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Physical-Optics Analysis of Optical Interferometers
S. Zhang*, H. Zhong**, Rui Shi**, C. Hellmann*, F. Wyrowski***University of Jena, ** LightTrans GmbH
DGaO 2019, Darmstadt, June 13, 2019
Applied Computational Optics Group
2
Applied Computational Optics Group and LightTrans
3
Connecting Optical Technologies
Physical-Optics System Modeling by Connecting Field Solvers
4
Field Solver Lenses, …
Field Solver Prisms, …
Field Solver Gratings, …
Field Solver Micro-and Nano-structures Field Solver Fibers,
…
Physical-Optics System Modeling: Regional Field Solvers
free space
prisms, plates, cubes, ...
lenses & freeforms
apertures & boundaries
gratings
diffractive, Fresnel, meta lenses
HOE, CGH, DOEmicro lens & freeform
arrays
SLM & adaptive components
diffractive beam splitters
diffusers
scatterer
waveguides & fibers
crystals & anisotropic components
nonlinear components
5
Field Solvers
Physical-Optics System Modeling by Connecting Field Solvers
6
Connection of solvers via I/O channel concept which enables non-sequential physical-optics
system modeling
free space prisms,
plates, cubes, ...
lenses & freeforms
apertures & boundaries
gratings
diffractive, Fresnel, meta
lensesHOE, CGH,
DOEmicro lens &
freeform arrays
SLM & adaptive
components
diffractive beam
splitters
diffusers
scatterer
waveguides & fibers
crystals & anisotropic components
nonlinear components
Field Solvers
Surface Channels: Example of Plate/Etalon
7
+/+
Surface +/+ +/- -/- -/+1st ×2nd ×
Setting A
Surface Channels: Example of Plate/Etalon
8
+/+
Surface +/+ +/- -/- -/+1st ×2nd ×
Setting Afree
space prisms, plates,
cubes, ...
lenses & freeforms
apertures & boundaries
gratings
diffractive, Fresnel, meta
lensesHOE, CGH,
DOEmicro lens &
freeform arrays
SLM & adaptive
components
diffractive beam
splitters
diffusers
scatterer
waveguides & fibers
crystals & anisotropic components
nonlinear components
Field Solvers
12
1 112
2
Surface Channels: Example of Plate/Etalon
9
+/+
Surface +/+ +/- -/- -/+1st ×2nd ×
Setting A
Surface Channels: Example of Plate/Etalon
10
+/+
Surface +/+ +/- -/- -/+1st ×2nd ×
+/+
Surface +/+ +/- -/- -/+1st × ×2nd ×
Surface +/+ +/- -/- -/+1st × ×2nd ×
Setting A Setting CSetting B
Surface Channels: Example of Plate/Etalon
11
Surface +/+ +/- -/- -/+1st × × ×2nd ×
+/+
Surface +/+ +/- -/- -/+1st × × × ×2nd × × × ×
+/+
+/+
Setting D Setting E
Surface Channels: Example of Plate/Etalon
12
planar-planar (parallel) - varying thickness from 100 to 99 µm
+/+
Surface +/+ +/- -/- -/+1st × × × ×2nd × × × ×
+/+
+/+
Surface Channels: Example of Plate/Etalon
13
planar-planar (parallel) - varying thickness from 100 to 99 µm
+/+
Surface +/+ +/- -/- -/+1st × × × ×2nd × × × ×
+/+
+/+
Constructive and destructive interference alternatively shows up when the thickness of plate varies.
Surface Channels: Example of Plate/Etalon
14
planar-planar (non-parallel) - center thickness 100 µm- tilt of first surface
+/+
Surface +/+ +/- -/- -/+1st × × × ×2nd × × × ×
+/+
+/+
no tilt
x [mm]
y[m
m]
Surface Channels: Example of Plate
15
planar-planar (non-parallel) - center thickness 100 µm- tilt of first surface
Linear interference fringes appear due to linear change
of etalon thickness.
tilt by 0.1°
x [mm]
y[m
m]+/+
Surface +/+ +/- -/- -/+1st × × × ×2nd × × × ×
+/+
+/+
x [mm]
y[m
m]
input polarization along x1.35
x [mm]
y[m
m]
input polarization along y
1.0
Surface Channels: Example of Plate
16
cylindrical-planar- center thickness 100 µm- cylindrical surface radius 1 m
Polarization-dependent effect on the interference is considered.
