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www.iap.uni-jena.de
Optical Design with Zemax
for PhD - Basics
Lecture 12: Tolerancing I
2020-01-29
Herbert Gross
Speaker: Dennis Ochse
Winter term 2019
Preliminary Schedule
No Date Subject Detailed content
1 23.10. Introduction
Zemax interface, menus, file handling, system description, editors, preferences, updates,
system reports, coordinate systems, aperture, field, wavelength, layouts, diameters, stop
and pupil, solves
2 30.10.Basic Zemax
handling
Raytrace, ray fans, paraxial optics, surface types, quick focus, catalogs, vignetting,
footprints, system insertion, scaling, component reversal
3 06.11.Properties of optical
systems
aspheres, gradient media, gratings and diffractive surfaces, special types of surfaces,
telecentricity, ray aiming, afocal systems
4 13.11. Aberrations I representations, spot, Seidel, transverse aberration curves, Zernike wave aberrations
5 20.11. Aberrations II Point spread function and transfer function
6 27.11. Optimization I algorithms, merit function, variables, pick up’s
7 04.12. Optimization II methodology, correction process, special requirements, examples
8 11.12. Advanced handling slider, universal plot, I/O of data, material index fit, multi configuration, macro language
9 08.01. Imaging Fourier imaging, geometrical images
10 15.01. Correction I Symmetry, field flattening, color correction
11 22.01. Correction II Higher orders, aspheres, freeforms, miscellaneous
12 29.01. Tolerancing I Practical tolerancing, sensitivity
13 19.02. Tolerancing II Adjustment, thermal loading, ghosts
14 26.02. Illumination I Photometry, light sources, non-sequential raytrace, homogenization, simple examples
15 04.03. Illumination II Examples, special components
16 11.03. Physical modeling I Gaussian beams, Gauss-Schell beams, general propagation, POP
17 18.03. Physical modeling II Polarization, Jones matrix, Stokes, propagation, birefringence, components
18 25.03. Physical modeling III Coatings, Fresnel formulas, matrix algorithm, types of coatings
19 01.04. Physical modeling IVScattering and straylight, PSD, calculation schemes, volume scattering, biomedical
applications
20 08.04. Additional topicsAdaptive optics, stock lens matching, index fit, Macro language, coupling Zemax-Matlab /
Python
2
1. Introduction
2. Tolerances
3. System integration
4. Sensitivity
5. Statistics
6. Tolerance analysis
Content
3
▪ Specifications are usually defined for the as-built system
▪ Optical designer has to develop an error budget that cover all influences on
performance degradation as
- design imperfections
- manufacturing imperfections
- integration and adjustment
- environmental influences
▪ No optical system can be manufactured perfectly (as designed)
- Surface quality, scratches, digs, micro roughness
- Surface figure (radius, asphericity, slope error, astigmatic contributions, waviness)
- Thickness (glass thickness and air distances)
- Refractive index (n-value, n-homogeneity, birefringence)
- Abbe number
- Homogeneity of material (bubbles and inclusions)
- Centering (orientation of components, wedge of lenses, angles of prisms, position of
components)
- Size of components (diameter of lenses, length of prism sides)
- Special: gradient media deviations, diffractive elements, segmented surfaces,...
