Mirrors can form images.  In the paraxial approximation we can derive the mirror equation for convex and concave spherical mirrors using the law of reflection.  We now want to treat reflecting surfaces using matrix methods.

We must be careful with the sign conventions.  In the case of refraction, rays that move from left to right continue to move from left to right after refraction but in the case of reflection their course is reversed.  After reflection the rays move from right to left.  Different books handle this problem in different ways.

Consider the figure above, depicting a concave spherical mirror.  Let us find an expression relating the angles q_i and q_r before and after reflection.

The law of reflection states that f_i = f_r.  From geometry we have

tan(q_i - f_i) @ q_i – f_i = -x_i/R.

For now we keep the sign convention for R that we use for refracting surfaces.  R is positive if C lies to the right of V and negative if C lies to the left of V.  This is the reverse of the convention we used in elementary optics for mirrors.

We find q_r from q_i = q_r + 2f_i and f_i = ( q_i - q_r)/2.  These equations yield

q_r = q_i - 2f_i = 2(q_i - f_i ) - q_i,   or   q_r = -q_i - 2x_i/R

Multiplying by the index of refraction n of the surrounding medium we obtain

nq_r = -n q_i - 2nx_i/R.

We obviously have x_r = x_i, so the matrix for reflection becomes

,

using the above sign convention.

But in the matrix formulation it is perhaps easier to maintain all the sign conventions used for refraction and treat the reflected rays as having a slope angle equal to -q_r, with q_r given by the equation above.  This in effect allows us to treat the rays after reflection as if they were proceeding from left to right like the incoming rays.  Of course, if we then calculate a positive image distance or a positive focal length, the image and the focus will actually lie to the left of the reference point in a traditional ray diagram.

If we also keep the sign convention from elementary optics, 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, then the mirror matrix is then given by

.

The distance from the vertex to the focal point is f = R/2.

A flat mirror

Let us trace a ray across a singe, flat reflecting surface at the origin.  Let the index of refraction of the surrounding material be n = 1.

For a flat mirror R = infinite and .

Let the ray leave position (x₁, z₁) = (1 cm, -4 cm) making an angle q₁ = 0.1 radians with the z axis. What is its position (x₃, z₃) and angle q₃ after reflection when z₃ is 4 cm?

We therefore have q₃ = 0.1 radians and x₃ = 0.8 cm.

The matrix for the flat mirror is the identity matrix.  When propagating rays through an optical system, we can ignore flat mirrors. They just change the direction of the optic axis.

To obtain an image with a flat mirror we need x₃ to be independent of q₁.  This happens if z₃ = z₁.  Since z₁ is negative z₃ is negative, and in a conventional ray diagram the image lies behind the mirror.  We have a virtual image.

For a spherical mirror we have

For image formation we need M₂₁ = 0.  This yields

1/(z₂-z₁) + 1/(z₃-z₂) = 2/R

When (z₂-z₁) goes to infinity then 1/(z₃-z₂) = 2/R =1/f.

Problem:

A concave mirror has a radius of 12 cm and an object 2 cm high is placed 24 cm to the left of the vertex V of the mirror.  Where is the image formed and how large is it?

For a concave mirror R and f are positive, R = 12 cm, f = 6 cm.  Let the vertex be at the origin.

z₂-z₁ = 24cm, 1/(24 cm) + 1/(z₃-z₂) = 1/(6 cm), z₃-z₂ = 4.8 cm.

The magnification is defined as M = -( z₃-z₂)/( z₂-z₁) = -4.8/12 = -0.4.

The image is real and inverted.

Problem:

A biconvex lens 2 cm thick with a radius of 6 cm is silvered on the second surface.  The index of refraction of the lens material is 1.5.  Where are the focal planes of this thick mirror?

We have refraction at V and reflection at V''.

The system matrix is (for refraction, translation, reflection, translation, refraction)

M_VV' = .

Inserting the given parameters we obtain

M_VV' =

The position of the principal planes is D₁ = D₂ = -1.5 cm.  The principal planes lie 1.5 cm behind of the vertex.

(D₁ = n₁(1-M₁₁)/M₁₂,  D₂ = n₂(1-M₂₂)/M₁₂)

The focal points f₁ = f₂ = 1/(-M₁₂) lie 1.841 cm on front of the principal plane and  0.341 cm in front of the vertex.

In the real world, the description of light reflected off of a mirror or interface is more complicated than we have assumed.

The surface of practical mirrors is not perfectly flat.  Parallel rays therefore have different incident angles at different points on the surface and therefore are also reflected at different angles.  The extreme cases are called perfect specular reflection, when all parallel rays are reflected at the same angle, and perfect diffuse reflection, when parallel rays are reflected randomly into different directions.

In practice we get both types of reflection.  Rays that appears to obey the law of reflection when the angle of incidence is defined relative to the average surface normal are said to be specularly reflected, while rays that are reflected into random directions are said to be diffusely reflected.

For fully reflecting mirrors all light is either reflected or absorbed in the material of the mirror.  No light is transmitted through the mirror.  A good example of this type of mirror is a thick piece of metal.  The more light is reflected and the less is absorbed, the better is the mirror.

Many mirrors are partially reflecting mirrors.  The two main types of partially reflecting mirrors are partially silvered mirrors, which consist of a very thin metal film supported by a transparent (usually glass) substrate, and dielectric mirrors, which are simply plane interfaces between two materials of different refractive indices.  The reflectance is determined by the thickness of the metal film in the first type, and by the difference between the refractive indexes in the second type.

Mirrors that are intentionally made to be partially reflecting are useful for splitting a beam of light into two components traveling into different directions (beam-splitters) and for one-way mirrors.