+/+
Surface +/+ +/- -/- -/+1st × × × ×2nd × × × ×
+/+
+/+
Surface Channels: Example of Plate
17
planar-spherical- center thickness 100 µm- spherical surface radius -1 m
+/+
Surface +/+ +/- -/- -/+1st × × × ×2nd × × × ×
+/+
+/+
Non-sequential simulation time of etalon with curved surfaces: few seconds and less
Collimation System: Sequential Simulation
18
#1 #2#3 #4 #5 #6
Perfect AR coating is assumed, so sequential modeling is used.
collimating objective lens (NA=0.63)
high-NAlaser diode
Laser ComponentsWSLD-1064-050m-1-PD- fundamental Gaussian- wavelength 1064 nm- divergence (FWHM) 20° x 10°- astigmatism 11.6 µm between
x- and y-plane
Collimation System: Non-Sequential Simulation
19
high-NAlaser diode
Laser ComponentsWSLD-1064-050m-1-PD- fundamental Gaussian- wavelength 1064 nm- divergence (FWHM) 20° x 10°- astigmatism 11.6 µm between
x- and y-plane
#1 #2#3 #4 #5 #6
multiple reflections between uncoated surfaces #4 and #5, other surfaces are well
coated.
Physical-Optics Modeling of Interferometer
20
Maximum flexibility in source modeling: profile, multimode, polarization, coherence
Maximum flexibility to select detector, e.g. e.m. field components, irradiance, intensity, polarization
He-Ne laser- fundamental Gaussian- wavelength 632.8 nm
3x beam expanderbeam splitter
beam splitter
2 mm
2 mm
BK7BK7
reference path
test path(test object may tilt and/or shift)
?
How to calculate interference fringe with the possible shift and tilt of components considered?
Flexible inclusion of different types of components
free space prisms,
plates, cubes, ...
lenses & freeforms
apertures & boundaries
gratings
diffractive, Fresnel, meta
lensesHOE, CGH,
DOEmicro lens &
freeform arrays
SLM & adaptive
components
diffractive beam
splitters
diffusers
scatterer
waveguides & fibers
crystals & anisotropic components
nonlinear components
Connecting Field Solvers: Example
21
Field Solvers
13
2
2
1
3
# idealized component
2
1 1
1
2
2
3 3
High flexibility for different interferometer setups.
Modeling and Design Examples in VirtualLab Fusion
• Michelson interferometer• Mach-Zehnder interferometer• Fizeau interferometer• Coherence measurement• White light interferometry• Coherence scanning interferometry• Spectroscopy with Etalon • Locally polarized fields by interference• Gouy phase shift demonstration
22
Laser-Based Michelson Interferometer and Interference Fringe Exploration
Modeling Task
24
monomode laser- wavelength: 635nm
- half-angle divergence: 2°
?interference
pattern
detector How does the interference fringe change with respect to the shift and tilt of the movable mirror?
movablemirror
fixed mirror
1 mm 1 mm
shift
tilt
shift
tilt
beam splitter
Result with Equivalent Optical Path
25
detector
movablemirror
fixed mirror
beam splittermonomode
laser
Coherent superposition of two fields reflected from planar mirrors at equivalent-path
positions gives no interference fringes.
Result with Shifted Movable Mirror
26
detector
movablemirror
fixed mirror
beam splittermonomode
laser
Shift of the movable mirror introduces defocus, and therefore ring fringes are
observed at the detector plane.
Result with Tilted Movable Mirror
27
detector
movablemirror
fixed mirror
beam splittermonomode
laser
Tilt of the movable mirror leads to parallel striped interference fringes are
seen at the detector plane.
Result with Shifted and Tilted Movable Mirror
28
detector
movablemirror
fixed mirror
beam splittermonomode
laser
Combination of both shift and tilt of the movable mirror gives rise to shifted ring
pattern in the interference fringes.
Modeling and Design Examples in VirtualLab Fusion
• Michelson interferometer• Mach-Zehnder interferometer• Fizeau interferometer• Coherence measurement• White light interferometry• Coherence scanning interferometry• Spectroscopy with Etalon • Locally polarized fields by interference• Gouy phase shift demonstration
29
Mach-Zehnder Interferometer
Modeling Task
31
He-Ne laser- fundamental Gaussian- wavelength 632.8 nm
3x beam expanderbeam splitter
beam splitter
2 mm
2 mm
BK7BK7
reference path
test path(test object may tilt and/or shift)
?
How to calculate interference fringe with the possible shift and tilt of components considered?
