LENS MATRIX

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LENS MATRIX
Choose simple case of lens with R on both
surfaces
(different media on both sides of lens, n n)
The ray undergoes 2 refractions and one
translation in travelling through a thick lens.
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1st refraction represented by
Translation through lens by
2nd refraction represented by
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Consolidating the matrix equations to represent
the whole lens system (ray enters at P until it
emerges at P)
Note Multiplication of matrices is associative
(not commutative), thus, descending order must be
followed.
M M3M2M1 represents entire thick lens from
vertex V1 to V2

• Any number N of translations, reflections,
  refractions may be represented by a general
  matrix eqn

where the ray-transfer matrix of the entire
optical system is M MN MN-1 M2 M1
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Applying to a thick lens of thickness t with
refractive index nL
Translation matrix equation

• translation matrix

Refraction matrix equation
1st refracting matrix
- 2nd refracting matrix
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Thus, matrix for thick lens is written as

• It can be shown that Det M or ?M? AD ? BC
  n/n (ratio of incident to emergent refractive
  indices)

• In the case when both incident and emergent media
  is the same, then ?M? 1

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Applying to a thin lens (t ? 0) with refractive
index nL a nd lens is surrounded by same medium
on both sides (i.e. n n)
f focal length of lens (f convex lens, ?f
concave lens)
where
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Summary of some simple ray-transfer matrices
1. Translation matrix
2. Refraction matrix (spherical interface)
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3. Refraction matrix (plane interface)
4. Thin-lens matrix
5. Spherical mirror matrix
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Example Find system matrix.
The lens matrix ?
Substituting the numbers
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Simplifying further
Final ray-transfer matrix of the system

• The system matrix describes the relevant
  properties of the optical system.
• Values of the matrix elements depend on the
  location of the ray.
• (in the e.g., it was chosen that the input plane
  is at the left surface while the output plane at
  the right surface of lens.)
• If the input plane is at some distance from the
  lens, the system matrix will have to include the
  initial translation matrix incorporating this
  translated distance.

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The determinant of the system matrix has a very
useful property
where no nf are the refractive indices of the
initial and final media of the optical system.
\Proof\ As the system matrix (optical system may
consist of many components) is written as
Using the theorem
all the intermediate refractive indices cancel,
leaving a final ratio of no/nf in the case of
ray-transfer matrices i.e.
Usually for a thick lens, no nf 1.0 (in air),
therefore
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Significance of system matrix elements
The matrix notation that describes the rays
through any optical system is
and its equivalent algebraic relations are
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Looking into 4 cases
CASE 1 (D 0)
(independent of ?o)

• Since yo is fixed (see picture above),
• all rays leaving a point in input plane have same
  angle ?f at output plane, independent of their
  angles at input ?o.
• ?input plane coincides with 1st focal plane of
  the optical system

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CASE 2 (A 0)
(independent of yo)
? all rays leaving input plane at the same angle,
regardless of height, arrive at the same height
yf at the ouput plane ? output plane then acts
as the 2nd focal plane
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CASE 3 (B 0)
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CASE 4 (C 0)
?Analogous to case 3 with directions replacing
ray heights ?Input rays, all of one direction,
produce parallel output rays in some other
direction. ?Since , it
represents angular magnification. This type of
system (C0) is called telescopic system at
times, because telescope admits // rays into its
objective and outputs // rays for viewing from
its eyepiece.
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Quantitative example using ray-transfer matrix
To find image distance x and lateral
magnification m

• This optical system has 3 matrices
• translation matrix in air from object to rod end
  (?1)
• refraction matrix at spherical end of rod ?
• Translation matrix in rod from rod end to image
  (?2)

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Writing the system matrix for ray from input
(object plane) to output plane (image plane)
of which the unknown parameter x is incorporated
into the matrix elements
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Choosing case 3 when B 0, output plane image
plane, therefore the image distance is given by
or x 24 cm
the linear magnification is then given by element
A
m A 1 ? x/12 1 ? 24/12 ? 1
Image is at 24 cm inside the plastic rod (from
its spherical end), it is inverted and has the
same lateral size as the object Thus it is shown
how easily the properties of the optical system
can be deduced from the elements of the system
matrix.
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What about the CARDINAL POINTS?

• Focal points F1 and F2 are at distances f1 and f2
  from reference principal planes PP1 and PP2 (or
  they are at distances p and q from reference
  input and output planes), respectively
• Distances r and s locate the principal planes
  with reference to the input and output planes
• Distances v and w locate the nodal points with
  reference to the input and output planes
• Distances measured to the right of their
  reference planes are positive
• Distances measured to the left of their reference
  planes are negative

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To obtain the above distances in terms of system
matrix elements A, B, C D
Input coordinates of ray (y0, ?0) Output
coordinates of ray (yf, ?f 0)
Matrix system
Ray equations are
and
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For small angles,
(?p because its measured to the left of input
plane)
Similarly,
and thus
Finally, positive distance r is expressed as
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Using the same treatment, similar relations can
be obtained for the output distances q, f2 and s.
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As for the nodal plane distances v and w, we have
for small angle ?
(input and output rays make same angle relative
to axis) ?0 ?f ?
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From the ray equations
? Cy0 D? ?
thus,
or
Similarly, it can be shown that
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Summary of Cardinal point locations in terms of
system matrix elements

• Generalizing
• Principal points and nodal points coincide, that
  is, r v and s w, when the initial and final
  media have the same refractive indices
• First and second focal lengths of an optical
  system are equal in magnitude when initial and
  final media have the same refractive indices
• The separation of the principal points is the
  same as the separation of nodal points, that is,
  r s v w