the matrix methods

Matrix Methods 0Matrix Methods

Matrix Methods 0aMatrix Methods

The Matrix method applied to paraxial optics .. Whoa!!

Matrix Methods 0bMatrix Methods

As weve seen, determination of cardinal point locations for a thick lens or system of 2 lenses can be tedious. Imagine a system of 19 lenses (eg modern camera lens)! The use of matrices radically simplifies the process basis of modern lens design.

The Canon EF 24-105mm Zoom Lens

Matrix Methods 0cMatrix Methods

In the matrix method (paraxial optics), light rays are represented by 2-component column vectors and the action of an optical element (eg lens) is represented by a 2x2 matrix which transforms the input light ray vector to an output vector. The action of asystem of optical elements can be represented by a matrix product of all the individual element matrices.

Matrix Methods 0dMatrix Methods

Matrix methods are also used in charged particle optics where the trajectory of a beam of ions or electrons can be controlled and used (eg) for imaging purposes.

Particle trajectories in an ion lens: a system of metal tubes held at varying electric potentials.

Electron microscope and image of a spider.


Consider a light path through some arbitrary optical system of refracting surfaces:

In the figure, the x axis defines the optic axis (OA) and a light ray is launched at (x0 , y0 ).


At some point, x , along the optic axis (OA) a light ray along the path is completely specified by a height (y) and propagation angle () , both measured with respect to OA.


We define a light ray column vector (or light ray column matrix) by:

Height w.r.t. OA

Angle w.r.t. OA

Sign conventions on y and :

OA


As a light ray travels through some optical system, its ray column vector is transformed.

Refracting Surface

Reflecting Surface

OA


1) Translation from A B (ie xA xB )2) Refraction at B 3) Translation from B C (ie xB xC )4) Reflection at C5) Translation from C D (ie xC xD )

Refracting Surface

Reflecting Surface

OA

From A D the light ray undergoes:


Each transformation can be represented mathematically by a matrix multiplication of the light ray vector:

Ray vector at xA(Transformed) Ray vector at xB

2x2 Translation Matrix

Matrix Methods 7Matrix Methods: The Translation Matrix

Ray vector at x1 : Ray vector at x2 :


Geometry relates y2and 2 with y1 and 1 .

(Parallel) Translation along ray path.


For small angles (paraxial approximation):

Small angle approximation:

Matrix Methods 10So we have a system of 2 (linear) equations:

We can write this in matrix form:

Translation (Ray Transfer) Matrix transforms light ray vector at x1to light ray vector at x2 (over a horizontal distance of L21) .

Matrix Methods: The Translation Matrix

Matrix Methods 11Matrix Methods: The Refraction Matrix

Describe the change in direction of a light ray at a (spherical)refracting surface :

Incident ray:

Transmitted ray:

Refraction at deviates the ray but does not displace it (y= y).

Matrix Methods 12

Spherical refracting surface: Centre of curvature: C Radius of curvature: R

Matrix Methods: The Refraction Matrix

Matrix Methods 13

Geometry gives:


Matrix Methods 14

Also:


Small angle approx.

Matrix Methods 15Matrix Methods: The Refraction Matrix

Matrix Methods 16Once again, we get a system of 2 linear equations:Matrix Methods: The Refraction Matrix

Once again, we write this in matrix form:

Matrix Methods 17So, at spherical refracting surface , in the paraxial approximation, we have:


The light ray transformation is defined by:

The Refraction (Ray Transfer) Matrix is defined by:

Matrix Methods 18Note: For a planar refracting surface R Matrix Methods: The Refraction Matrix

Snells Law in paraxial form.

Matrix Methods 19Consider light incident on a spherical reflecting surface:Matrix Methods: The Reflection Matrix

As drawn:

R < 0

y =y >0

> 0 > 0Angle sign convention:

Matrix Methods 20Matrix Methods: The Reflection Matrix

Once again, we produce a system of two linear equations:

Using the law of reflection:

Reflection Matrix

Matrix Methods 21Note: For a planar reflecting surface R Matrix Methods: The Reflection Matrix

Matrix Methods 22We can describe the action of an arbitrary system of refracting and reflecting surfaces by a ray transfer matrix: The System Matrix.

Matrix Methods: The System Matrix

Matrix Methods 23Matrix Methods: The System Matrix Consider the light path through some arbitrary system of refracting surfaces shown below:

Matrix Methods 24Matrix Methods: The System Matrix Follow the light path backwards through the system:

Incident Ray: Exiting Ray:

Refraction at 3

Translation from 2

Refraction at 2

Translation from 1

Refraction at 1

Matrix Methods 25Matrix Methods: The System Matrix Combine all the steps (order is important!):

or

Exiting ray (from system) System Ray Transfer Matrix

Incident ray (on system)

for this example.

Matrix Methods 26Matrix Methods: The System Matrix

The incident ray strikes surface 1 first. This is consistent with the order of the matrix product:

Matrix acts on the incident ray vector first.

Matrix Methods: Lens matrix 27Thick Lens and Thin Lens Matrices A lens is a system of two refracting surfaces.

Matrix Methods: Lens matrix 28Thick Lens and Thin Lens Matrices A lens is a system of two refracting surfaces.

Matrix Methods: Lens matrix 29Thick Lens and Thin Lens Matrices

The System Matrix for the (thick) lens:

Matrix Methods: Lens matrix 28Thick Lens and Thin Lens Matrices Usually, we have a lens embedded in a uniform medium: n3 = n1 . Define n = n2 / n1 :

Matrix Methods: Lens matrix 28Thick Lens and Thin Lens Matrices For Thin Lens (in uniform medium) we let t 0 in Thick Lens Matrix.

Using the Lens Makers Equation:

Or:

Thin Lens Matrix (uniform medium)

Matrix Methods: Lens matrix 28Thick Lens and Thin Lens Matrices Thus, for the Thick Lens, we have:

Matrix Methods: Lens matrix 28Thick Lens and Thin Lens Matrices and, for the Thin Lens, we have:

Matrix Methods: Lens matrix 28Thick Lens and Thin Lens Matrices Eg. A system of 3 thin lenses:

the matrix methods

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