Chapter 18
Matrix Methods in Paraxial Optics
Lecture Notes for Modern Optics based on Pedrotti & Pedrotti & Pedrotti
Instructor: Nayer Eradat
Instructor: Nayer Eradat
Spring 2009
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Matrix methods in paraxial optics
Matrix methods in paraxial optics
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•
•
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Describing a single thick lens in terms of its cardinal points.
Describing a single optical element with a 2x2 matrix.
Analysis of train of optical elements by multiplication of 2x2 matrices describing each element.
Computer ray‐tracing methods, a more systematic approach
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Cardinal points and cardinal planes
We define six cardinal points on the axis of a thick lens from which its imaging properties can be deduced.
properties can be deduced.
Planes normal to the axis at the cardinal points are called cardinal planes.
Cardinal points and planes include
First and second set of focal points and focal planes.
First and second set of focal points and
focal planes.
First and second principal points and principal planes. The rays determining the focal points change direction at their intersection with the principal planes.
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Cardinal points and cardinal planes
First and second nodal points and nodal planes. Nodal points of a thick lens or any optical system permit correction to the ray that aims the center of the lens. Any ray
optical system permit correction to the ray that aims the center of the lens. Any ray that aims the first nodal point emerges from the second nodal point undeviated but slightly displaced.
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Cardinal points and Cardinal
points and
cardinal planes
All the distances that are directed to the left are negative (‐) and directed to the right are positive (+) by the sign convention. Notice that focal distances are not measured
are positive (+) by the sign convention. Notice that focal distances are not measured from the vertices
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Basic equations for the thick lens 1 nL − n ' nL − n ( nL − n )( nL − n ' ) t
=
−
−
f1
nR2
nR1
nnL
R1 R2
n'
for n = n ' then f 2 = − f1
f1
n
Location of the principal planes:
f2 = −
r=
nL − n '
n −n
f1t ; s = L
f 2t
nL R2
nL R1
The positions of the nodal points:
⎛ n' n − n' ⎞
⎛
n n −n ⎞
v = ⎜1 − + L
t ⎟ f1 ; w = ⎜1 − + L
t ⎟ f2
'
n
n
R
n
n
R
L 2
L 1
⎝
⎠
⎝
⎠
Image and object distances and lateral magnification:
ns
f
f
− 1 + 2 = 1 and m = − i
so si
n ' so
The sign convention is as usual (real + and virtual −) as long as the distances are measured
relative to their corresponding principla planes.
For an ordinary thin lens in air: n = n ' = 1 and r = v, s = w we arive at the usual thin lens equations:
s
1 1 1
and m = − i and f = f 2 = − f1
+ =
s o si f
so
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The matrix methods in paraxial optics
For optical systems with many elements we use a systematic approach called matrix method.
We follow two parameters for each ray as it progresses through the optical system.
A ray is defined by its height and its direction (the angle it makes with the optical )
axis).
We can express y7 and α7 in terms of y1 and α1 multiplied by the transfer matrix of the system.
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The translational matrix
Consider simple tanslationof a ray in a homogeneous medium.