▪ Tolerancing and development of alignment concepts are essential parts of the optical
design processRef: K. Uhlendorf
Introductionto tolerancing
4
IntroductionData sheet
5
▪ Data for mechanical design, development and manufacturing:
▪ Data sheet with standard data/numbers of system and tolerances
▪ Additional support data (optional) :
1. Prisms, plano components, test procedures
2. Adjustment and system integration
3. Centering for cementing
4. Centering for mechanics
5. Coatings
6. Geometrical dimensions / folding mirrors
7. Test procedures and necessary accuracies
8. Auxiliary optics for testing
9. Combination of tolerances
10.Zoom curves and dependencies
11.Adaptive control data
12.Interface to connected systems
IntroductionSpecification of system data
6
▪ Standard ISO-10110
Ref.: M. Peschka
ISO 10110-2 Material imperfections: stress birefringence (old)
ISO 10110-3 Material imperfections: bubbles & inclusions (old)
ISO 10110-4 Material imperfections: inhomogeneity & striae (old)
ISO 10110-5 Surface shape tolerances
ISO 10110-6 Centering tolerances
ISO 10110-7 Surface imperfection tolerances
ISO 10110-8 Surface texture
ISO 10110-9 Surface treatment and coating
ISO 10110-18 Material imperfections: bubbles & inclusions, stress
birefringence, inhomogeneity & striae (new)
IntroductionTolerance standards
7
Tolerance of the surface shape:
▪ specification in fringes
▪ interferometric measurement
▪ irregularity: deviation from spherical shape
▪ typical specification:
5(1) means: 5 rings spherical deviation
1 ring asymmetry/astigmatism
R'
R
Ref: C. Menke
TolerancesSurface shape tolerances
8
▪ Typical impact of spatial frequency
ranges on PSF
▪ Low frequencies:
loss of resolution
classical Zernike range
▪ High frequencies:
Loss of contrast
statistical
▪ Large angle scattering
▪ Mid spatial frequencies:
complicated, often structured
false light distributions
log A2
Four
low spatial
frequency
figure errormid
frequency
range micro roughness
1/
oscillation of the
polishing machine,
turning ripple
10/D1/D 50/D
larger deviations in K-
correlation approach
ideal
PSF
loss of
resolution
loss of
contrast
large
angle
scattering
special
effects
often
regular
TolerancesPSD Ranges
9
figure error micro
roughness
midfrequency
errors
classical interferometer
white light interferometer
atomic force microscope
polynomial fit
TolerancesPSD Ranges
10
▪ Tolerances of thickness or distances
▪ in case of glass thickness: effect of tolerance depends on the mounting setup,
the difference usually is added or subtracted in the neighboring air distance
▪ usually, the overall length of the system remains constant
▪ the thickness tolerance is far less sensitive as curvature errors
Ref: C. Menke
TolerancesThickness tolerances
11
▪ Wedge error:
- tilt of a single surface relative to the optical or mechanical axis.
- Specification as tilt error of total indicator runout (TIR) in [mm]
- angle value in rad: TIR / D (D diameter)
▪ The optical axis is the straight line through the
two centers of curvature of the two spherical
surfaces
▪ Mechanical axis:
defined by the cylindrical boundary of the lens
▪ Usually the optical and the mechanical axes
do no coincide
▪ A wedge error must be specified only for one of
the surfaces
TIR = A - B
Ref: C. Menke
C1
C2
A
B
TolerancesLens wedge error
12
▪ Equivalence of decenter (offset)
and tilt angle
▪ Small change in sag
(vertex position) in 2nd order
optical axis
surface axis
r
C
v
surface
decentered
S
vertex
z
decenter/offset
tilt angle
