Optical Design with Zemax · 2012. 10. 22. · 3 30.10. Properties of optical systems II Types of...
Transcript of Optical Design with Zemax · 2012. 10. 22. · 3 30.10. Properties of optical systems II Types of...
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Optical Design with Zemax
Lecture 2: Properties of optical systems II
2012-10-30
Herbert Gross
Winter term 2012
2 2 Properties of Optical Systems II
Preliminary time schedule
1 16.10. Introduction
Introduction, Zemax interface, menues, file handling, preferences, Editors, updates, windows,
Coordinate systems and notations, System description, Component reversal, system insertion,
scaling, 3D geometry, aperture, field, wavelength
2 23.10. Properties of optical systems I Diameters, stop and pupil, vignetting, Layouts, Materials, Glass catalogs, Raytrace, Ray fans
and sampling, Footprints
3 30.10. Properties of optical systems II Types of surfaces, Aspheres, Gratings and diffractive surfaces, Gradient media, Cardinal
elements, Lens properties, Imaging, magnification, paraxial approximation and modelling
4 06.11. Aberrations I Representation of geometrical aberrations, Spot diagram, Transverse aberration diagrams,
Aberration expansions, Primary aberrations,
5 13.11. Aberrations II Wave aberrations, Zernike polynomials, Point spread function, Optical transfer function
6 20.11. Optimization I Principles of nonlinear optimization, Optimization in optical design, Global optimization
methods, Solves and pickups, variables, Sensitivity of variables in optical systems
7 27.11. Optimization II Systematic methods and optimization process, Starting points, Optimization in Zemax
8 04.12 Imaging Fundamentals of Fourier optics, Physical optical image formation, Imaging in Zemax
9 11.12. Illumination Introduction in illumination, Simple photometry of optical systems, Non-sequential raytrace,
Illumination in Zemax
10 18.12. Advanced handling I
Telecentricity, infinity object distance and afocal image, Local/global coordinates, Add fold
mirror, Scale system, Make double pass, Vignetting, Diameter types, Ray aiming, Material
index fit
11 08.01. Advanced handling II Report graphics, Universal plot, Slider, Visual optimization, IO of data, Multiconfiguration,
Fiber coupling, Macro language, Lens catalogs
12 15.01. Correction I
Symmetry principle, Lens bending, Correcting spherical aberration, Coma, stop position,
Astigmatism, Field flattening, Chromatical correction, Retrofocus and telephoto setup, Design
method
13 22.01. Correction II Field lenses, Stop position influence, Aspheres and higher orders, Principles of glass
selection, Sensitivity of a system correction, Microscopic objective lens, Zoom system
14 29.01. Physical optical modelling I Gaussian beams, POP propagation, polarization raytrace, polarization transmission,
polarization aberrations
15 05.02. Physical optical modelling II coatings, representations, transmission and phase effects, ghost imaging, general straylight
with BRDF
1. Types of surfaces
2. Aspheres
3. Gratings and diffractive surfaces
4. Gradient media
5. Cardinal elements
6. Lens properties
7. Imaging
8. Magnification
9. Paraxial approximation and modelling
3 2 Properties of Optical Systems II
Contents 3rd Lecture
Setting of surface properties
local tilt
and
decenterdiameter
surface type
additional drawing
switches
coatingoperator and
sampling for POP
scattering
options
2 Properties of optical systems II
Surface properties and settings
Ideale lens
Principal surfaces are spheres
The marginal ray heights in the vortex plane are different for larger angles
Inconsistencies in the layout drawings
2 Properties of optical systems II
Ideal lens
P P'
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 tan
6 2 Properties of optical systems II
Aspherical surface types
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 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 )
7 2 Properties of optical systems II
Conic sections
Conic aspherical surface
Variation of the conical parameter
2 Properties of optical systems II
Aspherical shape of conic sections
z
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
y
22
2
111 yc
cyz
Equation
c : curvature 1/Rs
: eccentricity ( = -1 )
radii of curvature :
22
2
)1(11 cy
ycz
2
tan 1
s
sR
yRR
2
32
tan 1
s
sR
yRR
vertex circle
parabolic
mirror
F
f
z
y
R s
C
Rsvertex circle
parabolic
mirror
F
y
z
y
ray
Rtan
x
Rsag
tangential circle
of curvature
sagittal circle of
curvature
2 Properties of optical systems II
Parabolic mirror
Equation
c: curvature 1/R
: Eccentricity
22
2
)1(11 cy
ycz
ellipsoid
F'
F
e
a
b
oblate
vertex
radius Rso
prolate
vertex
radius Rsp
2 Properties of optical systems II
Ellipsoid mirror
Sag z at height y for a spherical
surface:
Paraxial approximation:
quadratic term
22 yrrz
r
yz p
2
2
z
y
height
y
z
zp = y2/(2r)
sphere
parabola
sag
axis
2 Properties of optical systems II
Sag of a surface
Maximum intensity:
constructive interference of the contributions
of all periods
Grating equation
2 Properties of optical systems II
Grating Diffraction
g mo sin sin
grating
g
incident
light
+ 1.
