Flatness De-Mystified

8.1 m primary mirror of the Gemini North telescope, Mauna Kea, Hawaii
3/12/2024

Flatness De-Mystified

1

Contents•••••

3/12/2024

Overview• Definitions• Why Flatness matters• What’s Going on?
Basic physics/math
Conversions
TWD & Wedge
ExamplesFlatness De-Mystified

2

Definitions•

Flatness–––•

RWD - Reflected Wavefront Deviation (or Distortion)–––•

Also called TWE (“error” for “deviation”)
A measure of a wavefront’s deviation from perfectly planar after passing through the optic
Related to Centering, Wedge or Parallelism and Beam Deviation (although TWD metric is with “tilt”removed)

Wedge
––––

•

Also called RWE (“error” for “deviation”)
A measure of a wavefront’s deviation from perfectly planar after reflecting off the surface of an optic
Related to Flatness or Surface Form Tolerance

TWD - Transmitted Wavefront Deviation (or Distortion)–––•

Also called Surface Form Tolerance
A measure of a surface’s deviation from perfectly flat
Related to RWD or RWE

Also called Parallelism
The angle made between two surfaces of an optic, or degree to which the two surfaces are parallel
For lenses, the angle between the optical and mechanical axes
Related to Centering, TWD or TWE and Beam Deviation

Beam Deviation–––

3/12/2024

How much the beam deviates due to wedge angle or non-parallel surfaces
BeamDev = Wedge * (substrate refractive index - 1); ex. 10arc-sec * (1.46 – 1) = 4.6arc-sec
Related to Centering, TWD or TWE and Wedge or Parallelism
Flatness De-Mystified3

Why Flatness Matters•

Non-flatness, or non-ideal surface form leads to distorted images in an
optical system.

•In the imaging path this can lead toblurry images.

•In the excitation path it can lead to
distortions of the illumination beam –for example, in laser-based systems thelaser spot size could be larger orsmaller than needed.

3/12/2024

Flatness De-Mystified

4What’s Going on?
•Substrates end up non-flat due to their manufacture, i.e., grinding andpolishing, which can add surface curvature or in general, surface
inaccuracy.

•The thinner the substrate relative to it’s diameter (that is,the larger the aspect ratio), the harder it is to make flat.
Ideal starting substrate
Real starting substrate•

Sputtered coatings apply compressive stress due to film density,“pulling” on the surface, bending (or straining) the substrate further.
Coated substrate•

The thicker the substrate,
the less deformation(but stress is the same)

A sputtered anti-stress coatings applied to the opposite side applies
compressive stress to compensate, effectively “flattening” the part.
Flattened (anti-stressed) substrate

3/12/2024

Flatness De-Mystified5

What’s Going on?
•Light incident on this part, assuming it’s perfect to begin with, will takeon the shape of the surface it’s bouncing off of.

•The wavefront, if a perfect plane wave, will get bent out of shape uponreflection in the form of the surface it’s reflecting off of:

Perfect plane-waves Distorted waves

A wavefront is justa snap-shot
representation of
the continuous
wave of light:
3/12/2024

Flatness De-Mystified

6

What is a Wavefront?
parallel beam=plane wavefront

3/12/2024

converging beam=spherical wavefront

Flatness De-Mystified

7

What is the Wavefront?
parallel beam=plane wavefront

3/12/2024

ideal wavefront
defocused wavefront

Flatness De-Mystified

8What is the Wavefront?
parallel beam=plane wavefront

3/12/2024

ideal wavefront
aberrated beam=irregular wavefront

Flatness De-Mystified

9What is the Wavefront?
aberrated beam=irregular wavefront

diverging beam=spherical wavefront

ideal wavefront
3/12/2024

Flatness De-Mystified

10What is the Wave Aberration?
diverging beam=spherical wavefront

3/12/2024

wave aberration

Flatness De-Mystified

11

What’s Going on?
•We can characterize the wavefront, or we can characterize the surface;since they share a shape, you can measure one and figure out the other.

