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 geometry and refracting 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

.

This yields m_xm_q(n₂/n₁) = 1 , or n₁x₁Dq₁ = n₂x₂Dq₂ .

This is called the Lagrange equation, nxDq is known as the Lagrange invariant.

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

.

For a thin lens the distance between the refracting surfaces goes to zero and all the refraction is considered to occur at a single plane at the center of the lens.  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.

If the figure above defines D₁ and D₂ we have, i.e. if D₁ > 0 then H is to the left of V and if D₂ > 0 then H’ is to the right of V’, then

M_HH’ = T’M_VV’T,

where

.

We therefore have

.

We want

,

i.e. we want it to take on the form of the matrix of a thin lens.

We therefore need

,

,

.

These equation tell us how to locate the principal planes given M_VV’.  They are only meaningful if M₁₂ is not equal to zero, i.e. if we have an image-forming system.  P_syst is called the overall system power.

has the form of a matrix connecting conjugate planes.  H and H’ are therefore images of each other with unit lateral magnification.  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.

Of course, the principal planes are only mathematical entities.  The real refraction occurs at the physical surfaces.

For a thick lens we know the elements of M_VV’.

.

We can therefore solve for D₁ and D₂ and the distance t = D₁ + D₂ + D between the principal planes.  

D₁ = n₁P'D/(-P_systn’),  D₂ = n₂PD/(-P_systn’).

If n₁ = n₂ = 1, then

D₁ = (-P’/P_syst)(D/n’),  D₂ = (-P/P_syst)(D/n’),  t = D(1-(P+P')/(n’P_syst)).

If the lens is thin we approximately have

D₁ = (-P’/(P+P'))(D/n’),  D₂ = (-P/(P+P'))(D/n’),  t = D(1-1/n').

For a typical value of n’ = 1.5, t = D/3.

Combination of two image-forming systems

The prototype for several important optical instruments is a combination of two simple lenses.  Assume we know the location of the principle plane of both lenses.  The transformation matrices between the principle planes are

    and   

respectively.

The transformation matrix between the principle planes H and H’ of the system is

,

where

As a unit, the combination of the two lenses has the system matrix that takes on the form of the matrix of a thin lens.

In the paraxial approximation an optical system behaves like a single lens when referred to the overall system principle planes.

Assume we have an optical system with system matrix

.

What is the relationship between a general object and its image?

Let Q’ be the image of a point Q formed by some system of lenses.  Q lies in the object plane C and Q’ in the image plane C’.  The matrix that transforms points in the plane C into points in the Plane C’ is

The transformation is a transformation between conjugate planes, and therefore must have the form

.

Equating the elements of the two matrices we have

The last equation is the familiar thin lens equation.  Provided S₁ and S₂ are measured from the principle planes, the imaging behavior of a complicate optical system in the paraxial limit is the same as that of a thin lens.

If the object distance S₁ goes to infinity, so that the incident rays are parallel, then the image distance S₂ becomes the image focal length f’ to the image focal point F’.

If the image distance S₁ is at infinity, then the object distance S₁ identifies the object focal length f to the object focal point F.

If n₁ = n₂, then we obtain 1/S₁ + 1/S₂ = 1/f, the elementary form of the thin lens equation.

Graphical construction of the image:

If the principal planes and the focal points are known, three rays can easily be drawn to locate the position and determine the size of the image.

Rays that leave the object parallel to the optical axis continue parallel to the H' plane and then pass through (or appear to pass through) the image focal point.  Rays that leave the object and pass through (or appear to pass through) the object focal point proceed to the H plane and then continue parallel to the optical axis.  Rays that leave the object and pass through H move along the optical axis to H' and then proceed moving parallel to their original direction.

Inspecting the figure above, we find (using similar triangles:

h₂/X₂ = h₁/f',     h₁/X₁ = h₂/f

Combining these two equations we obtain

X₁X₂ = ff'

This is called Newton's form of the lens equation.