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Design and Correction of Optical
Systems
Lecture 1: Basics
2017-04-07
Herbert Gross
Summer term 2017
2
Preliminary Schedule - DCS 2017
1 07.04. Basics Law of refraction, Fresnel formulas, optical system model, raytrace, calculation
approaches
2 14.04. Materials and Components Dispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors,
aspheres, diffractive elements
3 21.04. Paraxial Optics Paraxial approximation, basic notations, imaging equation, multi-component
systems, matrix calculation, Lagrange invariant, phase space visualization
4 28.04. Optical Systems Pupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry,
photometry
5 05.05. Geometrical Aberrations Longitudinal and transverse aberrations, spot diagram, polynomial expansion,
primary aberrations, chromatical aberrations, Seidels surface contributions
6 12.05. Wave Aberrations Fermat principle and Eikonal, wave aberrations, expansion and higher orders,
Zernike polynomials, measurement of system quality
7 19.05. PSF and Transfer function Diffraction, point spread function, PSF with aberrations, optical transfer function,
Fourier imaging model
8 26.05. Further Performance Criteria Rayleigh and Marechal criteria, Strehl definition, 2-point resolution, MTF-based
criteria, further options
9 02.06. Optimization and Correction Principles of optimization, initial setups, constraints, sensitivity, optimization of
optical systems, global approaches
10 09.06. Correction Principles I Symmetry, lens bending, lens splitting, special options for spherical aberration,
astigmatism, coma and distortion, aspheres
11 16.06. Correction Principles II Field flattening and Petzval theorem, chromatical correction, achromate,
apochromate, sensitivity analysis, diffractive elements
12 23.06. Optical System Classification Overview, photographic lenses, microscopic objectives, lithographic systems,
eyepieces, scan systems, telescopes, endoscopes
13 30.06. Special System Examples Zoom systems, confocal systems
14 07.07. Further Topics New system developments, modern aberration theory,...
1. Refraction
2. Fresnel formulas
3. Optical systems
4. Raytrace
5. Calculation approaches
3
Contents
Law of Refraction
Angle deviation at skew incidence
Change of magnification at curved surfaces, lensing effect
surface
normal
interface
plane
n
Q
n'
P
Oi
i'
e
ray
s'
s
4
Law of Refraction
Scalar law of refraction
(Snells law)
Vectorial form
Special case reflection
All vectors in the plane of incidence
Fundamental basis:
Principle of Fermat
Invariance of field components
'sin'sin inin
seess
2'
esen
nse
n
ns
n
ns
2
2
1'
1''
'
n n'incidence
reflection
refraction
interface
normal direction
i
i'i
s
s's'
e
5
Law of Refraction
i
i'
n
n'
a
a'
d
'' anan
'sin'
sin
ida
ida
'sin'sin inin
Simple derivation of the law of refraction:
constant optical path length for two rays,
optical path length:
product of index of refraction times geometrical path length
Geometrical condition of triangles
Insertion delivers the law of
refraction
6
Law of Refraction
n i n i sin ' sin '
Scalar law of refraction
Sine is limited by -1…+1:
1. gracing incidence at i=90°
2. total internal reflection with
i‘ = 90°
for
i0 10 20 30 40 50 60 70 80 90
0
10
20
30
40
50
60
70
80
90
i'
n/n' = 1.9
n/n' = 1.5
n/n' = 1.2
n/n' = 1
n/n' = 1/1.2
n/n' = 1/1.5
n/n' = 1/1.9
total internal reflection
grazing