•The relationship between the surface form and the wavefront is simple:The wavefront feature is 2x the surface feature.
RWD = 2 x Flatness
Flatness = 1 / 2 x RWD

NOTE: flatness (and RWD) ONLY matters when a filter will be used in a reflection
3/12/2024

Flatness De-Mystified12

Reflective Wavefront Distortion (RWD) is simply the deviation of a wavefront
reflected off of a component relative to a perfect wavefront reflected off of a perfectly
plane surface•Flatness is the actual physical deviation of a component surface from a perfectly
plane surface

At normal incidence, the flatness and RWD
are simply related by a factor of 2

RWD = 2 x Flatness
Flatness = 1 / 2 x RWD

NOTE: flatness (and RWD) ONLY matters when a filter will be used in a reflection
3/12/2024

Flatness De-Mystified13

At non-normal incidence, the relationship between flatness and RWD depends on
the angle of incidence θRWD = 2 x Flatness
Flatness = 1 / 2 x RWD

RWD = 2 x cosθ x Flatness
Flatness = 1 / 2 x cosθ x RWD
45° AOI

0° AOINOTE: flatness (and RWD) ONLY matters when a filter will be used in a reflection
3/12/2024

Flatness De-Mystified14

Reflective Wavefront Distortion (RWD) may also be separated into
different contributions:• Power (or spherical error) – best fit of the reflected wavefront to a perfectsphere• Irregularity – the remaining deviation from a perfectly flat wavefront

Surface Flatness also can be separated into these components.
RWD or FlatnessTotal

3/12/2024=RWD or Flatness
PowerFlatness De-Mystified+RWD or Flatness
Irregularity

15

What’s Going on?

Zygo•Wait, so the Zygo actually measures the (0deg AOI) reflected wavefront distortion,
but then we just divide by 2?! …. Yup! Zygo calls this the ISF, or interference scaling
factor … set to 0.5 it measures surface form … set to 1 it measured (0deg AOI) RWD!
3/12/2024

Flatness De-Mystified

16

Flatness – what are we talking about?
• Forget for a moment that we’re talking about glass and optics, orwavefronts and waves.
• Perfectly “flat” can be represented as a line (1D, or no height):• Non-flat, a 2D shape, i.e. a “line” with height (relative to some “bottom”):
E:\FILMCAD\Calculators_and_worksheets\X-Y_plot.xlsx

ex. an arc:
ex. a squiggle:
ex. a box:
3/12/2024

Flatness De-Mystified

17

Flatness – what are we talking about?
• For any of these shapes you can measure the height (relative to somebottom reference) in a few different ways:– Average height– Total height (peak-to-valley)– RMS height•

You might also want to know how much of that height is due to purely
spherical curvature (power, or SAG), and how much is due to everything else(irregularity).
– With “MIL” specs, we might say “half-wave flat”, or “2 wave power, 1 wave irregularity”.
– With ISO 10110-5, we have the “3/” plus some coded language (see table below)
PV term
PVABC

3/12/2024

type of distortion
totalpowerirregularity (power removed)
irregularity (rotationally invariant, power removed)

Flatness De-Mystified

RMS term
RMSt
RMSiRMSa18

Flatness – what are we talking about?
• Peak to valley total, or just simply “PV”, is the maximum measurement and theworst-case scenario, taking into account the difference between the surface’s
lowest and highest points. It is by far the most widespread flatness specification
used today.
• A more accurate measurement of surface flatness is rms, as it takes into accountthe entire optic and calculates deviation from the ideal form.
• Traditionally, optical flats with monochromatic light (ex. 546.07nm) havemeasured surface flatness in fringes; today, however, laser interferometers at632.8 nm measure most optical components and can output fringes or waves(or sensible units, like microns).

3/12/2024

Flatness De-Mystified

19

Conversions•What makes this whole subject difficult (and annoying) is all the different conversions.
Fringes •

To convert from a per inch to a per CA, multiply by the ratio of the diameters (orlongest lengths, for example diagonal if rectangular piece) squared:–Convert from 1 wave/CA PV @ 632.8nm to waves/inch PV @ 632.8nm given CA is:• 25.4 diameter: 1 wave x (25.4/25.4)2 = 1 wave
• 10mm diameter: 1 wave x (25.4/10)2 = 6.45 waves• 35 x 27 rectangle: 1 wave x (25.4/44.2)2 = 0.33 waves• 35 x 27 ellipse: 1 wave x (25.4/35)2 = 0.53 waves

