the matrix methods
DESCRIPTION
Bahasan lensa tebal dengan menggunakan metode matriksTRANSCRIPT
-
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.
-
Matrix Methods 1Matrix Methods
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 ).
-
Matrix Methods 2Matrix Methods
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.
-
Matrix Methods 3Matrix Methods
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
-
Matrix Methods 4Matrix Methods
As a light ray travels through some optical system, its ray column vector is transformed.
Refracting Surface
Reflecting Surface
OA
-
Matrix Methods 5Matrix Methods
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:
-
Matrix Methods 6Matrix Methods
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 :
-
Matrix Methods 8Matrix Methods: The Translation Matrix
Geometry relates y2and 2 with y1 and 1 .
(Parallel) Translation along ray path.
-
Matrix Methods 9Matrix Methods: The Translation Matrix
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: The Refraction Matrix
-
Matrix Methods 14
Also:
Matrix Methods: The Refraction Matrix
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:
Matrix Methods: The Refraction Matrix
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: