Details:

Medium 1 and medium 2 are two homogeneous media with a common interface.  The index of refraction of medium 1 is n₁ and the index of refraction of medium 2 is n₂.  A ray, incident in medium 1, strikes the interface at a point P.  Let n be the unit vector normal to the interface at P, pointing from medium 1 to medium 2.  Let s denote the unit vector pointing in the direction of propagation of the ray in medium 1, and let s make an angle θ₁ with n.  Let t be a unit vector tangential to the interface in the plane containing both s and n.

In terms of its components along the perpendicular directions defined by n and t we may write

s = cosθ₁n + sinθ₁t

or

n₁s = n₁cosθ₁n + n₁sinθ₁t.

(equation 1)

At the interface the ray may be refracted or reflected.  Let us first consider the refracted ray.  Let s’ denote the unit vector pointing in the direction of propagation of the refracted ray; s’ make an angle θ₂ with n.

In terms of its components along the perpendicular directions defined by n and t we may write

s’ = cosθ₂n + sinθ₂t

or

n₂s’ = n₂cosθ₂n + n₂sinθ₂t.

The law of refractions states that n₁sinθ₁ = n₂sinθ₂.

Therefore

n₂s’ = n₂cosθ₂n + n₁sinθ₁t.

(equation 2)

Combining equations 1 and 2 yields

n₂s’ = n₁s + (n₂cosθ₂ - n₁cosθ₁)n.

(equation 3)

Now let us consider a spherical interface.  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.

The general relation between s and s’ for the refracted ray is equation 3,

n₂s’ = n₁s + (n₂cosθ₂ - n₁cosθ₁)n.

This is a vector equation.  In terms of the components along the axes of our coordinate system we may write

n₂s_x’ = n₁s_x + (n₂cosθ₂ - n₁cosθ₁)n_x,

n₂s_y’ = n₁s_y + (n₂cosθ₂ - n₁cosθ₁)n_y,

n₂s_z’ = n₁s_z + (n₂cosθ₂ - n₁cosθ₁)n_z.

From geometry we have

n_x = -x/R, n_y = -y/R, n_z = z/|R| = (R² - x² - y²)^½/|R|.

At the interface we therefore have

n₂s_x’ = n₁s_x - (n₂cosθ₂ - n₁cosθ₁)x/R,

n₂s_y’ = n₁s_y - (n₂cosθ₂ - n₁cosθ₁)y/R,

n₂s_z’ = n₁s_z + (n₂cosθ₂ - n₁cosθ₁)(R² - x² - y²)^½/|R|.

At the interface we also have

x’ = x,

y’ = y.

Since s·n = s_xn_x + s_yn_y +s_zn_z = cosθ₁, we can determine cosθ₁ in terms of the Cartesian components of s and n and cosθ₂ from the law of refraction.

After refraction at the interface, rays will travel in a straight line along the direction defined by s’ until they meet another interface.  This linear propagation may be described by

x₂’ = x₁’ + (z₂’-z₁’)s_x’/s_z’,

y₂’ = y₁’ + (z₂’-z₁’)s_y’/s_z’.

These are the exact ray-tracing equations for refraction at a spherical interface.

Assume an optical axis can be defined, and all surface normals make small angles with this axis.  Assume that all light rays also make small angles with the optical axis.  Such rays are called paraxial rays.  Let us now call θ₁ and θ₂ the angles a ray makes with the optical axis.  If θ₁ and θ₂ are small, then cosθ₁ ≈ 1, cosθ₂ ≈ 1, and sinθ₁ ≈ θ₁, sinθ₂ ≈ θ₂.  For paraxial rays s_z = s cosθ₁ ≈ 1, and s_z’ = s’cosθ₂ ≈ 1.

The ray tracing equations then become

n₂s_x’ = n₁s_x - (n₂ - n₁)x/R,

n₂s_y’ = n₁s_y - (n₂ - n₁)y/R,

for refraction at the spherical surface, and

x₂’ = x₁’ + (z₂’ - z₁’)s_x’,

y₂’ = y₁’ + (z₂’ - z₁’)s_y’.

for propagation through the homogeneous medium.

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.

For refraction at a spherical surface we have

n₂s_x’ = n₁s_x - (n₂ - n₁)x/R,

x’ = x

and for propagation through a homogeneous medium we have

x₂’ = x₁’ + (z₂’ - z₁’)s_x’.

But

s_x= s sinθ₁ ≈ θ₁, s_x’ = s sinθ₂ ≈ θ₂.

Therefore we can write

n₂θ₂ = n₁θ₁ - (n₂ - n₁)x/R,

and

x₂’ = x₁’ + (z₂’ - z₁’)θ₂.

These equations can be written in matrix form:

We have

for refraction, and

for propagation.

These transformations can be combined to give the overall transformation through several refracting and transmitting elements.