3/12/2024

Flatness De-Mystified

20

Conversions•

What makes this whole subject difficult (and annoying) is all the different conversions.
Fringes ••

To convert from 546.07nm to 632.8nm, simply multiply by 0.86
To reverse, multiply by 1.16

3/12/2024

Flatness De-Mystified

21

Conversions•

What makes this whole subject difficult (and annoying) is all the different conversions.
Fringes ••A fringe is a half a wave
Or 1 wave is 2 fringes

3/12/2024

Flatness De-Mystified

22

Conversions•

What makes this whole subject difficult (and annoying) is all the different conversions.
Fringes 

E:\FILMCAD\Calculators_and_worksheets\Flatness_Calc_ja2.xlsx

3/12/2024

Flatness De-Mystified

23

TWD
• Transmitted Wavefront Distortion (TWD) is typically the error left over after wedge(tilt) is removed.
• TWD is caused by deviations in thickness across the part and/or inhomogeneityof the index of refraction.

Total TWD

=Wedge (tilt)+Spherical+Irregularity

NOTE: Wedge (or tilt) is removed intentionally from Zygo measurements when measuring for TWD.
3/12/2024

Flatness De-Mystified24

Understanding wedge in a lens
• “wedge” in a lens refers to an asymmetry between the “mechanical axis”,
defined by the outer edge and the “optical axis” defined by the opticalsurfaces.
• The optical axis of a lens defined by line connecting centers of curvature of the
optical surfaces• The mechanical axis defined by outer edge, used for mounting.

3/12/2024

Flatness De-Mystified

25

Understanding wedge in a lens

3/12/2024

Flatness De-Mystified

26

Understanding wedge in a lens
• “wedge” in a lens refers to an asymmetry between the “mechanical axis”,defined by the outer edge and the “optical axis” defined by the opticalsurfaces.
• The optical axis of a lens defined by line connecting centers of curvature of the
optical surfaces• The mechanical axis defined by outer edge, used for mounting.
Wedge angle: α = Edge Thickness Difference (ETD)/Diameter
Beam Deviation: δ = α(n-1) defined by light going through the lensSame for flats!
ETD = max - min
α= ETD / D
δ = α(n – 1)
Example:Wedge, α = 5 arc-sec
n = 1.46 (fused silica)δ = 5 (1.46 -1) = 5 x 0.46
δ = 2.3 arc-sec
3/12/2024

Flatness De-Mystified

27

Thanks!

3/12/2024

Flatness De-Mystified

28

Flatness, Etc.

•Chroma’s Measurement with the Zygo Interferometer–Typically given as Peak-to-Valley in number of waves (@633nm) per inch, at 0deg AOI …only on a single surface, and with any
tilt (or wedge) factored out.
“Interferometers function by dividing a wavefront into two or more parts, principally a reference wavefront and a test
wavefront, which travel different paths and then combine to form an interference fringe pattern. The geometrical propertiesof the interference fringe pattern are determined by the difference in optical path traveled by the recombining wavefronts.”
http://www.zygo.com/?sup=/resource/faqs.cgi?id=60–Basically, at 0deg AOI, Flatness is RWD/2 … OR … RWD is 2x Flatness–––•

The general formula is RWD = 2 x cosθ x Flatness, so at 45deg AOI, RWD = 2 x cos45 x Flatness = 2 x 0.707 x Flatness … (forflatness of 1 wave, then, RWD = 1.4 waves).
Or Flatness = RWD/(2 x cosθ)

Wait, Flatness is measured using reflected wavefront deviation analysis!
–

So when we want to measure flatness we actually measure RWD and then convert to our in-house Flatness? Why?

– The answer is because “Flatness” is an industry-standard term that both our substrate
suppliers and many of our customers use.
•Covered by ISO 10110-5 Standard , called “Surface Form Tolerance” Which the Zygo Measures directly–––––

Indicated by the drawing code “3/”Units are in fringes @ 546.07nm, where one fringe is 2x waves
To go from this to Flatness we convert measured P-V fringes to waves and Waves @ 633nm = Fringes @ 546.07nm *(546.07/632.8) / 2(NOTE: we say “633”, but it’s actually 632.8nm)
See slides 20-25: http://www.visualopticslab.com/opti515r/notes/section5slides.pdf

3/12/2024

Flatness De-Mystified

29