Interference Fringe Due to Component Tilt
32
tilt angle
Calculation of interference pattern including element tilt
takes less than 2 seconds!
0° tilt
x [mm]
y[m
m]
3° tilt
x [mm]
y[m
m]
5° tilt
x [mm]
y[m
m]
10° tilt
x [mm]
y[m
m]
0 shift
x [mm]
y[m
m]
300µm shift
x [mm]
y[m
m]
500µm shift
x [mm]
y[m
m]
1000µm shift
x [mm]
y[m
m]
Interference Fringe Due to Component Shift
33
shiftCalculation of interference
pattern including element shift takes less than 2 seconds!
Modeling and Design Examples in VirtualLab Fusion
• Michelson interferometer• Mach-Zehnder interferometer• Fizeau interferometer• Coherence measurement• White light interferometry• Coherence scanning interferometry• Spectroscopy with Etalon • Locally polarized fields by interference• Gouy phase shift demonstration
34
Fizeau Interferometer for Optical Testing
Modeling Task
36
spherical wave- wavelength 532nm- half-opening angle
26.6°
collimation lens
beam splitter
imaging lens
referenceflat
test flat
detector ?How does the interference fringe change for different test flats?
Tilted Planar Surface under Observation
37
reference flat
detector
tilted planar surface
Reflection from the test planar surface remain as plane waves, but only with slightly different direction, and therefore leading to parallel striped fringes.
Cylindrical Surface under Observation
38
reference flat
detector
cylindrical surface
Reflected wavefront from the test cylindrical surface gets curved in one direction, therefore leading to parallel striped fringes but with varying pitch.
Spherical Surface under Observation
39
reference flat
detector
spherical surface
Spherical surface changes the reflected wavefront in radial direction, thus the interference fringes appears as concentric rings.
Modeling and Design Examples in VirtualLab Fusion
• Michelson interferometer• Mach-Zehnder interferometer• Fizeau interferometer• Coherence measurement• White light interferometry• Coherence scanning interferometry• Spectroscopy with Etalon • Locally polarized fields by interference• Gouy phase shift demonstration
40
Coherence Measurement Using Michelson Interferometer and Fourier Transform Spectroscopy
Modeling Task
42
fundamental Gaussian(central wavelength 635 nm)
a) bandwidth 50 nmb) bandwidth 100 nm(24 samples frequency)
detector
change of the lateral interference fringes for different d values
movablemirror
fixed mirror
shift distance d
point-wise measurement with respect to d values
…
Lateral Interference Fringes – 50 nm Bandwidth
43
fundamental Gaussian(central wavelength 635 nm)
a) bandwidth 50 nm shift distance d
d=0 d=1µm d= 2µm
Lateral Interference Fringes – 100 nm Bandwidth
44
shift distance d
d=0 d=1µm d= 2µm
fundamental Gaussian(central wavelength 635 nm)b) bandwidth: 100 nm
Broader spectral bandwidth leads to shorter coherent length; and therefore the interference fringe starts to vanish sooner in comparison to the case with
narrower bandwidth.
Pointwise Measurement
45
detector
movablemirror
fixed mirror
shift distance d
50nm bandwidth
100nm bandwidthfundamental Gaussian
(central wavelength 635 nm)a) bandwidth 50 nm
b) bandwidth 100 nm
Modeling and Design Examples in VirtualLab Fusion
• Michelson interferometer• Mach-Zehnder interferometer• Fizeau interferometer• Coherence measurement• White light interferometry• Coherence scanning interferometry• Spectroscopy with Etalon • Locally polarized fields by interference• Gouy phase shift demonstration
46
White-Light Michelson Interferometer
Modeling Task
48
Xenon lampblack body spectrum6200K temperature
41 frequency samples
achromatEdmund optics
No. 49-664
fixed mirror
movable mirror (slightly tilted)
?interference
fringe
How to calculate the interference fringes with consideration of the finite coherence length of the source and the mirror movement?
detector
power spectrum
Change in Interference Fringes
49
no shift 1µm shift 2µm shift
When the movable mirror moves from 0 µm to 2 µm, the contrast of interference pattern decreases, due to
limited coherence length of the light source.
shift
fixed mirror
detector
Xenon lamp
Modeling and Design Examples in VirtualLab Fusion
• Michelson interferometer• Mach-Zehnder interferometer• Fizeau interferometer• Coherence measurement• White light interferometry• Coherence scanning interferometry• Spectroscopy with Etalon • Locally polarized fields by interference• Gouy phase shift demonstration
50
Full-Field Optical Coherence Scanning Interferometry
Modeling Task
52
?interference
fringe
detector
beamsplitter
fixed mirror
specimen(movable)
How to calculate the interference pattern and even to derive the height profile of the specimen under test?