incidence
nni /'sin
7
Fresnel Formulas
n n'
incidence
reflection transmission
interface
E
B
i
i
E
B
r
r
E
Bt
t
normal to the interface
i
i'i
a) s-polarization
n n'
incidence
reflection
transmission
interface
B
E
i
i
B
E
r
r
B
E t
t
normal to the interface
i'i
i
b) p-polarization
Schematical illustration of the ray refraction ( reflection at an interface
The cases of s- and p-polarization must be distinguished
8
Fresnel Formulas
Electrical transverse polarization
TE, s- or s-polarization, E perpendicular to incidence plane
Magnetical transverse polarization
TM, p- or p-polarization, E in incidence plane
Boundary condition of Maxwell equations
at a dielectric interface:
continuous tangential component of E-field
Amplitude coefficients for
reflected field
transmitted field
Reflectivity and transmission
of light power
TEe
rTE
E
Er
1 TE
TEe
tTE r
E
Et 1
' TMTM r
n
nt
nn EE 2211
tt EE 21
TMe
rTM
E
Er
2rP
PR
e
r 2
cos
'cos't
in
in
P
PT
e
t
||
9
Fresnel Formulas: Stokes Relations
1 rt
1cos
'cos|||| r
i
it
Relation between the amplitude coefficients for reflection/transmission:
1. s-components:
field components additive
minus sign due to phase jump
2. p-components:
energy preservation but change of
area size due to projection,
correction factor, no additivity of
intensities
field
amplitudes
incidence reflection
transmission
Ei
Er
Et
cross section
area
continuous
tangential
component
10
Fresnel Formulas
Coefficients of amplitude for reflected rays, s and p
Coefficients of amplitude for transmitted rays, s and p
tzez
tzezE
kk
kk
inin
inin
innin
innin
ii
iir
'cos'cos
'cos'cos
sin'cos
sin'cos
)'sin(
)'sin(
222
222
tzez
tzezE
knkn
knkn
inin
inin
innnin
innnin
ii
iir
22
22
2222
2222
||'
'
'coscos'
'coscos'
sin'cos'
sin'cos'
)'tan(
)'tan(
tzez
ezE
kk
k
inin
in
innin
in
inin
int
2
'cos'cos
cos2
sin'cos
cos2
'cos'cos
cos2
222
tzez
ezE
knkn
kn
inin
in
innnin
inn
inin
int
22
2
2222||
'
'2
'coscos'
cos2
sin'cos'
cos'2
'coscos'
cos2
11
Fresnel Formulas
Typical behavior of the Fresnel amplitude coefficients as a function of the incidence angle
for a fixed combination of refractive indices
i = 0
Transmission independent
on polarization
Reflected p-rays without
phase jump
Reflected s-rays with
phase jump of p
(corresponds to r<0)
i = 90°
No transmission possible
Reflected light independent
on polarization
Brewster angle:
completely s-polarized
reflected light
r
r
t
t
i
Brewster
12
Fresnel Formulas: Energy vs. Intensity
Fresnel formulas, different representations:
1. Amplitude coefficients, with sign
2. Intensity cefficients: no additivity due to area projection
3. Power coefficients: additivity due to energy preservation
r , t
0 10 20 30 40 50 60 70 80 90-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
i
t
r
r
t
R , T
i0 10 20 30 40 50 60 70 80 90
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T(In)
T(In)
R(In)
R(In)
R , T
i0 10 20 30 40 50 60 70 80 90
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T
T
R
R
13
Fresnel Formulas
Reflectivity and transmittivity of power
Arbitrary azimuthal angle t of polarization: decompositioin of components
In case of vanishing absorption:
Energy preservation
Special case of normal incidence
Typical values for some glasses and optical materials in air
)'(sin
)'(sin2
2
ii
iiR
)'(tan
)'(tan2
2
||ii
iiR
)'(sin
2cos'2sin2 ii
iiT
)'(cos)'(sin
'2sin2cos22||
iiii
iiT
ee RRR tt 22
|| sincos ee TTT tt 22