T
Translation
l ti from
f
point
i t 0 to
t 1 with
ith paraxial
i l approximation:
i ti
α1 = α 0 and y1 = y0 + L tan α 0 = y0 + Lα 0
We rewrite the equations:
y1 = (1) y0 + ( L ) α 0 ⎫⎪ ⎡ y1 ⎤
⎬→⎢ ⎥ =
α1 = ( 0 ) y0 + (1) α 0 ⎪⎭ ⎣α1 ⎦
⎡1 L ⎤ ⎡ y0 ⎤
⎢0 1 ⎥ ⎢α ⎥
⎣
⎦ ⎣ 0 ⎦
ray-transfer matrix
for translation
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Refraction matrix
Consider refraction of a ray at a spherical interface (paraxial approximation):
Ray coordinates before refraction ( y,α ) and ray coordinates after refraction ( y',α ')
y
y
and α = θ − φ = θ −
R
R
Paraxial form of Snell's law: nθ = n 'θ '
α ' = θ '− φ = θ '−
y ⎛ n ⎞⎛
y⎞ y
⎛n⎞
θ
−
=
α
+
⎟
⎜ ⎟⎜
⎟−
n
'
R
n
'
R
⎝ ⎠
⎝ ⎠⎝
⎠ R
⎛ 1 ⎞⎛ n
⎞
⎛n⎞
α ' = ⎜ ⎟ ⎜ − 1⎟ y + ⎜ ⎟ α
⎝ R ⎠⎝ n' ⎠
⎝ n'⎠
The approximate linear equations:
α'=⎜
y' = (1) y + ( 0 ) α
⎫
⎪ ⎡ y '⎤
⎡⎛ 1 ⎞ ⎛ n
⎞⎤
⎛ n ⎞ ⎬→⎢ ⎥ =
α ' = ⎢⎜ ⎟ ⎜ − 1⎟ ⎥ y + ⎜ ⎟ α ⎪ ⎣α '⎦
⎝ n' ⎠ ⎭
⎣⎝ R ⎠ ⎝ n ' ⎠ ⎦
1
0⎤
⎡
⎢
⎥
n
n
1
⎛
⎞
⎢ ⎜ − 1⎟
⎥
⎢⎣ R ⎝ n ' ⎠ n ' ⎥⎦
⎡y⎤
⎢α ⎥
⎣ ⎦
Ray-transfer matrix for refraction
⎡ y '⎤
If R → ∞ we have transfer matrix for refration by plane interface: ⎢ ⎥ =
⎣α '⎦
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⎡1 0 ⎤
⎢
⎥
n
⎢0
⎥
n
'⎦
⎣
Refraction by a plane
⎡y⎤
⎢α ⎥
⎣ ⎦
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The reflection matrix
Consider refraction of a ray at a spherical interface (paraxial approximation):
Ray coordinates before refraction ( y,α ) and ray coordinates after refraction ( y',α ')
y
y
and α = θ − φ = θ −
R
R
Goal: connect ( y',α ') to ( y,α ) by a ray transfer matrix for reflection by a concave mirror
α ' = θ '− φ = θ '−
Sign convention for the angles: ( + ) pointing upward and ( − ) pointing downward
y
y
and α ' = θ '− φ = θ '−
−R
−R
To eliminate θ and θ ' we use θ =θ ' we get
α =θ +φ =θ +
y
2y
=α +
R
R
q
become:
The desired equations
α ' =θ +
y' = (1) y + ( 0 ) α
⎫
⎪ ⎡ y '⎤
⎬→⎢ ⎥ =
⎛2⎞
α ' = ⎜ ⎟ y + (1) α ⎪ ⎣α '⎦
⎝R⎠
⎭
⎡ 1 0⎤
⎢2
⎥ ⎡y⎤
⎢ ⎥
⎢
1 ⎥ ⎣α ⎦
R
⎣
⎦
Ray-transfer matrix
for reflection
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The thick lens and thin lens matrices The general Ray‐transfer matrix
G l construct a matrix
Goal:
i that
h represents a thick
hi k llens with
i h two diff
different material
i l on eachh side
id off it.
i
In traversing the lens the ray undergoes two refractions and one translation for which we have
derived the matrices. The radii of curvature are ( + ) in this example. The symbolic equations are:
⎫
⎡ y0 ⎤
⎡ y1 ⎤
=
M
for
the
first
reflection
⎪
1⎢
⎥
⎢α ⎥
α
⎣ 1⎦
⎣ 0⎦
⎪
⎪ ⎡y ⎤
⎡ y0 ⎤
⎡ y2 ⎤
⎡ y1 ⎤
⎪
3
⎬→⎢ ⎥ = M
3M 2 M1 ⎢
⎥
⎢α ⎥ = M 2 ⎢α ⎥ for the translation
α
⎣ 2⎦
⎣ 1⎦
⎪ ⎣ 3 ⎦ M : Transfer matrix ⎣α 0 ⎦
of the entire lens
⎪
⎡ y3 ⎤
⎡ y2 ⎤
⎢ ⎥ = M 3 ⎢ ⎥ for the second reflectiont ⎪
⎪⎭
⎣α 2 ⎦
⎣α 3 ⎦
Th individual
The
i di id l matrix
i operates on the
h ray in
i the
h same order
d in
i which
hi h the
h optical
i l actin
i s influense
i fl
the
h ray.