sag
change
r
v−=sin
( )cos1−= rz
TolerancesCentering of spherical surfaces
13
▪ Lens
1. Radial offset
2. Shearing
3. Wedge
▪ Lens group
1. Group tilt
2. Group offset
Ref.: M. Peschka
TolerancesCentering error of lenses and groups
14
▪ Discrete tolerance steps
▪ Rectangular tolerance areas in n-- plane
▪ Large influence of annealing rate on
achievable tolerance
Grade (tolerance step)
n /
1 +/– 0.0002 +/– 0.2%
2 +/– 0.0003 +/– 0.3%
3 +/– 0.0005 +/–0.5%
4 +/– 0.001 +/–0.8%
Ref.: M. Peschka
TolerancesTolerances of glass
15
Different classes of homogeneity
of glasses Class ISO 10110 n in the
sample
0 50 10-6
1 20 10-6
H 2 2 5 10-6
H 3 3 2 10-6
H 4 4 1 10-6
H 5 5 0.5 10-6
Ref.: M. Peschka
TolerancesIndex homogeneity of glasses
16
▪ Layered structure in the refractive index due to imperfect mixing of glass chemicals
▪ Wood like appearance in shadow image
▪ Interferometric measurement with angles 0°and 90°
axial
50mm
shadow image
transverse
TolerancesStriae
17
▪ Birefringence after first cooling
step
▪ Bubbles
Ref: P. Hartmann
TolerancesFurther glass properties
18
Diameter tolerance in mm 0.1
100 %
0.05
100 %
0.025
103 %
0.0125
115 %
0.0075
150 %
Thickness tolerance in mm 0.2
100 %
0.1
105 %
0.05
115 %
0.025
150 %
0.0125
300 %
Centering tolerance in minutes 6'
100 %
3'
103 %
2'
108 %
1'
115 %
30"
140 %
15"
200 %
Shape tolerance as ring number in
10 / 5
100 %
5 / 2
105 %
3 / 1
120 %
2 / 0.5
140 %
2 / 0.25
175 %
1 / 0.12
300 %
ratio diameter vs thickness 9
100 %
15
120 %
20
150 %
30
200 %
40
300 %
50
500 %
Scratches and dots
( MIL-Norm )
80 / 50
100 %
60 / 40
110 %
40 / 30
125 %
20 / 10
175 %
10 / 5
350 %
Coating without
100 %
1 Layer
115 %
3 Layer
150 %
Multilay.
< 500 %
Tolerancesand additional cost
19
System IntegrationDrawing of microscope lens with housing
20
System IntegrationMechanical design of photographic lens
21
▪ Different opportunities to mount lenses
Ref.: J. Bentley
System IntegrationMounting technologies
22
▪ Centering carrier lens
▪ Adjusting second lens
Light
source
Cross-hair
Beam splitter
Adjustable
test optics
Centering motion
of lens
Surface
being centered
Collimator
Rotating chuckEyepiece
Reading
scale
Eye / detector
Image of
Cross-hair
Adjustable
test optics
Centering motion
of lens Surface
being centered
Cross-hair
Beam splitter
Collimator
Light
source
Rotating chuck
Eyepiece
Reading
scale
Eye / detector
Image of
Cross-hair
Ref.: M. Peschka
System IntegrationCentering in bonding process
23
▪ Adjustment turning
▪ Adjustment gluing
Ref.: M. Peschka
System IntegrationHigh precision mountings
24
▪ Filling of lenses into mounting cylinder
with spacers
▪ Accumulation of centering errors by
transportation of reference
▪ Definition of lens positions by:
1. mechanical play inside mounting
2. fixating ring screw
3. planarity of spacers
mounting cylinderfixating screw
spacer
System IntegrationMechanical mounting geometry
25
▪ Mechanical Play
▪ Wedge errors
Ref.: M. Peschka
System IntegrationTolerances of mounting assembly
26
▪ Reality:
- as-designed performance: not reached in reality
- as-built-performance: more relevant
▪ Possible criteria for relaxation:
1. Incidence angles of refraction
2. Squared incidence angles
3. Surface powers
4. Seidel surface contributionsperformance
parameter
best
as
built
tolerance
interval
local optimal
design
optimal
design
SensitivitySensitivity and relaxation
27
▪ Quantitative measure for relaxation: normalized power distribution
with normalization
▪ Non-relaxed surfaces:
1. Large incidence angles