|| sincos
1TR
2
'
'
nn
nnR
2'
'4
nn
nnT
n R
1.4 2.778 %
1.5 4.0 %
1.8 8.16 %
2.4 16.96 %
14
Transmission in Optical Systems
Residual reflectivity of the (identical) surfaces in an optical system with n surfaces:
Overall transmission of energy:
Transmission decreases
nonlinear
Practical consequences:
1. loss of signal energy
2. contrast reduction in
case of imaging
3.occurence of ghost images
n0 10 20 30 40 50 60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T
R = 1 %
R = 2 %
R = 4 %
R = 6 %R = 10 %
n
ges RT 1
15
Brewster Angle
Brewster case of reflection:
reflected and transmitted ray are perpendicular
Condition of Brewster angle
The reflected light is completely
s polarized
Application:
stack of plates under Brewster angle
as polarizer
n
niB
'tan
90'ii
n n'
incidence
reflectiontransmission
interface
normal to the interface
Brewster case
i
90°
i'
i
16
Total Internal Reflection in Fibers
Total internal reflection between core and cladding in a step index fiber
Ref: M. Kaschke
17
Total Internal Reflection
Limiting angle ic of total internal
reflection:
no light leaves the medium with
the higher index
Condition:
n
nic
'sin
1|| RR
n
n'
i
i'
iic
2. limiting case of total internal reflection
3. total internal reflection1. refraction
refracted rays
total internal
reflection
medium n
source point
plane of interface
total internal
reflection
18
Total Internal Reflection
In case of total internal reflection, the Fresnel formulae reads
The wave penetrates the boundary
and generates an evanescent wave
which propagates
along the boundary
22223
22223
||
'sin1'cos
'sin1'cos
ninnin
ninninr
222
222
'sin1cos
'sin1cos
ninin
nininr
x
phase surfaces
evanescent :
parallel
incident
reflected
z
19
20
Evanescent Field
Visualization of the evanescent wave
Evanescent field: finite penetration depth
propagation parallel to the interface plane
Data: n = 1.5 , n‘ = 1.0 ; q = 45°
Ref: Peatros
Description of Optical Systems
Interface surfaces
- mathematical modelled surfaces
- planes, spheres, aspheres, conics, free shaped surfaces,…
Size of components
- thickness and distances along the axis
- transversal size,circular diameter, complicated contours
Geometry of the setup
- special case: rotational symmetry
- general case: 3D, tilt angles, offsets and decentrations, needs vectorial approach
Materials
- refractive indices for all used wavelengths
- other properties: absorption, birefringence, nonlinear coefficients, index gradients,…
Special surfaces
- gratings, diffractive elements
- arrays, scattering surfaces
21
Modelling of Optical Systems
Principal purpose of calculations:
1. Solving the direct problem of
understanding the properties:
analysis
2. Solving the inverse problem:
Finding the concret system data
for a required functionality:
synthesis
System, data of the structure(radii, distances, indices,...)
Function, data of properties,
quality performance(spot diameter, MTF, Strehl ratio,...)
Analysisimaging
aberration
theory
Synthesislens design
inverse
problem
Ref: W. Richter
22
Approximation of Optical Models
Imaging model with levels
of refinement
Paraxial model
(focal length, magnification, aperture,..)
approximation
l à 0
no description of
short pulses
Geometrical
optics
Analytical approximation
(3rd order aberrations,..)
exact geometry
Wave optics
no time dependence
Maxwell equations
Scalar approximation
Helmholtz equation
(PSF, OTF,...)
