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