No comutative propery for multiplication of matricies. Only associative property holds.
M=M 3 M 2 M 1 = ( M 3 M 2 ) M 1 = M 3 ( M 2 M 1 ) ≠ M 2 M 3 M 1
Generalizing the matrix relationship for any number of translating, reflecting, refracting surfaces:
⎡yf ⎤
⎡ y0 ⎤
with M = M N M N −1 " M 2 M 1 ray transfer matrix for the optical system.
⎢ ⎥=M
N M N −1 " M 2 M 1 ⎢
⎣α 0 ⎥⎦
⎢⎣α f ⎥⎦ M : Transfer matrix
of the entire lens
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The thick lens and thin lens matrices Goal: Applying the results for a thick lens
Let R represent a reflaction matrix and T represent tsranslation
M=R 2 TR1 the ray-transfer matrix for a thick lens can be written as:
0⎤
0⎤
⎡ 1
⎡ 1
1
t
⎡
⎤⎢
⎥
M = ⎢ nL − n ' nL ⎥ ⎢
n
−
n
n
⎥
L
⎥ ⎣0 1⎦ ⎢
⎥
⎢
n ' ⎥⎦
⎢⎣ n ' R2
⎣⎢ nL R1 nL ⎦⎥
y
matrix becomes
For a thin lens t → 0 in one environment ( n = n ') the ray-transfer
1
0⎤
0⎤
0⎤ ⎡
⎡ 1
⎡ 1
⎡1 0 ⎤ ⎢
⎥
⎥= ⎢
M = ⎢ nL − n nL ⎥ ⎢
n
n
−
n
⎛
⎞
n
n
−
1
1
L
⎢
⎥ ⎣0 1 ⎥⎦ ⎢
⎥ ⎢ L
− ⎟ 1⎥
⎜
n ⎥⎦
⎢⎣ nR2
⎢⎣ nL R1 nL ⎥⎦ ⎢⎣ n ⎝ R2 R1 ⎠ ⎥⎦
We exprss the lower left hand element in terms of the focal length:
1 nL − n ⎛ 1 1 ⎞
=
⎜ − ⎟ the lansmaker's formula
f
n ⎝ R2 R1 ⎠
⎡ 1 0⎤
⎥ = ⎡A B⎤
M =⎢ 1
⎢−
1 ⎥ ⎣⎢C D ⎦⎥
⎢⎣ f
⎥⎦
the ray-transfer matrix for a thin lens also known as the ABCD matrix.
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0
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Significance of system matrix elements
⎡ y f ⎤ ⎡ A B ⎤ ⎡ y0 ⎤
⎢ ⎥=⎢
⎥⎢ ⎥
⎣⎢α f ⎦⎥ ⎣C D ⎦ ⎣α 0 ⎦
a) If D = 0 → α f = Cy0 independent of α 0
All the rays leavoing the input plane will
have the same angle at the output plane.
Input plane is on the first focal plane.
b) If A = 0 → y f = Bα 0
means yf is independent of y0 that means
all the rays departing input plane have the
same height at the output plane. This means
output plane is the second focal plane.
c) If B = 0 → y f = Ay0 All the points
leaving the input plane at hight y0 will
arrive the output plane at height yf output
plane is image of the input plane.
A=y f / y0 corresponds to linear magnification.
d) If C = 0 → a f = Dα 0 independent of y0
p rays
y of all in one direction will
Input
produce output rays all in another direction.
This is called (thelescopic system).
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Example 18.3
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Location of cardinal points for an optical system
Since the system ray-transfer matix explains the optical properties of an optical system we expect a
relationship between the system matrix and location of the cardinal points.
Input and output planes define limits of an optical system.
We define distances locating six cardinal planes with respect to the input and output planes.
F1 and F2 are at f1 and f 2 from the principal points at H1 and H 2
F1 and F 2 are at p and q from the reference input and output planes
r and s are disances of the reference input
p and output
p planes
p
from the principal
p
p p
points at H1 and H 2
v and w are disances of the reference input and output planes from the nodal points at N1 and N 2
Sign convention:
( + ) distance measured to the
right of a reference plane
( - ) distance measured to the
left of a reference plane
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Location of cardinal points
Input ray ( y 0 , α 0 ) and output ray ( y f , 0 ) from figure (a)
y f = Ay0 + Bα 0 ⎫⎪
⎛D⎞
⎬ → y0 = − ⎜ ⎟ α 0
0 = Cy0 + Dα 0 ⎪⎭
⎝C⎠
For small angles α 0 =
y0
y
D
→ p=− 0 =
−p
α0 C
p is negative that means it is to the left of the
input reference plane.