diffraction
order
s =
in-phase
grating
constant
Ideal diffraction grating:
monochromatic incident collimated
beam is decomposed into
discrete sharp diffraction orders
Constructive interference of the
contributions of all periodic cells
Only two orders for sinusoidal
2 Properties of optical systems II
Ideale diffraction grating
grating
g = 1 / s
incidentcollimated
light
grating constant
-1.
-2.
-3.
0.
-4.
+1.
+2.
+3.
+4.
diffraction orders
Interference function of a finite number
N of periods
Finite width of every order depends on N
Sharp order direction only in the limit of
2 Properties of optical systems II
Finite width of real grating orders
sinsin
sinsin
2
2
g
Ng
I
N = 15 N = 50
-30 -20 -10 0 10 20 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-30 -20 -10 0 10 20 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-30 -20 -10 0 10 20 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
N = 5
Ng
42sin 2/1
N
z
h2
hred(x) : wrapped
reduced profile
h(x) :
continuous
profile
3 h2
2 h2
1 h2
hq(x) : quantized
profile
2 Properties of optical systems II
Diffractive Elements
Original lens height profile h(x)
Wrapping of the lens profile: hred(x) reduction on
maximal height h2
Digitalization of the reduced profile: hq(x)
egdn
gms
n
ns
ˆ''
'
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
16 2 Properties of optical systems II
Diffracting surfaces
17 2 Properties of optical systems II
Diffracting surfaces
Local micro-structured surface
Location of ray bending :
macroscopic carrier surface
Direction of ray bending :
local grating micro-structure
Independent degrees of freedom:
1. shape of substrate determines the
point of the ray bending
2. local grating constant determines
the direction of the bended ray
macroscopic
surface
curvature
local
grating
g(x,y)
lens
bending
angle
m-th
order
thin
layer
Discrete topography on the surface
Phase unwrapped to get a smooth surface
Only one desired order can be
calculated with raytrace
Model approximation
according to Sweatt:
same refraction by small height
and high index
numerical optimized DOE
phase unwrapped
ERL-model( equivalent refractive lens )
black box modelSweatt-model( thin lens with high index )
2 Properties of optical systems II
Modellierung diffractive elements
Phase function redistributed:
large index / small height
typical : n = 10000
Calculation in conventional software with raytrace possible
2 Properties of optical systems II
Diffractive Optics: Sweatt Model
),(**2),(2),( yxznyxznyx
equivalent refractive
aspherical lens
real index n synthetic
large index n*
Sweatt lens
zz*
the same ray
bending
ray bending
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
20 2 Properties of optical systems II
Raytracing in 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
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
n n c c h c c h c c h c c h c c ho , ( ) ( ) ( ) ( ) ( ) 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 , ( ) ( ) ( ) ( ) 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 , , , , , , , , 1
2
2
4
3
6
4
8
5 6
2
7
3
21 2 Properties of optical systems II
Description of GRIN media
Curved ray path in inhomogeneous media
Different types of profiles
2 Properties of optical systems II
Gradient Lens Types
n(x,y,z)
non
i
nentrance
(y)
y
z
nexit
(y)
radial gradient
rod lens
axial gradient
rod lens
radial and axial
gradient
rod lens
radial gradient
lens
axial gradient
lens
radial and axial
gradient lens
2 Properties of optical systems II