linear
approximation
no description of
small structures
and polarization
effects
no diffraction
no higher order
aberrations
no aberrations
exact
23
Five levels of modelling:
1. Geometrical raytrace with analysis
2. Equivalent geometrical quantities,
classification
3. Physical model:
complex pupil function
4. Primary physical quantities
5. Secondary physical quantities
Blue arrows: conversion of quantities
Modelling of Optical Systems
ray
tracing
optical path
length
wave
aberration W
transverse
aberrationlongitudinal aberrations
Zernike
coefficients
pupil
function
point spread
function (PSF)
Strehlnumber
optical
transfer function
geometricalspot diagramm
rms
value
intersectionpoints
final analysis reference ray in
the image space
referencesphere
orthogonalexpansion
analysis
sum of
coefficientsMarechal
approxima-
tion
exponentialfunction
of the
phase
Fourier
transformLuneburg integral
( far field )
Kirchhoffintegral
maximum
of the squared
amplitude
Fouriertransformsquared amplitude
sum of
squaresMarechalapproxima-
tion
integration ofspatial
frequencies
Rayleigh unit
equivalencetypes of
aberrationsdifferen
tiationinte-
gration
full
aperture
single types of aberrations
definition
geometricaloptical
transfer function
Fouriertransform
approximation
auto-correlationDuffieux
integral
resolution
threshold value spatial frequency
threshold value spatial
frequency approximationspot diameter
approximation diameter of the
spot
Marechalapproximation
final analysis reference ray in the image planeGeometrical
raytrace
with Snells law
Geometrical
equivalents
classification
Physical
model
Primary
physical
quantities
Secondary
physical
quantities
24
Scheme of Raytrace
zoptical
axis
y j
u'j-1
ij
dj-1
ds j-1
ds j
i'j
u'j
n j
nj-1
mediummedium
surface j-1
surface j
ray
dj
vertex distance
oblique thickness
rr
Ray: straight line between two intersection points
System: sequence of spherical surfaces
Data: - radii, curvature c=1/r
- vertex distances
- refractive indices
- transverse diameter
Surfaces of 2nd order:
Calculation of intersection points
analytically possible: fast
computation
25
Workflow Raytrace
transport of the ray
over a distance in the
medium
homogeneous medium
equation of a straight line
inhomogeneous medium
solution of the eikonal equation
e.g. with Runge-Kutta method
calculation of the
intersection of the ray
with the next surface
sequential raytrace :
next surface index
non-sequential raytrace : correct
index from the smallest distance of all
alternatives
surface of second order :
analytical solution for intersection
point with the ray
aspherical surface :
numerical iterative calculation of the
intersection point
check if intersection
point is correct
vignetting of the ray due to the size of
the surface
ray does not hit the surface
calculation of the new
ray direction
transition of the ray
into the new medium
refraction of the ray
reflection of the ray
new direction due to diffraction at a
diffractive surface
change of the direction due to
scattering at the surface
special case : total reflection
Step
No j
Step
No j+1
Two step process:
1. Transition to next surface
intersection point,
test of plausibility
2. Dielectric interface
refraction and new direction
test of plausibility
26
Raytrace Formulas
Different sets of formulas:
1. Paraxial formulas, for reference of ideal imaging
2. Meridional formulas, for circular symmetry
3. Vectorial formulas, 3D systems, state of the art today
4. Differential formulas, transfer of neighbourhood, ray density, astigmatism
Special aspects:
1. Aspherical surfaces, numerical iterative calculation of intersection points
2. Gradient media, eikonal differentail equation, Runge-Kutta numerical stepwise
3. Diffractive elements, local grating equation
4. Non-sequential raytrace, illumination and straylight
5. Scattering surfaces, Monte-Carlo decision for new direction
6. Photometric correct raytracing, transfer of relative weighting factor