α0 =
f1 =
f1 =
yf
− f1
−yf
α0
=
− ( Ay0 + Bα 0 )
α0
=
AD
−B
C
AD − BC Det ( M ) ⎛ n0 ⎞ 1
=
= ⎜ ⎟ = f1
⎜ nf ⎟ C
C
C
⎝ ⎠
We used: Det ( M ) = AD − BC =
r = p − f1 =
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n
1⎛
⎜⎜ D − 0
C⎝
nf
n0
nf
⎞
⎟⎟ = r
⎠
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Location of cardinal points
U i figure
Using
fi
(b) we can fi
find
d q, f 2 , s
A
1− A
1
; s=
; f2 = q − s = −
C
C
C
Using figure (c) we can find v, w
q=−
y0
v
Notice y 0 is negative i.e. below the optical axis
α0 = α f = α = −
α = Cy0 + Dα →
y0
α
=
1− D
= −v
C
n0 / n f ) − A
(
D −1
v=
and w =
C
C
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1)) When initial and final material have the same index of refraction then r = v and s = w i.e.
principal points and nodal points coinside
2) When initial and final material have the same index of refraction then first and second
focal lengths are equal f1 = f 2
3) The separation of the principal points is the same as separation of the nodal points or r − s = v − w
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Examples: tow thin lenses in air separated by a distance L 2/20/2009
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Examples: tow thin lenses in air separated by a distance L Focal lengths of the lenses f A , f B ,
Assume the input and output reference planes are located on the lenses
lenses.
The system transfer matrix includes two thin-lens matrices and a translation matrix.
L
⎡
⎤
1
−
L ⎥
0⎤ ⎢
0⎤
⎡ 1
⎡ 1
fA
⎥ ⎡1 L ⎤ ⎢ 1
⎥ =⎢
⎥
M =⎢ 1
⎢
⎥
⎢−
⎥
⎢
⎥
⎢
⎥
1 ⎣0 1 ⎦ −
1
⎞ 1
L
1 ⎛ 1
1− ⎥
⎢⎣ f B
⎥⎦
⎢⎣ f B
⎥⎦ ⎢ ⎜ − 1⎟ −
f B ⎥⎦
⎢⎣ f B ⎝ f A ⎠ f A
1
1
st focal
oca length
e gt of
o the
t e system:
syste : f1 = and
a d the
t e second
seco d focal
oca length
e gt of
o the
t e system:
syste : f 2 = − = f eq
First
C
C
L
1
1
1
=
+
−
f eq f A f B f A f B
⎛ f eq ⎞
The first principal point and nodal point: r = v = ⎜
⎟L
f
⎝ B⎠
The second principal point and
⎛ f eq ⎞
nodal
d l point:
i t s=w=⎜
⎟L
f
⎝ A⎠
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Example
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Ray tracing
Limiting analysis of optical systems to paraxial rays is an over simplification of the problem Limiting
anal sis of optical s stems to para ial ra s is an o er simplification of the problem
and ignores effect of aberrations. Ray tracing is following the actual path of each ray through the system using laws of reflection and refraction. Traditionally it is done by hand and graphically but today it is all computerized.
We introduce a ray‐tracing technique that is often limited to meridional rays.
Meridional rays are the rays that pass through the optical axis of the system.
Meridional rays tend to stay in the meridional planes as the laws of refraction/reflection
Meridional rays tend to stay in the meridional planes as the laws of refraction/reflection require them.
This limits our treatment to a 2‐dimensional space.
Skew rays are the ones that contribute to the image and do not pass the optical axis.
Analysis of the skew rays require a 3‐dimensional treatment.
Understanding aberrations require analysis of the non‐paraxial rays and skew rays. Design of the complicated lens systems require knowledge and experience with ray‐tracing techniques and optimizing performance of the system by changing system parameters and
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Ray tracing
Goal: followin a meridional ray through a single spherical refracting surface.
n, n ' : Indexes of refraction; R radius of curvature; A origin of the ray.