2. Large ray bending
3. Large surface contributions of aberrations
4. Significant occurence of higher aberration orders
5. Large sensitivity for centering
▪ Internal relaxation can not be easily recognized in the total performance
▪ Large sensitivities can be avoided by incorporating surface contribution of aberrations
into merit function during optimization
Fh
Fh
F
FA
jjj
jj
==
1
11
==
k
j
jA
SensitivityPower distribution
28
29SensitivityComa and decenter
𝐶𝑟3 cos 𝜃 𝑦 − 𝜎 = 𝐶𝑟3 cos 𝜃 𝑦 − 𝐶𝑟3 cos 𝜃 𝜎𝐶𝑟3 cos 𝜃 𝑦
▪ Decenter/Tilt of a surface shifts its aberration contributions in the image by
some amount s
▪ This causes additional field constant coma
▪ Seidel coma:
Additional constant coma
proportional to Seidel surface
contribution
LENS.ZMX
Configuration 1 of 1
0.2 Waves
Full field coma Z7/Z8 (Standard coefficients)
Average over field 0.0993 Waves
Peak value 0.1010 Waves
+ +
= =
0
Ref.: K. Thompson; JOSA A, Vol. 22 (2005) p.1389
▪ Comparison of performance with / without tolerances relaxed / stressed
Ref.: H. Zügge
a) cemented
b) splitted achromate
surface 3:
tilt 0.02°
surface 3:
tilt 0.02°
Seidel / y - wave
surface contribution
SensitivityExample as-built performance
30
▪ Gaussian
▪ Truncated Gaussian
▪ Uniform
▪ “Ping-pong”
(special case of binomial distribution)
2
20
2
)(
2
1)( s
s
tt
etp
−−
=
+−
= 00,2
1)( ttttp
)()(1
)( 00 −−++−+
= ttbttaba
tp
Ref.: M. Peschka
StatisticsTolerance distributions
31
▪ Statistics of refractive index tolerances:
nearly normal distributed
▪ Tolerances of lens thickness:
- biased statistical distribution with mean at 0.3 d
- less pronounced offset for small intervals
of tolerance
- width of the distribution depends on the
hardness of the material
probability
nominal value no no - n
probability
do - d
Knoophardness
HK = 550
HK = 450
HK = 350
0.275 do + d
probability
0
d = 0.10
-1 +1
0.03 0.23 0.40
d = 0.02
d = 0.01
d
-do
tolerancerange
StatisticsThickness and index distributions
32
33StatisticsCentral limit theorem
▪ The sum of n independent random variables with the same distribution converges to
a normal distribution for large n
▪ Therefore it is often reasonable to assume normal distribution for a function that
depends on many statistical parameters (2s confidence = 95%)
𝑋1 𝑋1 + 𝑋2
𝑋1 + 𝑋2 + 𝑋3 𝑋1 + 𝑋2 + 𝑋3 + 𝑋4
Ref: Wikipedia „Central limit theorem“Ref: Wikipedia „Binomial distribution“
𝐵 𝑘 𝑝, 𝑛 + 𝐵 𝑘 𝑝,𝑚 ~𝐵(𝑘|𝑝, 𝑛 + 𝑚)
▪ Evaluation of the complete system:
additive effect of all tolerances, taking partial compensations due to sign and statistics
into account
▪ Worst case superposition:
- adding all absolute amounts of degradations
- usually gives to costly and tight tolerances
- no compensations considered, too pessimistic
▪ RSS mean superposition:
- approach with ideal statistics and quadratic summation
- compensations are taken into account approximately
- real world statistics is more complicated
▪ Calculation of Monte Carlo statistics with
deterministic adjustment steps
- best practice approach
- statistical distribution can be adapted
to experience
- problems with small number manufacturing
f f jj
=
f f jj
= 2
probability
real values
nominal value
yield 90%
StatisticsModels and statistics in tolerancing
34
▪ Idea:
- calculating many sample systems
large numbers assumed due to statistics
- known statistics of every tolerance
assumed
- defining an allowed decrease in quality
relative to the nominal design value
- sorting the results gives the yield
▪ Example cases:
a) uncritical: every sample system below