7. Polarization raytrace, transfer of Jones vector on a ray
8. Geometrical approximated edge diffraction, ray deviation depends on edge distance
27
Paraxial y-U-Method
Paraxial ray trace
formulas
Parameters of ray
description:
1. ray height y
2. ray angle U
Transfer to next surface
index j -à j+1
height
angle of incidence
refraction
new direction angle
y y d Uj j j j 1 1 1
1 jjjj Uyci
in
nij
j
j
j''
''' 1 jjjjjjj icyiiUU
i
i'
rsinu-U
-U'-U
QQ'
-rsinur
L'
L
M
28
Normal Distance of Rays to Surface Vertex
Projection of ray to surface vertex point
length: Q
In the paraxial regime identical with ray intersection height y
yo
z
object
plane
starting
plane
entrance
pupil
marginal
raychief ray
yp
U1
W1
QRS1
QHS1
normals to
the ray
29
Exact Meridional Q-U-Raytrace Method
Exact raytrace scheme in the meridional plane
Ray description parameters:
- angle u with optical axis
- distance Q between ray and surface vertex point
Set of formuals:
(1) angle of incidence
(2) refraction
(3) new angle of ray
(4) auxiliary parameter
(5) distance to vertex
(6) intersection length
(7) final distance
uQci sinsin cos sini i 1 2
sin ''sini
n
ni cos ' sin 'i i 1 2
u u i i' '
GQ
u i
cos cos
'cos'coscos
'cos'sin'sin'
iu
G
u
uQ
c
uiQ
Q Q d uj j 1 sin '
L L dj j 1
Q L u1 sin
30
Exact Meridional Q-U-Raytrace Method (2)
Set of formuals (2):
(8) final intersection length
(9) intersection point
(10) failure condition 1:
ray don‘t hit surface
(11) failure condition 2:
total internal reflection
LQ
u'
'
sin '
)cos(1)sin(
iuGc
iuy
)sin()cos(1
iuGc
iuz
sini 1
sin 'i 1
31
Vectorial Raytrace
yj
z
Pj+1
sj
xj
yj+1
xj+1
Pj
surface
No j
surface
No j+1
dj sj+1
intersection
point
ej
normal
vector
ej+1
ray
distance
intersection
point
normal
vector
General 3D geometry
Tilt and decenter of surfaces
General shaped free form surfaces
Full description with 3 components
Global and local coordinate systems
32
Vectorial Raytrace Formulas
Restrictions:
- surfaces of second order, fast analytical calculation of intersection point possible
- homogeneous media
Direction unit vector of the straight ray
Vector of intersection point on a surface
Ray equation with skew thickness dsj
index j of the surface and the space behind
Equation of the surface 2.order
The coefficients H, F, G contains the surface shape parameters
s j
j
j
j
r
x
y
z
j
j
j
j
11,1 jjsjj sdrr
02 1,
2
1, jjsjjsj GdFdH
33
Vectorial Raytrace Formulas (2)
Special case spherical surface with
curvature c = 1/R
Coefficients H, G, F
Unit vector normal to the surface
Insertion of the ray equation into surface equation:
skew thickness
Angle of incidence
Refraction
or
reflection
jj cH
jjjjjj zzyxcG 2222
jjjjjjjjj zyxcF
jj
jj
jj
j
zc
yc
xc
e
1
jjjj
j
js
GHFF
Gd
21,
cosi s ej j j
cos ' cosin
nij
j
j
j
1 11
2
2
cos ' cosi ij j
34
Vectorial Raytrace Formulas (3)
Auxiliary parameter
New ray direction vector
j j j j jn i n i 1 cos ' cos
s
n
ns
nej
j
j
j
j
j
j
1
1 1
35
Conic Sections
222
22
111 yxc
yxcz
1
2
b
a
2a
bc
1
1
cb
1
1
ca
Explicite surface equation, resolved to z
Parameters: curvature c = 1 / R
conic parameter
Influence of k on the surface shape
Relations with axis lengths a,b of conic sections
Parameter Surface shape
= - 1 paraboloid
< - 1 hyperboloid
= 0 sphere
> 0 oblate ellipsoid (disc)
0 > > - 1 prolate ellipsoid (cigar )
36
Aspherical Surface Types
q
22 yxz
222
22
111 yxc
yxcz
22
22 yxRRRRz xxyy
Conic section
Special case spherical
Cone
Toroidal surface with
radii Rx and Ry in the two
section planes
Generalized onic section without
circular symmetry
Roof surface
2222
22
1111 ycxc
ycxcz
yyxx
yx
z y tanq
37
Aspherical Surfaces with Rotational Symmetry
22
122
2
111)(
k
k
k ycyc
ycyz
z
y
height
y
z(y)
deviationz
aspherical
surface
spherical
surface
Classical representation of an aspheric surface with rotational symmetry:
- basic shape of a conic section
- correction of real sag height with Taylor expansion
Aspherical constants cj
Usually only even orders, no cusp on axis