Xenon lampblack body spectrum6200K temperature
41 frequency samples
1.0
0.9
0.8
380nm 750nm
power spectrum
achromatEdmund optics
No. 49-664
height contour0
6 µm
25 µm
Simulated Interference Fringes
53
Contour lines of interference fringes corresponds to the height contour of the specimen under measurement.
height contour0
6 µm
25 µm
1 µm shift of specimenno shift of specimen 2 µm shift of specimen
Xenon lamp
detector
fixed mirror
specimen(movable)
Modeling and Design Examples in VirtualLab Fusion
• Michelson interferometer• Mach-Zehnder interferometer• Fizeau interferometer• Coherence measurement• White light interferometry• Coherence scanning interferometry• Spectroscopy with Etalon • Locally polarized fields by interference• Gouy phase shift demonstration
54
Examination of Sodium D Lines with Etalon
Modeling Task
56
input spherical wave- sodium D lines
@ 588.995 nm & 589.592 nm- linearly polaried
along x direction- half divergent angle is 2.3°
d1 = 70mm
d2 = 1.686mm
d3 = 10mm d5 = f = 100mm
d4 = 4.3mm
fused silica
silica-spaced etalonHR - coating- reflectance ≈ 90 % - thickness ≈ 700 nm- material: Silicon Dioxide
& Titanium Dioxide
spherical lens- type: plano – convex- radius = 51.94mm- material: N-BK7 ?
Visualization of Both Spectrum Lines
57
input spherical wave- sodium D lines
@ 588.995 nm & 589.592 nm
588.995nm
589.592nm
field in front of the etalon
field on the focal plane
Finesse vs. Coating Reflectance
58
input spherical wave- single spectrum line
@ 589.592 nm
HR - coating- reflectance ≈90%,
80%, 60%
R ≈ 90% R ≈ 80% R ≈ 60%
Sharpness of the interference fringes depends on the reflectance
of the coatings on the etalon.
Finesse vs. Coating Reflectance
59
extracting 1D data along the diagonal direction
R ≈ 90%
R ≈ 80%
R ≈ 60%
the higher reflectance, the higher finesse
Finesse vs. Coating Reflectance
60
extracting 1D data along the diagonal direction
R ≈ 90%
R ≈ 80%
R ≈ 60%
the higher reflectance, the higher finesse
Modeling and Design Examples in VirtualLab Fusion
• Michelson interferometer• Mach-Zehnder interferometer• Fizeau interferometer• Coherence measurement• White light interferometry• Coherence scanning interferometry• Spectroscopy with Etalon • Locally polarized fields by interference• Gouy phase shift demonstration
61
Generation of Spatially Varying Polarization byInterference with Polarized Light
Modeling Task
63
He-Ne laser- fundamental Gaussian- wavelength 632.8 nm- circularly polarized
3x beam expanderbeam splitter
beam splitter
2 mm
2 mm
BK7BK7
?
interference
polarizer(fixed)
polarizer(rotatable)
How does the interference pattern change with respect
to the polarization states of two arms?
Interference Pattern Changes with Polarizer Rotation
64
polarizer(rotatable)
polarizer(fixed)
polarizer rotation by 0° polarizer rotation by 45°
polarizer rotation by 75° polarizer rotation by 90°
Interference fringes start to disappear, when polarizer
rotates from parallel to orthogonal orientation.
Interference Pattern Changes with Polarizer Rotation
65
polarizer(rotatable)
polarizer(fixed)
polarizer rotation by 0°
polarizer rotation by 75°
Fringe contrast changes with
polarizer rotation.
Interference Pattern
66
polarizer(x-direction)
polarizer(x-direction)
polarizer(y-direction)
parallel polarizers
crossed polarizers
Interference information is encoded
in polarization state
Modeling and Design Examples in VirtualLab Fusion
• Michelson interferometer• Mach-Zehnder interferometer• Fizeau interferometer• Coherence measurement• White light interferometry• Coherence scanning interferometry• Spectroscopy with Etalon • Locally polarized fields by interference• Gouy phase shift demonstration
67
Observation of Gouy Phase Shift in a Mach-Zehnder Interferometer
Modeling Task
69
Gaussian beam- waist radius 1 mm- wavelength 680 nm
beam splitter beam combiner1 mm
1 mm
BK7BK7
?Interference
(in front of focus)
?Interference
(behind focus)
9.5 mm 9.5 mm
Interference Pattern
70
By observing the center of the fringes, the change from
constructive to destructive interference implies a 𝜋𝜋 −shift in the
phase of the spherical wave.
constructive interference
destructive interference
Interference Pattern
71
Conclusion
• Connecting field solvers enables fast physical-optics modeling • Channel concept allows non-sequential physical-optics modeling
72
High flexibility for physical-optics analysis of optical interferometers!