α , α ' : angle with optical axis before and after refraction.
O point of intersection with the optical axis; P with the refracting surface; I with the optical axis after refraction.
I & O are conjugate poits with distances s , s ' from the vertices.
Q: perpendicular distance from the vertex, V, to the incident ray.
θ ,θ ' : angles of incidence and refraction.
Sign convention: distances to the left of vertex − and to the right of V are + above the optical axis + and below are − .
From left to right, the angles have the same sign as the slopes.
Input
put parameters
pa a ete s for
o each
eac ray:
ay: h : elevation,
e evat o , α : aangle,
g e, and
a d
D: distance from the vertex parallel to the optical axis.
From figure we write:
s = D−
h
Q
; in ΔOBV → sin α =
→ Q = − s sin α
tan α
−s
a+Q
R
a
in ΔVNC → sin α =
R
in ΔPMC → sin θ =
⎫
⎪⎪
R sin α + Q
⎬ sin θ =
R
⎪
⎪⎭
⎛Q
⎞
+ sin α ⎟
⎝R
⎠
θ = sin −1 ⎜
⎛ n sin θ ⎞
At P → n sin θ = n 'sin θ ' → θ ' = sin −1 ⎜
⎟
⎝ n' ⎠
in ΔCPI → θ − α = θ '− α ' → α ' = θ '− θ + α
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Ray tracing
Q': perpendicular distance from the vertex, V, to the refracted ray.
a' ⎫
R ⎪⎪ Q ' = R sin θ '− sin α '
(
)
⎬
Q '− a ' ⎪
In ΔPLC → sin (θ ' ) =
R ⎪⎭
In ΔCMV → sin ( −α ') =
In ΔITV → sin ( −α ') =
Q'
−Q '
→ s' =
sin α '
s'
Now we have the new values for the refracted ray α ', Q ', s ' which prepare us for the next refraction in the sequence.
But before that we need to calculate effect of the transfer byy t in the material with index n '.
V M Q' −Q
In ΔV2 MV1 → sin ( −α 2 ) = 1 = 1 2 → Q2 = Q '1 + t sin α 2
t
t
And α 2 =α '
W needd to
We
t modify
dif th
the equations
ti
for the special cases:
1) Incident ray is parallel
to the optical axis.
2) Surface is plane with an
infinite radius of curvature.
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The rays are parallel to the axis so we use the second column of the table
Example ray tracing
Do a ray ttrace ffor ttwo rays through
D
th
h
a rapid landscape photographic lens
of three elements. The parallel rays
enter the lens from a distant object at
altitudes of 1 and 5 mm above the
optical axis. The lens specifications
are:
R1 = −120.8
R2 = −34.6
t1 = 6
n1 = 1.521
R3 = −96.2
t2 = 2
n2 = 1.581
R4 = −51.2
51 2
t3 = 3
n3 = 1.514
1 514
Input
First surface :
n = 1, n ' = 1.521
α =0
h = 1 or 5
R = -120.8
Results ray at h=5
Q =1
Q=5
α ' = 0.16250
α ' = 0.81280
s ' = −352.66
Q ' = 1.0000
s ' = −352.53
Q ' = 5.0010
Q = 1.0170
Q = 5.0861
α ' = 0.22020
α ' = 1.10410
s ' = −264.59
264 59
Q ' = 1.0170
s ' = −264.03
264 03
Q ' = 5.0876
Second surface :
t =6
n = 1.581
R = -34.6
34 6
Third surface :
t=2
n = 1.514
R = -96.2
Final surface :
t =3
n = 1.581
R = -51.2
51 2
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Results ray at h=1
Q = 1.0247
Q = 5.1261
α ' = 0.20300
α ' = 1.01780
s ' = −289.26
Q ' = 1.0247
s ' = −288.58
Q ' = 5.1260
Q = 1.0353
Q = 5.1793
α ' = −0.28830
α ' = −1.45200
s ' = −205.72
Q ' = 1.0353
s ' = −203.91
Q ' = 5.1672
There is no common focus Δs ' = 1.8mm
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