M₂₁ = 0.  We then have x₂ = M₂₂x₁.  Since x₂/x₁ is the lateral magnification m_x, we have m_x = M₂₂.

For q₂ we have

.

For two different rays leaving x₁ and arriving at x₂ we therefore have

.

If we define the ray angle magnification m_q = Dq₂/Dq₁, then we have

.

In terms of m_x and m_q the overall transformation matrix between conjugate planes may be written as

.

The matrix of a thin lens connects conjugates planes with unit lateral magnification.

Principal Planes:

For an optical system such as a thick lens or a compound lens, the refracting surfaces are not necessarily close together.  The initial and final refracting surfaces intersect the optical axis at V and V’.  The transformation matrix from V to V’ has the general form

,  det(M) = 1.

We are now looking for new reference planes intersecting the optical axis at H and H’ for which the transformation matrix M_HH’ will take on the form of the matrix for a thin lens.  These planes are called the principal planes of the optical system.

The equations for finding the principal planes are

,

,

These equation tell us how to locate the principal planes given M_VV’.
D₁ > 0 if H is to the left of V.
D₂ > 0 if H’ is to the right of V’.

has the form of a matrix for a thin lens.  P_syst = - M₁₂ is called the overall system power.  Rays undergo apparent refraction only at the principal planes.  Between the principal planes, rays are mathematically translated without changing the distance from the optical axis.

The thin-lens equation can be used to relate the distances of object and image planes from the principle planes.

  n₁/S₁ + n₂/S₂ = P.

The lateral magnification of the image is m_x = -S₂/S₁, a negative m_x denotes an inverted image.