EOS Topical Meeting on Diffractive Optics 2019
16. – 19. September 2019
Jena, Germany
Submit your paper to EOS Topics Meeting
EOS Topical Meeting on Diffractive Optics 2019
16. – 19. September 2019
Jena, Germany
Guest of Honor & Plenary Speaker:Bernard Kress (Microsoft, USA)
Invited SpeakersBenfeng Bai, Tsinghua University, China
Sven Burger, Zuse Institute Berlin, Germany
Andreas Erdmann, Fraunhofer IISB Erlangen, Germany
Patrice Genevet, Université côted’azur, France
Michael A. Golub, Tel Aviv University, Israel
Tommi Hakala, University of Eastern Finland, Finland
Jürgen Jahns, Fernuniversität in Hagen, Germany
Uwe Zeitner, Fraunhofer IOF Jena, Germany
EOS Topical Meeting on Diffractive Optics 2019
16. – 19. September 2019
Jena, Germany
Submit your paper to EOS Topics Meeting
Thank You!
76
Physical-Optics Analysis of Optical InterferometersApplied Computational Optics Group Applied Computational Optics Group and LightTrans Physical-Optics System Modeling by Connecting Field SolversPhysical-Optics System Modeling: Regional Field SolversPhysical-Optics System Modeling by Connecting Field SolversSurface Channels: Example of Plate/EtalonSurface Channels: Example of Plate/EtalonSurface Channels: Example of Plate/EtalonSurface Channels: Example of Plate/EtalonSurface Channels: Example of Plate/EtalonSurface Channels: Example of Plate/EtalonSurface Channels: Example of Plate/EtalonSurface Channels: Example of Plate/EtalonSurface Channels: Example of PlateSurface Channels: Example of PlateSurface Channels: Example of PlateCollimation System: Sequential SimulationCollimation System: Non-Sequential SimulationPhysical-Optics Modeling of InterferometerConnecting Field Solvers: ExampleModeling and Design Examples in VirtualLab FusionLaser-Based Michelson Interferometer and Interference Fringe ExplorationModeling TaskResult with Equivalent Optical Path Result with Shifted Movable MirrorResult with Tilted Movable MirrorResult with Shifted and Tilted Movable MirrorModeling and Design Examples in VirtualLab FusionMach-Zehnder InterferometerModeling TaskInterference Fringe Due to Component TiltInterference Fringe Due to Component ShiftModeling and Design Examples in VirtualLab FusionFizeau Interferometer for Optical TestingModeling TaskTilted Planar Surface under ObservationCylindrical Surface under ObservationSpherical Surface under ObservationModeling and Design Examples in VirtualLab FusionCoherence Measurement Using Michelson Interferometer and Fourier Transform SpectroscopyModeling TaskLateral Interference Fringes – 50 nm BandwidthLateral Interference Fringes – 100 nm BandwidthPointwise Measurement Modeling and Design Examples in VirtualLab FusionWhite-Light Michelson InterferometerModeling TaskChange in Interference FringesModeling and Design Examples in VirtualLab FusionFull-Field Optical Coherence Scanning InterferometryModeling TaskSimulated Interference FringesModeling and Design Examples in VirtualLab FusionExamination of Sodium D Lines with EtalonModeling TaskVisualization of Both Spectrum LinesFinesse vs. Coating ReflectanceFinesse vs. Coating ReflectanceFinesse vs. Coating ReflectanceModeling and Design Examples in VirtualLab FusionGeneration of Spatially Varying Polarization by Interference with Polarized LightModeling TaskInterference Pattern Changes with Polarizer RotationInterference Pattern Changes with Polarizer RotationInterference PatternModeling and Design Examples in VirtualLab FusionObservation of Gouy Phase Shift in a Mach-Zehnder InterferometerModeling TaskInterference Pattern Interference Pattern ConclusionFoliennummer 73Foliennummer 74Foliennummer 75Thank You!