30% degredation,
maybe tolerencing too tight
b) too sensitive: yield below 40% for 50%
allowed deterioration
c) feasible tolerancing for 50% allowed
degradation: approx. 95% yield
Ref: E. Kasperkiewicz master thesis 2017
Tolerance analysisMonte Carlo simulation
35
36StatisticsEstimation of statistical error
▪ Linearization of performance function f 𝑓(𝑥0 + Δ𝑥) ≈ 𝑓 𝑥0 +𝜕𝑓
𝜕𝑥𝑖Δ𝑥𝑖
E𝑓 = E𝑓 𝑥0 +𝜕𝑓
𝜕𝑥𝑖EΔ𝑥𝑖
Var 𝑓 =𝜕𝑓
𝜕𝑥𝑖
2
Var(Δ𝑥𝑖)
𝜎(𝑓) ≈ 𝜕𝑓
𝜕𝑥𝑖
2
𝜎 Δ𝑥𝑖2
▪ Expected value is a linear operation
▪ Variance for independent random variables
xi yields the RSS formula for the
estimated standard deviation of the
performance function
▪ Problem: Not all performance criteria have a good linear approximation
Spot size for example is typically close to a minimum, therefore not perfectly linear
➔ Results need to be verified using Monte Carlo simulation
▪ Sources of errors:
- materials
- manufacturing
- integration/adjustment/mounting
- enviromental influences
- residual design aberrations
▪ Performance evaluation:
− selection of proper criterion
− fixation of allowed performance level
− calculation of sensitivity of individual tolerances and combined effects (groups,
dependent errors)
− balancing of overall tolerance limits for complete system
Ref: C. Menke
IntegrationManufacturing EnvironmentDesignsavety
adding
Performance criteria
(spot size, RMS, Strehl, MTF, …)
Tolerance analysisError budget
37
▪ Selection of the performance criterion,
spot size, rms wavefront, MTF, Strehl,...
▪ Choice of the allowed degradation of performance, limiting maximum value of the criterion
▪ Definition of compensators for the adjustments
image location, intermediate air distances, centering lenses, tilt mirrors,...
▪ Calculation of the sensitivities of all tolerances:
influence of all tolerances on all performance numbers
▪ Starting the tolerance balancing with proper default values,
alternatively inverse sensitivity: largest amount of deviation for the accepted degradation
▪ Tolerance balancing:
calculating all tolerances individually to keep the overall performance with technical
realistic accuracies of the parameter
Ref: C. Menke
Tolerance analysisTolerance analysis
38
▪ Systematic finding of
tolerances
Optical system design
Set of assignedtolerances
Sensitivity analysis(evaluation of performancemeasure changes for each
tolerance. Optionally with adjustment)
Estimation of overallperformance degradation
and tolerance cost
Further performancesimulations with
assigned tolerances
Define/changeadjustment steps
and compensators
Change assignedtolerances
and / or
Adjustment stepsand compensators
Redesign ofoptical system
Performancedegradationacceptable ?
Cost acceptable ?
YES
NO
NO
YES
System toosensitive ?
YES
NO
Ref.: M. Peschka
Tolerance analysisAnalysis process
39
▪ Tolerance data editor / Tolerance wizard
Tolerance analysisTolerancing in Zemax
40
▪ Specification of tolerances with tolerance data editor operands
Operands : TRAD Radius
TFRN Number of fringes
TTHI Thickness
TEDX Element decenter x
TETX Element tilt x
TSDX Surface decenter x
TSTX Surface tilt x
TIRR Surface irregularity
TIND Refractive index
TABB Abbe number
....
Tolerance analysisTolerancing in Zemax
41
▪ Specifying options:
- statistics
- model mode
- criteria
- compensators
- ...
Tolerance analysisTolerancing in Zemax
42
▪ Results
▪ Sensitivity and total performance
Tolerance analysisTolerancing in Zemax
43
▪ Graphical overlay of
tolerance influence
Tolerance analysisTolerancing in Zemax
44