Gradium Lenses
Axial profile 'Gradium'
Focussing effect only for oblique rays
Combined effect of front surface curvature
and index gradient
Special cylindrical blanks with given profile allows choice of individual z-interval for the lens
k
k
kz
znzn
max
)(
n
1.75
1.70z
zmaxz
lens
2 Properties of optical systems II
Collecting radial selfoc lens
L
P P'
F
F'
L
P P'
F'
Thick Wood lens with parabolic index
profile
Principal planes at 1/3 and 2/3 of
thickness
n2 > 0 : collecting lens
n2 < 0 : negative lens
2
20)( rnnrn
Types of lenses with parabolic profile
Pitch length
2 Properties of optical systems II
Gradient Lenses
ymarginal
ycoma
2
0
2
0
2
20
2
11
1)(
rAn
rnnrnnrn r
rnn
np
2
2
22
2
0
0.25 Pitch
Object at infinity
0.50 Pitch
Object at front surface
0.75 Pitch
Object at infinity
1.0 Pitch
Object at front surface
Pitch 0.25 0.50 0.75 1.0
Focal points:
1. incoming parallel ray
intersects the axis in F‘
2. ray through F is leaves the lens
parallel to the axis
Principal plane P:
location of apparent ray bending
y
f '
u'P' F'
sBFL
sP'
principal
plane
focal plane
nodal planes
N N'
u
u'
Nodal points:
Ray through N goes through N‘
and preserves the direction
2 Properties of optical systems II
Cardinal elements of a lens
P principal point
S vertex of the surface
F focal point
s intersection point
of a ray with axis
f focal length PF
r radius of surface
curvature
d thickness SS‘
n refrative index
O
O'
y'
y
F F'
S
S'
P P'
N N'
n n n1 2
f'
a'
f'BFL
fBFL
a
f
s's
d
sP
s'P'
u'u
2 Properties of optical systems II
Notations of a lens
Main notations and properties of a lens:
- radii of curvature r1 , r2
curvatures c
sign: r > 0 : center of curvature
is located on the right side
- thickness d along the axis
- diameter D
- index of refraction of lens material n
Focal length (paraxial)
Optical power
Back focal length
intersection length,
measured from the vertex point
2
2
1
1
11
rc
rc
'tan',
tan
'
u
yf
u
yf F
'
'
f
n
f
nF
'' ' HF sfs
2 Properties of optical systems II
Main properties of a lens
Different shapes of singlet lenses:
1. bi-, symmetric
2. plane convex / concave, one surface plane
3. Meniscus, both surface radii with the same sign
Convex: bending outside
Concave: hollow surface
Principal planes P, P‘: outside for mesicus shaped lenses
P'P
bi-convex lens
P'P
plane-convex lens
P'P
positive
meniscus lens
P P'
bi-concave lens
P'P
plane-concave
lens
P P'
negative
meniscus lens
2 Properties of optical systems II
Lens shape
Ray path at a lens of constant focal length and different bending
The ray angle inside the lens changes
The ray incidence angles at the surfaces changes strongly
The principal planes move
For invariant location of P, P‘ the position of the lens moves
P P'
F'
X = -4 X = -2 X = +2X = 0 X = +4
2 Properties of optical systems II
Lens bending und shift of principal plane
Optical Image formation:
All ray emerging from one object point meet in the perfect image point
Region near axis:
gaussian imaging
ideal, paraxial
Image field size:
Chief ray
Aperture/size of
light cone:
marginal ray
defined by pupil
stop
image
object
optical
system
O2field
point
axis
pupil
stop
marginal
ray
O1 O'1
O'2
chief
ray
2 Properties of optical systems II
Optical imaging
Single surface
imaging equation
Thin lens in air
focal length
Thin lens in air with one plane
surface, focal length
Thin symmetrical bi-lens
Thick lens in air
focal length
'
1'
'
'
fr
nn
s
n
s
n
21
111
'
1
rrn
f
1'
n
rf
12'
n
rf
21
2
21
1111
'
1
rrn
dn
rrn
f