Problems with this representation:
- deviation z not perpendicular to surface
- Taylor expansion terms not orthogonal
- oscillations of higher order terms
38
Numerical Iterative Raytrace at Aspheres
Calculation of the intersection point with an aspherical surface:
- no fast analytical calculation possible
- iterative numerical computation is used
- often problems with stability for
steep aspheres
General scheme:
- intersection point with vertex plane Q0
- projection onto surface, point Q1
- determine the tangential plane in Q1
- intersection with tangential plane Q‘1
- projection onto surface, point Q2
- …
Qo
y
z
tangential plane in Q1
surface No j
ray
e2
unit vector in Q
Q1
Q2
Q'1
tangential plane in Q2
Q
sj
z-projection
vertex plane
39
egdn
gms
n
ns
l ˆ''
'
Diffracting Surfaces
grooves
s
s
e
d
gp
p
Surface with grating structure:
new ray direction follows the grating equation
Local approximation in the case of space-varying
grating width
Raytrace only into one desired diffraction order
Notations:
g : unit vector perpendicular to grooves
d : local grating width
m : diffraction order
e : unit normal vector of surface
Applications:
- diffractive elements
- line gratings
- holographic components
40
Raytracing in Grin Media
s
y
x
Strahl
Brechzahl :
n(x,y,z)
b
c s
b
c
y'
x'
z
nn
y
nn
x
nn
Dnndt
rd
2
2
Ray: in general curved line
Numerical solution of Eikonal equation
Step-based Runge-Kutta algorithm
4th order expansion, adaptive step width
Large computational times necessary for high accuracy
41
Description of Grin Media
3
15
2
1413
3
12
2
1110
4
9
3
8
2
76
8
5
6
4
4
3
2
21,
ycycycxcxcxc
zczczczchchchchchcnn o
l
Analytical description of grin media by Taylor expansions of the function n(x,y,z)
Separation of coordinates
Circular symmetry, nested expansion with mixed terms
Circular symmetry only radial
Only axial gradients
Circular symmetry, separated, wavelength dependent
8
19
6
18
4
17
2
1615
38
14
6
13
4
12
2
1110
2
8
9
6
8
4
7
2
65
8
4
6
3
4
2
2
1,
hchchchcczhchchchccz
hchchchcczhchchchcnn o
l
n n c c h c c h c c h c c h c c ho , ( ) ( ) ( ) ( ) ( )l 1 2 1
2
3 1
4
4 1
6
5 1
8
6 1
10
n n c c z c c z c c z c c zo , ( ) ( ) ( ) ( )l 1 2 1
2
3 1
4
4 1
6
5 1
8
n n c h c h c h c h c z c z c zo , , , , , , , ,l l l l l l l l1
2
2
4
3
6
4
8
5 6
2
7
3
42
Gradient Lenses
Refocusing in parabolic profile
Helical ray path in 3 dimensions
axis ray bundle
off axis ray bundle
waist
points
view
along z
perspectivic viewy
x
y
x
y'
x'
z
43
Non-Sequential Raytrace
Conventional raytrace:
- the sequence of surface hits of a ray is pre-given and is defined by the index vector
- simple and fast programming of the surface-loop of the raytrace
Non-sequential raytrace:
- the sequence of surface hits is not fixed
- every ray gets ist individual path
- the logic of the raytrace algorithm determines the next surface hit at run-time
- surface with several new directions of the ray are allowed:
1. partial reflection, especially Fresnel-formulas
2. statistical scattering surfaces
3. diffraction with several grating orders or ranges of deviation angles
Many generalizations possible:
several light sources, segmented surfaces, absorption, …
Applications:
1. illumination modelling
2. statistical components (scatter plates)
3. straylight calculation
44
Nonsequential Raytrace: Examples
Signal
1 2 3 4
Reflex 1 - 2
Reflex 3 - 2
1
2
3
1. Prism with total internal
reflection
2. Ghost images in optical systems
with imperfect coatings
45
Non-Sequential Raytrace: Examples
3. Illumination systems, here:
- cylindrical pump-tube of a solid state laser
- two flash lamps (A, B) with cooling flow tubes (C, D)
- laser rod (E) with flow tube (F, G)
- double-elliptical mirror
for refocussing (H)
Different ray paths
possible
A: flash lamp gas
H
4
B: glass tube of
lamp
C: water cooling
D: glass tube of cooling
5
6
3
2
1
7
E: laser rod
F: water cooling
G: glass tube of cooling
46
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