2 Properties of optical systems II
Formulas for surface and lens imaging
Lateral magnification for finite imaging
Scaling of image size
'tan'
tan'
uf
uf
y
ym
z f f' z'
y
P P'
principal planes
object
imagefocal pointfocal point
s
s'
y'
F F'
2 Properties of optical systems II
Magnification
Image in infinity:
- collimated exit ray bundle
- realized in binoculars
Object in infinity
- input ray bundle collimated
- realized in telescopes
- aperture defined by diameter
not by angle
object at
infinity
image in
focal
plane
lens acts as
aperture stop
collimated
entrance bundle
image at
infinity
stop
image
eye lens
field lens
2 Properties of optical systems II
Object or field at infinity
Imaging by a lens in air:
lens makers formula
Magnification
Real imaging:
s < 0 , s' > 0
Intersection lengths s, s'
measured with respective to the
principal planes P, P'
fss
11
'
1
s'
2f'
4f'
2f' 4f'
s-2f'- 4f'
-2f'
- 4f'
real object
real image
real object
virtual object
virtual image
virtual image
real image
virtual image
Imaging equation
s
sm
'
Afocal systems with object/image in infinity
Definition with field angle w
angular magnification
Relation with finite-distance magnification
''tan
'tan
hn
nh
w
w
w
w'
'f
fm
Angle Magnification
Paraxiality is given for small angles
relative to the optical axis for all rays
Large numerical aperture angle u
violates the paraxiality,
spherical aberration occurs
Large field angles w violates the
paraxiality,
coma, astigmatism, distortion, field
curvature occurs
Paraxial Approximation
Paraxial approximation:
Small angles of rays at every surface
Small incidence angles allows for a linearization of the law of refraction
All optical imaging conditions become linear (Gaussian optics),
calculation with ABCD matrix calculus is possible
No aberrations occur in optical systems
There are no truncation effects due to transverse finite sized components
Serves as a reference for ideal system conditions
Is the fundament for many system properties (focal length, principal plane, magnification,...)
The sag of optical surfaces (difference in z between vertex plane and real surface
intersection point) can be neglected
All waves are plane of spherical (parabolic)
The phase factor of spherical waves is quadratic
2 Properties of optical systems II
Paraxial approximation
'' inin
R
xi
eExE
2
0)(
Law of refraction
Taylor expansion
Linear formulation of the law of refraction
Error of the paraxial approximation
2 Properties of optical systems II
Paraxial approximation
i0 5 10 15 20 25 30 35 40
0
0.01
0.02
0.03
0.04
0.05
i'- I') / I'
n' = 1.9
n' = 1.7
n' = 1.5
...!5!3
sin53
xx
xx
'sin'sin InIn
1
'
sinarcsin
'
'
''
n
inn
in
I
Ii
'' inin
2 Properties of optical systems II
Modelling of Optical Systems
Principal purpose of calculations:
System, data of the structure(radii, distances, indices,...)
Function, data of properties,
quality performance(spot diameter, MTF, Strehl ratio,...)
Analysisimaging
aberration
theorie
Synthesislens design
Ref: W. Richter
Imaging model with levels of refinement
Analytical approximation and classificationl
(aberrations,..)
Paraxial model
(focal length, magnification, aperture,..)
Geometrical optics
(transverse aberrations, wave aberration,
distortion,...)
Wave optics
(point spread function, OTF,...)
with
diffractionapproximation
--> 0
Taylor
expansion
linear
approximation
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
2 Properties of optical systems II
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